(1) The functions *U* : [0, *a*] × [0, *a*] × R → R, *F* : [0, *a*] × [0, *a*] × R × R → R, and *G* : [0, *a*] × [0, *a*] × R × R → R are continuous, and there exist nonnegative constants *L*, *M* such that:

$$|L(l,h,0)| \le L\_\star \quad |F(l,h,M\_1,0)| \le M \quad \text{and} \quad |F(l,h,M\_2,0)| \le M\_\star$$

where *M*1, *M*2 ∈ R.

(2) Let *Ai* : [0, *a*] × [0, *a*] → R+ (*i* = 1, 2, 3, 4, 5) be continuous functions such that:

$$|\mathcal{U}(l,h,\mathbf{x}\_1) - \mathcal{U}(l,h,\mathbf{x}\_2)| \le A\_1(l,h) \left| \mathbf{x}\_1 - \mathbf{x}\_2 \right|,$$

$$|F(l,h,\mathbf{y},\mathbf{x}\_1) - F(l,h,\mathbf{y},\mathbf{x}\_2)| \le A\_2(l,h) \left| \mathbf{x}\_1 - \mathbf{x}\_2 \right|,$$

$$|G(l,h,\mathbf{y},\mathbf{x}\_1) - G(l,h,\mathbf{y},\mathbf{x}\_2)| \le A\_3(l,h) \left| \mathbf{x}\_1 - \mathbf{x}\_2 \right|,$$

$$|F(l,h,\mathbf{y}\_1,\mathbf{x}) - F(l,h,\mathbf{y}\_2,\mathbf{x})| \le A\_4(l,h) \left| \mathbf{y}\_1 - \mathbf{y}\_2 \right| $$

and:

$$|G(l, h, y\_1, \mathbf{x}) - G(l, h, y\_2, \mathbf{x})| \le A\_5(l, h) \left| y\_1 - y\_2 \right| \rho$$

where *l*, *h* ∈ [0, *a*] and *x*, *x*1, *x*2, *y*, *y*1, *y*2 ∈ R. Furthermore, let:

$$K = \max\{A\_i(l, h) : i = 1, 2, 3, 4, 5; l, h \in [0, a]\},$$

where *K* ≥ 0.

**Theorem 3.** *Under the hypotheses (1)–(4), Equation (2) has at least one solution in E* = *C*(*I* × *I*), *where I* = [0, *a*].

**Proof.** Let us consider the operators *F*ˆ and *G*ˆ defined on *E* by:

$$\mathcal{L}(\mathbf{f}\mathbf{x})(l,h) = \mathcal{U}(l,h,\mathbf{x}(l,h)) + F\left(l, h, \int\_0^l \int\_0^h \mathbf{P}(l,h,r,u,\mathbf{x}(r,u)) dr du, \mathbf{x}(l,h)\right)$$

and:

$$\mathcal{G}(\mathsf{G}x)(l,h) = \mathcal{G}\left(l,h,\int\_0^a \int\_0^a \mathcal{Q}\left(l,h,r,u,\mathsf{x}(r,u)\right) dr du, \mathsf{x}(l,h)\right), \text{ where } l, h \in [0,a].$$

From Assumptions (1)–(3), we get that *F*ˆ and *G*ˆ map *C*(*I* × *I*) into itself. Furthermore, let us define another operator *T*ˆ on *C*(*I* × *I*) as follows:

$$\mathfrak{T}\mathfrak{x} = (\mathfrak{F}\mathfrak{x})(\mathring{G}\mathfrak{x}).$$

It is obvious that *T*ˆ maps *C*(*I* × *I*) into itself.

Let:

$$I\_1(\mathbf{x}) = \int\_0^l \int\_0^h \mathbf{P}(l, h, r, u, \mathbf{x}(r, u)) dr du$$

and:

$$I\_2(\mathbf{x}) = \int\_0^a \int\_0^a \mathbb{Q}\left(l\_\prime h\_\prime r\_\prime u\_\prime \mathbf{x}(r\_\prime u)\right) dr d\prime \mu.$$

### Let *x* ∈ *C*(*I* × *I*) be fixed and *l*, *h* ∈ *I*. We get:

( ((*Tx*ˆ )(*l*, *h*) ( ( = ( ((*Fx*ˆ )(*l*, *h*) ( ( . ( ((*Gx*ˆ )(*l*, *h*) ( ( = |*U*(*l*, *h*, *x*(*l*, *h*)) + *F* (*l*, *h*, *I*1(*x*), *x*(*l*, *h*))| × |*G* (*l*, *h*, *I*2(*x*), *x*(*l*, *h*))| ≤ (|*U*(*l*, *h*, *x*(*l*, *h*)) − *U*(*l*, *h*, 0)| + |*U*(*l*, *h*, 0)| + |*F* (*l*, *h*, *I*1(*x*), *x*(*l*, *h*)) −*F* (*l*, *h*, *I*1(*x*), 0)| + |*F* (*l*, *h*, *I*1(*x*), 0)|) × (|*G*(*l*, *h*, *I*2(*x*), *x*(*l*, *h*)) −*G*(*l*, *h*, *I*2(*x*), 0)| + |*G*(*l*, *h*, *I*2(*x*), 0)|) ≤ (*A*1(*l*, *h*)|*x*(*l*, *h*)| + *L* + *A*2(*l*, *h*)|*x*(*l*, *h*)| + *M*) × (*A*3(*l*, *h*)|*x*(*l*, *h*)| + *M*) ≤ (2*K x* +*L* + *M*) (*K x* +*M*) ≤ (2*K x* +*L* + *M*) 2 .

Let *α* = 2*k*, *β* = *L* + *M*. Then, we have:

$$\|\|\hat{\mathcal{F}}\mathbf{x}\|\| \le \alpha \parallel \|\mathbf{x}\parallel\| + \beta\_{\prime}$$

$$\|\|\hat{\mathcal{G}}\mathbf{x}\parallel\| \le \alpha \parallel \|\mathbf{x}\parallel\| + \beta$$

$$\|\|\mathcal{T}\mathbf{x}\parallel\| \le \left(\alpha \parallel \|\mathbf{x}\parallel\| + \beta\right)^{2}\tag{3}$$

and:

for *x* ∈ *C*(*I* × *I*). From (3), we have that the operator *T*ˆ maps *Bd* ⊂ *C*(*I* × *I*) into *Bd*, where:

$$B\_d = \{ \mathfrak{x}(l, h) \in I : \| \mathfrak{x}(l, h) \| \le d \}$$

for *d*2 ≤ *d* ≤ *d*1, where:

$$d\_1 = \frac{1 - 2a\beta - \sqrt{1 - 4a\beta}}{2a^2}$$

and:

$$d\_2 = \frac{1 - 2\alpha\beta + \sqrt{1 - 4\alpha\beta}}{2\alpha^2}.$$

Furthermore, we have:

and:

$$\|\|\:\hat{G}B\_d\:\|\le\alpha d+\beta.\tag{5}$$

 *FB*ˆ *d* ≤ *αd* + *β* (4)

Let > 0 be fixed and *x*(*l*, *h*), *y*(*l*, *h*) ∈ *Bd* such that:

$$\|\|\|\mathbf{x} - \mathbf{y}\|\| \le \epsilon, \ (l, h \in I).$$

Then, we have:

( ((*Fx*ˆ )(*l*, *h*) − (*Fy*ˆ )(*l*, *h*) ( ( = |*U* (*l*, *h*, *x*(*l*, *h*)) + *F* (*l*, *h*, *I*1(*x*), *x*(*l*, *h*)) −*U* (*l*, *h*, *y*(*l*, *h*)) − *F* (*l*, *h*, *I*1(*y*), *y*(*l*, *h*)) ( ( ( ≤ |*U* (*l*, *h*, *x*(*l*, *h*)) − *U* (*l*, *h*, *y*(*l*, *h*))| + |*F* (*l*, *h*, *I*1(*x*), *x*(*l*, *h*)) − *F* (*l*, *h*, *I*1(*x*), *y*(*l*, *h*))| + |*F* (*l*, *h*, *I*1(*x*), *y*(*l*, *h*)) − *F* (*l*, *h*, *I*1(*y*), *y*(*l*, *h*))| ≤ *A*1(*l*, *h*)|*x*(*l*, *h*) − *y*(*l*, *h*)| + *A*2(*l*, *h*)|*x*(*l*, *h*) − *y*(*l*, *h*)| +*A*4(*l*, *h*)|*I*1(*x*) − *I*1(*y*)| ≤ 2*K x* − *y* +*K l* 0 *h* 0 |*P*(*l*, *h*,*r*, *u*, *x*(*r*, *u*)) − *P*(*l*, *h*,*r*, *u*, *y*(*r*, *u*))| *drdu* ≤ 2*K x* − *y* +*Ka*2*ω*(*P*, ),

where:

$$\omega(P,\mathfrak{e}) = \sup \left\{ \begin{array}{l} \left| P(l, h, r, \mathfrak{u}, \mathfrak{x}(r, \mathfrak{u})) - P(l, h, r, \mathfrak{u}, \mathfrak{y}(r, \mathfrak{u})) \right| : l, h, r, \mathfrak{u} \in I, \\ \mathfrak{x}, \mathfrak{y} \in [-d, d], \|\mathfrak{x} - \mathfrak{y}\| < \mathfrak{e} \end{array} \right\}.$$

Since *P* is continuous, so it is uniformly continuous on the compact set *I* × *I* × *I* × *I* × [−*d*, *d*]; therefore:

*ω*(*P*, ) → 0 as → 0.

Thus, *F*ˆ is continuous on *Bd*. Similarly, one can prove that *G*ˆ is continuous on *Bd*. Thus, we can conclude that *T*ˆ is continuous on *Bd*.

Let us consider a non-empty subset *X* of *Bd* and *x* ∈ *X*. Then, for a fixed > 0 and *l*1, *l*2, *h*1, *h*2 ∈ *I* such that *l*1 ≤ *l*2, *h*1 ≤ *h*2, *l*1 − *l*2 ≤ , *h*1 − *h*2 ≤ , one obtains:

( ((*Fx*ˆ )(*l*2, *h*2) − (*Fx*ˆ )(*l*1, *h*1) ( ( = ( ( ( ( ( *U* (*l*2, *h*2, *x*(*l*2, *h*2)) + *F l*2, *h*2, *l*2 0 *h*2 0 *P*(*l*2, *h*2,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) − *U* (*l*1, *h*1, *x*(*l*1, *h*1)) − *F l*1, *h*1, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*1, *h*1) ( ( ( ( ( ≤ |*U* (*l*2, *h*2, *x*(*l*2, *h*2)) − *U* (*l*2, *h*2, *x*(*l*1, *h*1))| + |*U* (*l*2, *h*2, *x*(*l*1, *h*1)) − *U* (*l*1, *h*1, *x*(*l*1, *h*1))| + ( ( ( ( ( *F l*2, *h*2, *l*2 0 *h*2 0 *P*(*l*2, *h*2,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) − *F l*2, *h*2, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) ( ( ( ( ( + ( ( ( ( ( *F l*2, *h*2, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) − *F l*1, *h*1, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) ( ( ( ( ( + ( ( ( ( ( *F l*1, *h*1, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) − *F l*1, *h*1, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*1, *h*1) ( ( ( ( ( ≤ *A*1(*l*, *h*)|*x*(*l*2, *h*2) − *x*(*l*1, *h*1)| + |*U* (*l*2, *h*2, *x*(*l*1, *h*1)) − *U* (*l*1, *h*1, *x*(*l*1, *h*1))| + *A*4(*l*, *h*) ( ( ( ( *l*2 0 *h*2 0 *P*(*l*2, *h*2,*r*, *u*, *x*(*r*, *u*))*drdu* − *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu* ( ( ( ( + ( ( ( ( ( *F l*2, *h*2, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) − *F l*1, *h*1, *l*1 0 *h*1 0 *P*(*l*1, *h*1,*r*, *u*, *x*(*r*, *u*))*drdu*, *x*(*l*2, *h*2) ( ( ( ( ( + *A*2(*l*, *h*)|*x*(*l*2, *h*2) − *x*(*l*1, *h*1)|

which yields:

$$\begin{aligned} &|(\hat{\mathbf{f}}\mathbf{x})(l\_2, h\_2) - (\hat{\mathbf{f}}\mathbf{x})(l\_1, h\_1)| \leq 2K \left| \mathbf{x}(l\_2, h\_2) - \mathbf{x}(l\_1, h\_1) \right| \\ &+ |\mathcal{U}\left(l\_2, h\_2, \mathbf{x}(l\_1, h\_1)\right) - \mathcal{U}\left(l\_1, h\_1, \mathbf{x}(l\_1, h\_1)\right)| \\ &+ K \left| \int\_0^{l\_2} \int\_0^{h\_2} P(l\_2, h\_2, r, u, \mathbf{x}(r, u)) dr du \right| \end{aligned}$$

$$\begin{aligned} & -\left| \int\_{0}^{l\_{1}} \int\_{0}^{l\_{1}} P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) dr du \right| \\ &+ \left| F \left( l\_{2}, h\_{2}, \int\_{0}^{l\_{1}} \int\_{0}^{l\_{1}} P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) dr du, \mathbf{x}(l\_{2}, h\_{2}) \right) \right| \\ &- F \left( l\_{1}, h\_{1}, \int\_{0}^{l\_{1}} \int\_{0}^{l\_{1}} P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) dr du, \mathbf{x}(l\_{2}, h\_{2}) \right) \right|. \end{aligned}$$

Let:

$$\omega(L,\varepsilon) = \sup\left\{ \begin{array}{l} \left| \operatorname{II}\left(l\_{2}, h\_{2}, \operatorname{x}(l\_{2}, h\_{2})\right) - \operatorname{II}\left(l\_{1}, h\_{1}, \operatorname{x}(l\_{1}, h\_{1})\right)\right| : l\_{1}, l\_{2}, h\_{1}, h\_{2} \in I\_{\prime} \\ \left| l\_{2} - l\_{1} \right| \leq \varepsilon, \left| h\_{2} - h\_{1} \right| \leq \varepsilon, \operatorname{x} \in \left[ -d, d \right] \end{array} \right\},$$

$$\omega(P, \varepsilon) = \sup\left\{ \begin{array}{l} \left| P\left(l\_{2}, h\_{2}, r, u, \operatorname{x}(r, u) \right) - P\left(l\_{1}, h\_{1}, r, u, \operatorname{x}(r, u) \right) \right| : l\_{1}, l\_{2}, h\_{1}, h\_{2}, r, u \in I\_{\prime} \\ \left| l\_{2} - l\_{1} \right| \leq \varepsilon, \left| h\_{2} - h\_{1} \right| \leq \varepsilon, \operatorname{x} \in \left[ -d, d \right] \end{array} \right\},$$

$$\bar{k} = \sup\left\{ \left| P\left(l\_{\prime}h\_{\prime}r, u, \operatorname{x}(r, u) \right) \right| : l\_{\prime}h\_{\prime}r, u \in I\_{\prime} \} \in \left[ -d, d \right] \right\}$$

and:

$$\omega(F,\varepsilon) = \sup \left\{ \begin{array}{l} \left| F\left(l\_2, h\_2, z, \mathbf{x}(l\_2, h\_2)\right) - F\left(l\_1, h\_1, z, \mathbf{x}(l\_1, h\_1)\right) \right| : l\_1, l\_2, h\_1, h\_2 \in I, \\\left| l\_2 - l\_1 \right| \le \varepsilon, \left| h\_2 - h\_1 \right| \le \varepsilon, \mathbf{x} \in \left[ -d, d \right], z \in \left[ -\bar{k}a^2, \bar{k}a^2 \right] \end{array} \right\}.$$

Furthermore:

$$\begin{aligned} & \left| \int\_{0}^{l\_{2}} \int\_{0}^{l\_{2}} P(l\_{2}, h\_{2}, r, u, \mathbf{x}(r, u)) dr du - \int\_{0}^{l\_{1}} \int\_{0}^{l\_{1}} P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) dr du \right| \\ & \leq \left| \int\_{0}^{l\_{2}} \int\_{0}^{l\_{2}} \left( P(l\_{2}, h\_{2}, r, u, \mathbf{x}(r, u)) - P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) \right) dr du \right| \\ & + \left| \int\_{l\_{1}}^{l\_{2}} \int\_{h\_{1}}^{h\_{2}} P(l\_{1}, h\_{1}, r, u, \mathbf{x}(r, u)) dr du \right| \\ & \leq a^{2} \omega(P, \boldsymbol{\varepsilon}) + \bar{k} \boldsymbol{\varepsilon}^{2} .\end{aligned}$$

Therefore:

$$\begin{aligned} \left| (\mathsf{f}\mathbf{x})(l\_2, h\_2) - (\mathsf{f}\mathbf{x})(l\_1, h\_1) \right| &\leq 2K \left| \mathbf{x}(l\_2, h\_2) - \mathbf{x}(l\_1, h\_1) \right| + \omega \left( \mathsf{U}, \boldsymbol{\varepsilon} \right) \\ &+ K \left( a^2 \omega(\boldsymbol{P}, \boldsymbol{\varepsilon}) + \bar{k} \boldsymbol{\varepsilon}^2 \right) + \omega(\boldsymbol{\mathcal{F}}, \boldsymbol{\varepsilon}). \end{aligned}$$

This gives:

$$
\omega(\mathsf{f}x,\mathfrak{e}) \le 2K\omega(x,\mathfrak{e}) + \omega(\mathsf{U},\mathfrak{e}) + K\left[a^2\omega(P,\mathfrak{e}) + \bar{k}a^2\right] + \omega(F,\mathfrak{e}).
$$

Since *U* and *F* are continuous on *I* × *I* × R and *I* × *I* × R × R, respectively, therefore we get:

$$
\omega(\mathcal{U}, \epsilon) \to 0, \; \omega(P, \epsilon) \to 0 \text{ and } \omega(F, \epsilon) \to 0 \text{ as } \epsilon \to 0.
$$

Thus:

$$
\omega\_0(\mathbb{P}X) \le 2\mathcal{K}\omega\_0(X). \tag{6}
$$

Similarly, we can show that:

$$
\omega\_0(\mathring{G}X) \le 2K\omega\_0(X). \tag{7}
$$

From (4)–(7) and Theorem 2 (for the details of this theorem, we refer to [8]), we get that *T*ˆ satisfies the Darbo condition on *Bd* with respect to measure *ω*0 with constant:

$$\begin{aligned} 2K(ad+\beta) + 2K(ad+\beta) &=& 4K(ad+\beta) \\ &=& 4K(ad\_1+\beta) \\ &=& 4K\left[a\left(\frac{1-2a\beta-\sqrt{1-4a\beta}}{2a^2}\right)+\beta\right] \\ &=& 2K\left(\frac{1-\sqrt{1-4a\beta}}{a}\right) \\ &<& 1. \end{aligned}$$

This implies that *T*ˆ is a contraction operator on *Bd* with respect to *ω*0. Thus, by Theorem 2, we have that *T*ˆ has at least one fixed point in *Bd*. Hence, Equation (2) has at least one solution in *Bd* ⊂ *C*([0, *a*] × [0, *a*]). This completes the proof.

#### **4. An Illustrative Example**

We construct the following example to illustrate the obtained result in the previous section.

**Example 1.** *Consider the following integral equation:*

$$\exp(l,h) = \left(\frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{1}{9}\int\_0^l \int\_0^h \frac{rue^{-lh}}{3+x^2(r,u)} dr du\right) \left(\frac{1}{8}\int\_0^1 \int\_0^1 \frac{lh}{6+|x(r,u)|} dr du\right) \tag{8}$$

*for l*, *h* ∈ [0, 1] = *I*. *Here, we have:*

$$\mathcal{U}(l, h, \mathbf{x}(l, h)) = \frac{1}{6} \cos\left(\frac{l + h}{2}\right),$$

$$\mathcal{F}(l, h, \mathbf{y}, \mathbf{x}(l, h)) = \frac{\mathcal{Y}}{9'} $$

$$\mathcal{G}(l, h, \mathbf{y}, \mathbf{x}(l, h)) = \frac{\mathcal{Y}}{8'} $$

$$\mathcal{P}(l, h, \mathbf{r}, \mathbf{u}, \mathbf{x}(\mathbf{r}, \mathbf{u})) = \frac{r u e^{-l h}}{3 + \mathbf{x}^2(r, u)},$$

$$\mathcal{Q}(l, h, \mathbf{r}, \mathbf{u}, \mathbf{x}(\mathbf{r}, u)) = \frac{l h}{6 + |\mathbf{x}(r, u)|}$$

*and a* = 1*; x*, *y* ∈ R.

*It is obvious that all the functions U*, *F*, *G*, *P, and Q are continuous. We have:*

$$|\mathcal{U}(l,h,\mathbf{x}\_1) - \mathcal{U}(l,h,\mathbf{x}\_2(l,h))| = 0. \left| \mathbf{x}\_1(l,h) - \mathbf{x}\_2(l,h) \right|,$$

$$|F(l,h,\mathbf{y},\mathbf{x}\_1(l,h)) - F(l,h,\mathbf{y},\mathbf{x}\_2(l,h))| = 0. \left| \mathbf{x}\_1(l,h) - \mathbf{x}\_2(l,h) \right|,$$

$$|G(l,h,\mathbf{y},\mathbf{x}\_1(l,h)) - G(l,h,\mathbf{y},\mathbf{x}\_2(l,h))| = 0. \left| \mathbf{x}\_1(l,h) - \mathbf{x}\_2(l,h) \right|,$$

$$|F(l,h,\mathbf{y}\_1,\mathbf{x}(l,h)) - F(l,h,\mathbf{y}\_2,\mathbf{x}(l,h))| = \frac{1}{9} \left| \mathbf{y}\_1 - \mathbf{y}\_2 \right|.$$

*and:*

$$|\left|\mathbf{G}(l,h,\mathcal{Y}\_{l},\mathbf{x}(l,h)) - \mathbf{G}(l,h,\mathcal{Y}\_{\mathcal{Y}},\mathbf{x}(l,h))\right| = \frac{1}{8} \left|y\_{1} - y\_{2}\right|.$$

*It follows that:*

$$A\_1(l,h) = A\_2(l,h) = A\_3(l,h) = 0,\ A\_4(l,h) = \frac{1}{9} \text{ and } A\_5(l,h) = \frac{1}{8}.$$

*Consequently, we get K* = 1 8 .

*Furthermore,*

$$\begin{aligned} \left| \mathcal{U}(l, h, 0) \right| &\le \frac{1}{6}, \\\\ \left| F(l, h, y\_1, 0) \right| &\le \frac{1}{27\varepsilon^2} \approx \frac{1}{198.29}, \\\\ \left| G(l, h, y\_2, 0) \right| &\le \frac{1}{48}. \\\\ M &= \frac{1}{48'} \quad L = \frac{1}{6} \end{aligned}$$

*and:*

*Thus:*

*and:*

4*αβ* = 9 48 < 1.

6

*Hence, all the assumption from (1)–(4) are satisfied. Thus, by applying Theorem 3, we conclude that Equation (8) has at least one solution in the Banach algebra C*([0, 1] × [0, 1]).

#### **5. An Iterative Algorithm Created by a Coupled Semi-Analytic Method to Find the Solution of the Integral Equation**

To find an approximation of solution for Equation (8), we make an iterative algorithm by a coupled method created by modified homotopy perturbation and the Adomian decomposition method in the case of two-dimensional functions. Applications of the modified homotopy perturbation method to solve nonlinear integral equations, nonlinear singular integral equations, and nonlinear differential equations can be seen in [25–27], respectively. The Adomian decomposition method to solve physical problems was used in [28] and also to solve integro-differential equations system in [29]. However, in this article, we introduce a modified homotopy perturbation method in terms of a function with two variables, and for simplification of nonlinear terms, we use the Adomian decomposition method in the suitable form; therefore, we make an effective algorithm by the above process. Equation (8) can be shown in a general form of the two-dimensional nonlinear problem:

$$A\left(\mathbf{x}(l,h)\right) - f(l,h) = 0$$

with (*l*, *h*) ∈ *I* × *I*, where *A* is a general nonlinear operator and *f* is a known analytic function. Similar to [26,27], we divide the general operator *A* into two nonlinear operators as *M*1 and *M*2. Of course, *M*1 or *M*2 can be linear operators in the special case that also *f* is converted to *f*1 and *f*2 functions; in other words, we have:

$$M\_1\left(\mathbf{x}(l,h)\right) - f\_1(l,h) + M\_2\left(\mathbf{x}(l,h)\right) - f\_2(l,h) = 0.$$

A modified homotopy perturbation for the above problem can be introduced as follows:

$$H\left(u(l,h),p\right) = M\_1\left(u(l,h)\right) - f\_1(l,h) + p\left[M\_2\left(u(l,h)\right) - f\_2(l,h)\right] = 0, \ p \in [0,1],\tag{9}$$

where *p* is an embedding parameter and *u* is an approximation of *x*. According to the variations of *p* = 0 to *p* = 1, it can be observed that *M*1 (*u*(*l*, *h*)) = *f*1(*l*, *h*) to *A* (*u*(*l*, *h*)) = *f*(*l*, *h*). This implies that for *p* = 1 in (9), we get the solution of (5).

We consider the above solution as the series:

$$\mathfrak{x}(l,h) \simeq \mathfrak{u}(l,h) = \sum\_{k=0}^{\infty} p^k \mathfrak{u}\_k(l,h) \tag{10}$$

and:

$$\mathfrak{x}(l,h) = \lim\_{p \to 1} \mathfrak{u}(l,h). \tag{11}$$

To solve Equation (8), *M*1, *M*2 and *f* can be defined as follows:

$$M\_1(\mathbf{x}(l,h)) = \mathbf{x}(l,h),\tag{12}$$

$$M\_2(\mathbf{x}(l,h)) = -\left(\frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{1}{9}\int\_0^l \int\_0^h \frac{r u e^{-lh}}{3 + \mathbf{x}^2(r,u)} dr du\right) \left(\frac{1}{8}\int\_0^1 \int\_0^1 \frac{lh}{6 + |\mathbf{x}(r,u)|} dr du\right), \tag{13}$$

and:

$$f(l,h) = f\_1(l,h) + f\_2(l,h). \tag{14}$$

Since in (8) *f*(*l*, *h*) = 0, therefore *f*1(*l*, *h*) = *f*2(*l*, *h*) = 0. From (9)–(14), we have:

$$\begin{aligned} &\sum\_{k=0}^{\infty} p^k u\_k(l, h) \\ & - p \left( \frac{1}{6} \cos \left( \frac{l+h}{2} \right) + \frac{1}{9} \int\_0^l \int\_0^h \frac{r u e^{-lh}}{3 + \left( \sum\_{k=0}^{\infty} p^k u\_k(r, u) \right)^2} dr du \right), \\ & - \left( \frac{1}{8} \int\_0^1 \int\_0^1 \frac{lh}{6 + \left| \sum\_{k=0}^{\infty} p^k u\_k(r, u) \right|} dr du \right) = 0. \end{aligned}$$

Now, we use Adomain polynomials for simplicity for the nonlinear terms:

$$\frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{1}{9}\int\_0^l \int\_0^h \frac{r u e^{-lh}}{3 + \left(\sum\_{k=0}^\infty p^k u\_k(r,u)\right)^2} dr du = \sum\_{k=0}^\infty p^k A\_k(l,h)$$

and:

$$\frac{1}{8} \int\_0^1 \int\_0^1 \frac{lh}{6 + \left| \sum\_{k=0}^\infty p^k u\_k(r, u) \right|} dr du = \sum\_{k=0}^\infty p^k A\_k(l, h),$$

where the Adomain polynomials are given by:

$$A\_k(l,h) = \frac{1}{k!} \left[ \frac{d^k}{dp^k} \left\{ \frac{1}{6} \cos\left(\frac{l+h}{2}\right) + \frac{1}{9} \int\_0^l \int\_0^h \frac{r u e^{-lh}}{3 + \left(\sum\_{k=0}^\infty p^k u\_k(r,u)\right)^2} dr du \right\} \right]\_{p=0}$$

and:

$$\mathcal{A}\_k(l,h) = \frac{1}{k!} \left[ \frac{d^k}{dp^k} \left\{ \frac{1}{8} \int\_0^1 \int\_0^1 \frac{lh}{6 + \left| \sum\_{k=0}^\infty p^k \mu\_k(r,u) \right|} dr du \right\} \right]\_{p=0}$$

Therefore, we have:

$$\sum\_{k=0}^{\infty} p^k u\_k(l, h) - p \left\{ \sum\_{k=0}^{\infty} p^k A\_k(l, h) \right\} \left\{ \sum\_{k=0}^{\infty} p^k \hat{A}\_k(l, h) \right\} = 0. \tag{15}$$

By rearranging the terms in powers in *p* of (15) and using modified homotopy perturbation (9), the coefficients of *p* powers must be equal to zero, so we obtain an iterative algorithm (Algorithm 1) to solve for the numerical solution of Equation (8).

Calculating the sequence {*u*0(*l*, *h*), *u*1(*l*, *h*), ...}, we can obtain a closed form of the solution for (8) using the above algorithm.

We compute the Adomain polynomial for *k* = 0,

$$\begin{aligned} &A\_0(l,h) \\ &= \frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{1}{9}\int\_0^l \int\_0^h \frac{rue^{-lh}}{3+u\_0^2(r,u)} drdu \\ &= \frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{l^2h^2e^{-lh}}{108} \end{aligned}$$

and:

$$\hat{A}\_0(l,h) = \frac{1}{8} \int\_0^1 \int\_0^1 \frac{lh}{6 + |u\_0(r,u)|} dr du \;= \frac{lh}{48}.$$

Therefore, we obtain by the algorithm:

$$u\_1(l,h) = \left\{\frac{1}{6}\cos\left(\frac{l+h}{2}\right) + \frac{l^2h^2\varepsilon^{-lh}}{108}\right\}\frac{lh}{48}.$$

We use (10) to approximate *x*(*l*, *h*) by a few term of *uk*(*l*, *h*) as follows:

$$\ln x\_1(l,h) = u\_0(l,h) + u\_1(l,h) = \left\{ \frac{1}{6} \cos \left(\frac{l+h}{2}\right) + \frac{l^2 h^2 e^{-lh}}{108} \right\} \frac{lh}{48}.$$

#### **6. Conclusions**

In our present investigation, we have established the existence of the solution of a functional integral equation of two variables, which is of the form of the product of two operators in the Banach algebra *C*([0, *a*] × [0, *a*]), *a* > 0 and illustrated our results with the help of an example. We also constructed an iteration algorithm to get the solution of Equation (8). Further, one can solve Equation (8) using different numerical, as well as analytical methods in the setting of Banach sequence spaces and Banach algebra. Moreover, due our existence theorem for Equation (8) of two variables, we therefore conclude that our existence result is more general than the one obtained earlier by Deepmala and Pathak [9].

**Author Contributions:** All authors contributed equally in this work.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations**

**Ndolane Sene 1 and Gautam Srivastava 2,3,\***

Received: 4 April 2019; Accepted: 24 April 2019; Published: 1 May 2019

**Abstract:** The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in many stability problems. In this paper, we present a new stability notion of the fractional differential equations with exogenous input. Motivated by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, we present our work here. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also very common. During the last two decades, this class of functions has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, to name just a few. Moreover, we propose the generalized Mittag-Leffler input stability conditions. The left generalized fractional differential equation has been used to help create this new notion. We investigate in depth here the Lyapunov characterizations of the generalized Mittag-Leffler input stability of the fractional differential equation with input.

**Keywords:** fractional differential equations with input; Mittag-Leffler stability; left generalized fractional derivative; *ρ*-Laplace transforms

#### **1. Introduction**

The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in stability problems [1]. There exist many stability notions introduced in fractional calculus. Some examples are asymptotic stability, global asymptotic uniform stability, synchronization problems, stabilization problems, Mittag-Leffler stability and fractional input stability. In this paper, we extend the Mittag-Leffler input stability in the context of the fractional differential equations described by the left generalized fractional derivative. We note here that the left generalized fractional derivative is the generalization of the Liouville-Caputo fractional derivative and the Riemann-Liouville fractional derivative [2]. There exist many works related to stability problems. In [3], Souahi et al. propose some new Lyapunov characterizations of fractional differential equations described by the conformable fractional derivative. In [4], Sene proposes a new stability notion and introduce the Lyapunov characterization of the conditional asymptotic stability. In [5,6], Sene proposes some applications of the fractional input stability to the electrical circuits described the Liouville-Caputo fractional derivative and the Riemann-Liouville fractional derivative. In [7], Li et al. introduce the Mittag-Leffler stability of the fractional differential equations described by the Liouville-Caputo fractional derivative [8]. In [9], Song et al. analyze the stability of the fractional differential equations with time variable impulses. In [10], Tuan et al. propose a novel methodology for studying the stability of the fractional differential equations using the Lyapunov

direct method. In [11], Makhlouf studies the stability with respect to part of the variables of nonlinear Caputo fractional differential equations. In [12], Alidousti et al. propose a new stability analysis of the fractional differential equation described by the Liouville-Caputo fractional derivative. Many other works related to the stability analysis exist in literature, we direct our readers to the References section for more related literature.

The generalized Mittag-Leffler input stability is a new stability notion. This new stability notion studies the behavior of the analytical solution of the fractional differential equations with exogenous input described by the left generalized fractional derivative [13]. We know from previous work in stability problems, it is not trivial to get analytical solutions. The issue is to propose a method to analyze the stability of the fractional differential equations with exogenous input. Classically, the most popular method is the Lyapunov direct method as given in [14–18]. We propose the Lyapunov characterization of the generalized Mittag-Leffler input stability here in this work. As we will be able to show, the generalized Mittag-Leffler input stability generates three properties:

We note here that the fractional differential equation with exogenous input is said to be generalized Mittag-Leffler input stable when the Euclidean norm of its solution is bounded, by a generalized Mittag-Leffler function, plus a quantity which is proportional to the exogenous input bounded when the input is bounded and converging when the input converges in time. The fractional input stability and its consequences are a good compromise in stability problems of the fractional differential equations described by the fractional order derivative operators.

We organize the rest of the paper as follows. In Section 2, we recall the definition of the fractional derivative operators with or without singular kernels. In Section 3, we propose our motivations for studying the generalized Mittag-Leffler input stability. In Section 4, we give the Lyapunov characterizations for the generalized Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. In Section 5, we provide numerical examples for illustrating the main results of this paper. Finally, we finish with some concluding remarks in Section 6.

#### **2. Background on Fractional Derivatives**

Let us first recall the fractional derivative operators and the comparison functions [19]. We will use them throughout this paper. There exist many fractional derivative operators in fractional calculus. There exist two types of fractional derivative operators. The first is fractional derivatives with singular kernels and the second is fractional derivatives without singular kernels. With regards to fractional derivatives with singular kernels, we cite the Riemann-Liouville fractional derivative [2], the Liouville-Caputo fractional derivative [2], the Hilfer fractional derivative [20], the Hadamard fractional derivative [2], and Erdélyi-Kober fractional derivative [21]. We note here that all previous fractional derivatives are associated to their fractional integrals [2,20]. As fractional derivatives without singular kernels we cite the Atangana-Baleanu-Liouville-Caputo derivative [22], the Caputo-Fabrizio fractional derivative [23], and the Prabhakar fractional derivative [24]. We note here that all previous fractional derivatives are associated to their fractional integrals [21–24]. Recently, the generalization of the Riemann-Liouville and the Liouville-Caputo fractional derivative were introduced in the literature by Udita [25]. Namely, the generalized fractional derivative and the Liouville-Caputo generalized fractional derivative. Let us now observe the comparison functions used in this paper.

**Definition 1.** *The class* PD *function denotes the set of all continuous functions α* : R≥0 → R≥0 *satisfying α*(0) = 0*, and α*(*s*) > 0 *for all s* > 0*. A class* K *function is an increasing* PD *function. The class* K∞ *represents the set of all unbounded* K *functions [17].*

**Definition 2.** *A continuous function β* : R≥0 → R≥0 *is said to be of class* L *if β is non-increasing and tends to zero as its arguments tend to infinity [17].*

**Definition 3.** *Let the function f* : [0, +∞[−→ R*, the Liouville-Caputo derivative of the function f of order α is expressed in the form*

$$D\_a^c f(t) = \frac{1}{\Gamma(1-a)} \int\_0^t \frac{f'(s)}{(t-s)^a} ds,\tag{1}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [2,26–29].*

**Definition 4.** *Let the function f* : [0, +∞[−→ R*, the Riemann-Liouville derivative of the function f of order α is expressed in the form*

$$D\_{\alpha}^{RL}f(t) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int\_{0}^{t} \frac{f(s)}{(t-s)^{\alpha}}ds,\tag{2}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [2,26–30].*

**Definition 5.** *Let the function f* : [0, +∞[−→ R*, the Liouville-Caputo generalized derivative of the function f of order α is expressed in the form*

$$\left(D\_c^{a,\rho}f\right)(t) = \frac{1}{\Gamma(1-a)} \int\_0^t \left(\frac{t^\rho - s^\rho}{\rho}\right)^{-a} f'(s) \frac{ds}{s^{1-\rho}}\tag{3}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [2,26,28,29,31].*

**Definition 6.** *Let the function f* : [0, +∞[−→ R*, the left generalized derivative of the function f of order α is expressed in the form*

$$(D^{a,\rho}f)\left(t\right) = \frac{1}{\Gamma(1-a)} \left(\frac{d}{dt}\right) \int\_0^t \left(\frac{t^\rho - s^\rho}{\rho}\right)^{-a} f(s) \frac{ds}{s^{1-\rho}}\,. \tag{4}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [2,26,28,29,31].*

**Definition 7.** *Let us take the function f* : [0, +∞[−→ R*, the Caputo-Fabrizio fractional derivative of the function f of order α is expressed in the form*

$$D\_{\mathfrak{a}}^{CF}f(t) = \frac{M(\mathfrak{a})}{1-\mathfrak{a}} \int\_{0}^{t} f'(s) \exp\left(-\frac{\mathfrak{a}}{1-\mathfrak{a}}(t-s)\right) ds,\tag{5}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [22].*

**Definition 8.** *Let the function f* : [0, +∞[−→ R*, the Caputo-Fabrizio fractional derivative of the function f of order α is expressed in the form*

$$D\_{\alpha}^{ABC} f(t) = \frac{AB(\alpha)}{1 - \alpha} \int\_{0}^{t} f'(s) E\_{a} \left( -\frac{\alpha}{1 - \alpha} (t - s)^{a} \right) ds,\tag{6}$$

*for all t* > 0, *where the order α* ∈ (0, 1) *and* Γ(.) *is the gamma function [22,30].*

**Definition 9.** *Let us consider the function f* : [0, +∞[−→ R*, the Erdélyi-Kober fractional integral of the function f of order α* > 0*, η* > 0 *and γ* ∈ R *is expressed in the form*

$$M\_{\eta}^{\gamma,a}f(t) = \frac{t^{-\eta\left(\gamma+a\right)}}{\Gamma(a)} \int\_0^t \tau^{\eta\gamma} \left(t^{\eta} - \tau^{\eta}\right)^{a-1} f(\tau) d(\tau^{\eta}),\tag{7}$$

*for all t* > 0, *and* Γ(.) *is the gamma function [21].*

**Definition 10.** *Let us consider the function f* : [0, +∞[−→ R*, the Erdélyi-Kober fractional derivative of the function f of order α* > 0*, η* > 0 *and γ* ∈ R *is expressed in the form*

$$D\_{\eta}^{\gamma,\mathfrak{a}}f(t) = \prod\_{j=1}^{n} \left(\gamma + j + \frac{1}{\eta}\frac{d}{dt}\right) \left(I\_{\eta}^{\gamma+a,n-\mu}f(t)\right),\tag{8}$$

*for all t* > 0, *and where n* − 1 < *α* ≤ *n [21].*

Some special cases can be recovered with the above definitions. In Definition 8, when *ρ* = 1, we recover the Liouville-Caputo fractional derivative. In Definition 9, when *ρ* = 1, we recover the Riemann-Liouville fractional derivative. In Definition 10, when *γ* = −*α* and *η* = 1, we obtain the relation existing between Erdélyi-Kobar fractional derivative and Riemann-Liouville fractional derivative expressed in the form *D*−*α*,*α* 1 *f*(*t*) = *t αDα*,1 *f*(*t*).

The Laplace transform will be used for solving a class of the fractional differential equations. The *ρ*-Laplace transform was recently introduced by Fahd et al. in order to solve differential equations in the frame of conformable derivatives to extend the possibility of working in a large class of functions [2]. We encourage readers to refer to [2] for more detailed information about *ρ*-Laplace transforms and their applications.

The *ρ*-Laplace transform of the generalized fractional derivative in the Liouville-Caputo sense is expressed in the following form

$$\mathcal{L}\_{\rho}\left\{ \left( D\_{\varepsilon}^{a,\rho} f \right)(t) \right\} = s^{a} \mathcal{L}\_{\rho} \left\{ f(t) \right\} - s^{a-1} f(0), \tag{9}$$

The *ρ*-Laplace transform of the function *f* is given in the form

$$\mathcal{L}\_{\mathcal{P}}\left\{f(t)\right\}(s) = \int\_0^\infty e^{-s\frac{t^{\rho}}{\rho}} f(t) \frac{dt}{t^{1-\rho}}.\tag{10}$$

**Definition 11.** *The Mittag–Leffler function with two parameters is defined as the following series*

$$E\_{\mathfrak{a}, \mathfrak{F}}\left(z\right) = \sum\_{k=0}^{\infty} \frac{z^k}{\Gamma(\mathfrak{a}k + \beta)}\tag{11}$$

*where α* > 0, *β* ∈ R *and z* ∈ C*. The classical exponential function is obtained with α* = *β* = 1. *Here we see that when α and β are strictly positive, the series is convergent [14].*

#### **3. New Stability Notion of the Fractional Differential Equations**

In this section, we introduce a new stability notion for the fractional differential equation with exogenous input described by the left generalized fractional derivative. Historically, the fractional input stability and the Mittag-Leffler input stability of the fractional differential equation represented by the Liouville-Caputo fractional derivative were stated in previous works [5,18]. Moreover, the idea of a discrete version of fractional derivatives is studied in the seminal work [32]. The Lyapunov characterizations of these new stability notions have been provided in [15,18]. In this section, we extend the Mittag-Leffler input stability involving the left generalized fractional derivative. We provide some modifications in the structure of the definitions, however the idea is not modified. The new stability notion addressed in this paper is called the generalized Mittag-Leffler input stability. In the literature there exist many stability notions related to the fractional differential equations without exogenous inputs such as the asymptotic stability [7,14], the practical stability [12,33], the Mittag-Leffler stability [7] and many others notions. Let us consider the fractional differential equations with exogenous inputs. In fractional calculus, we have not seen a lot of work related to the stability of the

fractional differential equations with inputs. The stabilization problems [3] of the fractional differential equations with exogenous inputs is one of the most popular notion existing in the known literature. The challenge consists of finding possible values of the input under which the trivial solution of the obtained fractional differential equation is asymptotically stable. In this paper, we adopt a new method. Let us consider the fractional differential equation with exogenous input described by the left generalized fractional derivative

$$D^{\alpha,\rho}\mathbf{x} = A\mathbf{x} + Bu,\tag{12}$$

where *x* ∈ R*n* is a state variable, the matrix *A* ∈ R*n*×*n* satisfies the property |arg(*λ*(*A*))| > *απ* 2 , the matrix *B* ∈ R*n*×*n* and *u* ∈ R*n* represents the exogenous input. The initial boundary condition is defined by *I*1−*α*,*ρx* (0) = *x*0. Firstly, we give the analytical solution of the fractional differential equation with exogenous input described by the left generalized fractional derivative defined by Equation (12). Applying the *ρ*-Laplace transform to both sides of Equation (12), we obtain

$$\begin{aligned} \mathcal{L}\_{\rho} \left( D^{a,\rho} \mathbf{x}(t) \right) - \left( I^{1-a,\rho} \mathbf{x} \right)(0) &=& A \mathcal{L}\_{\rho} \left( \mathbf{x}(t) \right) + \mathcal{L}\_{\rho} \left( Bu \right) \\ \mathbf{s}^{a} \vec{\mathbf{x}}(s) - \mathbf{x}\_{0} &=& A \vec{\mathbf{x}}(s) + B \vec{u}(s) \end{aligned}$$

$$\begin{aligned} \vec{\mathbf{x}}(s) - \mathbf{x}\_{0} \left( \mathbf{s}^{a} I\_{n} - A \right)^{-1} &=& \left( s^{a} I\_{n} - A \right)^{-1} B \vec{u}(s), \end{aligned} \tag{13}$$

where *x*¯ denotes the Laplace transform of the function *x* and *u*¯ denotes the Laplace transform of the function *u*. Applying the inverse of the *ρ*-Laplace transform to both sides of Equation (13), we obtain

$$\begin{split} x(t) &= \quad x\_0 \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a-1} E\_{\mathfrak{a}, \mathfrak{a}} \left( A \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a} \right) \\ &+ \int\_{t\_0}^{t} \left( \frac{t^{\rho} - s^{\rho}}{\rho} \right)^{a-1} E\_{\mathfrak{a}, \mathfrak{a}} \left( A \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a} \right) B u(s) \frac{ds}{s^{1-\rho}}. \end{split} \tag{14}$$

Applying the Euclidean norm to both sides of Equation (14), we obtain the following relationship

$$\begin{split} \|\mathbf{x}(t)\| &\leq \quad \|\mathbf{x}\_{0}\| \left\| \left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)^{\alpha-1} E\_{\mathbf{a},\mathbf{a}}\left(A\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)^{\alpha}\right) \right\| \\ &+ \quad \|\|B\|\|\,\|\mathbf{u}\|\| \int\_{t\_{0}}^{t} \left\| \left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1} E\_{\mathbf{a},\mathbf{a}}\left(A\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)^{\alpha}\right) \frac{ds}{s^{1-\rho}} \right\|. \end{split} \tag{15}$$

From assumption |arg(*λ*(*A*))| > *απ* 2 , there exist a positive number *M* > 0 [4,18,34] such that, we have

$$\int\_{t\_0}^t \left\| \left( \frac{t^\rho - s^\rho}{\rho} \right)^{a-1} E\_{a,a} \left( A \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^a \right) \frac{ds}{s^{1-\rho}} \right\| \leq M. \tag{16}$$

This inequality is a classic condition in stability analysis of fractional derivatives shown in [34]. Finally, the solution of the fractional differential Equation (12) described by the left generalized fractional derivative with exogenous input satisfies the following relationship

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{x}\_0\| \left\| \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^{a-1} E\_{\mathbf{z},a} \left( A \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^a \right) \right\| + \|\|B\|\| \|\mathbf{u}\|\|\,\mathcal{M}.\tag{17}$$

We first notice, when the exogenous input of the fractional differential Equation (12) described by the left generalized fractional derivative is null *u* = 0. The solution obtained in Equation (17) becomes

$$\|\mathbf{x}(t)\| \le \|\mathbf{x}0\| \left\| \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a-1} E\_{\mathbf{a}, \mathbf{a}} \left( A \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a} \right) \right\|. \tag{18}$$

It corresponds to the classical Mittag-Leffler stability of the trivial solution of the fractional differential equation without input *Dα*,*ρx* = *Ax* described by the left generalized fractional derivative.

Secondly, let us consider the exogenous input converging to zero when *t* tends to infinity. We know when the identity |arg(*λ*(*A*))| > *απ* 2 is held, we have

$$E\_{a,a}\left(A\left(\frac{t^{\rho}-t\_0^{\rho}}{\rho}\right)^a\right)\longrightarrow 0.\tag{19}$$

From which we obtain *x*(*t*) −→ 0. Summarizing, we have the following

$$\|\|\mathbf{u}\|\|\longrightarrow 0 \implies \|\mathbf{x}(t)\|\|\longrightarrow 0. \tag{20}$$

In other words, a converging input generates a converging state. This property is called the CICS property, derived in [15,18].

Finally, let us consider the exogenous input bounded ( *u* ≤ *η*). The solution of the fractional differential Equation (12) described by the left generalized fractional derivative satisfies the following relationship

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{x}\_0\| \left\| \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^{a-1} E\_{\mathbf{a},a} \left( A \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^a \right) \right\|\,\tag{21}$$

Furthermore, we consider the function *t ρ*−*t ρ* 0 *ρ α*−1 *Eα*,*α A t ρ*−*t ρ* 0 *ρ α* ∈ L, thus there exist *σ* > 0 such that we have the following relationship

$$\left(\frac{t^{\rho} - t\_0^{\rho}}{\rho}\right)^{a-1} E\_{a,a}\left(A\left(\frac{t^{\rho} - t\_0^{\rho}}{\rho}\right)^a\right) \le \sigma. \tag{22}$$

Thus Equation (21) can be expressed in the following form

$$\|\|x(t)\|\| \le \|\|x\_0\|\|\sigma + \|\|B\|\|\eta M. \tag{23}$$

Thus, the solution of the fractional differential given in Equation (12) described by the left generalized fractional derivative is bounded as well. A bounded input for Equation (12), we obtain a bounded state for Equation (12). This property is called the BIBS property, created in [15,18]. The objective in this paper is to introduce a new stability notion taking into account a few things; namely the converging input, the converging state, the bounded input bounded state and the generalized Mittag-Leffler stability of the trivial solution of the unforced fractional differential equation. This stability notion we refer to as the generalized Mittag-Leffler input stability. In other words, the fractional differential equation described by the Left generalized fractional derivative is said generalized Mittag-Leffler stable, when its solution is bounded by a class KL function (contain a Mittag-Leffler function) plus a class K∞ function proportional to the input of the given fractional differential equation. A similar derivation leading to Equation (23) has also recently been applied to the study of fixed-time stability in [35].

In this section, we introduce new stability notion in the context of the fractional differential equations described by the left generalized fractional derivative operator. The fractional differential equation under consideration is expressed in the following form

$$D^{\mathfrak{a},\rho}\mathfrak{x} = f(t, \mathfrak{x}, \mathfrak{u})\tag{24}$$

where the function *f* : R+ × R × R*m* → R*n* is a continuous locally Lipschitz function, the function *x* ∈ R*n* is a state variable, and furthermore the condition *f*(*t*, 0, 0) = 0 is held. Given an initial condition *x*0 ∈ R*n*, the solution of the fractional differential Equation (24) starting at *x*0 at time *t*0 is represented by *x*(.) = *x*(., *x*0, *u*).

**Definition 12.** *The solution x* = 0 *of the fractional differential equation described by the left generalized fractional derivative defined by Equation (24) is said to be generalized Mittag-Leffler stable if, for any initial condition x*0 *and initial time t*0*, its solution satisfies the following condition*

$$||\mathbf{x}(t, \|\mathbf{x}\_0\|)|| \le \left[ m(\|\mathbf{x}\_0\|) \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^{a-1} E\_{a,a} \left( \eta \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^a \right) \right]^a,\tag{25}$$

*where a* > 0*, η* < 0 *and the function m is locally Lipschitz on a domain contained in* R*n and satisfies m*(0) = 0 *[7,14].*

In the following definition, we introduce the definition of the generalized Mittag-Leffler input stability in the context of the fractional differential equation described by the left generalized fractional derivative operator.

**Definition 13.** *The fractional differential equation described by the left generalized fractional derivative defined by Equation (24) is said to be generalized Mittag-Leffler input stable if, there exist a class γ* ∈ K∞ *function such that for any initial condition x*0 *, its solution satisfies the following condition*

$$\|\|\mathbf{x}(t, \|\mathbf{x}\_0\|)\|\| \le \left[ m(\|\mathbf{x}\_0\|) \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^{a-1} E\_{a,a} \left( \eta \left( \frac{t^\rho - t\_0^\rho}{\rho} \right)^a \right) \right]^a + \gamma \left( \|\|\boldsymbol{u}\|\|\_\infty \right), \tag{26}$$

*where a* > 0 *and η* < 0*.*

From the condition *γ* ∈ K∞, we get *γ*(0) = 0. We recover Definition 13. That is, the generalized Mittag-Leffler input stability of the fractional differential given in Equation (24) implies the generalized Mittag-Leffler stability of the trivial solution of the fractional differential equation with no input defined by *Dα*,*ρx* = *f*(*t*, *x*, 0). From the fact *γ* ∈ K∞, when the input is bounded implies the function *γ* ( *u* ∞) is bounded as well. Thus the state of the fractional differential Equation (24) is bounded too. We thus recover BIBS. From the fact *γ* ∈ K∞, a converging input causes *γ* ( *u* ∞) to converge. Thus the state of the fractional differential Equation (24) is converging as well. We thus recover CICS. In conclusion we can say that Definition 12 is well posed.

#### **4. Lyapunov Characterizations of the Generalized Mittag-Leffler input Stability**

In this section, we give the Lyapunov characterization of the generalized Mittag-Leffler input stability of the fractional differential equation. We know, it is not always trivial to get the analytical solution of the fractional differential equation with exogenous inputs. An alternative is to propose a method of analyzing the Mittag-Leffler input stability. The method consist of calculating the fractional energy of the fractional differential equation along the trajectories. In other words, we use the Lyapunov direct method.

**Theorem 1.** *Let us consider that there exists a positive function V* : R+ × R*n* −→ R *continuous and differentiable, and a class* K∞ *function χ*1 *and class* K *functions χ*2*, χ*3 *satisfying the following assumptions:*

*Then the fractional differential Equation (24) described by the left generalized fractional derivative is generalized Mittag-Leffler input stable.*

**Proof.** Summarizing [18], combining Assumption (1) and Assumption (2), we have

$$\begin{array}{rcl} \left\|\mathbf{x}\right\|^{a} &\leq& V(\mathbf{x},t) \leq \boldsymbol{\alpha}\_{1} \circ \boldsymbol{\alpha}\_{2} \left(\left\|\boldsymbol{\mu}\right\|\right) \\\left\|\mathbf{x}\right\| &\leq& \left(\boldsymbol{\alpha}\_{1} \odot \boldsymbol{\alpha}\_{2} \left(\left\|\boldsymbol{\mu}\right\|\right)\right)^{1/a} \\\left\|\mathbf{x}\right\| &\leq& \gamma \left(\left\|\boldsymbol{\mu}\right\|\right), \end{array} \tag{27}$$

where the function *γ* ( *u* ) = (*α*1 ◦ *α*2 ( *u* ))1/*a* ∈ K∞.

From Assumption (2), using an exponential form of the Lyapunov function in, there exist positive constant such that, we have

$$\begin{split} \|\mathbf{x}\| \ge \chi\_2((|\boldsymbol{u}| )) &\implies D\_\mathbf{c}^{\mathbf{a},\boldsymbol{\rho}} V(t, \mathbf{x}) \le -\chi\_3((\|\mathbf{x}\| )) \\ &\implies D\_\mathbf{c}^{\mathbf{a},\boldsymbol{\rho}} V(t, \mathbf{x}) \le -\chi\_3((\|\mathbf{x}\| )) \le -kV(\mathbf{x}, t). \end{split} \tag{28}$$

It follows from Equation (28), the following inequality

$$\|\mathbf{x}\|\!^{a}\leq&V(t,\mathbf{x})\leq V(\|\mathbf{x}\_{0}\|\|)\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)^{a-1}E\_{a,a}\left(-k\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)\right)$$

$$\|\mathbf{x}\|\;^{a}\leq&\left\{V(\|\mathbf{x}\_{0}\|\|)\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)^{a-1}E\_{a,a}\left(-k\left(\frac{t^{\rho}-t\_{0}^{\rho}}{\rho}\right)\right)\right\}^{1/a}.\tag{29}$$

Finally, combining Equations (27) and (29), we obtain

$$\|\mathbf{x}\| \le \max \left\{ \left\{ V(\|\mathbf{x}\|) \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right)^{a-1} E\_{\mathbf{a}, \mathbf{a}} \left( -k \left( \frac{t^{\rho} - t\_0^{\rho}}{\rho} \right) \right) \right\}^{1/a} : \gamma \left( \|\mathbf{u}\| \right) \right\}. \tag{30}$$

Thus the fractional differential equation defined by Equation (24) is generalized Mittag-Leffler input stable.

The second characterization is a consequence of the first theorem. It is more simplest to be applied in many cases. We have the following characterization.

**Theorem 2.** *Let there exist a positive function V* : R+ × R*n* −→ R *continuous and differentiable, and a class* K∞ *of functions χ*1 *and class* K∞ *function γ satisfying the following assumption:*

*Then fractional differential Equation (24) described by the left generalized fractional derivative is generalized Mittag-Leffler input stable stable.*

**Proof.** From Assumption (2), we have the following relationships

$$\begin{array}{llll}D\_{\varepsilon}^{a,\rho}V(t,x) & \leq & -kV(\mathbf{x},t) + \gamma\left(\|\mathbf{u}\|\right) \\ D\_{\varepsilon}^{a,\rho}V(t,x) & \leq & -\left(1-\theta\right)kV(\mathbf{x},t) - \theta kV(\mathbf{x},t) + \gamma\left(\|\mathbf{u}\|\right), \end{array} \tag{31}$$

where *θ* ∈ (0, 1). We have

$$-\theta kV(\mathbf{x},t) + \gamma \left(\|\mathbf{u}\|\right) \le 0 \implies D\_c^{\mathbf{a},\rho} V(t,\mathbf{x}) \le -\left(1-\theta\right) kV(\mathbf{x},t)$$

$$V(\mathbf{x},t) \ge \frac{\gamma \left(\|\mathbf{u}\|\right)}{\theta k} \implies D\_c^{\mathbf{a},\rho} V(t,\mathbf{x}) \le -\left(1-\theta\right) kV(\mathbf{x},t). \tag{32}$$

From first assumption, it yields that

$$\theta k \chi\_1 \left( \|\mathbf{x}\| \right) \ge \gamma \left( \|\mathbf{u}\| \right) \implies D\_\mathbf{c}^{a,\rho} V(t, \mathbf{x}) \le -\left(1 - \theta\right) k V(\mathbf{x}, t).$$

Thus the fractional differential equation described by the left generalized fractional derivative is Mittag-Leffler input stable.

#### **5. Practical Applications**

In this section, we give many practical applications of the Mittag-Leffler input stability of the fractional differential equation described by the generalized fractional derivative using the Lyapunov characterizations.

Let us consider the fractional differential equation described by the left generalized fractional differential equation defined by

$$\begin{cases} \begin{array}{c} D\_c^{a,\rho} \mathbf{x}\_1 = -\mathbf{x}\_1 + \frac{1}{2} \mathbf{x}\_2 + \frac{1}{2} \boldsymbol{\mu}\_1 \\\ D\_c^{a,\rho} \mathbf{x}\_2 = -\mathbf{x}\_2 + \frac{1}{2} \boldsymbol{\mu}\_2 \end{array} \tag{33}$$

where *x* = (*x*1, *x*2) ∈ R2 and *u* = (*u*1, *u*2) ∈ R2 represents the exogenous input. Let us take the Lyapunov function defined by *V*(*x*) = 1 2 *x*2 1 + *x*2 2 . The left generalized fractional derivative of the Lyapunov function along the trajectories is given by

$$\begin{aligned} D\_\varepsilon^{\mu,\rho} V(t, \mathbf{x}) &= \left| -\mathbf{x}\_1^2 + \frac{1}{2} \mathbf{x}\_1 \mathbf{x}\_2 + \frac{1}{2} \mathbf{x}\_1 u\_1 - \mathbf{x}\_2^2 + \frac{1}{2} \mathbf{x}\_2 u\_2 \right| \\ &\leq \left| -\frac{1}{2} \mathbf{x}\_1^2 - \frac{1}{2} \mathbf{x}\_2^2 + \frac{1}{4} \left\| \boldsymbol{u} \right\|^2 \\ &\leq \left| -V(\mathbf{x}) + \frac{1}{4} \left\| \boldsymbol{u} \right\|^2 \end{aligned} \right. \tag{34}$$

Consider *γ*( *u* ) = 1 4 *u* 2 ∈ K∞. It follows from Theorem 2, the fractional differential equation described by the left generalized fractional derivative given in Equation (33) is Mittag-Leffler input stable. Thus, the origin of the unforced fractional differential equation obtained with *u* = (*u*1, *u*2)=(0, 0)

$$\begin{cases} \begin{array}{c} D\_{\varepsilon}^{a,\rho} \mathbf{x}\_{1} = -\mathbf{x}\_{1} + \frac{1}{2} \mathbf{x}\_{2} \\\ D\_{\varepsilon}^{a,\rho} \mathbf{x}\_{2} = -\mathbf{x}\_{2} \end{array} \end{cases} \tag{35}$$

where *x* = (*x*1, *x*2) ∈ R2, is Mittag-Leffler stable.

Let us consider the fractional differential equation described by the left generalized fractional differential equation defined by

$$\begin{cases} \begin{array}{c} D\_{\varepsilon}^{a,\rho} \mathbf{x}\_{1} = -\mathbf{x}\_{1} + \mathbf{x}\_{2} + \boldsymbol{\mu}\_{1} \\\ D\_{\varepsilon}^{a,\rho} \mathbf{x}\_{2} = -\mathbf{x}\_{2} + \boldsymbol{\mu}\_{2} \end{array} \end{cases} \tag{36}$$

where *x* = (*x*1, *x*2) ∈ R2 and *u* = (*u*1, *u*2) ∈ R2 represents the exogenous input. Let the Lyapunov function defined by *V*(*x*) = 1 2 *x*2 1 + *x*2 2 . The left generalized fractional derivative of the Lyapunov function along the trajectories is given by

$$\begin{aligned} \left| D\_{\varepsilon}^{\mu,\rho} V(t, \mathbf{x}) \right| &= \left| -\mathbf{x}\_{1}^{2} + \mathbf{x}\_{1} \mathbf{x}\_{2} + \mathbf{x}\_{1} \mu\_{1} - \mathbf{x}\_{2}^{2} + \mathbf{x}\_{2} \mu\_{2} \right| \\ &\leq \left| -\mathbf{x}\_{1}^{2} + \frac{1}{2} \mathbf{x}\_{1}^{2} + \frac{1}{2} \mathbf{x}\_{1}^{2} - \mathbf{x}\_{2}^{2} + \frac{1}{2} \mathbf{x}\_{2}^{2} + \frac{1}{2} \mathbf{x}\_{2}^{2} + \left| \left| \mu \right| \right|^{2} \\ &\leq \left\| \left| u \right| \right\|^{2}. \end{aligned} \tag{37}$$

Let *γ*( *u* ) = 1 4 *u* 2 ∈ K∞. It follows from Theorem 2, the fractional differential equation described by the left generalized fractional derivative in Equation (35) is bounded as well [36].

Let us consider the electrical RL circuit described by the left generalized fractional differential equation defined by

$$D\_c^{a, \rho} \mathbf{x} = -\frac{\sigma^{1-a} R}{L} \mathbf{x} + \mathbf{u} \tag{38}$$

with the initial boundary condition defined by *x*(0) = *x*0. The parameter *σ* is associated with the temporal components in the differential equation. *u* represents the exogenous input. Let us take the Lyapunov function defined by *V*(*x*) = 1 2 *x* 2 . The left generalized fractional derivative of the Lyapunov function along the trajectories is given by

$$\begin{array}{rcl}D\_c^{a,\rho}V(t, \mathbf{x}) &=& \frac{\sigma^{1-a}R}{L}\mathbf{x}^2 + \mathbf{x}u\\ &\leq& -\frac{\sigma^{1-a}R}{L}||\mathbf{x}||^2 + \frac{1}{2}||\mathbf{x}||^2 + \frac{1}{2}||u||\\ &\leq& -\left(\frac{\sigma^{1-a}R}{L} - \frac{1}{2}\right)||\mathbf{x}||^2 + \frac{1}{2}||u||\,. \end{array} \tag{39}$$

Let us consider *k* = *σ*1−*αR L* − 1 2 and *θ* ∈ (0, 1). We have the following relationship

$$\left|D\_{\varepsilon}^{\mu,\rho}V(t,\mathbf{x})\right| \le \left|-(1-\theta)k\left\|\mathbf{x}\right\|^2 + k\theta\left\|\mathbf{x}\right\|^2 + \frac{1}{2}\left\|\mathbf{u}\right\|\tag{40}$$

From Theorem 1, if *x* ≥ *u* 2*kθ* , we have *Dα*,*ρ c V*(*t*, *x*) ≤ −(1 − *θ*)*k x* 2 . Thus, the electrical RL circuit (36) is Mittag-Leffler input stable form.

Let us consider the fractional differential equation described in [4] by the left generalized fractional differential equation defined by

$$D\_c^{a, \rho} \mathbf{x} = -\mathbf{x} + \mathbf{x}u,\tag{41}$$

where *x* ∈ R*n* is a state variable. *u* represents the exogenous input. Let's the Lyapunov function defined by *V*(*x*) = 1 2 *x* 2 . The left generalized fractional derivative of the Lyapunov function along the trajectories is given by

$$\begin{array}{rcl}D\_c^{a,\rho}V(t, \mathbf{x}) &=& -\mathbf{x}^2 + \mathbf{x}^2 u\\ &\leq& -\left\|\mathbf{x}\right\|^2 + \left\|\mathbf{x}\right\|^2 \left\|u\right\|\\ &\leq& -\left(1 - \left\|u\right\|\right)\left\|\mathbf{x}\right\|^2. \end{array} \tag{42}$$

We can observe, when we pick *u* > 1, using *α*-integration, the state *x* of Equation (42) diverge as *t* tends to infinity. Then the fractional differential equation is not BIBS. Thus, the fractional differential Equation (41) is not, in general, Mittag-Leffler input stable.

#### **6. Conclusions**

In this paper, the Mittag-Leffler input stability has been thoroughly investigated. We have tried to motivate this study with its connection to many real world applications known to use Mittag-Leffler functions. We also address the Lyapunov characterization of the fractional differential equations. In doing so, we have created a further Lyapunov characterization which is more useful. Finally, we give some numerical examples to help illustrate the work that was accomplished in this paper. Analyzing the generalized Mittag-Leffer input stability of the fractional differential equations without decomposing it can be non trivial. The possible issue is to decompose it as a cascade of triangular equations and to find a method to analyze the generalized Mittag-Leffer input stability of the obtained fractional differential equation. In other words, finding the conditions for the generalized Mittag-Leffer input stability of the fractional differential cascade equations will be subject of future works.

**Author Contributions:** N.S. was responsible for methodology, validation, conceptualization and formal analysis. G.S. was responsible for analysis, writing, review and draft preparation.

**Funding:** This research received no external funding. We thank MDPI publishers for waiving our APC fee.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function**

#### **Lei Shi 1, Hari Mohan Srivastava 2,3, Muhammad Arif 4,\* and Shehzad Hussain 4 and Hassan Khan 4**

Received: 27 March 2019 ; Accepted: 22 April 2019; Published: 26 April 2019

**Abstract:** In the current article, we consider certain subfamilies S∗ *e* and C*e* of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family S∗ *e* in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.

**Keywords:** subordinations; exponential function; Hankel determinant

#### **1. Introduction and Definitions**

Let the collection of functions *f* that are holomorphic in Δ = {*z* ∈ C : |*z*| < 1} and normalized by conditions *f* (0) = *f* (0) − 1 = 0 be denoted by the symbol A. Equivalently; if *f* ∈ A, then the Taylor-Maclaurin series representation has the form:

$$f(z) = z + \sum\_{k=2}^{\infty} a\_k \, z^k \, \big( z \in \Delta \big) \,. \tag{1}$$

Further, let we name by the notation S the most basic sub-collection of the set A that are univalent in Δ. The familiar coefficient conjecture for the function *f* ∈ S of the form (1) was first presented by Bieberbach [1] in 1916 and proved by de-Branges [2] in 1985. In 1916-1985, many mathematicians struggled to prove or disprove this conjecture and as result they defined several subfamilies of the set S of univalent functions connected with different image domains. Now we mention some of them, that is; let the notations S∗, C and K, shows the families of starlike, convex and close-to-convex functions respectively and are defined as:

$$\begin{array}{rcl} \mathcal{S}^{\*} &=& \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec \frac{1+z}{1-z}, \ (z \in \Delta) \right\}, \\\mathcal{C}^{\*} &=& \left\{ f \in \mathcal{S} : \frac{(zf'(z))'}{f'(z)} \prec \frac{1+z}{1-z}, \ (z \in \Delta) \right\}, \\\mathcal{K}^{\*} &=& \left\{ f \in \mathcal{S} : \frac{f'(z)}{g'(z)} \prec \frac{1+z}{1-z}, \ \text{for } g'(z) \in \mathcal{C}, \ (z \in \Delta) \right\}. \end{array}$$

,

where the symbol " ≺ " denotes the familiar subordinations between analytic functions and is define as; the function *h*1 is subordinate to a function *h*2, symbolically written as *h*1 ≺ *h*2 or *h*1 (*z*) ≺ *h*2 (*z*), if we can find a function *w*, which is holomorphic in Δ with *w* (0) = 0 & |*w*(*z*)| < 1 such that *h*1 (*z*) = *h*2 (*w* (*z*)) (*z* ∈ Δ). Thus, *h*1(*z*) ≺ *h*2(*z*) implies *h*1(Δ) ⊂ *h*2(Δ). In case of univalency of *h*1 in Δ, then the following relation holds:

$$h\_1(z) \prec h\_2(z) \quad (z \in \Delta) \quad \Longleftrightarrow \quad h\_1(0) = h\_2(0) \quad \text{and} \quad h\_1(\Delta) \subset h\_2(\Delta).$$

In [3], Padmanabhan and Parvatham in 1985 defined a unified families of starlike and convex functions using familiar convolution with the function *z*/ (1 − *z*) *a* , for all *a* ∈ R. Later on, Shanmugam [4] generalized the idea of paper [3] and introduced the set

$$\mathcal{S}\_h^\*\left(\phi\right) = \left\{ f \in \mathcal{A} : \frac{z\left(f \ast h\right)'}{\left(f \ast h\right)} \prec \phi\left(z\right), \quad (z \in \Delta) \right\},$$

where "∗" stands for the familiar convolution, *φ* is a convex and *h* is a fixed function in A. We obtain the families S∗ (*φ*) and C (*φ*) when taking *z*/ (1 − *z*) and *z*/ (1 − *z*) 2 instead of *h* in S∗ *h* (*φ*) respectively. In 1992, Ma and Minda [5] reduced the restriction to a weaker supposition that *φ* is a function, with *Reφ* > 0 in Δ, whose image domain is symmetric about the real axis and starlike with respect to *φ*(0) = 1 with *φ* (0) > 0 and discussed some properties. The set S∗ (*φ*) generalizes various subfamilies of the set A, for example:

$$\mathcal{S}\_{\varepsilon}^{\*} = \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f'(z)} \prec c^{z}, \ (z \in \Delta) \right\},\tag{2}$$

or, equivalently

$$\mathcal{S}\_{\varepsilon}^{\*} = \left\{ f \in \mathcal{S} : \left| \log \frac{zf'(z)}{f(z)} \right| < 1, \ (z \in \Delta) \right\}. \tag{3}$$

They investigated some interesting properties and also links these classes to the familiar subfamilies of the set S. In [9], the authors choose the function *f* (*z*) = *z* + 1 4 *z*2 (Figure 1) and then sketch the following figure of the function class S∗ *e* by using the form (3) as:

**Figure 1.** The figure of the function class S∗ 1 for *f* (*z*) = *z* + 1 4 *z*2.

Similarly, by using Alexandar type relation in [9], we have;

$$\mathcal{C}\_{\mathfrak{c}} = \left\{ f \in \mathcal{S} : \frac{\left( zf'(z)\right)'}{f'\left(z\right)} \prec \mathfrak{c}^z, \ (z \in \Delta) \right\}. \tag{4}$$

From the above discussion, we conclude that the families S∗ *e* and C*e* considered in this paper are symmetric about the real axis.

For given parameters *q*, *n* ∈ N = {1, 2, . . .}, the Hankel determinant *Hq*,*n* (*f*) was defined by Pommerenke [10,11] for a function *f* ∈ S of the form (1) as follows:

$$H\_{q,n}\left(f\right) = \begin{vmatrix} a\_{n} & a\_{n+1} & \dots & a\_{n+q-1} \\ a\_{n+1} & a\_{n+2} & \dots & a\_{n+q} \\ \vdots & \vdots & \dots & \vdots \\ a\_{n+q-1} & a\_{n+q} & \dots & a\_{n+2q-2} \end{vmatrix} \cdot \tag{5}$$

The concept of Hankel determinant is very useful in the theory of singularities [12] and in the study of power series with integral coefficients. For deep insight, the reader is invited to read [13–15]. Specifically, the absolute sharp bound of the functional *H*2,2 (*f*) = *a*2*a*4 − *a*2 3 for each of the sets S∗ and C were proved by Janteng et al. [16,17] while the exact estimate of this determinant for the family of close-to-convex functions is still unknown (see, [18]). On the other side for the set of Bazileviˇc functions, the sharp estimate of |*H*2,2 (*f*)| was given by Krishna et al. [19]. Recently, Srivastava and his coauthors [20] found the estimate of second Hankel determinant for bi-univalent functions involving symmetric *q*-derivative operator while in [21], the authors discussed Hankel and Toeplitz determinants for subfamilies of *q*-starlike functions connected with a general form of conic domain. For more literature see [22–29]. The determinant with entries from (1)

$$H\_{3,1}(f) = \begin{vmatrix} 1 & a\_2 & a\_3 \\ a\_2 & a\_3 & a\_4 \\ a\_3 & a\_4 & a\_5 \end{vmatrix}$$

is known as Hankel determinant of order three and the estimation of this determinant |*H*3,1 (*f*)| is very hard as compared to derive the bound of |*H*2,2 (*f*)|. The very first paper on *H*3,1 (*f*) visible in 2010 by Babalola [30] in which he got the upper bound of *H*3,1 (*f*) for the families of S∗ and C. Later on, many authors published their work regarding |*H*3,1 (*f*)| for different sub-collections of univalent functions, see [8,31–36]. In 2017, Zaprawa [37] upgraded the results of Babalola [30] by giving

$$|H\_{3,1}(f)| \le \begin{cases} 1, & \text{for} \quad f \in \mathcal{S}^\*,\\ \frac{49}{540}, & \text{for} \quad f \in \mathcal{C}, \end{cases}$$

and claimed that these bounds are still not best possible. Further for the sharpness, he examined the subfamilies of S∗ and C consisting of functions with *m*-fold symmetry and obtained the sharp bounds. Moreover this determinant was further improved by Kwon et al. [38] and proved |*H*3,1 (*f*)| ≤ 8/9 for *f* ∈ S∗, yet not best possible. The authors in [39–41] contributed in similar direction by generalizing different classes of univalent functions with respect to symmetric points. In 2018, Kowalczyk et al. [42] and Lecko et al. [43] got the sharp inequalities

$$|H\_{3,1}(f)| \le 4/135, \quad \text{and} \quad |H\_{3,1}(f)| \le 1/9,$$

for the recognizable sets K and S∗ (1/2) respectively, where the symbol S∗ (1/2) indicates to the family of starlike functions of order 1/2. Also we would like to cite the work done by Mahmood et al. [44] in which they studied third Hankel determinant for a subset of starlike functions in *q*-analogue. Additionally Zhang et al. [45] studied this determinant for the set S∗ *e* and obtained the bound |*H*3,1 (*f*)| ≤ 0.565.

In the present article, our aim is to investigate the estimate of |*H*3,1 (*f*)| for both the above defined classes S∗ *e* and C*e*. Moreover, we also study this problem for *m*-fold symmetric starlike and convex functions associated with exponential function.

#### **2. A Set of Lemmas**

Let P denote the family of all functions *p* that are analytic in D with (*p*(*z*)) > 0 and has the following series representation

$$p(z) = 1 + \sum\_{n=1}^{\infty} c\_n z^n \left( z \in \Delta \right). \tag{6}$$

**Lemma 1.** *If p* ∈ P *and has the form , then*

$$|c\_n| \quad \le \quad \text{2} \text{ for } n \ge 1,\tag{7}$$

$$\left|\mathfrak{c}\_{n+k} - \mu \mathfrak{c}\_n \mathfrak{c}\_k\right| \quad \prec \quad \text{2, for } 0 \le \mu \le 1,\tag{8}$$

$$|c\_m c\_n - c\_k c\_l| \quad \le \quad 4 \text{ for } m+n = k+l,\tag{9}$$

$$\left| c\_{n+2k} - \mu c\_n c\_k^2 \right| \quad \le \quad 2(1+2\mu); \text{ for } \mu \in \mathbb{R}, \tag{10}$$

$$\left| c\_2 - \frac{c\_1^2}{2} \right| \quad \le \quad 2 - \frac{\left| c\_1 \right|^2}{2}, \tag{11}$$

*and for complex number λ, we have*

$$\left|c\_2 - \lambda c\_1^2\right| \le 2 \max\left\{1, |2\lambda - 1|\right\}.\tag{12}$$

*For the inequalities* (7)*,* (11)*,* (8)*,* (10)*,* (9) *see [46] and* (12) *is given in [47]*.

#### **3. Improved Bound of** |*H*3,1 (*f*)| **for the Set** S∗ *e*

**Theorem 1.** *If f belongs to* S∗ *e , then*

$$|H\_{3,1}(f)| \le 0.50047781... $$

**Proof.** Let *f* ∈ S∗ *e* . Then we can write (2), in terms of Schwarz function as

$$\frac{zf'(z)}{f'(z)} = e^{w(z)}\_{-}$$

If *h* ∈ P, then it can be written in form of Schwarz function as

$$h\left(z\right) = \frac{1+w\left(z\right)}{1-w\left(z\right)} = 1 + c\_1 z + c\_2 z^2 + \dots + \dotsb$$

From above, we can get

$$w\left(z\right) = \frac{h\left(z\right) - 1}{h\left(z\right) + 1} = \frac{c\_1 z + c\_2 z^2 + c\_3 z^3 + \cdots}{2 + c\_1 z + c\_2 z^2 + c\_3 z^3 + \cdots}.$$

$$\frac{z f'\left(z\right)}{f\left(z\right)} = 1 + a\_2 z + \left(2a\_3 - a\_2^2\right) z^2 + \left(3a\_4 - 3a\_2 a\_3 + a\_2^3\right) z^3$$

$$+ \left(4a\_5 - 2a\_3^2 - 4a\_2 a\_4 + 4a\_2^2 a\_3 - a\_2^4\right) z^4 = 1 + p\_1 z + p\_2 z^2 + \cdots \ . \tag{13}$$

and from the series expansion of *w* along with some calculations, we have

$$e^{w(z)} = 1 + w\left(z\right) + \frac{\left(w\left(z\right)\right)^2}{2!} + \frac{\left(w\left(z\right)\right)^3}{3!} + \frac{\left(w\left(z\right)\right)^4}{4!} + \frac{\left(w\left(z\right)\right)^5}{5!} + \dotsb \dotsb$$

After some computations and rearranging, it yields

$$\begin{aligned} e^{w(z)} &= 1 + \frac{1}{2}c\_1 z + \left(\frac{c\_2}{2} - \frac{c\_1^2}{8}\right)z^2 + \left(\frac{c\_1^3}{48} + \frac{c\_3}{2} - \frac{c\_1 c\_2}{4}\right)z^3 \\ &+ \left(\frac{1}{384}c\_1^4 + \frac{1}{2}c\_4 - \frac{1}{8}c\_2^2 + \frac{1}{16}c\_1^2 c\_2 - \frac{1}{4}c\_1 c\_3\right)z^4 + \dotsb \end{aligned} \tag{14}$$

Comparing (13) and (14), we have

$$a\_2 \quad = \begin{array}{c} c\_1 \\ 2 \end{array} \tag{15}$$

$$a\_3 \quad = \quad \frac{1}{4} \left( c\_2 + \frac{c\_1^2}{4} \right),\tag{16}$$

$$a\_4 = -\frac{1}{6} \left( c\_3 + \frac{c\_1 c\_2}{4} - \frac{c\_1^3}{48} \right),\tag{17}$$

$$a\_5 \quad = \quad \frac{1}{4} \left( \frac{c\_1^4}{288} + \frac{c\_4}{2} + \frac{c\_1 c\_3}{12} - \frac{c\_1^2 c\_2}{24} \right). \tag{18}$$

From (5), the Third Hankel determinant can be written as

$$H\_{3,1}(f) = -a\_2^2 a\_5 + 2a\_2 a\_3 a\_4 - a\_3^3 + a\_3 a\_5 - a\_4^2.$$

Using (15), (16), (17) and (18), we get

$$H\_{3,1} \left( f \right) = \frac{35}{27648} \mathfrak{c}\_1^4 \mathfrak{c}\_2 + \frac{53}{6912} \mathfrak{c}\_1^3 \mathfrak{c}\_3 + \frac{\mathfrak{c}\_2 \mathfrak{c}\_4}{32} + \frac{19}{576} \mathfrak{c}\_1 \mathfrak{c}\_2 \mathfrak{c}\_3 - \frac{211}{331776} \mathfrak{c}\_1^6 - \frac{\mathfrak{c}\_2^3}{64} - \frac{3}{128} \mathfrak{c}\_1^2 \mathfrak{c}\_4 - \frac{13}{2304} \mathfrak{c}\_1^2 \mathfrak{c}\_2^2 - \frac{\mathfrak{c}\_3^2}{36} \mathfrak{c}\_2^2 \mathfrak{c}\_4$$

After rearranging, it yields

$$\begin{split} H\_{3,1}\left(f\right) &=& \frac{211}{165888}c\_{1}^{4}\left(c\_{2}-\frac{c\_{1}^{2}}{2}\right)+\frac{3}{64}c\_{4}\left(c\_{2}-\frac{c\_{1}^{2}}{2}\right)-\frac{c\_{1}c\_{3}}{96}\left(c\_{2}-\frac{c\_{1}^{2}}{2}\right)+\frac{1}{165888}c\_{1}^{3}\left(c\_{3}-c\_{1}c\_{2}\right) \\ &+& \frac{407}{165888}c\_{1}^{2}\left(c\_{1}c\_{3}-c\_{2}^{2}\right)-\frac{c\_{3}}{36}\left(c\_{3}-c\_{1}c\_{2}\right)-\frac{c\_{2}}{64}\left(c\_{4}-c\_{1}c\_{3}\right)-\frac{529}{165888}c\_{1}^{2}c\_{2}^{2}-\frac{c\_{2}^{3}}{64}.\end{split}$$

Using triangle inequality along with (7), (11), (8) and (9), provide us

$$\begin{split} |H\_{3,1}(f)| &\leq \ \ & \ \_{165988}^{211} |c\_1|^4 \left( 2 - \frac{|c\_1|^2}{2} \right) + \frac{3}{32} \left( 2 - \frac{|c\_1|^2}{2} \right) + \frac{|c\_1|}{48} \left( 2 - \frac{|c\_1|^2}{2} \right) + \frac{1}{82944} \left| c\_1 \right|^3 \\ & \quad + \frac{407}{41472} \left| c\_1 \right|^2 + \frac{1}{9} + \frac{1}{16} + \frac{529}{41472} \left| c\_1 \right|^2 + \frac{1}{8} . \end{split}$$

If we substitute |*c*1| = *x* ∈ [0, 2], we obtain a function of variable *x*. Therefore, we can write

$$\begin{array}{rcl}|H\_{3,1}\left(f\right)| &\leq& \frac{211}{165888}x^{4}\left(2-\frac{\mathbf{x}^{2}}{2}\right)+\frac{3}{32}\left(2-\frac{\mathbf{x}^{2}}{2}\right)+\frac{\mathbf{x}}{48}\left(2-\frac{\mathbf{x}^{2}}{2}\right)+\frac{1}{82944}x^{3}\\ &+\frac{407}{41472}\mathbf{x}^{2}+\frac{1}{9}+\frac{1}{16}+\frac{529}{41472}\mathbf{x}^{2}+\frac{1}{8}.\end{array}$$

The above function attains its maximum value at *x* = 0.64036035, which is

$$|H\_{3,1}(f)| \le 0.50047781... $$

Thus, the proof is completed.

**4. Bound of** |*H*3,1 (*f*)| **for the Set** C*e*

**Theorem 2.** *Let f has the form* (1) *and belongs to* C*e. Then*

$$|a\_2| \le \frac{1}{\frac{2}{1}} \tag{19}$$

$$|a\_3| \le \frac{1}{4'} \tag{20}$$

$$|a\_4| \le \underbrace{17}\_{\text{\\_}} \tag{21}$$

$$|a\overline{s}| \le \frac{7}{96}.\tag{22}$$

*The first three inequalities are sharp.*

**Proof.** If *f* ∈ C*e*, then we can write (4), in form of Schwarz function as

$$1 + \frac{z f'''(z)}{f'(z)} = \mathfrak{e}^{w(z)}.$$

From (1), we can write

$$\begin{aligned} 1 + \frac{z f^{\prime\prime}(z)}{f^{\prime}(z)} &= 1 + 2a\_2 z + \left(6a\_3 - 4a\_2^2\right) z^2 + \left(12a\_4 - 18a\_2 a\_3 + 8a\_2^3\right) z^3 \\ &+ \left(20a\_5 - 18a\_3^2 - 32a\_2 a\_4 + 48a\_2^2 a\_3 - 16a\_2^4\right) z^4 + \cdots \end{aligned} \tag{23}$$

By comparing (23) and (14), we get

$$a\_2 \quad = \begin{array}{c} c\_1 \\ \frac{c\_1}{4} \\ \ddots \end{array} \tag{24}$$

$$a\_3 = \ \frac{1}{12} \left( c\_2 + \frac{c\_1^2}{4} \right) ,\tag{25}$$

$$a\_4 = \frac{1}{24} \left( \frac{c\_1 c\_2}{4} + c\_3 - \frac{c\_1^3}{48} \right) . \tag{26}$$

$$a\_5 \quad = \ \frac{1}{20} \left( \frac{c\_1^4}{288} + \frac{c\_4}{2} + \frac{c\_1 c\_3}{12} - \frac{c\_1^2 c\_2}{24} \right). \tag{27}$$

Implementing (7), in (24) and (25), we have

$$|a\_2| \le \frac{1}{2} \quad \text{and} \quad |a\_3| \le \frac{1}{4} \cdot \frac{1}{2}$$

Reshuffling (26), we have

$$|a\_4| = \frac{1}{24} \left| \frac{5}{24} c\_1 c\_2 + \frac{c\_1}{24} \left( c\_2 - \frac{c\_1^2}{2} \right) + c\_3 \right|.$$

Application of triangle inequality and (7) and (11) leads us to

$$|a\_4| \le \frac{1}{24} \left\{ \frac{5}{12} \left| c\_1 \right| + \frac{\left| c\_1 \right|}{24} \left( 2 - \frac{\left| c\_1 \right|^2}{2} \right) + 2 \right\}.$$

If we insert |*c*1| = *x* ∈ [0, 2], then we get

$$|a\_4| \le \frac{1}{24} \left\{ \frac{5}{12}x + \frac{x}{24} \left( 2 - \frac{x^2}{2} \right) + 2 \right\} \dots$$

The overhead function has a maximum value at *x* = 2, thus

$$|a\_4| \le \frac{17}{144}.$$

Reordering (27), we have

$$|a\_{5}| = \frac{1}{20} \left| \frac{1}{2} \left( c\_{4} - \frac{c\_{1}^{2} c\_{2}}{48} \right) - \frac{c\_{1}^{2}}{96} \left( c\_{2} - \frac{c\_{1}^{2}}{3} \right) + \frac{c\_{1}}{12} \left( c\_{3} - \frac{c\_{1} c\_{2}}{4} \right) \right|. $$

By using triangle inequality along with (7), and (8), we get

$$|a\_5| \le \frac{7}{96}.$$

Equalities are obtain if we take

$$f(z) = \int\_0^z e^{l(t)} dt = z + \frac{1}{2}z^2 + \frac{1}{4}z^3 + \frac{17}{144}z^4 + \frac{19}{360}z^5 + \cdots \tag{28}$$

where

$$J(t) = \int\_0^t \frac{e^x - 1}{x} d\mathbf{x}.$$

**Theorem 3.** *If f is of the form* (1) *belongs to* C*e, then*

$$\left| a\_3 - \gamma a\_2^2 \right| \le \frac{1}{6} \max \left\{ 1, \frac{3}{2} \left| \gamma - 1 \right| \right\},\tag{29}$$

*where γ is a complex number.*

**Proof.** From (24) and (25), we get

$$\left| a\_3 - \gamma a\_2^2 \right| = \left| \frac{c\_2}{12} + \frac{c\_1^2}{48} - \frac{\gamma}{16} c\_1^2 \right|.$$

By reshuffling it, provides

$$\left| a\_3 - \gamma a\_2^2 \right| = \frac{1}{12} \left| \left( c\_2 - \frac{1}{2} \left( \frac{3\gamma - 1}{2} \right) c\_1^2 \right) \right| \dots$$

Application of (12), leads us to

$$\left| a\_3 - \gamma a\_2^2 \right| \le \max \left\{ \frac{1}{6}, \frac{1}{12} \left| 3\gamma - 3 \right| \right\}.$$

Substituting *γ* = 1, we obtain the following inequality.

**Corollary 1.** *If f* ∈ C*e and has the series represntaion* (1)*, then*

$$\left| a\_3 - a\_2^2 \right| \le \frac{1}{6}.\tag{30}$$

**Theorem 4.** *If f has the form* (1) *belongs to* C*e, then*

$$|a\_2 a\_3 - a\_4| \le \frac{31}{288}.\tag{31}$$

**Proof.** Using (24), (25) and (26), we have

$$|a\_{2}a\_{3} - a\_{4}| = \left| \frac{c\_{1}c\_{2}}{96} + \frac{\mathcal{T}}{1152}c\_{1}^{3} - \frac{c\_{3}}{24} \right|.$$

By rearranging it, gives

$$|a\_2 a\_3 - a\_4| = \left| -\frac{1}{48} \left( c\_3 - \frac{c\_1 c\_2}{2} \right) - \frac{1}{48} \left( c\_3 - \frac{7}{24} c\_1^3 \right) \right|.$$

By applying triangle inequality plus (8) and (10), we get

$$|a\_2a\_3 - a\_4| \le \left\{\frac{1}{24} + \frac{19}{288}\right\} = \frac{31}{288}.$$

**Theorem 5.** *Let f* ∈ C*e be of the form* (1)*. Then*

$$\left| a\_2 a\_4 - a\_3^2 \right| \le \frac{3}{64}.\tag{32}$$

**Proof.** From (24), (25) and (26), we have

$$\left| a\_2 a\_4 - a\_3^2 \right| = \left| \frac{c\_1 c\_3}{96} - \frac{c\_1^4}{1536} - \frac{c\_1^2 c\_2}{1152} - \frac{c\_2^2}{144} \right|.$$

By reordering it, yields

$$\left| a\_2 a\_4 - a\_3^2 \right| = \left| \frac{c\_1}{576} \left( c\_3 - \frac{c\_1 c\_2}{2} \right) + \frac{c\_1}{576} \left( c\_3 - \frac{3}{8} c\_1^3 \right) + \frac{1}{144} \left( c\_1 c\_3 - c\_2^2 \right) \right|.$$

Application of triangle inequality plus (7), (11), (10) and (9), we obtain

$$\left| a\_2 a\_4 - a\_3^2 \right| \le \frac{4}{576} + \frac{7}{576} + \frac{4}{144} = \frac{3}{64}.$$

**Theorem 6.** *If f* ∈ C*e and has the form* (1)*, then*

$$|H\_{3,1} \left( f \right)| \le 0.0234598.$$

**Proof.** Using (5), the Hankel determinant of order three can be formed as;

$$H\_{3,1}(f) = -a\_2^2 a\_5 + 2a\_2 a\_3 a\_4 - a\_3^3 + a\_3 a\_5 - a\_4^2.$$

Using (24), (25), (26) and (27), gives us

$$H\_{3,1}\left(f\right) = \frac{7}{5760}c\_1c\_2c\_3 - \frac{c\_3^2}{576} - \frac{c\_2^3}{1728} - \frac{173}{6635520}c\_1^6 + \frac{23}{276480}c\_1^4c\_2 + \frac{c\_2c\_4}{480} - \frac{13}{46980}c\_1^2c\_2^2 - \frac{c\_1^2c\_4}{960} + \frac{23}{69120}c\_1^3c\_3.$$

Now, rearranging it provides

$$\begin{split} H\_{3,1}\left(f\right) &= \ \frac{173}{3317760}c\_1^4\left(c\_2 - \frac{c\_1^2}{2}\right) - \frac{103}{1658880}c\_1^2c\_2\left(c\_2 - \frac{c\_1^2}{2}\right) + \frac{c\_4}{480}\left(c\_2 - \frac{c\_1^2}{2}\right) \\ &+ \frac{11}{17280}c\_1c\_2\left(c\_3 - \frac{365}{1056}c\_1c\_2\right) + \frac{c\_2}{1728}\left(c\_1c\_3 - c\_2^2\right) - \frac{c\_3}{576}\left(c\_3 - \frac{23}{120}c\_1^3\right). \end{split}$$

Application of triangle inequality plus (7), (11), (8), (10) and (9), leads us to

$$|H\_{3,1}\left(f\right)| \le \frac{173}{3317760} |c\_1|^4 \left(2 - \frac{|c\_1|^2}{2}\right) + \frac{103}{829440} |c\_1|^2 \left(2 - \frac{|c\_1|^2}{2}\right) + \frac{1}{4320} |c\_1| + \frac{83}{8640} + \frac{1}{216}.$$

Now, replacing |*c*1| = *x* ∈ [0, 2], then, we can write

$$|H\_{3,1}\left(f\right)| \le \frac{173}{3317760} \mathbf{x}^4 \left(2 - \frac{\mathbf{x}^2}{2}\right) + \frac{103}{829440} \mathbf{x}^2 \left(2 - \frac{\mathbf{x}^2}{2}\right) + \frac{1}{240} \left(2 - \frac{\mathbf{x}^2}{2}\right) + \frac{11}{4320} \mathbf{x} + \frac{41}{2880} \mathbf{x}^2$$

The above function gets its maximum at *x* = 0.7024858, Therefore, we have

$$|H\_{3,1} \left( f \right)| \le 0.02345979... $$

Thus the proof is completed.

#### **5. Bounds of** |*H*3,1 (*f*)| **for 2-Fold and 3-Fold Functions**

Let *m* ∈ N = {1, 2, . . .} . If a rotation about the origin through an angle 2*π*/*m* carries on itself, then such a domain is called *m*-fold symmetric. An analytic function *f* is *m*-fold symmetric in Δ, if

$$f\left(e^{2\pi i/m}z\right) = e^{2\pi i/m}f\left(z\right), \left(z \in \Delta\right).$$

By S(*m*), we define the set of *m*-fold univalent functions having the following Taylor series form

$$f\left(z\right) = z + \sum\_{k=1}^{\infty} a\_{mk+1} z^{mk+1}, \ \left(z \in \Delta\right). \tag{33}$$

The sub-families S∗(*m*) *e* and C(*m*) *e* of S(*m*) are the sets of *m*-fold symmetric starlike and convex functions respectively associated with exponential functions. More intuitively, an analytic function *f* of the form (33), belongs to the families S∗(*m*) *e* and C(*m*) *e* , if and only if

$$\frac{z f'(z)}{f(z)} = -\exp\left(\frac{p\left(z\right) - 1}{p\left(z\right) + 1}\right), \; p \in \mathcal{P}^{(m)},\tag{34}$$

$$1 + \frac{z f''(z)}{f'(z)} \quad = \ \exp\left(\frac{p(z) - 1}{p'(z) + 1}\right), \ p \in \mathcal{P}^{(m)}.\tag{35}$$

where the set P(*m*) is defined by

$$\mathcal{P}^{(m)} = \left\{ p \in \mathcal{P} : p\left(z\right) = 1 + \sum\_{k=1}^{\infty} c\_{mk} z^{mk} \; , \; \left(z \in \Delta\right) \right\}. \tag{36}$$

Here we prove some theorems related to 2-fold and 3-fold symmetric functions.

**Theorem 7.** *If f* ∈ S∗(2) *e and has the form* (33)*, then*

$$|H\_{3,1}(f)| \le \frac{1}{8}.$$

**Proof.** Let *f* ∈ S∗(2) *e* . Then, there exists a function *p* ∈ P(2), such that

$$\frac{z f'(z)}{f(z)} = \exp\left(\frac{p\left(z\right) - 1}{p\left(z\right) + 1}\right).$$

Using the series form (33) and (36), when *m* = 2 in the above relation, we can get

$$a\mathbf{y}\_1 = \begin{array}{c} c\_2 \\ \hline 4 \end{array} \tag{37}$$

$$a\_{\mathbb{S}^1} = \begin{array}{c} \frac{c\_4}{8} . \end{array} . \tag{38}$$

Now,

$$H\_3\left(f\right) = a\_3a\_5 - a\_3^3.$$

Utilizing (37) and (38), we get

$$H\_{3,1}\left(f\right) = -\frac{c\_2^3}{64} + \frac{c\_2 c\_4}{32}.$$

$$c\_2 \left(\int\_{-\infty}^{\infty} c\_2^2 \right)$$

By rearranging, it yields

$$H\_{3,1}\left(f\right) = \frac{c\_2}{32} \left(c\_4 - \frac{c\_2^2}{2}\right).$$

Using triangle inequality long with (8) and (7), gives us

$$|H\_{3,1}(f)| \le \frac{1}{8}$$

.

Hence, the proof is done.

**Theorem 8.** *If f* ∈ S∗(3) *e and has the series form* (33)*, then*

$$|H\_{3,1}(f)| \le \frac{1}{9}.$$

*This result is sharp for the function*

$$f\left(z\right) = \exp\left(\int\_0^z \frac{e^{x^3}}{x} dx\right) = z + \frac{1}{3}z^4 + \frac{5}{36}z^7 + \cdots \tag{39}$$

**Proof.** As, *f* ∈ S∗(3) *e* , therefore there exists a function *p* ∈ P(3), such that

$$\frac{z f'(z)}{f(z)} = \exp\left(\frac{p\left(z\right) - 1}{p\left(z\right) + 1}\right).$$

Utilizing the series form (33) and (36), when *m* = 3 in the above relation, we can obtain

$$a\_4 = \frac{c\_3}{6}.$$

Then,

$$H\_{3,1} \left(f\right) = -a\_4^2 = -\frac{c\_3^2}{36}$$

Utilizing (7) and triangle inequality, we have

$$|H\_{3,1}(f)| \le \frac{1}{9}.$$

Thus the proof is ended.

**Theorem 9.** *Let f* ∈ C(2) *e and has the form given in* (33). *Then*

$$|H\_{3,1}(f)| \le \frac{1}{120}.$$

**Proof.** As, *f* ∈ C(2) *e* , then there exists a function *p* ∈ P(2), such that

$$1 + \frac{z f''(z)}{f'(z)} = \exp\left(\frac{p\left(z\right) - 1}{p\left(z\right) + 1}\right).$$

Utilizing the series form (33) and (36), when *m* = 2 in the above relation, we can obtain

$$a\_3 \quad = \begin{array}{c} c\_2 \\ \hline 12' \end{array} \tag{40}$$

$$a\overline{s}\_{-}=\frac{c\_{4}}{40}.\tag{41}$$

$$H\_{3,1}(f) = a\_3 a\_5 - a\_3^3.$$

Using (40) and (41), we have

$$H\_{3,1}\left(f\right) = -\frac{c\_2^3}{1728} + \frac{c\_2 c\_4}{480}.$$

Now, reordering the above equation, we obtain

$$H\_3\left(f\right) = \frac{c\_2}{480} \left(c\_4 - \frac{5}{18}c\_2^2\right).$$

Application of (7), (8) and triangle inequality, leads us to

$$|H\_{3,1}(f)| \le \frac{1}{120}.$$

Thus, the required result is completed.

**Theorem 10.** *If f* ∈ C(3) *e and has the form given in* (33)*, then*

$$|H\_{3,1}\left(f\right)| \le \frac{1}{144}.\tag{42}$$

*This result is sharp for the function*

$$f\left(z\right) = \int\_0^z e^{I(t)}dt = z + \frac{1}{12}z^4 + \frac{5}{252}z^7 + \dotsb \tag{43}$$

*where*

$$I(t) = \int\_0^t \frac{e^{x^3} - 1}{x} dx.$$

**Proof.** Let, *f* ∈ C(3) *e* . Then there exists a function *p* ∈ P(3), such that

$$1 + \frac{z f''(z)}{f'(z)} = \exp\left(\frac{p\left(z\right) - 1}{p\left(z\right) + 1}\right).$$

Utilizing the series form (33) and (36), when *m* = 3 in the above relation, we can obtain

$$a\_4 = \frac{c\_3}{24}.$$

Then,

$$H\_{3,1}\left(f\right) = -\frac{c\_3^2}{576}.$$

Implementing (7) and triangle inequality, we have

$$|H\_{3,1}(f)| \le \frac{1}{144}.$$

Hence, the proof is done.

#### **6. Conclusions**

In this article, we studied Hankel determinant *H*3,1 (*f*) for the families S∗ *e* and C*e* whose image domain are symmetric about the real axis. Furthermore, we improve the bound of third Hankel determinant for the family S∗ *e* . These bounds are also discussed for 2-fold symmetric and 3-fold symmetric functions.

**Author Contributions:** Conceptualization, L.S. and H.K.; Methodology, M.A.; Software, H.M.S.; Validation, H.K. and M.A.; Formal Analysis, L.S.; Investigation, M.A. and H.M.S; Resources, H.K. and S.H.; Data Curation, S.H.; Writing—Original Draft Preparation, S.H.; Writing—Review and Editing, H.K., M.A. and L.S.; Visualization, M.A.;Supervision, M.A., L.S.; Project Administration, L.S.; Funding Acquisition, L.S.

**Funding:** This research was funded by School of Mathematics and Statistics, Anyang Normal University, Anyan 455002, Henan, China

**Conflicts of Interest:** The authors have no conflict of interest.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **On Periodic Solutions of Delay Differential Equations with Impulses**

#### **Mostafa Bachar**

Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia<EMAIL_ADDRESS>
Received: 18 February 2019; Accepted: 8 April 2019; Published: 11 April 2019

**Abstract:** The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces R([−*r*, 0], R*n*). The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schäffer fixed point theorem in regulated space R([−*r*, 0], R*n*).

**Keywords:** delay differential equations; integral operator; periodic solutions

**MSC:** primary 06F30; 46B20; 47E10; 34K13; 34K05

#### **1. Introduction**

In this paper, we will investigate the existence of periodic solutions for vectorial distributed delay differential equations with impulses in regulated Banach spaces. More precisely, the prototype of this delay differential equations with impulses, is of the form

$$\frac{d\mathbf{x}(t)}{dt} = -\lambda \mathbf{x}(t) + f(t, \mathbf{x}\_t),\\a.e.t \in [0, \omega + \tau], \lambda > 0, \omega > 0,\tag{1}$$

$$\mathbf{x}(t\_j) \quad = \quad \mathbf{x}(t\_j^-), \text{ and } \mathbf{x}(t\_j^+) - \mathbf{x}(t\_j) = h\_j(\mathbf{x}(t\_j)), \forall j = 1, \dots, l,\tag{2}$$

$$\mathbf{x}\_0(\theta) \quad = \quad \varphi(\theta), \ \theta \in [-\tau, 0], \tag{3}$$

with *xt*(*θ*) = *x*(*t* + *θ*), *θ* ∈ [−*τ*, 0], *τ* > 0 and where *x* and *ϕ* are R*n*-valued functions on [−*τ*, *ω*], and [−*τ*, 0], respectively. The Equation (1) is a nonlinear delay differential equation. More details about this type of equations can be found in [1]. Moreover, we assume that

$$0 < \delta \le t\_{j+1} - t\_j \le T < \infty.$$

We call (2) the impulses equation where, *x*(*t* − *j* ) (resp. *x*(*t* + *j* )) denotes the limit from the left (resp. from the right) of *x*(*t*), as *t* tends to *tj*. This type of differential equations without delay was initiated in 1960's by Milman and Myshkis [2,3]. This problem started to be popular mostly in Eastern Europe in the years 1960–1970, with special attention during the seventies of the last century. Later on, several investigations and important monographs appeared with more details, which show the importance of studying such systems, see for example [4–11]. In recent years, many investigations have arisen with applications to life sciences, such that the periodic treatment of some biomedical applications, where the impulses correspond to administration of a drug treatment at certain given times [12–15]. However, comparatively speaking, not much has been done in the study of impulsive functional

differential equations in regulated vectorial space, taking into account the general theory of functional analysis and having an acceptable hypothesis that can be used in real life applications, see [12] for more details.

Let us first introduce for each *τ* > 0, the regulated Banach space R = R([−*τ*, 0], R*n*), given by:

R = *ϕ* : [−*τ*, 0] → R*n* : *ϕ* has left and right limits at every points of [−*τ*, 0] ,

endowed with the following norm

$$\|\|\boldsymbol{\varrho}\|\|\_{\mathcal{R}} = \sup\_{\theta \in [-\pi,0]} \|\|\boldsymbol{\varrho}(\theta)\|\|.$$

We will make the following assumptions

(I) The map *f* : [0, *ω* + *τ*] × R*n* → R*n*, *ω* > 0, satisfies

$$\begin{array}{ll} \bullet & \|f(t,\boldsymbol{\varrho}) - f(t,\boldsymbol{\psi})\|\_{\mathcal{R}} \leq \underset{\boldsymbol{\omega}}{K} \|\boldsymbol{\varrho} - \boldsymbol{\psi}\|\_{\mathcal{R}\_{\boldsymbol{\omega}}}, \forall t \in [0, \boldsymbol{\omega} + \boldsymbol{\tau}], \boldsymbol{\varrho}, \boldsymbol{\psi} \in \mathcal{R}\_{\boldsymbol{\omega}},\\ \bullet & \exists \mathcal{M} > 0, \, \|f(t,\boldsymbol{0})\|\_{\mathcal{R}} \leq \mathcal{M}, \, \forall t \in [0, \boldsymbol{\omega} + \boldsymbol{\tau}]. \end{array}$$

We set the initial value problem as follows

**Problem 1.** *Let ϕ be an element of* R*. We want to find a function x defined on* [−*τ*, *ω* + *τ*] *such that x satisfies (1)–(3).*

We consider the nonlinear impulsive delay differential equation in R as

$$\begin{cases} \begin{array}{lcl} \frac{d\mathbf{x}(t)}{dt} &=& -\lambda \mathbf{x}(t) + f(t, \mathbf{x}\_{l}), a.e. \ t \in [0, \omega + \tau], \lambda > 0, \omega > 0, \\\ \mathbf{x}(t\_{j}) &=& \mathbf{x}(t\_{j}^{-}), \text{ and } \mathbf{x}(t\_{j}^{+}) - \mathbf{x}(t\_{j}) = h\_{j}(\mathbf{x}(t\_{j})), \forall j = 1, \dots, l, \\\ \mathbf{x}\_{0}(\theta) &=& \boldsymbol{\varrho}(\theta), \boldsymbol{\theta} \in [-\tau, 0] \text{ and } \mathbf{x}(0^{+}) = \boldsymbol{\xi} \in \mathbb{R}^{n}. \end{array} \end{cases}$$

The aim of this paper is to extend the main results related to the existence of the *ω*-periodic solutions for ordinary differential equations with impulses presented by Li et al. [16] and Nieto [17]. These papers contain references which provide additional reading on this topic, i.e., differential equations with impulses by using the fixed point theory.

#### **2. Existence and Uniqueness of Solution**

Let us start first by introducing some related definitions and lemmas.

**Definition 1.** *A function x* : [−*τ*, *ω* + *τ*] → R*n is called a solution of (1)–(3) if:*

**Lemma 1.** *Let f* : [0, *ω* + *τ*] ×R→ R*n be a map satisfying (I) and (II) and t*1 ∈ [0, *ω* + *τ*]*. Then, for each* (*ϕ*, *ξ*) ∈R× R*n*, *the problem*

$$\frac{d\mathbf{x}(t)}{dt} = -\lambda \mathbf{x}(t) + f(t, \mathbf{x}\_t), \mathbf{a.e.}\ t \in [0, t\_1] \tag{4}$$

$$(\mathbf{x}\_0, \mathbf{x}(0^+)) \quad = \quad (\boldsymbol{\varrho}, \boldsymbol{\xi}) \in \mathbb{R} \times \mathbb{R}^n,\tag{5}$$

*has a unique solution.*

**Proof.** We set *S* = {*y* ∈ *C*([0, *t*1], R*n*), *y*(0) = *x*(0+) = *ξ*}. Let us define the operator *T* by

$$T(\mathbf{x})(t) \quad = \; \; \xi + \int\_0^t \left( f(\mathbf{s}, \mathbf{x}\_s) - \lambda \mathbf{x}(\mathbf{s}) \right) ds, 0 \le t \le t\_1. \tag{6}$$

For each *y* ∈ *S*, we consider the Nemytski operator *F*, defined by

$$F(y)(t) \quad = \quad f(t, z\_l), \tag{7}$$

where

$$z\_t(\theta) := \begin{cases} \
y(t+\theta), & \text{if } t+\theta \ge 0, \\
\
\varphi(t+\theta), & \text{if } t+\theta \le 0. \end{cases} \tag{8}$$

Then, we get

$$T(y)(t) = \,\_0^\pi + \int\_0^t \left( F(y)(s) - \lambda y(s) \right) ds. \tag{9}$$

Define, the norm of any function *y* in *S* by

$$\|y\|\_{S} = \sup\_{0 \le t \le t\_1} \left\{ \|y(t)\| e^{-\rho t} \right\},\tag{10}$$

where *ρ* is a fixed positive constant greater than *K* + *λ*. We have for each *y*1(*t*) and *y*2(*t*) in *S*,

$$\begin{aligned} \|\|T(\mathcal{y}\_1)(t) - T(\mathcal{y}\_2)(t)\|\| &\leq \ (K + \lambda) \int\_0^t \|\|\mathcal{y}\_1(s) - \mathcal{y}\_2(s)\|\| \, ds, \\ &\leq \ (K + \lambda) \int\_0^t \epsilon^{\rho s - \rho s} \, \|\|\mathcal{y}\_1(s) - \mathcal{y}\_2(s)\|\| \, ds, \\ &\leq \ (K + \lambda) \ \|\|\mathcal{y}\_1 - \mathcal{y}\_2\|\|\_{\mathcal{S}} \int\_0^t \epsilon^{\rho s} \, ds, \\ &\leq \ \frac{(K + \lambda)}{\rho} \ \|\|\mathcal{y}\_1 - \mathcal{y}\_2\|\|\_{\mathcal{S}} \epsilon^{\rho t} \end{aligned}$$

and hence

$$\|\|T(y\_1) - T(y\_2)\|\|\_{\mathcal{S}} \le \frac{(K + \lambda)}{\rho} \|\|y\_1 - y\_2\|\|\_{\mathcal{S}}.$$

Since *K*+*λ ρ* < 1, then, *T* is a contraction on *S*, and the result follows immediately.

**Lemma 2.** *[18] Let f* : [0, *ω* + *τ*] ×R→ R*n be a map satisfying (I) and (II) and hj, for j* = 1, ··· , *l, satisfy the condition (III). Then the problem (1)–(3) has a unique solution.*

**Proof.** The proof follows by using the last lemma.

**Lemma 3.** *[18] Under the assumptions (I) and (II), if x*(*ϕ*)(*t*) *is the unique solution of (4) and (5), then one has:*

$$\|\|\mathbf{x}(\varrho)(t)\|\| \le \varepsilon^{Kt} \left( \|\|\varrho\|\| + \int\_0^\omega \|\|f(s, 0)\|\| \, ds \right). \tag{11}$$

The next Lemma, gives a similar, key representation formula for the solutions of the delay differential equations with impulses (1)–(3) in regulated Banach space R, see [4] for more details.

**Lemma 4.** *The problem (1)–(3) can be written as*

$$\begin{aligned} \mathbf{x}\_t &= \ & \mathbf{q}\_t^0 + H\_t^0 \otimes \left( (\not\!\!\! \mathbf{e}^{-\boldsymbol{\lambda}\,\mathbf{max}(0,\bullet)})\_t - \not\!\!\!\mathbf{q}(0) \right) \\ &+ \left( \int\_0^{\max(0,\bullet)} f(\mathbf{s}, \mathbf{x}\_s) \mathbf{e}^{-\boldsymbol{\lambda}(\bullet-s)} ds + \sum\_{0 \le t\_j < \bullet} \mathbf{e}^{-\boldsymbol{\lambda}(\bullet-t\_j)} u\_j \right)\_{\mathbf{t}'} \end{aligned}$$

*where*

*ϕ*0(*θ*) = *ϕ*(*θ*), *if θ* ≤ 0, *ϕ*(0), *if θ* > 0*,* (12)

*H*0 *is the Heaviside function*

$$H^0(\theta) := \begin{cases} \begin{array}{ll} 0, & \text{if } \theta \le 0, \\ 1, & \text{if } \theta > 0, \end{array} \tag{13}$$

*and the sequence*

$$\mu\_k = \mathfrak{x}(t\_k^+) - \mathfrak{x}(t\_k)\_\prime k \ge 1$$

*is determined by the following non-autonomous recurrence equation*

$$u\_k = \left. \left. h\_k \right| \left( \mathfrak{z} e^{-\lambda t\_k} + \int\_0^{t\_k} f(s, \mathfrak{x}\_s) e^{-\lambda \left( t\_k - s \right)} ds + \sum\_{0 \le t\_j < t\_k} e^{-\lambda \left( t\_k - t\_j \right)} u\_j \right), k \ge 1, \ldots$$

*starting from*

$$\|u\_1\|\_{\lambda} = \|h\_1\left(\xi e^{-\lambda t\_1} + \int\_0^{t\_1} f(s, x\_s) e^{-\lambda \left(t\_1 - s\right)} ds\right)\|$$

**Proof.** Let us consider *z*(*t*) = *eλt x*(*t*), ∀*t* ∈ [0, *ω* + *τ*], then the problem (1)–(3) becomes

$$\frac{dz(t)}{dt} = \, \_1f(t, e^{-\lambda(t+\theta)}z\_t)e^{\lambda t}, \text{a.e.} \, t \in [0, \omega + \tau], \lambda > 0, \omega > 0,\tag{14}$$

$$z(t\_j) = \;z(t\_j^-), \; \text{and} \; z(t\_j^+) - z(t\_j) = e^{\lambda t\_j} h\_j(e^{-\lambda t\_j} z(t\_j)), \forall j = 1, \ldots, l,\tag{15}$$

$$z\_0(\theta) \quad = \quad \varepsilon^{\lambda\theta}\varrho(\theta) = \hat{\varrho}(\theta), \theta \in [-\tau, 0], \text{ and } z(0^+) = \xi \in \mathbb{R}^n. \tag{16}$$

Let us consider *t* ∈ [*tj*, *tj*+1), *j* = 1, . . . , *l* − 1, with *t*0 = 0, then we get

$$z(t) := \left. z(t\_j^+) + \int\_{t\_j}^t f(s\_\prime e^{-\lambda(s+\theta)} z\_s) e^{\lambda s} ds \right|$$

By passing to the limit as *t* goes to *t* − *j* , and by solving the delay differential Equation (14) on the interval [*tj*−1, *tj*), we have

$$z(t\_j) := z(t\_{j-1}^+) + \int\_{t\_{j-1}}^{t\_j} f(s\_\prime e^{-\lambda(s+\theta)} z\_\ast) e^{\lambda s} ds.$$

Then, by taking into account the impulses condition (15), we have

$$z(t) = \;z(t\_{j-1}^+) + \int\_{t\_{j-1}}^t f(s, e^{-\lambda(s+\theta)}z\_s) e^{\lambda s} ds + e^{\lambda t\_j} h\_j(e^{-\lambda t\_j} z(t\_j)),$$

for all *t* ∈ [*tj*, *tj*+1), for *j* = 1, ··· , *l* − 1. Consequently, we can rewrite the last equations in more general form for all *t* > 0

$$z(t) \quad = \; \zeta + \int\_0^t f(s, e^{-\lambda(s+\theta)} z\_s) e^{\lambda s} ds + \sum\_{0 \le t\_j < t} e^{\lambda t\_j} u\_{j'} t \notin \{t\_k\}\_{k \ge 1'} \tag{17}$$

where *z*(0+) = *x*(0+) = *ξ*, and

$$
\mu\_k \quad = \quad z(t\_k^+) - z(t\_k) = h\_k(e^{-\lambda t\_k} z(t\_k)), \\
k \ge 1. \tag{18}
$$

Now, we will try to involve the *u ks*. To this end, we will take the limit from the left of the Formula (17) as *t* tends to *tk* > 0, we obtain

$$z(t\_k) = \,\_0^\mathbb{F} + \int\_0^{t\_k} f(s, e^{-\lambda(s+\theta)} z\_s) e^{\lambda s} ds + \sum\_{0 \le t\_j < t\_k} e^{\lambda t\_j} u\_j \dots$$

Substituting the last expression into (18), we have

$$\|u\_k\| = \|u\_k\|\left(\mathfrak{e}^{-\lambda t\_k}\mathfrak{g} + \int\_0^{t\_k} f(s, \mathfrak{e}^{-\lambda(s+\theta)}z\_s)\mathfrak{e}^{\lambda(s-t\_k)}ds + \sum\_{0 \le t\_j < t\_k} \mathfrak{e}^{\lambda(t\_j - t\_k)}u\_j\right).$$

In particular, we have {*j* : 0 ≤ *tj* < *t*1} = ∅, and therefore

$$\mu\_1 \quad = \quad h\_1 \Big( \mathfrak{e}^{-\lambda t\_1} \mathfrak{k} + \int\_0^{t\_1} f(s, \mathfrak{e}^{-\lambda(s+\theta)} z\_s) \mathfrak{e}^{\lambda s} ds \Big) . $$

By using, the Equation (16), we can rewrite the Equation (17) as

$$\begin{split} z\_{l}(\theta) &= \, \_{\xi} + \int\_{0}^{t+\theta} f(s, e^{-\lambda(s+\theta)} z\_{s}) e^{\lambda s} ds \\ &+ \sum\_{0 \le t\_{j} < t+\theta} e^{\lambda t\_{j}} u\_{j}, t+\theta \notin \{t\_{k}\}\_{k \ge 0}, \text{ and } t+\theta \ge 0, \end{split} \tag{19}$$

and by using *x*(*t*) = *e*−*λt z*(*t*), we have for *t* + *θ* ∈ { / *tk*}*k*≥1, and *t* + *θ* ≥ 0

$$\text{tr}\_t(\varphi(\theta)) \;= \; \_0^\pi \mathcal{E}^{-\lambda(t+\theta)} + \int\_0^{t+\theta} f(s, \mathbf{x}\_s) e^{-\lambda(t+\theta-s)} ds + \sum\_{0 \le t\_j < t+\theta} e^{-\lambda(t+\theta-t\_j)} u\_j \dots u\_{j-1}$$

Using (12) and (13), we get

$$\begin{aligned} \left(\mathbf{x}\_{t}(\boldsymbol{\varrho})\right)\_{t} &= \left.\boldsymbol{\varrho}^{0}\_{t} + H^{0}\_{t} \otimes \left( (\boldsymbol{\zeta}^{\mathbf{s}-\lambda\,\max(0,\bullet)})\_{t} - \boldsymbol{\varrho}(0) \right) \right. \\ &\left. + \left( \int\_{0}^{\max(0,\bullet)} f(s,\mathbf{x}\_{s}) e^{-\lambda(\bullet-s)} ds + \sum\_{0 \le t\_{j} < \bullet} e^{-\lambda(\bullet-t\_{j})} u\_{j} \right)\_{t} \right) \end{aligned}$$

where

$$\mu\_k = \quad h\_k \left( \mathbf{j} \, e^{-\lambda t\_k} + \int\_0^{t\_k} f(s, \mathbf{x}\_s) e^{-\lambda (t\_k - s)} ds + \sum\_{0 \le t\_j < t\_k} e^{-\lambda (t\_k - t\_j)} u\_j \right), k \ge 1.$$

starting from

$$\mu\_1 \quad = \quad h\_1 \left( \mathfrak{J} e^{-\lambda t\_1} + \int\_0^{t\_1} f(s, \mathbf{x}\_s) e^{-\lambda \left( t\_1 - s \right)} ds \right).$$

**Remark 1.** *Taking into account the conditions (II)–(III), we have ut* ∈ R*,* ∀*t* ∈ [0, *ω* + *τ*]*, and t* → *xt is a regulated function, because the functions t* → *ϕ*0 *t , and t* → *H*0 *t are regulated.*

In the next section, we will investigate the existence of the periodic solution(s) for the delay differential equation with impulses (1)–(3) using Schäffer's fixed point theorem [19].

#### **3. Existence of Periodic Solutions**

Let us consider the Poincaré operator, given by:

$$\begin{array}{rcl} I: \mathcal{R} & \to & \mathcal{R} \\ \varphi & \to & \mathbf{x}\_{\omega}(\varphi) \end{array}$$

where *xω*(*ϕ*) is the solution of the delay differential equation with impulses (1)–(3). It is clear that if the Poincaré operator *J* admit a fixed point, then (1)–(3) has a *ω*-periodic solution. The following lemma is useful to prove the main theorem.

**Lemma 5.** *The problem (1)–(3) has a ω-periodic solution in* R *if and only if the integral equation*

$$\mathbf{x}\_t(\boldsymbol{\varrho})(\theta) \; \;= \begin{cases} \displaystyle \varepsilon^{-\lambda\theta} \int\_{t+\theta}^{t+\omega+\theta} \mathbf{G}(t,\mathbf{s}) f(\mathbf{s},\mathbf{x}\_s) ds + \operatorname{\!\!s}^{-\lambda\theta} \sum\_{t+\theta \le t\_j < t+\omega+\theta} \mathbf{G}(t,t\_j) u\_{j\boldsymbol{\varrho}} & \text{if } 0 \le t+\theta \le \omega, \\\ \boldsymbol{\varrho}(t+\theta), & \text{if } -\boldsymbol{\tau} \le t+\theta \le 0, \end{cases}$$

*has a solution* ∀*t* ∈ [0, *ω* + *τ*] *and ω* ≥ *τ, where*

$$G(t,s) \quad = \ \frac{e^{-\lambda(t-s)}}{e^{\lambda\omega}-1},\tag{20}$$

*and the sequence*

$$u\_k = \mathfrak{x}(t\_k^+) - \mathfrak{x}(t\_k), k \ge 1$$

*is determined by the following non-autonomous recurrence equation*

$$\mu\_k = \nu\_k \left( \int\_{t\_k}^{t\_k + \omega} G(t, s) f(s, x\_s) ds + \sum\_{t\_k \le t\_j < t\_k + \omega} G(t, t\_j) u\_j \right), k \ge 1, \omega$$

*starting from*

$$\mu\_1 \quad = \quad h\_1 \left( \mathfrak{E} e^{-\lambda t\_1} + \int\_0^{t\_1} f(s, \mathfrak{x}\_s) e^{-\lambda (t\_1 - s)} ds \right).$$

**Proof.** Using the expression (19) for *t* + *ω* + *θ*, where *t* ≥ 0, and *ω* ≥ *τ*, we have for all *t* + *θ* ≥ 0

*zt*+*ω*(*θ*) = *ξ* + *t*+*ω*+*θ* 0 *f*(*s*,*e* −*λ*(*s*+*θ*) *zs*)*e λs ds* + ∑ 0≤*tj*<*t*+*ω*+*θ e λtjuj*, = *ξ* + *t*+*θ* 0 *f*(*s*,*e* −*λ*(*s*+*θ*) *zs*)*e λs ds* + ∑0≤*tj*<*t*+*θ e λtjuj* + *t*+*ω*+*θ t*+*θ f*(*s*,*e* −*λ*(*s*+*θ*) *zs*)*e λs ds* + ∑ *t*+*θ*≤*tj*<*t*+*ω*+*θ e λtjuj*, = *zt*(*θ*) + *t*+*ω*+*θ t*+*θ f*(*s*,*e* −*λ*(*s*+*θ*) *zs*)*e λs ds* + ∑ *t*+*θ*≤*tj*<*t*+*ω*+*θ e λtjuj*,

and, by using the *ω*-periodic condition *zt*+*ω*(*θ*) = *eλωzt*(*θ*), we get

$$z\_t(\theta) = \frac{1}{\varepsilon^{\lambda \omega} - 1} \int\_{t+\theta}^{t+\omega+\theta} f(s, e^{-\lambda(s+\theta)} z\_s) e^{\lambda s} ds + \frac{1}{\varepsilon^{\lambda \omega} - 1} \sum\_{t+\theta \le t\_j < t+\omega+\theta} e^{\lambda t\_j} u\_j.$$

Therefore, using *zt*(*θ*) = *eλ*(*t*+*θ*)*xt*(*θ*), we have

$$\mathbf{x}\_t(\theta) = \left. e^{-\lambda \theta} \int\_{t+\theta}^{t+\omega + \theta} G(t, s) f(s, \mathbf{x}\_s) ds + e^{-\lambda \theta} \sum\_{t+\theta \le t\_j < t+\omega + \theta} G(t, t\_j) \mathbf{u}\_j \omega \right|$$

where

$$G(t,s) \quad = \ \frac{e^{-\lambda(t-s)}}{e^{\lambda\omega}-1}.\tag{21}$$

Then

$$\mu\_k = \nu\_k \left( \int\_{t\_k}^{t\_k + \omega} G(t, s) f(s, \mathbf{x}\_s) ds + \sum\_{t\_k \le t\_j < t\_k + \omega} G(t, t\_j) u\_j \right), k \ge 1, \omega$$

starting from

$$\mu\_1 \quad = \quad h\_1 \left( \zeta e^{-\lambda t\_1} + \int\_0^{t\_1} f(s, x\_s) e^{-\lambda \left( t\_1 - s \right)} ds \right).$$

**Example 1.** *Let us consider the scalar delay differential equation with impulses:*

$$\frac{d\mathbf{x}(t)}{dt} = -\lambda \mathbf{x}(t) + f(t, \mathbf{x}(t-\tau)), a.e.t \in [0, 2\pi],\tag{22}$$

$$\mathbf{x}(\tau) \quad = \ x(\tau^-), \ and \ x(\tau^+) - \mathbf{x}(\tau) = c\mathbf{x}(\tau), \tag{23}$$

$$\mathbf{x}(\theta) \quad = \quad \varphi(\theta), \theta \in [-\pi, 0], \tag{24}$$

*where f* : [0, 2*τ*] ×R → R*n is a map satisfying (II). Let us investigate the existence of the τ-periodic solution of (22)–(24) such that xt*+*τ*(−*τ*) = *xt*(−*τ*), *τ* ≤ *t* ≤ 2*τ*. *The solution of the delay differential Equations (22)–(24), can be written as*

$$x(t) \quad = \begin{cases} \varphi(\theta), & \text{if } -\tau \le t \le 0, \\\varphi(0)e^{-\lambda t} + \int\_0^t e^{-\lambda(t-s)} f(s, x(s-\tau)) ds, & \text{if } 0 \le t \le \tau, \\\ x(\tau^+)e^{-\lambda(t-\tau)} + \int\_\tau^t e^{-\lambda(t-s)} f(s, x(s-\tau)) ds, & \text{if } \tau < t \le 2\tau. \end{cases} \tag{25}$$

*Using (23), we get*

$$\begin{aligned} \mathbf{x}(\tau^+) &= \mathbf{x}(\tau) + c\mathbf{x}(\tau), \\ &= (c+1)\mathbf{x}(0)e^{-\lambda\tau} + (c+1)\int\_0^\tau e^{-\lambda(\tau-s)}f(s, \mathbf{x}(s-\tau))ds. \end{aligned}$$

*Therefore, if τ* ≤ *t* < 2*τ*, *we have*

$$\begin{split} \mathbf{x}(t) &= \ & \left( (c+1)\rho(0)e^{-\lambda\tau} + (c+1) \int\_{0}^{\tau} e^{-\lambda(\tau-s)} f(s, \mathbf{x}(s-\tau)) ds \right) e^{-\lambda(t-\tau)} \\ & \quad + \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds, \\ &= & (c+1)\rho(0)e^{-\lambda\tau}e^{-\lambda(t-\tau)} + (c+1)e^{-\lambda\tau} \int\_{0}^{t-\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds \\ & \quad + (c+1)e^{-\lambda\tau} \int\_{t-\tau}^{\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds + \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds, \\ &= & (c+1)e^{-\lambda\tau} \Big( \mathbf{x}(t-\tau) + \int\_{t-\tau}^{\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds \Big) + \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds, \end{split}$$

*which implies*

$$\begin{aligned} \mathbf{x}\_{t+\tau}(-\tau) &= \begin{array}{c} (\mathbf{c}+1)e^{-\lambda\tau} \Big( \mathbf{x}\_{l}(-\tau) + \int\_{t-\tau}^{\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds \Big) \\ &+ \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds. \end{array} \end{aligned}$$

*Then, we have three cases.*

$$\int\_{t-\tau}^{\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds + \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds = 0, \tau$$

*then, we have the existence of infinitely many τ-periodic solutions. (3) If* 1 − (*c* + 1)*e*−*λτ* = 0*, and*

$$\int\_{t-\tau}^{\tau} e^{-\lambda(t-\tau-s)} f(s, \mathbf{x}(s-\tau)) ds + \int\_{\tau}^{t} e^{-\lambda(t-s)} f(s, \mathbf{x}(s-\tau)) ds \neq 0,$$

*then, there exists no τ-periodic solution.*

Now, we can consider for each *t* ≥ −*τ* and *ω* ≥ *τ*, the Poincaré operator *J* : R→R defined by

*Jϕ* = *e* −*λ*(•−*t*) •+*ω* • *G*(*t*,*s*)*f*(*s*, *ϕ*)*ds* + *e* −*λ*(•−*t*) ∑ •≤*tj*<•+*ω G*(*t*, *tj*)*uj t* ,

where

$$\mu\_k = \nu\_k \left( \int\_{t\_k}^{t\_k + \omega} G(t, s) f(s, x\_s) ds + \sum\_{t\_k \le t\_j < t\_k + \omega} G(t, t\_j) u\_j \right), k \ge 2, \nu$$

and, starting from

$$\mu\_1 \quad = \quad h\_1 \left( \zeta e^{-\lambda t\_1} + \int\_0^{t\_1} f(s, x\_s) e^{-\lambda \left( t\_1 - s \right)} ds \right).$$

It is clear, that, the *ω*-periodic solutions in R of (1)–(3) are exactly the fixed points of the Poincaré operator *J*, i.e., *Jϕ* = *ϕ*.

The following theorem, is known as the Schäffer's fixed point theorem [19], which can be found for example in Deimling's book [20].

**Theorem 1.** *[19–22] Let X be a normed space,* F *a continuous mapping of X into X, such that the closure of* F(*B*) *is compact for any bounded subset B of X. Then either:*

Before, we state the main theorem of our work, we will need the following lemma.

**Lemma 6.** *Let f* : [0, *ω* + *τ*] ×R→ R*n be a map satisfying (I) and (II), where ω* ≥ *τ, and hj*, *j* = 1, ... , *l are bounded and satisfy the condition (III). Then, the Poincaré operator J* : R→R *is completely continuous.*

**Proof.** Let *B* ⊂ R be a bounded set and *ϕ* ∈ *B*. Then by using the condition (I), we have

$$\|\|f(t,\boldsymbol{\varrho})\|\|\_{\mathcal{R}} \le \|\|f(t,0)\|\|\_{\mathcal{R}} + \|\|f(t,\boldsymbol{\varrho}) - f(t,0)\|\|\_{\mathcal{R}} \le M + K\|\|\boldsymbol{\varrho}\|\|\_{\mathcal{R}} < \infty.$$

Therefore, there exist two constants *M*7 and *M* such that

$$\begin{split} \|\parallel f\varphi(\theta)\parallel\| &=& \|\|\varepsilon^{-\lambda\theta}\int\_{t+\theta}^{t+\theta+\omega} f(s,\rho)G(t,s)ds + \epsilon^{-\lambda\theta}\sum\_{t+\theta \le t\_{j} < t+\theta+\omega} G(t,t\_{j})u\_{j}\|, \\ &\leq \|\|\varepsilon^{-\lambda\theta}\int\_{t+\theta}^{t+\theta+\omega} f(s,\rho)G(t,s)ds\| + \epsilon^{\lambda r} \|\sum\_{t+\theta \le t\_{j} < t+\theta+\omega} G(t,t\_{j})u\_{j}\|, \\ &\leq \|\varepsilon^{\lambda\tau}\omega\tilde{M} + \epsilon^{\lambda r}\sum\_{t+\theta \le t\_{j} < t+\theta+\omega} \overline{M}\_{t} \end{split} \tag{26}$$

where

$$\|\|u\_k\|\| = \left\|\left|h\_k\left(\int\_{t\_k}^{t\_k+\omega} G(t,s)f(s,\mathbf{x}\_s)ds + \sum\_{t\_k \le t\_j < t\_k + \omega} G(t,t\_j)u\_j\right)\right|\right\| < \infty, k \ge 2,\alpha$$

and starting from

$$\left\|\left\|u\_{1}\right\|\right\|\;=\left\|\left\|h\_{1}\left(\left\|e^{-\lambda t\_{1}}+\int\_{0}^{t\_{1}}f(s,\mathbf{x}\_{s})e^{-\lambda(t\_{1}-s)}ds\right\|\right\|\;<\infty,\right\|\;$$

and, we have

$$\|\|J\boldsymbol{\varrho}\|\|\_{\mathcal{R}} \leq \|\boldsymbol{e}^{\lambda\tau}\boldsymbol{\omega}\boldsymbol{\mathcal{M}} + \boldsymbol{e}^{\lambda r}\overline{\boldsymbol{\mathcal{M}}}\|\sum\_{t+\theta \leq t\_j < t+\theta+\omega} 1\_{\mathcal{I}\_j}$$

which imply that *J*(*B*) is uniformly bounded. For each *t* ≥ 0, there exists *n* ∈ N∗ such that *t* ∈ [*tn*, *tn*+1), and for any *θ*, *θ* 7 ∈ [−*r*, 0], one can obtain for any *ϕ* ∈ *B*

$$\begin{split} \|\|f\varrho(\theta) - f\varrho(\tilde{\theta})\|\| &\leq \|\|e^{-\lambda\theta}\int\_{t+\theta}^{t+\theta+\omega} f(s,\boldsymbol{\varrho})G(t,\boldsymbol{s})ds - e^{-\lambda\theta}\int\_{t+\tilde{\theta}}^{t+\tilde{\theta}+\omega} f(s,\boldsymbol{\varrho})G(t,\boldsymbol{s})ds\| \\ &\quad + \|\|e^{-\lambda\theta}\sum\_{t+\theta \leq t\_{j} < t+\theta+\omega} G(t,t\_{j})u\_{j} - e^{-\lambda\theta}\sum\_{t+\tilde{\theta} \leq t\_{j} < t+\tilde{\theta}+\omega} G(t,t\_{j})u\_{j}\|\| \\ &\leq \frac{e^{\lambda(r+\omega)}\left(M+K\|\|\boldsymbol{\varrho}\|\|\right)}{1 - e^{-\lambda\omega}} \parallel \int\_{t+\theta}^{t+\theta+\omega} e^{-\lambda(t-s)}ds - \int\_{t+\tilde{\theta}}^{t+\tilde{\theta}+\omega} e^{-\lambda(t-s)}ds \parallel \\ &\quad + \frac{e^{\lambda(r+\omega)}}{1 - e^{-\lambda\omega}} \parallel \sum\_{t+\theta \leq t\_{j} < t+\theta+\omega} u\_{j} - \sum\_{t+\tilde{\theta} \leq t\_{j} < t+\tilde{\theta}+\omega} u\_{j}\parallel . \end{split}$$

Therefore, for each *t* ∈ [*tn*, *tn*+1), we will have as | *θ* − *θ* 7 | goes to 0, *Jϕ*(*θ*) − *Jϕ*(*θ* 7) goes to 0, which imply that the Poincaré operator *J*(*B*) is equicontinuous. Using Arzelà-Ascoli's theorem, we conclude that the Poincaré operator *J* is completely continuous.

Now, we are ready to state the main result of our work, related to the existence of *ω*-periodic solution(s) of (1)–(3).

**Theorem 2.** *Let f* : [0, *ω* + *τ*] ×R→ R*n be a map satisfying (I) and (II), where ω* ≥ *τ, and hj*, *j* = 1, ... , *l are bounded and satisfy the condition (III). Then, the nonlinear impulsive problem (1)–(3), has at least one ω-periodic solution in* R*.*

**Proof.** Let us define *H*(*ϕ*, *μ*) : R × [0, 1] −→ R by

$$H(\mathfrak{q}, \mathfrak{\mu}) \quad = \quad \mathfrak{\mu} \mathfrak{l} \mathfrak{\mathfrak{q}}.\tag{27}$$

Then, by using (26), we have

$$\|\|H(\boldsymbol{\varrho},\boldsymbol{\mu})\|\|\_{\mathcal{R}} \leq \quad \mu\Big(\boldsymbol{\varepsilon}^{\lambda\tau}\omega\,\check{\boldsymbol{M}} + \boldsymbol{\varepsilon}^{\lambda\tau}\sum\_{t+\theta \leq t\_j \leq t+\theta+\omega} \overline{\boldsymbol{M}}\Big).$$

Then, for each *μ* ∈ (0, 1) the set *S* = {*ϕ* : *ϕ* = *H*(*ϕ*, *μ*)} is bounded. Since *J* is completely continuous, then by using Schäffer's fixed point theorem, the Poincaré operator *J* admits a fixed point.

Next, we give the conditions of the existence and uniqueness of a *ω*-periodic solution of (1)–(3).

**Theorem 3.** *Let f* : [0, *ω* + *τ*] ×R→ R*n be a map satisfying (I) and (II), where ω* ≥ *τ, and hj*, *j* = 1, ... , *l are bounded and satisfy the condition (III), and there exist constants Hj, j* = 1, . . . , *l, such that*

$$\|\|h\_j(\varphi(0)) - h\_j(\psi(0))\|\| \le \|H\_j \|\|\varphi - \Psi \|\|\_{\mathcal{R}}.$$

*If, there exists a constant C* < 1, *such that*

$$\frac{K\omega e^{\lambda r}}{1 - e^{-\lambda\omega}} + \frac{e^{\lambda r}}{1 - e^{-\lambda\omega}} \sum\_{t - r + \omega \le t\_j < t + \omega} \overline{H}\_j \quad \le \quad C\_{\prime \nu}$$

*then, the nonlinear impulsive problem (1)–(3), has a unique ω-periodic solution in* R*.*

**Proof.** Let *ϕ*, *ψ* ∈ R be two solutions of (1)–(3), i.e., *Jϕ* = *ϕ* and *Jψ* = *ψ*. Assume *φ* = *ψ*. We have

 *ϕ*(*θ*) − *ψ*(*θ*) = *Jϕ*(*θ*) − *Jψ*(*θ*) , ≤ *e λr t*+*θ*+*ω t*+*θ* | *G*(*t*,*s*) | *f*(*s*, *ϕ*) − *f*(*s*, *ψ*) R *ds* + *e λr* ∑ *t*−*r*+*ω*≤*tj*<*t*+*ω* | *G*(*t*, *tj*) | *hj*(*ϕ*(0) − *hj*(*ψ*(0)) , ≤ *Kωeλr* 1 − *e*−*λω* + *eλr* 1 − *e*−*λω* ∑ *t*−*r*+*ω*≤*tj*<*t*+*ω Hj ϕ* − *ψ* R, ≤ *C ϕ* − *ψ* R .

Hence

$$\begin{aligned} \parallel \; \varphi - \psi \; \parallel \; \mathbb{R} &\leq \quad \mathbb{C} \parallel \; \varphi - \psi \; \parallel \; \mathbb{R}\_{\mathcal{R}'} \\ &< \quad \parallel \; \varphi - \psi \; \mid \; \mathbb{R} \end{aligned} \tag{28}$$

This contradiction implies, the uniqueness of the *ω*-periodic solution of (1)–(3).

#### **4. Conclusions**

The method described in this work presents new challenges for more investigation on more realistic models; such as the extension of the ascorbic acid model [12] and *HIV* model [13,14]. Taking into account the delay effect on respective compartments [23–25]. This kind of work, will need more investigation on modeling validation effort, keeping a close eye on the real life data in order to have a more realistic model. The explicit solutions presented in the technical Lemma 4 and methods of proving the existence of periodic solutions are very useful for further future investigations.

**Funding:** This research was funded by Deanship of Scientific Research at King Saud University, Grant No. RG-1435-079.

**Acknowledgments:** The authors would like to express their deep appreciation to the Deanship of Scientific Research at King Saud University for supporting this Research group No. (RG-1435-079). The authors profusely thank the referees for their valuable comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

c 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Collaborative Content Downloading in VANETs with Fuzzy Comprehensive Evaluation**

**Tianyu Huang 1, Xijuan Guo 1,\*, Yue Zhang 1 and Zheng Chang 2**

Received: 7 February 2019; Accepted: 31 March 2019; Published: 6 April 2019

**Abstract:** Vehicle collaborative content downloading has become a hotspot in current vehicular ad-hoc network (VANET) research. However, in reality, the highly dynamic nature of VANET makes users lose resources easily, and the transmission of invalid segment data also wastes valuable bandwidth and storage of the users' vehicles. In addition, the individual need of each customer vehicle should also be taken into consideration when selecting an agent vehicle for downloading. In this paper, a novel scheme is proposed for vehicle selection in the download of cooperative content from the Internet, by considering the basic evaluation information of the vehicle. To maximize the overall throughput of the system, a collaborative content downloading algorithm is proposed, which is based on fuzzy evaluation and a customer's own expectations, in order to solve the problems of agent vehicle selection. With the premise of ensuring successful downloading and the selection preferences of customer vehicles, linear programming is used to optimize the distribution of agent vehicles and maximize customer's satisfaction. Simulation results show that the proposed scheme works well in terms of average quality of service, average bandwidth efficiency, failure frequency, and average consumption.

**Keywords:** vehicle collaborative content downloading; fuzzy comprehensive evaluation; VANET

#### **1. Introduction**

With the rapid development of the network, the demand of the network extends to all aspects of people's lives. As a platform, which provides a specific network service, the vehicular ad-hoc network (VANET) brings new technical challenges to the transmission of information while providing information services, including: how to improve the efficiency of the vehicle network and how to meet the continuous improvement of the users' needs [1,2].

Scholars have discussed different ways to modify VANET, in order to improve the performance of vehicle networking and meet the growing demand of users. In terms of enhancing the performance of the vehicle-to-infrastructure (V2I) connection in VANET, the possibility of constructing a network with the TV white space geolocation database for vehicle networking was discussed by some scholars. Then vehicular communication architectures were proposed to mitigate the resulting high spectrum demands and provide vehicular connectivity with wider communication range, higher transmission rate, and lower data transfer cost [3,4]. By analyzing the end-to-end transmission performance from individual vehicles to a road side unit (RSU), an efficient message routing scheme was put forward to balance the data traffic across the network and improve the network throughput [5]. In Reference [6], a collaborative download algorithm, namely maximum throughput and minimum delay collaborative download (MMCD) was proposed, which minimizes the average transmission delay of each user's request and maximizes the number of packets downloaded from an RSU. Reference [7] mainly studies

the cost of minimizing the download of a hybrid vehicle ad hoc network, and proposes the basic satisfaction algorithm (BMA) and heuristic algorithm (TSA) to solve the huge download delay caused by vehicle mobility in VANET. In order to solve the frequent collision among agent vehicles and customer vehicles, a transmission scheduling method was put forward to adjust the relationship between link routing and transmission time [8].

In terms of enhancing the vehicle-to-vehicle (V2V) collaborative download performance in VANET: ECDS gives an efficient collaborative downloading solution to popular content distribution in urban vehicle networks. Furthermore, a cross-domain relay selection strategy was proposed to build a peer-to-peer (P2P) network, which helps strengthen information dissemination [9]. In Reference [10], to solve the problems of popular content distribution (PCD) in a highway scene, the author modeled the problem as a coalition formation game with transferable utilities, and proposed a coalition formation algorithm that converges into a Nash-stable partition, adapting to environmental changes as a result of the VANET's rapid and unpredictable topological changes. In Reference [11], the design incentive mechanism is employed to propose a collaborative downloading method, which encourages cooperation between vehicles and helps users effectively obtain the required resources. The author designed a server-assisted key management scheme that promotes cooperation and ensures fairness and efficiency. In the scheme, vehicles with common interests form a cluster and take turns as the cluster head, which directly downloads data packets from the Internet and V2V shares the content with surrounding vehicles [12]. A delicate linear cluster formation scheme is proposed and applied to significantly enhance the probability of a successful file download in VANET [13]. In Reference [14], the author proposed a security incentive program (SIRC) to achieve reliable, fair, and secure collaborative downloading in VANET. SIRC stimulates vehicle users to help each other download and forward packets, encourages cooperation between users, and also punishes malicious vehicles to ensure the safety of vehicles. Efficient privacy-preserving cooperative data downloading for value-added services is used to solve the problems of limited communication range and high dynamics, which gains the access control in VANET [15].

The methods of improving the performance of the vehicle network are also discussed from other aspects. Digital fountain code (DFC) is proposed and applied in the field of cooperative downloading for VANET. As long as enough data packets encoded by hierarchical fountain code are available, the client can recover the raw data and avoid data transmission interruption [16,17]. In Reference [18], a fuzzy logic-based resource management (FLRM) scheme was proposed, and the lifetime of each storage resource was defined by the proposed fuzzy logic-based popularity evaluation algorithm.

#### *1.1. Related Work*