It is a fact supported by many researchers that fractional calculus (FC) establishes a flexible extension for the classical one to arbitrary orders. FC has attracted particular attention from many researchers of mathematics, applied sciences, and engineering because of the various important applications of this field in modeling certain scientific phenomena and complex physical systems. Modeling systems using fractional derivatives can provide a good interpretation of the physical behavior of the studied systems due to the nonlocality and memory effects that have been exhibited in some systems. Some studies have been conducted on the mathematical analysis of FC and its applications such as European option pricing models [1], p-Laplacian nonperiodic nonlinear boundary value problem [2], nonlocal Cauchy problem [3], economic models involving time fractal [4], complex integral [5], incompressible second-grade fluid models [6], complex-valued functions of a real variable [7], and separated homotopy method [8]. Likewise, quantum calculus is a corresponding field of the standard infinitesimal one without the concept of limits. In spite of the long history that they already have, both theories are in the field of mathematical analysis, the investigation of their properties has emerged not so long ago. The quantum fractional calculus (q-fractional calculus), considered as the fractional correspondence of the q-calculus, was initially proposed by Jackson [9–11]. Researchers such as Al-Salam [12] and Agarwal [13] gave a great boost to the fractional q-calculus and obtained important theoretical results. Based on these results, the fractional q-calculus has emerged as an

**Citation:** Rezapour, S.; Imran, A.; Hussain, A.; Martínez, F.; Etemad, S.; Kaabar, M.K.A. Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs. *Symmetry* **2021**, *13*, 469. https:// doi.org/10.3390/sym13030469

Academic Editor: Carlo Cattani

Received: 23 February 2021 Accepted: 10 March 2021 Published: 12 March 2021

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**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

instrument with great potential in the field of applications [14–17]. Even in recent years, many articles have been appeared on quantum integro-difference boundary value problems (BVPs), which are valuable abstract tools for modeling many phenomena in various fields of science [18–30].

Asawasamrit et al. [31] provided a multi-term q-integro-difference equation subject to nonlocal multi-quantum integral conditions displayed as

$$\begin{cases} \prescript{R}{q}{\mathfrak{D}}\_{0^{+}}^{\xi}\,\hbar(r) = \phi(r,\hbar(r),\prescript{R}{q}{\mathfrak{I}}\_{0^{+}}^{\sigma\_{1}}\hbar(r)), & (r \in [0,\mathbb{K}]), \\\\ \hbar(0) = 0, & \nu^{R}\_{q\_{3}}\mathfrak{I}^{\sigma\_{2}}\_{0^{+}}\hbar(\eta\_{1}) = \prescript{R}{q\_{4}}{\mathfrak{I}}\_{0^{+}}^{\sigma\_{3}}\hbar(\eta\_{2}), \end{cases}$$

where *q*1, *q*2, *q*3, *q*4 ∈ (0, 1), *ς* ∈ (1, 2), *σ*1, *σ*2, *σ*3 > 0, *η*1, *η*2 ∈ (0, *K*) and *ν* ∈ R. The approach implemented by them to arrive at the existence property of solutions for the suggested q-BVP is based on the fixed-point techniques [31]. After that in 2015, Etemad, Ettefagh and Rezapour [32] concerned the three-term q-difference FBVP

$$(\prescript{\mathbb{C}}{q}{\mathfrak{D}}\_{0^+}^\xi \hbar)(r) = w(r, \hbar(r), \prescript{\mathbb{C}}{q}{\mathfrak{D}}\_{0^+}^1 \hbar(r)),$$

with four-point q-integro-difference conditions

$$\begin{aligned} \lambda\_1 \hbar(0) + \zeta\_1 \, ^C\_q \mathfrak{D}\_{0^+}^1 \hbar(0) &= m\_1 \, ^R\_q \mathfrak{J}\_{0^+}^\beta \hbar(\mathfrak{J}\_1) = m\_1 \int\_0^{\mathfrak{T}\_1} \frac{(\mathfrak{F}\_1 - qv)^{(\beta - 1)}}{\Gamma\_q(\beta)} \hbar(v) d\_q v, \\\ \lambda\_2 \hbar(1) + \zeta\_2 \, ^C\_q \mathfrak{D}\_{0^+}^1 \hbar(1) &= m\_2 \, ^R\_q \mathfrak{J}\_{0^+}^\beta \hbar(\mathfrak{J}\_2) = m\_2 \int\_0^{\mathfrak{T}\_2} \frac{(\mathfrak{F}\_2 - qv)^{(\beta - 1)}}{\Gamma\_q(\beta)} \hbar(v) d\_q v, \end{aligned}$$

where 0 ≤ *r* ≤ 1, 1 < *ς* ≤ 2, *q* ∈ (0, 1), *β* ∈ (0, 2], *λ*1, *λ*2, *ζ*1, *ζ*2, *m*1, *m*2 ∈ R and *ξ*1, *ξ*2 ∈ (0, 1) with *ξ*1 < *ξ*2. Ntouyas and Samei [33] turned to studying the solutions' existence for the q-integro-difference FBVP

$$\prescript{\mathbb{C}}{}{\mathfrak{D}}\_{0^{+}}^{\xi}h(r) = w(r,h(r),(\phi\_{1}h)(r),(\phi\_{2}h)(r),\prescript{\mathbb{C}}{}{\mathfrak{D}}\_{0^{+}}^{\xi\_{1}}h(r),\prescript{\mathbb{C}}{}{\mathfrak{D}}\_{0^{+}}^{\xi\_{2}}h(r),\dots,\prescript{\mathbb{C}}{}{\mathfrak{D}}\_{0^{+}}^{\xi\_{n}}h(r)),$$

via boundary conditions *h*(0) + *ah*(1) = 0 and *h* 0 (0) + *bh*0 (1) = 0, in which *r* ∈ [0, 1], *q* ∈ (0, 1), 1 < *ς* < 2, *ςk* ∈ (0, 1) with *k* = 1, 2, . . . , *n*, *a*, *b* 6= −1, *φm* are defined by the rule (*φmh*)(*r*) = Z *r* 0 *µm*(*r*, *v*)*h*(*v*) d*qv* for *m* = 1, 2 and *w* : [0, 1] × R*n*+3 → R is assumed to be continuous with respect to all (*n* + 4) variables [33].

Stimulated by the above research studies, the following proposed nonlinear Caputo fractional quantum BVP is furnished with the fractional quantum integro-conditions:

$$\begin{cases} \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\xi} \boldsymbol{\hbar}(\boldsymbol{r}) = \boldsymbol{\varrho}\_{\*} (\boldsymbol{r}, \boldsymbol{\hbar}(\boldsymbol{r})), & (\boldsymbol{\xi} \in (\mathbf{2}, \mathbf{3}), \ q \in (0, 1)), \\ \boldsymbol{\hbar}(\boldsymbol{0}) + \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{1}{}^{R}{}\_{q} \mathfrak{D}\_{0^{+}}^{\boldsymbol{r}} \boldsymbol{\hbar}(\boldsymbol{1}), & (\ell\_{1} \in \mathbb{R}^{>0}), \\ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\boldsymbol{\theta}} \boldsymbol{\hbar}(\boldsymbol{0}) + \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\boldsymbol{\theta}} \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{2}{}^{R} \mathfrak{D}\_{0^{+}}^{\boldsymbol{r}} \left[ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\boldsymbol{\theta}} \boldsymbol{\hbar} \right] (1), & (\ell\_{2} \in \mathbb{R}^{>0}), \\ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar}(\boldsymbol{0}) + \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{3}{}^{R} \mathfrak{D}\_{0^{+}}^{\boldsymbol{r}} \left[ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar} \right] (1), & (\ell\_{3} \in \mathbb{R}^{>0}), \end{cases} (1)$$

along with its inclusion version given by

$$\begin{cases} \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\xi} \boldsymbol{\hbar}(\boldsymbol{r}) \in \operatorname{T}\_{\*} (\boldsymbol{r}, \boldsymbol{\hbar}(\boldsymbol{r})), & (\boldsymbol{\xi} \in (\mathbf{2}, \mathbf{3}), \ q \in (0, 1)), \\ \boldsymbol{\hbar}(\boldsymbol{0}) + \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{1}{}^{R}{}\_{q} \mathfrak{D}\_{0^{+}}^{\nu} \boldsymbol{\hbar}(\boldsymbol{1}), & (\ell\_{1} \in \mathbb{R}^{>0}), \\ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\theta} \boldsymbol{\hbar}(\boldsymbol{0}) + \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\theta} \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{2}{}^{R} \mathfrak{D}\_{0^{+}}^{\sigma} \left[ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{\theta} \boldsymbol{\hbar} \right](\boldsymbol{1}), & (\ell\_{2} \in \mathbb{R}^{>0}), \\ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar}(\boldsymbol{0}) + \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar}(\boldsymbol{\xi}) = \ell\_{3}{}^{R} \mathfrak{D}\_{0^{+}}^{\sigma} \left[ \prescript{\text{C}}{q} \mathfrak{D}\_{0^{+}}^{1} \boldsymbol{\hbar} \right](\boldsymbol{1}), & (\ell\_{3} \in \mathbb{R}^{>0}), \end{cases} (2)$$

where *r* ∈ [0, 1], *ξ* ∈ (0, 1), *\$* ∈ (1, 2) and *σ* > 0. Two operators *C q*D (·) 0+ and *R q* I (·) 0+ represent the Caputo quantum derivative (CpQD) and the Riemann-Liouville quantum integral (RLQI). Furthermore, continuous single-valued function *ϕ*∗ : [0, 1] × R → R and multivalued function T∗ : [0, 1] × R → P(R) are assumed to be arbitrary equipped with some required specifications that will be explained subsequently. In comparison to other researches on the quantum difference BVPs that were published in the literature, we here deal with two abstract and extended structures of new fractional quantum difference equations/inclusions via q-integro-difference conditions in which the existing property of the relevant solutions is derived by terms of new notions of the functional analysis such as the condensing maps and the measure of noncompactness and the approximate endpoint criterion. These procedures on the suggested q-difference-BVPs (1) and (2) have been implemented in a limited range of research studies on the quantum fractional modelings. This yields the novelty and our main motivation to finalize this manuscript.

This research scheme is outlined as follows: We present the main concepts of the quantum calculus in Section 2. Our main results caused by new fixed-point approaches about solutions' existence of quantum BVP (1) and (2) will be obtained in Section 3. In Section 4, two numerical examples will be provided to support and validate our obtained results. A conclusion about our research work will be stated in Section 5.

#### **2. Fundamental Preliminaries**

In this section, some important issues in the sense of q-calculus are discussed. We suppose that 0 < *q* < 1. On the function (*m*1 − *m*2) *n* given for *n* ∈ N0, its q-analogue is defined by (*m*1 − *m*2) (0) = 1, and

$$(m\_1 - m\_2)^{(n)} = \prod\_{k=0}^{n-1} (m\_1 - m\_2 q^k)\_{\prime}$$

so that *m*1, *m*2 ∈ R and N0 := {0, 1, 2, . . . } [17]. Now, *n* = *ς* is a constant which is assumed to be contained in R. Let us now display the follwoing q-analogue of the existing power mapping (*m*1 − *m*2) *n* in a q-fractional settings:

$$(m\_1 - m\_2)^{(\varsigma)} = m\_1^{\varsigma} \prod\_{n=0}^{\infty} \frac{1 - (\frac{m\_2}{m\_1}) q^n}{1 - (\frac{m\_2}{m\_1}) q^{\varsigma + n'}} \tag{3}$$

for *m*1 6= 0. We note that by having *m*2 = 0, an equality *m* (*ς*) 1 = *m ς* 1 is obtained immediately [17]. For the given real number *m*1 ∈ R, a q-number [*m*1]*q* is expressed as:

$$[m\_1]\_q = \frac{1 - q^{m\_1}}{1 - q} = q^{m\_1 - 1} + \dots + q + 1.$$

The q-Gamma function is illustrated using the following format:

$$\Gamma\_q(r) = \frac{(1-q)^{(r-1)}}{(1-q)^{r-1}},\tag{4}$$

so that *r* ∈ R \ {0, −1, −2, . . .} [9,17]. It is notable that Γ*q*(*r* + 1) = [*r*]*q*Γ*q*(*r*) is valid [9]. A pseudo-code inspired by (3) and (4) is proposed in Algorithm 1 for computing various Gamma function's values in the proposed quantum settings.

Given a real-valued continuous function }, the quantum derivative of this function can be formulated by:

$$(\prescript{}{q}{\mathfrak{D}}\_{0^{+}}\hbar)(r) = \frac{\hbar(r) - \hbar(qr)}{(1-q)r} \,. \tag{5}$$

and also ( *q*D0+ })(0) = lim*r*→0( *q*D0+ })(*r*) [34]. Given a function }, the quantum derivative of this function can be extended to an arbitrary higher order by ( *q*D*n* 0+ })(*r*) =

*q*D0+ ( *q*D*n*−1 0+ })(*r*) for any *n* ∈ N [34]. Obviously, we notice that ( *q*D0 0+ })(*r*) = }(*r*). Similarly, for computing this kind of q-derivative of }, in Algorithm 2, we propose a pseudo-code inspired by (5).

**Algorithm 1** Pseudo-code for Γ*q*(*ς*):

**Require:** *ς* ∈ R\{0} ∪ Z −, *q* ∈ (0, 1), *n* 1: *w* ← 1 2: **for** *l* = 0 to *n* **do** 3: *w* ← *w*((1 − *q l*+1 )/(1 − *q ς*+*l* )) 4: **end for** 5: Γ*q*(*ς*) ← *w*/(1 − *q*) *ς*−1 **Ensure:** Γ*q*(*ς*)

**Algorithm 2** Pseudo-code for *q*D0+ }(*r*):

**Require:** *q* ∈ (0, 1), }(*r*), *r* 1: syms *b* 2: **if** *r* = 0 **then** 3: *φ* ← lim((}(*b*) − }(*q* ∗ *b*))/((1 − *q*)*b*), *b*, 0) 4: **else** 5: *φ* ← (}(*r*) − }(*q* ∗ *r*))/((1 − *q*) ∗ *r*) 6: **end if Ensure:** *q*D0+ }(*r*)

Given continuous map } : [0, *m*2] → R, the quantum integral of this function can be expressed as:

$$(\prescript{}{q}{\mathfrak{I}}\_{0^{+}}\hbar)(r) = \int\_{0}^{r} \hbar(v) \, \mathrm{d}\_{q}v = r(1-q) \sum\_{k=0}^{\infty} \hbar(rq^{k})q^{k} \, \quad (r \in [0, m\_{2}]) \tag{6}$$

provided the absolute convergence of the existing series holds [34]. The quantum integral of } can be similarly extended like quantum derivative to an arbitrary higher order using an iterative rule ( *q*I *n* 0+ })(*r*) = *q*I0+ ( *q*I *n*−1 0+ })(*r*) for all *n* ≥ 1 [34]. Moreover, it is clear to note that ( *q*I 0 0+ })(*r*) = }(*r*). A pseudo-code caused by (6) is proposed in in Algorithm 3. We now suppose that *m*1 ∈ [0, *m*2]. This time, the similar q-operator of } from *m*1 to *m*2 can be defined in this case as follows:

$$\int\_{m\_1}^{m\_2} \hbar(v) \, \mathbf{d}\_q v = \, \_q\Im\_{0^+} \hbar(m\_2) - \, \_q\Im\_{0^+} \hbar(m\_1)$$

$$= \int\_0^{m\_2} \hbar(v) \, \mathbf{d}\_q v - \int\_0^{m\_1} \hbar(v) \, \mathbf{d}\_q v$$

$$= (1 - q) \sum\_{k=0}^\infty [m\_2 \hbar(m\_2 q^k) - m\_1 \hbar(m\_1 q^k)] q^k,\tag{7}$$

when the series exists [34]. A proposed pseudo-code caused by (7) is organized in Algorithm 4 for such a purpose.

If we assume that a function } is continuous at *r* = 0, then ( *q*I0+ *q*D0+ })(*r*) = }(*r*) − }(0) is obtained [34]. Moreover, the equality ( *q*D0+ *q*I0+ })(*r*) = }(*r*) holds for each *r*. By considering a real number *ς* ≥ 0 in this case such that *n* − 1 < *ς* < *n*, i.e., *n* = [*ς*] + 1, for given function } ∈ CR([0, +∞)), the RLQI of } is introduced by:

$$\prescript{R}{q}{\mathfrak{I}}\_{0^{+}}^{\xi}\,\hbar(r) = \frac{1}{\Gamma\_{q}(\xi)}\int\_{0}^{r} (r - qv)^{(\xi - 1)} \hbar(v) \,\mathbf{d}\_{q}v, \qquad \xi > 0.$$

provided that the above value is finite and *R q* I 0 0+ }(*r*) = }(*r*) [35,36]. Further, the semigroup specification for the mentioned q-operator occurs such that *R q* I *ς*1 0+ ( *R q* I *ς*2 0+ })(*r*) = *R q* I *ς*1+*ς*2 0+ }(*r*) for *σ*1, *σ*2 ≥ 0 [35]. For *θ* ∈ (−1, ∞),

$$\prescript{R}{q}{\mathfrak{I}}\_{0^{+}}^{\xi} r^{\theta} = \frac{\Gamma\_{\mathfrak{q}}(\theta + 1)}{\Gamma\_{\mathfrak{q}}(\theta + \mathfrak{q} + 1)} r^{\theta + \xi} \, \vert \quad (r > 0).$$

It is evident that if we take *θ* = 0, then *R q* I *ς* 0+ 1(*r*) = 1 Γ*q*(*ς* + 1) *r ς* for any *r* > 0. Given a function } ∈ C(*n*) R ([0, +∞)), the CpQD for this function is formulated by:

$$\prescript{C}{}{\mathfrak{D}}\_{0^{+}}^{\xi}\,\hbar(r) = \frac{1}{\Gamma\_{q}(n-\xi)}\int\_{0}^{r}(r-qv)^{(n-\xi-1)}\,\_{q}\mathfrak{D}\_{0^{+}}^{n}\,\hbar(v)\,\mathbf{d}\_{q}v\,\omega$$

if the integral exists [35,36]. The following property is valid:

$$\, \_q\mathfrak{D}\_{0^+}^{\xi} r^{\theta} = \frac{\Gamma\_q(\theta + 1)}{\Gamma\_q(\theta - \xi + 1)} r^{\theta - \xi} \, \_ { } \quad (r > 0).$$

It is evident that *C q*D *ς* 0+ 1(*r*) = 0 for any *r* > 0. For instance, by letting *θ* = 2, *q* = 0.5 and }(*r*) = *r* 2 , we have

$$\, \_{0.5}\mathfrak{D}\_{0^+}^{\xi}r^2 = \frac{\Gamma\_{0.5}(3)}{\Gamma\_{0.5}(3-\emptyset)}r^{2-\emptyset} \, . $$

In this direction, the graph of the CpQD for the function }(*r*) = *r* 2 for *q* = 0.5 is available in Figure 1.

> *l* ))

**Algorithm 3** Pseudo-code for *q*I *ς* 0+ }(*r*):

**Require:** *ς*, *n*, }(*r*), *r*, *q* ∈ (0, 1) 1: *P* ← 0 2: **for** *k* = 0 to *n* **do** 3: *φ* ← (1 − *q k*+1 ) *ς*−1 4: *P* ← *P* + *φ* ∗ *q k* ∗ }(*r* ∗ *q k* ) 5: **end for** 6: *ψ* ← (*r ς* ∗ (1 − *q*) ∗ *P*)/(Γ*q*(*r*)) **Ensure:** *q*I *ς* 0+ }(*r*)

*m*1

**Algorithm 4** Pseudo-code for Z *m*2 *m*1 }(*v*) d*qv*: **Require:** }(*r*), *m*1, *k*, *m*2, *q* ∈ (0, 1) 1: *P* ← 0 2: **for** *l* = 0 : *k* **do** 3: *P* ← *P* + *q l* ∗ (*m*2 ∗ }(*m*2 ∗ *q l* ) − *m*1 ∗ }(*m*1 ∗ *q* 4: **end for** 5: *φ* ← (1 − *q*) ∗ *P* **Ensure:** Z *m*2 }(*v*) d*qv*

**Figure 1.** The graph of the Caputo q-derivative of }(*r*) = *r* 2 for *q* = 0.5.

**Lemma 1** ([37])**.** *Assume that n* − 1 < *ς* < *n and* } ∈ C(*n*) R ([0, +∞))*. Then, we have:*

$$(\prescript{C}{q}{\mathfrak{I}}\_{0^{+}} \prescript{C}{q}{\mathfrak{I}}\_{0^{+}} \hbar)(r) = \hbar(r) - \sum\_{k=0}^{n-1} \frac{r^{k}}{\Gamma\_{q}(k+1)} (\prescript{}{q}{\mathfrak{I}}\_{0^{+}} \hbar)(0).$$

According to the above lemma, the given fractional quantum differential equation, *C q*D *ς* 0+ }(*r*) = 0, has a general solution which is obtained by }(*r*) = *µ*˜0 + *µ*˜1*r* + *µ*˜2*r* 2 + · · · + *µ*˜*n*−1*r n*−1 so that *µ*˜0, . . . , *µ*˜*n*−1 ∈ R, and *n* = [*ς*] + 1 [37]. It is worth noting that for each continuous }, according to Lemma 1, we get:

$$(\prescript{R}{q}{\mathfrak{I}}\_{0^{+}}^{\xi}\prescript{C}{q}{\mathfrak{I}}\_{0^{+}}^{\xi}\hbar)(r) = \hbar(r) + \sharp\_{0} + \sharp\_{1}r + \sharp\_{2}r^{2} + \cdots + \sharp\_{n-1}r^{n-1},$$

where *µ*˜0, . . . , *µ*˜*n*−1 illustrate constants contained in R, and *n* = [*ς*] + 1 [37].

Next, we recall some essential inequalities and concepts. The Kuratowski measure of noncompactness O is defined by

$$\mathbb{C}\mathbb{D}(\mathcal{H}) := \inf \{ \varepsilon > 0 : \mathcal{H} = \bigcup\_{k=1}^{n} \mathcal{H}\_{k} \text{ and } \text{diam } (\mathcal{H}\_{k}) \le \varepsilon \text{ for } k = 1, \dots, n \}.$$

where diam(H*k* ) = sup{|} − } 0 | : }, } 0 ∈ H*k*} and H is bounded subset of Banach space A. Moreover, it is identified that 0 ≤ O(H) ≤ diam (H) < +∞ [38].

**Lemma 2** ([38])**.** *Consider the bounded subsets* H, H1 *and* H2 *of an arbitrary real Banach space* A*. Then, the following conditions hold:*

(C5) ∀*κ* ∈ R, O(*κ*H) = |*κ*|O(H)*;*

(C6) O(H1 + H2) ≤ O(H1) + O(H2), *where* H1 + H2 = {}1 + }2; }1 ∈ H1, }2 ∈ H2}*;* (C7) O(H1 ∪ H2) ≤ max{O(H1) + O(H2)}*.*

**Lemma 3** ([39])**.** *Regard* A *as a Banach space. Then, for each bounded set* H ⊆ A*, a countable set* H0 ⊆ H *exists subject to* O(H) ≤ 2 O(H0)*.*

**Lemma 4** ([38])**.** *Regard* A *as a Banach space. Let* H *be bounded and equi-continuous set contained in* CA([*a*, *b*])*. Then,* O(H(*r*)) *is continuous on* [*a*, *b*]*, and we have* O(H) = sup*r*∈[*a*,*b*] O(H(*r*))*.*

**Lemma 5** ([38])**.** *Let* A *be a Banach space. Let* H = {}*n*}*n*≥1 ⊆ CA([*a*, *b*]) *be bounded and countable set. Then,* O(H(*r*)) *is Lebesgue integrable on* [*a*, *b*]*, and we have:*

$$\mathbb{D}\left(\left\{\int\_0^r \hbar\_n(v) \,dv\right\}\_{n\geq 1}\right) \leq 2 \int\_0^r \mathbb{D}(\{\hbar\_n(v)\}\_{n\geq 1}) \,dv.$$

**Definition 1** ([38])**.** *Regard* A *as a Banach space and ϕ*∗ : S ⊂ A → A *as a bounded and continuous operator. Then, the map ϕ*∗ *is termed condensing if for any bounded closed set* H ⊆ S*, the inequality* O(*ϕ*∗(H)) < O(H) *holds.*

**Theorem 1** ([38], Sadovskii's fixed point theorem)**.** *Regard* A *as a Banach space. Let* H *be a bounded, closed and convex set contained in* A*. Furthermore, assume that continuous mapping ϕ*∗ : H → H *is condensing. Then, there exists at least one fixed point for the map ϕ*∗ *in* H*.*

Let us denote the normed space by (A, k · kA). Regard P(A), P*bd*(A), P*cl*(A), P*cm*(A) and P*cx*(A) as a family of all non-empty, all bounded, all closed, all compact and all convex sets contained in A, respectively.

**Definition 2** ([40])**.** *An element* } ∈ A *is termed an endpoint of a multi-valued function* T∗ : A → P(A) *whenever we get* T∗(}) = {}}*.*

The multi-valued map T∗ has an approximate endpoint criterion (AEPC) if

$$\inf\_{\hbar\_1 \in \mathfrak{A}} \sup\_{\hbar\_2 \in \mathbb{T}\_\*(\hbar\_1)} d(\hbar\_1, \hbar\_2) = 0\_\prime$$

Ref. [40]. Next, a required theorem related to the proposed quantum boundary problem is recalled.

**Theorem 2** ([40], Endpoint theorem)**.** *Let's assume that* (A, *d*) *is a complete metric space, and ψ* : [0, ∞) → [0, ∞) *is u.s.c subject to for each r* > 0*,* lim inf*r*→∞(*r* − *ψ*(*r*)) > 0*, and ψ*(*r*) < *r. Assume that* T∗ : A → P*cl*,*bd*(A) *is a multi-valued map such that for each* }1, }2 ∈ A*, the following inequality holds:*

$$\mathbb{H}\_d(\mathbb{T}\_\*\hbar\_1, \mathbb{T}\_\*\hbar\_2) \le \psi(d(\hbar\_1, \hbar\_2)).$$

*Then, there is exactly one endpoint for* T∗ *iff* T∗ *has an approximate endpoint criterion.*

#### **3. Main Results**

We regard the family of continuous functions on [0, 1] by A = CR([0, 1]) and the defined sup-norm k}kA = sup*r*∈[0,1] |}(*r*)|, for all members } ∈ A, confirms that the space A becomes a Banach space. In the sequel, we will establish the existence results for quantum BVP (1) and (2). Before moving to the existence results, the following proposition will play an essential role:

**Proposition 1.** *Let ϕ*∗ ∈ A*, ς* ∈ (2, 3)*, \$* ∈ (1, 2)*, ξ* ∈ (0, 1)*,* `1, `2, `3 ∈ R>0 *and σ* > 0*. Then, the function* } ∗ *satisfies as a solution for the given quantum integro-difference FBVP (CpQFP) formulated by*

$$\begin{cases} \prescript{\text{C}}{q}{\mathfrak{D}}\_{0^{+}}^{\xi} \, \mathsf{h}^{\star}(r) = \varphi\_{\ast}(r), & (r \in [0, 1], q \in (0, 1)), \\ \boldsymbol{\hbar}(\mathsf{0}) + \boldsymbol{\hbar}(\xi) = \ell\_{1}{}^{R}{}\_{q} \mathfrak{I}\_{0^{+}}^{\sigma} \, \mathsf{h}(1), \\ \prescript{\text{C}}{q}{\mathfrak{D}}\_{0^{+}}^{\theta} \, \boldsymbol{\hbar}(\mathsf{0}) + \prescript{\text{C}}{q}{}\_{q} \mathfrak{D}\_{0^{+}}^{\theta} \, \boldsymbol{\hbar}(\xi) = \ell\_{2}{}^{R} \mathfrak{I}\_{0^{+}}^{\sigma} \left[ \prescript{\text{C}}{q}{}\_{q} \mathfrak{D}\_{0^{+}}^{\theta} \, \boldsymbol{\hbar} \right] (1), \\ \prescript{\text{C}}{q}{\mathfrak{D}}\_{0^{+}}^{1} \, \boldsymbol{\hbar}(\mathsf{0}) + \prescript{\text{C}}{q}{}\_{q} \mathfrak{D}\_{0^{+}}^{1} \, \boldsymbol{\hbar}(\xi) = \ell\_{3}{}^{R} \mathfrak{I}\_{0^{+}}^{\sigma} \left[ \prescript{\text{C}}{q}{}\_{q} \mathfrak{D}\_{0^{+}}^{1} \, \boldsymbol{\hbar} \right] (1), \end{cases} (8)$$

*iff* } ∗ *is a solution for the fractional quantum integral (FQI) equation given by*

$$h^\*(r) = \int\_0^r \frac{(r - q\mathbf{v})^{(\zeta - 1)}}{\Gamma\_q(\zeta)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v + \frac{\ell\_1}{\delta\_1} \int\_0^1 \frac{(1 - q\mathbf{v})^{(\zeta + \sigma - 1)}}{\Gamma\_q(\zeta + \sigma)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v \tag{9}$$

$$- \frac{1}{\delta\_1} \int\_0^{\xi} \frac{(\zeta - q\mathbf{v})^{(\zeta - 1)}}{\Gamma\_q(\zeta)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v + \ell\_3 \Lambda\_1(r) \int\_0^1 \frac{(1 - q\mathbf{v})^{(\zeta + \sigma - 2)}}{\Gamma\_q(\zeta + \sigma - 1)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v$$

$$- \Lambda\_1(r) \int\_0^{\xi} \frac{(\zeta - q\mathbf{v})^{(\zeta - 2)}}{\Gamma\_q(\zeta - 1)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v$$

$$+ \ell\_2 \Lambda\_2(r) \int\_0^1 \frac{(1 - q\mathbf{v})^{(\zeta + \sigma - \varrho - 1)}}{\Gamma\_q(\zeta + \sigma - \varrho)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v - \Lambda\_2(r) \int\_0^{\xi} \frac{(\xi - qv)^{(\zeta - \varrho - 1)}}{\Gamma\_q(\zeta - \varrho)} \boldsymbol{\varrho}\_\*(\mathbf{v}) \, \mathbf{d}\_q v. \tag{10}$$

**Proof.** Firstly, the given function } ∗ is regarded as a solution for (8). By virtue of *ς* ∈ (2, 3), taking the integral in the RL-settings of order *ς* to (8), we arrive at

$$\hbar^\*(r) = \int\_0^r \frac{(r - qv)^{(\varsigma - 1)}}{\Gamma\_q(\varsigma)} \varphi\_\*(v) \, \mathrm{d}\_q v + \sharp\_0 + \sharp\_1 r + \sharp\_2 r^2,\tag{11}$$

so that *µ*˜0, *µ*˜1, *µ*˜2 ∈ R are some constants that are needed to be obtained. By considering *\$* ∈ (1, 2), the following immediate results are obtained

$$\begin{split} \, ^C\_q \mathfrak{D}\_{0+}^1 \hbar^\*(r) &= \int\_0^r \frac{(r - qv)^{(\mathfrak{c} - 2)}}{\Gamma\_q(\mathfrak{c} - 1)} \varrho\_\*(v) \, \mathrm{d}\_q v + \bar{\mu}\_1 + \bar{\mu}\_2 (1 + q) r\_r \\ \, ^C\_q \mathfrak{D}\_{0+}^\theta \, \hbar^\*(r) &= \int\_0^r \frac{(r - qv)^{(\mathfrak{c} - \mathfrak{c} - 1)}}{\Gamma\_q(\mathfrak{c} - \mathfrak{g})} \varrho\_\*(v) \, \mathrm{d}\_q v + \bar{\mu}\_2 \frac{2r^{2 - \mathfrak{g}}}{\Gamma\_q(3 - \mathfrak{g})} , \\ \, ^R\_q \mathfrak{D}\_{0+}^\sigma \, \hbar^\*(r) &= \int\_0^r \frac{(r - qv)^{(\mathfrak{c} + \sigma - 1)}}{\Gamma\_q(\mathfrak{c} + \sigma)} \varrho\_\*(v) \, \mathrm{d}\_q v + \bar{\mu}\_0 \frac{r^\sigma}{\Gamma\_q(\sigma + 1)} + \bar{\mu}\_1 \frac{r^{\sigma + 1}}{\Gamma\_q(\sigma + 2)} , \\ & \quad + \bar{\mu}\_2 \frac{(1 + q) r^{\sigma + 2}}{\Gamma\_q(\sigma + 3)} . \end{split}$$

$$\prescript{R}{q}{\mathcal{D}}\_{0^{+}}^{\sigma} \left( \prescript{C}{q}{\mathfrak{D}}\_{0^{+}}^{1} \kappa^{\*}(r) \right) = \int\_{0}^{r} \frac{(r - qv)^{(\varsigma + \sigma - 2)}}{\Gamma\_{q}(\varsigma + \sigma - 1)} \varphi\_{\*}(v) \, \mathrm{d}\_{q}v + \tilde{\mu}\_{1} \frac{r^{\sigma}}{\Gamma\_{q}(\sigma + 1)} + \tilde{\mu}\_{2} \frac{(1 + q)r^{\sigma + 1}}{\Gamma\_{q}(\sigma + 2)}.$$

$$\Gamma\_q^{\mathbb{R}} \mathfrak{I}\_{0^+}^{\sigma} \left( {}^{\mathbb{C}}\_q \mathfrak{D}\_{0^+}^{\varrho} \hbar^\*(r) \right) = \int\_0^r \frac{(r - qv)^{(\varrho + \sigma - \varrho - 1)}}{\Gamma\_q(\varrho + \sigma - \varrho)} \wp\_\*(v) \, \mathrm{d}\_q v + \sharp \! 2 \frac{2r^{\sigma + 2 - \varrho}}{\Gamma\_q(\sigma + 3 - \varrho)}.$$

Now, by virtue of the given boundary conditions, we get

$$\mathfrak{f}\_0 = \frac{\ell\_1}{\delta\_1} \int\_0^1 \frac{(1 - qv)^{(\zeta + \sigma - 1)}}{\Gamma\_q(\zeta + \sigma)} \varphi\_\*(v) \, \mathbf{d}\_q v - \frac{1}{\delta\_1} \int\_0^\xi \frac{(\zeta - qv)^{(\zeta - 1)}}{\Gamma\_q(\zeta)} \varphi\_\*(v) \, \mathbf{d}\_q v$$

− `3Θ<sup>1</sup> Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) *ϕ*∗(*v*) d*qv* + Θ<sup>1</sup> Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) *ϕ*∗(*v*) d*qv* + `2Θ2 Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) *ϕ*∗(*v*) d*qv* − Θ2 Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) *ϕ*∗(*v*) d*qv*, *µ*˜1 = `3 ∆1 Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) *ϕ*∗(*v*) d*qv* − 1 ∆1 Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) *ϕ*∗(*v*) d*qv* − `2∆2 ∆1∆3 Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) *ϕ*∗(*v*) d*qv* + ∆2 ∆1∆3 Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) *ϕ*∗(*v*) d*qv*,

and

$$\tilde{\mu}\_2 = \frac{\ell\_2}{\Delta\_3} \int\_0^1 \frac{(1 - qv)^{(\zeta + \sigma - \varrho - 1)}}{\Gamma\_q(\zeta + \sigma - \varrho)} \varphi\_\*(v) \, \mathbf{d}\_q v - \frac{1}{\Delta\_3} \int\_0^\xi \frac{(\xi - qv)^{(\zeta - \varrho - 1)}}{\Gamma\_q(\zeta - \varrho)} \varphi\_\*(v) \, \mathbf{d}\_q v \, \mathbf{d}\_q$$

where we regard the constants

$$\begin{aligned} \delta\_1 &= \frac{2\Gamma\_q(\sigma+1)-\ell\_1}{\Gamma\_q(\sigma+1)}, \quad \delta\_2 = \frac{\xi\Gamma\_q(\sigma+2)-\ell\_1}{\Gamma\_q(\sigma+2)}, \quad \delta\_3 = \frac{\xi^2\Gamma\_q(\sigma+3)-\ell\_1(1+q)}{\Gamma\_q(\sigma+3)}, \\\\ \Delta\_1 &= \frac{2\Gamma\_q(\sigma+1)-\ell\_3}{\Gamma\_q(\sigma+1)}, \quad \Delta\_2 = \frac{(1+q)\left(\xi\Gamma\_q(\sigma+2)-\ell\_3\right)}{\Gamma\_q(\sigma+2)}, \\\\ \Delta\_3 &= \frac{2\xi^{2-q}\Gamma\_q(\sigma+3-q)-2\ell\_2\Gamma\_q(3-q)}{\Gamma\_q(3-q)\Gamma\_q(\sigma+3-q)}, \quad \Theta\_1 = \frac{\delta\_2}{\delta\_1\Delta\_1}, \quad \Theta\_2 = \frac{\delta\_2\Delta\_2-\delta\_3\Delta\_1}{\delta\_1\Delta\_1\Delta\_3}, \end{aligned}$$

along with the functions with respect to *r* as

$$\Lambda\_1(r) = \frac{r - \Theta\_1 \Delta\_1}{\Delta\_1}, \qquad \qquad \Lambda\_2(r) = \frac{r^2 \Delta\_1 - r \Delta\_2 + \Theta\_2 \Delta\_1 \Delta\_3}{\Delta\_1 \Delta\_3}. \tag{12}$$

By substituting the values of *µ*˜0, *µ*˜1 and *µ*˜2 in (11), integral solution (9) is obtained. The converse part can be easily deduced.

**Remark 1.** *Note that for simplicity in the subsequent computations, we set the following upper bounds by virtue of the functions displayed in* (12)*:*

$$|\Lambda\_1(r)| \le \frac{1 + |\Theta\_1||\Delta\_1|}{|\Delta\_1|} := \Lambda\_1^\* > 0,$$

$$|\Lambda\_2(r)| \le \frac{|\Delta\_1| + |\Delta\_2| + |\Theta\_2||\Delta\_1||\Delta\_3|}{|\Delta\_1||\Delta\_3|} := \Lambda\_2^\* > 0. \tag{13}$$

**Theorem 3.** *Let ϕ*∗ : [0, 1] × A → R *be continuous. In addition, assume that there exists a continuous ϑ* : [0, 1] → R>0 *along with a nondecreasing continuous map* ℘ : [0, ∞) → (0, ∞) *such that for each r* ∈ [0, 1] *and* } ∈ A*,*

$$|\varphi\_\*(r,\hbar(r))| \le \theta(r)\wp(\|\hbar\|\_{\mathfrak{A}}).\tag{14}$$

*We suppose that there exists a function mϕ*∗ : [0, 1] → R *such that for each bounded set* H ⊆ A *and r* ∈ [0, 1]*,*

$$\mathbb{G}(\varrho\_\*(r,\mathcal{H})) \le m\_{\varrho\_\*}(r)\mathbb{G}(\mathcal{H}).\tag{15}$$

*Then, at least one solution of the given Caputo fractional quantum BVP* (1) *exists on* [0, 1] *if*

$$
\left[\frac{\dot{m}\_{\varphi\_{\*}}}{\Gamma\_{q}(\boldsymbol{\varsigma}+1)} + \frac{\dot{m}\_{\varphi\_{\*}}}{|\delta\_{1}|} \left(\frac{\ell\_{1}}{\Gamma\_{q}(\boldsymbol{\varsigma}+\sigma+1)} + \frac{\xi^{\boldsymbol{\varsigma}(\boldsymbol{\varsigma})}}{\Gamma\_{q}(\boldsymbol{\varsigma}+1)}\right) + \dot{m}\_{\varphi\_{\*}}\Lambda\_{1}^{\star} \left(\frac{\ell\_{3}}{\Gamma\_{q}(\boldsymbol{\varsigma}+\sigma)} + \frac{\xi^{\boldsymbol{\varsigma}(\boldsymbol{\varsigma}-1)}}{\Gamma\_{q}(\boldsymbol{\varsigma})}\right) $$

$$
+ \dot{m}\_{\varphi\_{\*}}\Lambda\_{2}^{\star} \left(\frac{\ell\_{2}}{\Gamma\_{q}(\boldsymbol{\varsigma}+\sigma-\varrho+1)} + \frac{\xi^{\boldsymbol{\varsigma}(\boldsymbol{\varsigma}-\varrho)}}{\Gamma\_{q}(\boldsymbol{\varsigma}-\varrho+1)}\right) \right] < \frac{1}{4},\tag{16}
$$

*where m*˜ *ϕ*∗ = sup*r*∈[0,1] |*mϕ*∗ (*r*)|*.*

**Proof.** Introduce the mapping G : H → H defined as:

$$\begin{split} \mathcal{G}(\hbar)(r) &= \int\_{0}^{\tau} \frac{(r - qv)^{(\zeta - 1)}}{\Gamma\_{q}(\varsigma)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v \\ &+ \frac{\ell\_{1}}{\delta\_{1}} \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - 1)}}{\Gamma\_{q}(\varsigma + \sigma)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v - \frac{1}{\delta\_{1}} \int\_{0}^{\zeta} \frac{(\xi - qv)^{(\zeta - 1)}}{\Gamma\_{q}(\varsigma)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v \\ &+ \ell\_{3} \Lambda\_{1}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - 2)}}{\Gamma\_{q}(\varsigma + \sigma - 1)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v \\ &- \Lambda\_{1}(r) \int\_{0}^{\zeta} \frac{(\xi - qv)^{(\zeta - 2)}}{\Gamma\_{q}(\varsigma - 1)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v \\ &+ \ell\_{2} \Lambda\_{2}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - \varrho - 1)}}{\Gamma\_{q}(\varsigma + \sigma - \varrho)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v \\ &- \Lambda\_{2}(r) \int\_{0}^{\zeta} \frac{(\xi - qv)^{(\zeta - \sigma - 1)}}{\Gamma\_{q}(\varsigma - \varrho)} \varphi\_{\*}(v, \hbar(v)) \, \mathrm{d}\_{q}v, \end{split} \tag{19}$$

where H = {} ∈ A : k}kA ≤ *ε*∗,*ε*∗ ∈ R>0} ⊆ A and is classified as a convex bounded closed space. Obviously, the fixed point of the proposed operator G is the quantum fractional BVP's solution (1).

Firstly, we verify the continuity of G on H. Take the sequence {}*n*}*n*≥1 in H such that }*n* → } for each } ∈ H. Since *ϕ*∗ is continuous on [0, 1] × A, so we can write lim*n*→∞ *ϕ*∗(*r*, }*n*(*r*)) = *ϕ*∗(*r*, }(*r*)). Now, with the aid of Lebesgue dominated convergence theorem, we obtain:

lim*n*→∞ (G}*n*)(*r*) = Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) lim*n*→∞ *ϕ*∗(*v*, }*n*(*v*)) d*qv* + `1 *δ*1 Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) lim*n*→<sup>∞</sup> *ϕ*∗(*v*, }*n*(*v*)) d*qv* − 1 *δ*1 Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) lim*n*→<sup>∞</sup> *ϕ*∗(*v*, }*n*(*v*)) d*qv* + `3Λ1(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) lim*n*→∞ *ϕ*∗(*v*, }*n*(*v*)) d*qv* − Λ1(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) lim*n*→∞ *ϕ*∗(*v*, }*n*(*v*)) d*qv* + `2Λ2(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) lim*n*→∞ *ϕ*∗(*v*, }*n*(*v*)) d*qv*

$$\begin{aligned} & -\Lambda\_2(r) \int\_0^\xi \frac{(\mathfrak{J} - qv)^{(\mathfrak{c} - \varrho - 1)}}{\Gamma\_q(\mathfrak{g} - \varrho)} \lim\_{n \to \infty} \varrho\_\*(v, \hbar\_n(v)) \, \mathrm{d}\_q v \\ & = (\mathcal{G}\hbar)(r), \end{aligned}$$

for each *r* ∈ [0, 1]. Thus, we get lim*n*→∞(G}*n*)(*r*) = (G})(*r*). Hence, the continuity of G on H is proved. Now, we want to examine uniform boundedness of G on H. To accomplish this goal, consider } ∈ H. In view of inequalities (13) and (14), we have:

Set

$$
\Omega = \frac{1}{\Gamma\_q(\mathfrak{c} + 1)} + \frac{1}{|\delta\_1|} \left( \frac{\ell\_1}{\Gamma\_q(\mathfrak{c} + \sigma + 1)} + \frac{\mathfrak{z}^{(\mathfrak{c})}}{\Gamma\_q(\mathfrak{c} + 1)} \right) + \Lambda\_1^\* \left( \frac{\ell\_3}{\Gamma\_q(\mathfrak{c} + \sigma)} + \frac{\mathfrak{z}^{(\mathfrak{c} - 1)}}{\Gamma\_q(\mathfrak{c})} \right)
$$

$$
+ \Lambda\_2^\* \left( \frac{\ell\_2}{\Gamma\_q(\mathfrak{c} + \sigma - \varrho + 1)} + \frac{\mathfrak{z}^{(\mathfrak{c} - \varrho)}}{\Gamma\_q(\mathfrak{c} - \varrho + 1)} \right). \tag{20}
$$

Consequently, we can declare that kG}kA ≤ Ωˆ *ϑ* ∗℘(*ε*) < ∞, and this implies uniform boundedness of G on H. Next, we ensure the equi-continuity of G. In order to check this, consider *r*1,*r*2 ∈ [0, 1] such that *r*1 < *r*2 and } ∈ H. Then, we get:

$$\begin{split} \left| (\mathcal{G}h)(r\_{2}) - (\mathcal{G}h)(r\_{1}) \right| &\leq \int\_{0}^{r\_{1}} \frac{[(r\_{2}-qv)^{(\zeta-1)} - (r\_{1}-qv)^{(\zeta-1)}]}{\Gamma\_{q}(\zeta)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v \\ &\quad + \int\_{r\_{1}}^{r\_{2}} \frac{(r\_{2}-qv)^{(\zeta-1)}}{\Gamma\_{q}(\zeta)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v \\ &\quad + \ell\_{3} |\Lambda\_{1}(r\_{2}) - \Lambda\_{1}(r\_{1})| \int\_{0}^{1} \frac{(1-qv)^{(\zeta+\sigma-2)}}{\Gamma\_{q}(\zeta+\sigma-1)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v \\ &\quad + [\Lambda\_{1}(r\_{2}) - \Lambda\_{1}(r\_{1})] \int\_{0}^{\xi} \frac{(\xi-qv)^{(\zeta-2)}}{\Gamma\_{q}(\zeta-1)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v \\ &\quad + \ell\_{2} [\Lambda\_{2}(r\_{2}) - \Lambda\_{2}(r)] \int\_{0}^{1} \frac{(1-qv)^{(\zeta+\sigma-q-1)}}{\Gamma\_{q}(\zeta+\sigma-q)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v \\ &\quad + [\Lambda\_{2}(r\_{2}) - \Lambda\_{2}(r)] \int\_{0}^{\xi} \frac{(\xi-qv)^{(\zeta-q-1)}}{\Gamma\_{q}(\zeta-q)} |q\_{\*}(v,h(v))| \, \mathbf{d}\_{q}v. \end{split}$$

Note that the above inequality's right hand side goes to zero as *r*1 → *r*2 (independent of }). Hence, it is evident that k(G})(*r*2) − (G})(*r*1)kA → 0 as *r*1 → *r*2, and this confirms that G is an equi-continuous. Consequently, we conclude that G is a compact operator on H in view of the famous Arzela–Ascoli theorem.

At this point, we will check that G is condensing operator on H. By Lemma 3, it is obvious that a countable set H0 = {}*n*}*n*≥1 ⊂ H exists for each bounded subset H ⊂ H such that O(G(H)) ≤ 2 O(G(H0)) holds. Hence, in the light of Lemmas 2, 4 and 5, the following is obtained

O(G(H)(*r*)) ≤ 2 O(G({}*n*}*n*≥1)) ≤ 2 Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2`1 |*δ*1 | Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2 |*δ*1 | Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2`3Λ1(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2Λ1(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2`2Λ2(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv* + 2Λ2(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) O(*ϕ*∗(*v*, {}*n*(*v*)}*n*≥1)) d*qv*

≤ 4 Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4`1 |*δ*1 | Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4 |*δ*1 | Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4`3Λ1(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4Λ1(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4`2Λ2(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* + 4Λ2(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) *mϕ*∗ (*v*)O({}*n*(*v*)}*n*≥1) d*qv* ≤ 4*m*˜ *ϕ*∗O(H) Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) d*qv* + 4`1*m*˜ *<sup>ϕ</sup>*∗O(H) |*δ*1 | Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) d*qv* + 4*m*˜ *<sup>ϕ</sup>*∗O(H) |*δ*1 | Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) d*qv* + 4`3Λ ∗ 1*m*˜ *ϕ*∗O(H) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) d*qv* + 4Λ ∗ 1*m*˜ *ϕ*∗O(H) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) d*qv* + 4`2Λ ∗ 2*m*˜ *ϕ*∗O(H) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) d*qv* + 4Λ ∗ 2*m*˜ *ϕ*∗O(H) Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) d*qv* ≤ 4*m*˜ *ϕ*∗O(H) Γ*q*(*ς* + 1) + 4`1*m*˜ *<sup>ϕ</sup>*∗O(H) |*δ*1 |Γ*q*(*ς* + *σ* + 1) + 4*ξ* (*ς*)*m*˜ *<sup>ϕ</sup>*∗O(H) |*δ*1 |Γ*q*(*ς* + 1) + 4`3Λ∗ 1*m*˜ *ϕ*∗O(H) Γ*q*(*ς* + *σ*) + 4*ξ* (*ς*−1)Λ∗ 1*m*˜ *ϕ*∗O(H) Γ*q*(*ς*) + 4`2Λ∗ 2*m*˜ *ϕ*∗O(H) Γ*q*(*ς* + *σ* − *\$* + 1) + 4*ξ* (*ς*−*\$*)Λ∗ 2*m*˜ *ϕ*∗O(H) Γ*q*(*ς* − *\$* + 1) .

Hence,

$$\begin{split} \mathbb{G}(\mathcal{G}(\mathcal{H})) &\leq 4\left[\frac{\tilde{m}\_{\rho\_{\ast}}}{\Gamma\_{q}(\boldsymbol{\xi}+1)} + \frac{\tilde{m}\_{\rho\_{\ast}}}{|\boldsymbol{\xi}\_{1}|} \left(\frac{\ell\_{1}}{\Gamma\_{q}(\boldsymbol{\xi}+\boldsymbol{\sigma}+1)} + \frac{\tilde{\xi}^{(\boldsymbol{\xi})}}{\Gamma\_{q}(\boldsymbol{\xi}+1)}\right) \right. \\\\ &\left. + \tilde{m}\_{\rho\_{\ast}}\Lambda\_{1}^{\*} \left(\frac{\ell\_{3}}{\Gamma\_{q}(\boldsymbol{\xi}+\boldsymbol{\sigma})} + \frac{\tilde{\xi}^{(\boldsymbol{\xi}-1)}}{\Gamma\_{q}(\boldsymbol{\xi})}\right) \right. \\\\ &\left. + \tilde{m}\_{\rho\_{\ast}}\Lambda\_{2}^{\*} \left(\frac{\ell\_{2}}{\Gamma\_{q}(\boldsymbol{\xi}+\boldsymbol{\sigma}-\boldsymbol{\varrho}+1)} + \frac{\tilde{\xi}^{(\boldsymbol{\xi}-\boldsymbol{\varrho})}}{\Gamma\_{q}(\boldsymbol{\xi}-\boldsymbol{\varrho}+1)}\right) \right] \mathbb{G}(\mathcal{H}). \end{split}$$

By applying condition (16), we get O(G(H)) < O(H). This clearly implies that G is condensing operator on H. Ultimately, by employing Theorem 1, we can infer that the map G possesses one fixed point leastwise in H. Thus, it is found at least one solution for the supposed quantum-integro-difference FBVP (1) and finally the proof process is terminated.

Now, we set up an existence criterion for the given fractional quantum inclusion BVP (2). The inclusion problem's solution (2) is determined by an absolutely continuous function } : [0, 1] → R whenever it satisfies the given fractional quantum integro-difference conditions, and a function z ∈ L1 ([0, 1], R) exists such that the inclusion z(*r*) ∈ T∗(*r*, }(*r*)) holds for almost all *r* ∈ [0, 1], and we have:

$$\begin{split} \boldsymbol{\hbar}(\boldsymbol{r}) &= \int\_{0}^{\boldsymbol{r}} \frac{(\boldsymbol{r} - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} - 1)}}{\Gamma\_{q}(\boldsymbol{\varepsilon})} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q} \boldsymbol{v} + \frac{\ell\_{1}}{\delta\_{1}} \int\_{0}^{1} \frac{(\boldsymbol{1} - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} + \boldsymbol{\varepsilon} - 1)}}{\Gamma\_{q}(\boldsymbol{\varepsilon} + \boldsymbol{\sigma})} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q} \boldsymbol{v} \\ &- \frac{1}{\delta\_{1}} \int\_{0}^{\boldsymbol{\xi}} \frac{(\boldsymbol{\xi} - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} - 1)}}{\Gamma\_{q}(\boldsymbol{\varepsilon})} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q} \boldsymbol{v} \\ &+ \ell\_{3} \Lambda\_{1}(\boldsymbol{r}) \int\_{0}^{1} \frac{(1 - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} + \boldsymbol{\sigma} - 2)}}{\Gamma\_{q}(\boldsymbol{\varepsilon} + \boldsymbol{\sigma} - 1)} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q} \boldsymbol{v} - \Lambda\_{1}(\boldsymbol{r}) \int\_{0}^{\boldsymbol{\xi}} \frac{(\boldsymbol{\xi} - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} - 2)}}{\Gamma\_{q}(\boldsymbol{\varepsilon} - 1)} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q} \boldsymbol{v} \\ &+ \ell\_{2} \Lambda\_{2}(\boldsymbol{r}) \int\_{0}^{1} \frac{(1 - q\boldsymbol{v})^{(\boldsymbol{\varepsilon} + \boldsymbol{\sigma} - q - 1)}}{\Gamma\_{q}(\boldsymbol{\varepsilon} + \boldsymbol{\sigma} - q)} \mathfrak{z}(\boldsymbol{v}) \, \mathrm{d}\_{q$$

for each *r* ∈ [0, 1]. Let ST∗,} represents the collection of all selections of T∗ for each } ∈ A and is defined as

$$\mathfrak{S}\_{\mathbb{T}\_\*,\hbar} = \{ \mathfrak{z} \in \mathcal{L}^1([0,1]) : \mathfrak{z}(r) \in \mathbb{T}\_\*(r,\hbar(r)) \text{ for almost all } r \in [0,1] \}.$$

Construct a multi-valued map J : A → P(A) which is defined as

$$\mathcal{J}(\hbar) = \{ \hbar \in \mathfrak{A} : \mathfrak{h}(r) = \mathfrak{a}(r) \}, \tag{21}$$

where

*v*(*r*) = Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) z(*v*) d*qv* + `1 *δ*1 Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) z(*v*) d*qv* − 1 *δ*1 Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) z(*v*) d*qv* + `3Λ1(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) z(*v*) d*qv* − Λ1(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) z(*v*) d*qv* + `2Λ2(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) z(*v*) d*qv* − Λ2(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) z(*v*) d*qv*, z ∈ ST∗,}.

**Theorem 4.** *Let* T∗ : [0, 1] × A → P*cm*(A) *be a multi-valued map. Suppose that*

(A3) *ζ* ∈ C([0, 1], [0, ∞)) *exists subject to*

$$\mathbb{H}\_d(\mathbb{T}\_\*(r, \hbar\_1(r)), \mathbb{T}\_\*(r, \hbar\_2(r))) \le \zeta(r)\psi(|\hbar\_1(r) - \hbar\_2(r)|) \frac{1}{\mathcal{Q}'} $$

*for each r* ∈ [0, 1] *and* }1, }2 ∈ A*, where* sup*r*∈[0,1] |*ζ*(*r*)| = k*ζ*k *and*

$$\mathcal{Q} = \left[ \frac{1}{\Gamma\_q(\zeta + 1)} + \frac{1}{|\delta\_1|} \left( \frac{\ell\_1}{\Gamma\_q(\zeta + \sigma + 1)} + \frac{\xi^{(\varsigma)}}{\Gamma\_q(\varsigma + 1)} \right) + \Lambda\_1^\* \left( \frac{\ell\_3}{\Gamma\_q(\varsigma + \sigma)} + \frac{\xi^{(\varsigma - 1)}}{\Gamma\_q(\varsigma)} \right) \right]$$

$$+ \Lambda\_2^\* \left( \frac{\ell\_2}{\Gamma\_q(\varsigma + \sigma - \varrho + 1)} + \frac{\xi^{(\varsigma - \varrho)}}{\Gamma\_q(\varsigma - \varrho + 1)} \right) \|\zeta\|;\tag{22}$$

(A4) *the multi-valued map* J : A → P(A) *formulated in* (21) *satisfies approximate endpoint criterion.*

*Then, a solution is found for the given quantum-difference inclusion FBVP* (2)*.*

**Proof.** We are going to determine that an endpoint exists for the multifunction J : A → P(A) given by (21). Since the map *r* → T∗(*r*, }(*r*)) is measurable and closed-valued setvalued mappingl therefore, it has a measurable selection. As a result, ST∗,} 6= ∅. Firstly, we show that J (}) is closed for every } ∈ A. Consider the sequence {}*n*}*n*≥1 in J (}) such that }*n* converges to }. For each n, there exists z*n* ∈ ST∗,} such that

}*n*(*r*) = Z *r* 0 (*r* − *qv*) (*ς*−1) Γ*q*(*ς*) z*n*(*v*) d*qv* + `1 *δ*1 Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) z*n*(*v*) d*qv* − 1 *δ*1 Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) z*n*(*v*) d*qv* + `3Λ1(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) z*n*(*v*) d*qv* − Λ1(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) z*n*(*v*) d*qv* + `2Λ2(*r*) Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) z*n*(*v*) d*qv* − Λ2(*r*) Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) z*n*(*v*) d*qv*,

for almost all *r* ∈ [0, 1]. Since the multi-valued function T∗ is compact, we have a subsequence {z*n*}*n*≥1 converging to z ∈ L1 ([0, 1]). Thus, z ∈ ST∗,} and

$$\begin{split} \lim\_{n \to \infty} \hbar\_{\mathfrak{n}}(r) &= \int\_{0}^{r} \frac{(r - q\underline{v})^{(\varepsilon - 1)}}{\Gamma\_{q}(\xi)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v \\ &+ \frac{\ell\_{1}}{\delta\_{1}} \int\_{0}^{1} \frac{(1 - q\underline{v})^{(\varepsilon + \sigma - 1)}}{\Gamma\_{q}(\xi + \sigma)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v - \frac{1}{\delta\_{1}} \int\_{0}^{\xi} \frac{(\xi - q\underline{v})^{(\varepsilon - 1)}}{\Gamma\_{q}(\xi)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{3} \Lambda\_{1}(r) \int\_{0}^{1} \frac{(1 - q\underline{v})^{(\varepsilon + \sigma - 2)}}{\Gamma\_{q}(\xi + \sigma - 1)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v - \Lambda\_{1}(r) \int\_{0}^{\xi} \frac{(\xi - q\underline{v})^{(\varepsilon - 2)}}{\Gamma\_{q}(\xi - 1)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{2} \Lambda\_{2}(r) \int\_{0}^{1} \frac{(1 - q\underline{v})^{(\varepsilon + \sigma - q - 1)}}{\Gamma\_{q}(\xi + \sigma - q)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v - \Lambda\_{2}(r) \int\_{0}^{\xi} \frac{(\xi - q\underline{v})^{(\varepsilon - q - 1)}}{\Gamma\_{q}(\xi - q)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v \\ &= \hbar(r), \end{split}$$

for almost all *r* ∈ [0, 1]. This indicates that } ∈ J and therefore, J is closed-valued. Since T∗ is compact multi-valued function, it is simple to check that J (}) is bounded for all } ∈ A. At last, we prove that H*d* (J (}1), J (}2)) ≤ *ψ*(k}1 − }2k) holds. Let }1, }2 ∈ A and *τ*1 ∈ J (}2). Select z1 ∈ ST∗,} such that

$$\begin{split} \mathfrak{r}\_{1}(\tau) &= \int\_{0}^{\tau} \frac{(r - qv)^{(\zeta - 1)}}{\Gamma\_{q}(\varsigma)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v \\ &+ \frac{\ell\_{1}}{\delta\_{1}} \int\_{0}^{1} \frac{(1 - qv)^{(\varsigma + \sigma - 1)}}{\Gamma\_{q}(\varsigma + \sigma)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v - \frac{1}{\delta\_{1}} \int\_{0}^{\frac{\varsigma}{\varsigma}} \frac{(\xi - qv)^{(\varsigma - 1)}}{\Gamma\_{q}(\varsigma)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{3} \Lambda\_{1}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\varsigma + \sigma - 2)}}{\Gamma\_{q}(\varsigma + \sigma - 1)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v - \Lambda\_{1}(r) \int\_{0}^{\frac{\varsigma}{\varsigma}} \frac{(\xi - qv)^{(\varsigma - 2)}}{\Gamma\_{q}(\varsigma - 1)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{2} \Lambda\_{2}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\varsigma + \sigma - \varrho - 1)}}{\Gamma\_{q}(\varsigma + \sigma - \varrho)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v - \Lambda\_{2}(r) \int\_{0}^{\frac{\varsigma}{\varsigma}} \frac{(\xi - qv)^{(\varsigma - \varrho - 1)}}{\Gamma\_{q}(\varsigma - \varrho)} \mathfrak{z}\_{1}(v) \, \mathrm{d}\_{q} v, \end{split}$$

for all *r* ∈ [0, 1]. Since

$$\mathbb{H}\_d(\mathbb{T}\_\*(r, \hbar\_1(r)), \mathbb{T}\_\*(r, \hbar\_2(r))) \le \zeta(r)\psi(|\hbar\_1(r) - \hbar\_2(r)|) \left| \frac{1}{\mathcal{Q}} \right.$$

for each *r* ∈ [0, 1], so there exists z ∗ ∈ T∗(*r*, }1(*r*)) such that

$$|\mathfrak{z}\_1(r) - \mathfrak{z}^\*| \le \zeta(r)\psi(|\hbar\_1(r) - \hbar\_2(r)|) \frac{1}{\mathcal{Q}'} $$

for each *r* ∈ [0, 1]. Now, the multi-valued map X : [0, 1] → P(A) is considered, which is characterized by

$$\mathfrak{X}(r) = \left\{ \mathfrak{z}^\* \in \mathfrak{A} : |\mathfrak{z}\_1(r) - \mathfrak{z}^\*| \le \zeta(r)\psi(|\hbar\_1(r) - \hbar\_2(r)|) \,\frac{1}{\mathcal{Q}} \right\}.$$

Since z1 and *η* = *ζ*(*ψ*(}1 − }2)) 1 Q are measurable, so it is obvious that the multifunction X ∩ T∗(·, }(·)) is measurable. Now, select z2(*r*) ∈ T∗(*r*, }(*r*)) such that

$$|\mathfrak{z}\_1(r) - \mathfrak{z}\_2(r)| \le \zeta(r) (\psi(|\hbar\_1(r) - \hbar\_2(r)|)) \frac{1}{\mathcal{Q}'} $$

for all *r* ∈ [0, 1]. Choose *τ*2 ∈ J (}1) such that

$$\begin{split} \mathfrak{r}\_{2}(r) &= \int\_{0}^{r} \frac{(r - qv)^{(\zeta - 1)}}{\Gamma\_{q}(\zeta)} \mathfrak{z}\_{2}(v) \, \mathrm{d}\_{q} v \\ &+ \frac{\ell\_{1}}{\delta\_{1}} \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - 1)}}{\Gamma\_{q}(\zeta + \sigma)} \mathfrak{z}\_{2}(v) \, \mathrm{d}\_{q} v - \frac{1}{\delta\_{1}} \int\_{0}^{\xi} \frac{(\xi - qv)^{(\zeta - 1)}}{\Gamma\_{q}(\zeta)} \mathfrak{z}\_{2}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{3} \Lambda\_{1}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - 2)}}{\Gamma\_{q}(\zeta + \sigma - 1)} \mathfrak{z}\_{2}(v) \, \mathrm{d}\_{q} v - \Lambda\_{1}(r) \int\_{0}^{\xi} \frac{(\xi - qv)^{(\zeta - 2)}}{\Gamma\_{q}(\zeta - 1)} \mathfrak{z}\_{2}(v) \, \mathrm{d}\_{q} v \\ &+ \ell\_{2} \Lambda\_{2}(r) \int\_{0}^{1} \frac{(1 - qv)^{(\zeta + \sigma - \varrho - 1)}}{\Gamma\_{q}(\zeta + \sigma - \varrho)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v - \Lambda\_{2}(r) \int\_{0}^{\xi} \frac{(\xi - qv)^{(\zeta - \varrho - 1)}}{\Gamma\_{q}(\zeta - \varrho)} \mathfrak{z}(v) \, \mathrm{d}\_{q} v. \end{split}$$

for any *r* ∈ [0, 1]. Then, we get

$$|\tau\_1(r) - \tau\_2(r)| \le \int\_0^r \frac{(r - qv)^{(\varsigma - 1)}}{\Gamma\_q(\varsigma)} |\mathfrak{z}\_1(v) - \mathfrak{z}\_2(v)| \, \mathrm{d}\_q v^{\varsigma}$$

+ `1 |*δ*1 | Z 1 0 (1 − *qv*) (*ς*+*σ*−1) Γ*q*(*ς* + *σ*) |z1(*v*) − z2(*v*)| d*qv* + 1 |*δ*1 | Z *ξ* 0 (*ξ* − *qv*) (*ς*−1) Γ*q*(*ς*) |z1(*v*) − z2(*v*)| d*qv* + `3|Λ1(*r*)| Z 1 0 (1 − *qv*) (*ς*+*σ*−2) Γ*q*(*ς* + *σ* − 1) |z1(*v*) − z2(*v*)| d*qv* + |Λ1(*r*)| Z *ξ* 0 (*ξ* − *qv*) (*ς*−2) Γ*q*(*ς* − 1) |z1(*v*) − z2(*v*)| d*qv* + `2|Λ2(*r*)| Z 1 0 (1 − *qv*) (*ς*+*σ*−*\$*−1) Γ*q*(*ς* + *σ* − *\$*) |z1(*v*) − z2(*v*)| d*qv* + |Λ2(*r*)| Z *ξ* 0 (*ξ* − *qv*) (*ς*−*\$*−1) Γ*q*(*ς* − *\$*) |z1(*v*) − z2(*v*)| d*qv* ≤ 1 Γ*q*(*ς* + 1) k*ζ*k*ψ*(k}1 − }2k) 1 Q + `1 |*δ*1 |Γ*q*(*ς* + *σ* + 1) k*ζ*k*ψ*(k}<sup>1</sup> − }2k) 1 Q + *ξ* (*ς*) |*δ*1 |Γ*q*(*ς* + 1) k*ζ*k*ψ*(k}<sup>1</sup> − }2k) 1 Q + `3Λ∗ 1 Γ*q*(*ς* + *σ*) k*ζ*k*ψ*(k}1 − }2k) 1 Q + Λ∗ 1 *ξ* (*ς*−1) Γ*q*(*ς*) k*ζ*k*ψ*(k}1 − }2k) 1 Q + `2Λ∗ 2 Γ*q*(*ς* + *σ* − *\$* + 1) k*ζ*k*ψ*(k}<sup>1</sup> − }2k) 1 Q + Λ∗ 2 *ξ* (*ς*−*\$*) Γ*q*(*ς* − *\$* + 1) k*ζ*k*ψ*(k}<sup>1</sup> − }2k) 1 Q = " 1 Γ*q*(*ς* + 1) + 1 |*δ*1 | `1 Γ*q*(*ς* + *σ* + 1) + *ξ* (*ς*) Γ*q*(*ς* + 1) ! + Λ ∗ 1 `3 Γ*q*(*ς* + *σ*) + *ξ* (*ς*−1) Γ*q*(*ς*) ! + Λ ∗ 2 `2 Γ*q*(*ς* + *σ* − *\$* + 1) + *ξ* (*ς*−*\$*) Γ*q*(*ς* − *\$* + 1) !#k*ζ*k*ψ*(k}1 − }2k) 1 Q = Q*ψ*(k}1 − }2k) 1 Q = *ψ*(k}1 − }2k).

Thus, we get k*τ*1 − *τ*2k ≤ *ψ*(k}1 − }2k). Hence, H*d* (J (}1), J (}1)) ≤ *ψ*(k}1 − }2k) for each }1, }2 ∈ A. By utilizing (A4), we realize that J has an approximate endpoint criterion. Now by employing Theorem 2, a member } ∗ ∈ A exists such that J (} ∗ ) = {} ∗}. This indicates that } ∗ is the solution of the fractional quantum-difference inclusion problem (2), hence, our proof is finally completed.

#### **4. Numerical Examples**

This section provides some interesting numerical examples to apply and validate our results in this research work.

**Example 1.** *Consider the following Caputo quantum-difference FBVP:*

$$\begin{cases} \begin{aligned} \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{2.5}\_{0^{+}} \hbar(r) &= \frac{3r+1}{8000e^{-r}} \sin(\hbar(r)), \\ \hbar(0) + \hbar(0.25) &= (0.1) \,^{\mathbb{R}}\_{0.5} \mathfrak{I}^{0.75}\_{0^{+}} \hbar(1), \\ \end{aligned} \\ \begin{aligned} \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1.5}\_{0^{+}} \hbar(0) + \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1.5}\_{0^{+}} \hbar(0.25) &= (0.2) \,^{\mathbb{R}}\_{0.5} \mathfrak{I}^{0.75}\_{0^{+}} \left[ \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1.5}\_{0^{+}} \hbar \right](1), \\ \end{aligned} \\ \begin{aligned} \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1}\_{0^{+}} \hbar(0) + \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1}\_{0^{+}} \hbar(0.25) &= (0.3) \,^{\mathbb{R}}\_{0.5} \mathfrak{I}^{0.75}\_{0^{+}} \left[ \,^{\mathbb{C}}\_{0.5} \mathfrak{D}^{1}\_{0^{+}} \hbar \right](1), \end{aligned} \end{cases} \end{aligned} \tag{23}$$

*such that q* = 0.5*,* `<sup>1</sup> = 0.1*, ς* = 2.5, *ξ* = 0.25*,* `2 = 0.2*, σ* = 0.75, *\$* = 1.5*,* `3 = 0.3 *and r* ∈ [0, 1]*. Furthermore, we consider a continuous function ϕ*∗(*r*, }(*r*)) : [0, 1] × R → R *constructed as:*

$$
\varphi\_\*(r, \hbar(r)) = \frac{3r+1}{8000e^{-r}} \sin(\hbar(r)).
$$

*The graph of this function is shown in Figure 2.*

**Figure 2.** Graph of the function *ϕ*∗(*r*, }) on [0, 1] × [0, 50].

*Then, for each* } ∈ R*, we have:*

$$|\varphi\_\*(r, \hbar(r))| = \frac{3r+1}{8000e^{-r}} |\sin(\hbar(r))| \le \frac{3r+1}{8000e^{-r}} = \theta(r)\wp(\|\hbar\|\_{\mathbb{R}})\_\*$$

*where ϑ* : [0, 1] → R>0 *is a continuous function defined by ϑ*(*r*) = 3*r*+1 8000*e*−*r and* ℘ : R≥0 → R>0 *is nondecreasing and continuous via* ℘(k}kR) = 1*. Now, for any* }1, }2 ∈ R*, we can write:*

$$|\varrho\_\*(r, \hbar\_1(r)) - \varrho\_\*(r, \hbar\_2(r))| = \frac{3r+1}{8000e^{-r}} |\sin(\hbar\_1(r)) - \sin(\hbar\_2(r))|$$

$$\leq \frac{3r+1}{8000e^{-r}} |\hbar\_1(r) - \hbar\_2(r)|.$$

*Hence, for any bounded set* H *contained in* R*, we reach*

$$\mathbb{G}(\varrho\_\*(r,\mathcal{H})) \le \frac{3r+1}{8000e^{-r}} \mathbb{G}(\mathcal{H}) := m\_{\varrho\_\*} \mathbb{G}(\mathcal{H}).$$

*We compute m*˜ *ϕ*∗ = sup*r*∈[0,1] |*mϕ*∗ | ' 0.001355*. Then, by taking into account the above calculations and the following inequality, we get*

$$\begin{split} \left[ \frac{\tilde{m}\_{\varphi\_{\*}}}{\Gamma\_{q}(\boldsymbol{\zeta}+1)} + \frac{\tilde{m}\_{\varphi\_{\*}}}{|\delta\_{1}|} \left( \frac{\ell\_{1}}{\Gamma\_{q}(\boldsymbol{\zeta}+\boldsymbol{\sigma}+1)} + \frac{\tilde{\xi}^{(\boldsymbol{\varepsilon})}}{\Gamma\_{q}(\boldsymbol{\zeta}+1)} \right) + \tilde{m}\_{\varphi\_{\*}} \Lambda\_{1}^{\*} \left( \frac{\ell\_{3}}{\Gamma\_{q}(\boldsymbol{\zeta}+\boldsymbol{\sigma})} + \frac{\tilde{\xi}^{(\boldsymbol{\varepsilon}-1)}}{\Gamma\_{q}(\boldsymbol{\zeta})} \right) \right. \\ \left. + \tilde{m}\_{\varphi\_{\*}} \Lambda\_{2}^{\*} \left( \frac{\ell\_{2}}{\Gamma\_{q}(\boldsymbol{\zeta}+\boldsymbol{\sigma}-\boldsymbol{\varrho}+1)} + \frac{\tilde{\xi}^{(\boldsymbol{\varepsilon}-\boldsymbol{\varrho})}}{\Gamma\_{q}(\boldsymbol{\zeta}-\boldsymbol{\varrho}+1)} \right) \right] \simeq 0.001741 < 0.25 = \frac{1}{4}. \end{split}$$

*We figure out that Theorem 3 is settled. As a result, at least one solution exists for Caputo fractional quantum-difference FBVP* (23)*.*

**Example 2.** *Consider the following Caputo fractional quantum-difference inclusion FBVP:*

$$\begin{cases} \begin{aligned} ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{2.75} \hbar(r) \in \left[0, \frac{5(r+1)\arctan(\hbar(r))}{256(4+3r^{2})}\right], \\ \hbar(0) + \hbar(0.9) = (0.11) \, ^{\mathcal{R}}\_{0.8} \mathfrak{D}\_{0^{+}}^{0.6} \hbar(1), \\ \end{aligned} \\ \begin{aligned} ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1.7} \hbar(0) + \, ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1.7} \hbar(0.9) = (0.12) \, ^{\mathcal{R}}\_{0.8} \mathfrak{D}\_{0^{+}}^{0.6} \left[\, ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1.7} \hbar\right](1), \\ \end{aligned} \\ \begin{aligned} ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1} \hbar(0) + \, ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1} \hbar(0.9) = (0.13) \, ^{\mathcal{R}}\_{0.8} \mathfrak{D}\_{0^{+}}^{0.6} \left[\, ^{\mathcal{C}}\_{0.8} \mathfrak{D}\_{0^{+}}^{1} \mu\right](1), \end{aligned} \end{cases} \end{aligned} \tag{24}$$

*where q* = 0.8, *ς* = 2.75, *ξ* = 0.9*,* `<sup>1</sup> = 0.11, `2 = 0.12, `3 = 0.13*, σ* = 0.6, *\$* = 1.7*, and r* ∈ [0, 1]*. Now, we introduce a multi-valued function* T∗ : [0, 1] × R → P(R) *as follows:*

$$\mathbb{T}\_\*(r, \hbar(r)) = \left[ 0, \frac{5(r+1)\arctan(\hbar(r))}{256(4+3r^2)} \right].$$

*Next, we regard ψ* : [0, ∞) → [0, ∞) *as increasing upper semi-continuous function defined by ψ*(*r*) = *r* 4 *for any r* > 0*. It can easily be noted that* lim inf*r*→∞(*r* − *ψ*(*r*)) > 0 *and ψ*(*r*) < *r for each r* > 0*. We select ζ* ∈ C([0, 1], [0, ∞)) *formulated by ζ*(*r*) = 5(*r*+1) 64(4+3*r* 2) *. Thus,* k*ζ*k ' 0.0390625*. For any* }, } ∗ ∈ R*, we have:*

$$\begin{aligned} \mathbb{H}\_d(\mathbb{T}\_\*(r, \hbar(r)) - \mathbb{T}\_\*(r, \hbar^\*(r))) &= \frac{5(r+1)}{256(4+3r^2)} |\arctan(\hbar(r)) - \arctan(\hbar^\*(r))| \\\\ &\le \frac{5(r+1)}{256(4+3r^2)} |\hbar(r) - \hbar^\*(r)| \\\\ &= \frac{5(r+1)}{64(4+3r^2)} \psi(|\hbar(r) - \hbar^\*(r)|) \\\\ &\le \xi(r)\psi(|\hbar(r) - \hbar^\*(r)|) \frac{1}{\mathcal{Q}'} \end{aligned}$$

*where*

$$\mathcal{Q} = \left[ \frac{1}{\Gamma\_q(\mathfrak{c} + 1)} + \frac{1}{|\delta\_1|} \left( \frac{\ell\_1}{\Gamma\_q(\mathfrak{c} + \sigma + 1)} + \frac{\mathfrak{z}^{(\mathfrak{c})}}{\Gamma\_q(\mathfrak{c} + 1)} \right) + \Lambda\_1^\* \left( \frac{\ell\_3}{\Gamma\_q(\mathfrak{c} + \sigma)} + \frac{\mathfrak{z}^{(\mathfrak{c} - 1)}}{\Gamma\_q(\mathfrak{c})} \right) \right]$$

$$+\Lambda\_2^\* \left( \frac{\ell\_2}{\Gamma\_q(\mathfrak{c} + \sigma - \varrho + 1)} + \frac{\mathfrak{f}^{(\mathfrak{c}-\mathfrak{q})}}{\Gamma\_q(\mathfrak{c} - \varrho + 1)} \right) \Big| \: \|\zeta\|\: \simeq 0.066907.$$

*The graphs of the functions:* Λ1(*r*) *and* Λ2(*r*) *for r* ∈ [0, 1] *are shown in Figure 3.*

**Figure 3.** Graphs of functions: Λ1(*r*) and Λ2(*r*) for *r* ∈ [0, 1].

*Next, consider the multifunction* J : A → P(A) *given by:*

$$\mathcal{I}(\hbar) = \{ \mathfrak{h} \in \mathfrak{A} : \text{there exists } \mathfrak{z} \in \mathfrak{S}\_{\mathbb{T}\_\*, \hbar} \text{ such that } \mathfrak{h}(r) = \mathfrak{a}(r) \text{ for all } r \in [0, 1] \} \lambda$$

*where*

*v*(*r*) = Z *r* 0 (*r* − *qv*) (2.75−1) Γ*q*(2.75) z(*v*) d*qv* + 0.11 1.8784 Z 1 0 (1 − *qv*) (2.75+0.6−1) Γ*q*(2.75 + 0.6) z(*v*) d*qv* − 1 1.8784 Z 0.9 0 (0.9 − *qv*) (2.75−1) Γ*q*(2.75) z(*v*) d*qv* + (0.13)Λ1(*r*) Z 1 0 (1 − *qv*) (2.75+0.6−2) Γ*q*(2.75 + 0.6 − 1) z(*v*) d*qv* − Λ1(*r*) Z 0.9 0 (0.9 − *qv*) (2.75−2) Γ*q*(2.75 − 1) z(*v*) d*qv* + (0.12)Λ2(*r*) Z 1 0 (1 − *qv*) (2.75+0.6−1.7−1) Γ*q*(2.75 + 0.6 − 1.7) z(*v*) d*qv* − Λ2(*r*) Z 0.9 0 (0.9 − *qv*) (2.75−1.7−1) Γ*q*(2.75 − 1.7) z(*v*) d*qv*,

*with δ*1 ' 1.8784 *and*

$$
\Lambda\_1(r) = 0.5387r - 0.2348 \quad \text{and} \quad \Lambda\_2(r) = 0.53002r^2 - 0.4133r - 0.02959...
$$

*Hence, by utilizing Theorem 4, it is found a solution for the quantum-difference inclusion FBVP* (24)*.*

#### **5. Conclusions**

The proposed nonlinear Caputo quantum-difference FBVP with fractional quantum integro-conditions along with its fractional quantum-difference inclusion BVP has been studied in this work. In this direction, we proved the existence of a solution for the first quantum-difference Equation (1) with the help of some notions in topological degree theory. In other words, we defined a new operator and checked its properties and finally showed that it is a condensing function. The existence of a fixed point for this operator ensured the existence of a solution for the mentioned quantum-difference Equation (1). In the next step, we considered the inclusion version of the above FBVP which had a form as (2). To

arrive at the main purpose this time for confirming the existence of solutions of (2), we used new techniques based on the approximate endpoint property and the existence of endpoints for a newly-defined multifunction. Numerical illustrative examples have been provided to display the validity and potentiality of our main results to be applied in future research works. We recommend that other researchers can study different generalizations of the proposed q-difference-FBVPs by using novel fractional difference-operators such as (*p*, *q*)-difference ones.

**Author Contributions:** Conceptualization, S.R., A.I., A.H. and S.E.; Formal analysis, S.R., A.I., A.H., F.M., S.E. and M.K. A.K.; Investigation, S.R., A.I., A.H., S.E. and M.K.A.K.; Methodology, S.R., A.I., S.E. and M.K.A.K.; Supervision, S.R., A.I., F.M., S.E. and M.K.A.K.; Validation, A.H., F.M., S.E. and M.K.A.K.; Writing—original draft, S.E.; Writing—review and editing, S.R., A.I., A.H., F.M., S.E. and M.K.A.K. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

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

#### **References**

## *Article* **Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems**

**Alberto Cabada 1,\* ,† , Nikolay D. Dimitrov 2,† and Jagan Mohan Jonnalagadda 3,†**

**Abstract:** In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green's function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray– Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results.

**Keywords:** nabla fractional difference; boundary value problem; separated boundary conditions; Green's function; existence of solutions

#### **1. Introduction**

Denote the set of all real numbers and positive real numbers by R and R+, respectively. Define by N*a* = {*a*, *a* + 1, *a* + 2, . . .} and N*b a* = {*a*, *a* + 1, *a* + 2, . . . , *b*} for any *a*, *b* ∈ R such that *b* − *a* ∈ N1.

In this article, we consider the following nabla fractional difference equation associated with separated boundary conditions:

$$\begin{array}{rcl}-\left(\nabla\_{a}^{\nu-1}(\nabla u)\right)(t) + \mathfrak{g}(t)u(t) &=& f(t, u(t)), \quad t \in \mathbb{N}\_{a+2\prime}^{b} \\ uu(a+1) - \mathfrak{f}(\nabla u)(a+1) &=& 0, \\ \gamma u(b) + \delta(\nabla u)(b) &=& 0. \end{array} \tag{1}$$

Here *a*, *b* ∈ R with *b* − *a* ∈ N1; 1 < *ν* < 2; *g* : N*b a* → R; *f* : N*b a* × R → R; ∇*ν*−1 *a* denotes the (*ν* − 1)-th order Riemann–Liouville backward (nabla) difference operator; ∇ denotes the first order backward (nabla) difference operator; *α*, *β*, *γ*, *δ* ∈ R such that *α* 2 + *β* 2 > 0 and *γ* 2 + *δ* 2 > 0.

Gray and Zhang [1], Atici and Eloe [2] and Anastassiou [3] initiated the study of nabla fractional sums and differences. The combined efforts of a number of researchers has resulted in a fairly strong foundation to the basic theory of nabla fractional calculus during the past decade. For a detailed discussion on the evolution of nabla fractional calculus, we refer to the recent monograph [4] and the references therein.

We point out that problem (1) is a discrete version of the second order ordinary differential Hill's equation, which has a lot of applications in engineering and physics. We can find, among others, several problems in astronomy, circuits, electric conductivity of metals and cyclotrons. Hill's equation is named after the pioneering work of the mathematical astronomer George William Hill (1838–1914), see [5]. There is a long literature

**Citation:** Cabada, A.; Dimitrov, N.D.; Jonnalagadda, J.M. Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems. *Symmetry* **2021**, *13*, 1101. https://doi.org/10.3390/sym13061101

Academic Editor: Juan Luis García Guirao

Received: 5 May 2021 Accepted: 16 June 2021 Published: 21 June 2021

#### **Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

in the study of the oscillation of the solutions of such an equation and the constant sign solutions. The reader can consult the monographs [6,7] and references therein. We note that the boundary conditions cover the Sturm–Liouville conditions, which include, as particular cases, the Dirichlet, Neumann and Mixed ones.

Recently, there has been a surge of interest in the development of the theory of nabla fractional boundary value problems. Brackins [8] initiated the study of boundary value problems for linear and nonlinear nabla fractional difference equations. Following this work, several authors have studied nabla fractional boundary value problems extensively. We refer to [9–18] and the references therein to name a few.

Brackins [8] showed that for all (*t*,*s*) ∈ N*b a* × N*b a*+1 (see Figure 1)

$$\mathcal{G}\_0(t,s) = \begin{cases} v\_1(t,s), & t \in \mathbb{N}\_a^{\rho(s)}, \\ v\_2(t,s), & t \in \mathbb{N}\_s^b \end{cases} \tag{2}$$

is the Green's function related to the following boundary value problem:

$$\begin{cases} -\left(\nabla\_a^{\nu-1} \left(\nabla u\right)\right)(t) = 0, & t \in \mathbb{N}\_{a+2\prime}^b \\ au(a+1) - \beta \left(\nabla u\right)(a+1) = 0, & \text{on } \begin{aligned} (3) \\ \gamma u(b) + \delta \left(\nabla u\right)(b) = 0. \end{aligned} \end{cases} \tag{3}$$

Here,

$$\begin{split} v\_{1}(t,s) &= \frac{1}{\tilde{\xi}} \Big[ a\gamma H\_{\nu-1}(t,a)H\_{\nu-1}(b,\rho(s)) + a\delta H\_{\nu-1}(t,a)H\_{\nu-2}(b,\rho(s)) \Big] \\ &+ (\beta - a)\gamma H\_{\nu-1}(b,\rho(s)) + (\beta - a)\delta H\_{\nu-2}(b,\rho(s)) \Big], \\ v\_{2}(t,s) &= v\_{1}(t,s) - H\_{\nu-1}(t,\rho(s)), \\ \tilde{\xi} &= (\beta - a)\gamma + a\gamma H\_{\nu-1}(b,a) + a\delta H\_{\nu-2}(b,a) \neq 0. \end{split}$$

**Figure 1.** Graphic of *G*0(*t*, 20) for *α* = *β* = *γ* = 1, *δ* = 0 (Dirichlet case), *µ* = 3/2, *a* = 5 and *b* = 40.

This result was obtained by expressing the general solution of the nabla fractional difference equation in (3), using the method of variation of constants. Notice that, for a non-constant function *g* the expression of the general solution does not exist and, as a consequence, the method used in [8] is not applicable for the following boundary value problem:

$$\begin{cases} -\left(\nabla\_a^{\nu-1} \left(\nabla u\right)\right)(t) + \mathcal{g}(t)u(t) = 0, \quad t \in \mathbb{N}\_{a+2\prime}^b\\ au(a+1) - \mathcal{\beta}\left(\nabla u\right)(a+1) = 0, \\ \gamma u(b) + \mathcal{\delta}\left(\nabla u\right)(b) = 0. \end{cases} \tag{4}$$

Due to this reason, Graef et al. [19] and Cabada et al. [20] followed a different approach. Graef et al. [19] studied the following Dirichlet problem:

$$\begin{cases} -\left(D\_0^\mu u\right)(t) + \mathbf{g}(t)u(t) = w(t)f(t, u(t)), \quad 0 < t < 1, \\ u(0) = u(1) = 0, \end{cases}$$

where 1 < *µ* < 2; *g* : [0, 1] → R, *w* : [0, 1] → R+ ∪ {0}, *f* : [0, 1] × R → R are continuous functions, and *D µ* 0 denotes the *µ* th-th order Riemann–Liouville fractional derivative. Cabada et al. [20] studied the following Dirichlet problem:

$$-\left(\Delta^{\mu}u\right)(t) + g(t+\mu-1)u(t+\mu-1) \quad = \quad w(t)f(t+\mu-1, u(t+\mu-1)),$$

$$u(\mu-2) = u(\mu+b+1) \quad = \quad 0,$$

where *t* ∈ N *b*+1 0 , *b* ∈ N5; 1 < *µ* < 2; *g*, *w* : N *b*+1 0 → R with *w* 6≡ 0 on N *b*+1 0 ; *f* : N *µ*+*b µ*−1 × R → R is a continuous function, and ∆ *µ* denotes the *µ*-th order Riemann–Liouville forward (delta) difference operator.

Similar to these works, we obtained the Green's function related to (4) as a series of functions by using the spectral theory. Then, under suitable conditions on *g*, *w* and *f* , we proved the existence of at least one solution of the boundary value problem (1). This work provides a new approach for constructing Green's functions for nabla fractional boundary value problems.

This article is organized as follows: In Section 2, we recall some definitions and preliminary results. In Section 3, we obtain the Green's function related to (4), and deduce some of its important properties. In Section 4, we establish a couple of existence results for the boundary value problem (1), using two different Leray–Schauder fixed point theorems and under different assumptions on the data of the problem. Finally, we give some examples to demonstrate the applicability of these results.

#### **2. Preliminaries**

In this section, we recall some elementary definitions and fundamental facts of nabla fractional calculus, which will be used throughout the article. Denote by N*a* = {*a*, *a* + 1, *a* + 2, . . .} and N*b a* = {*a*, *a* + 1, *a* + 2, . . . , *b*} for any *a*, *b* ∈ R such that *b* − *a* ∈ N1. The backward jump operator *ρ* : N*a*+1 → N*a* is defined by

$$\rho(t) = \max\left\{a, t - 1\right\}, \quad t \in \mathbb{N}\_a.$$

The Euler gamma function is defined by

$$
\Gamma(z) = \int\_0^\infty t^{z-1} e^{-t} dt, \quad \Re(z) > 0.
$$

Using its reduction formula, the Euler gamma function can also be extended to the halfplane <(*z*) ≤ 0 except for *z* ∈ {. . . , −2, −1, 0}.

If *t* ∈ {. . . , −2, −1, 0} and *r* ∈ R such that (*t* + *r*) ∈ R \ {. . . , −2, −1, 0}, then we find that *t r* = 0.

Let *µ* ∈ R \ {. . . , −2, −1}, define the *µ*-th order nabla fractional Taylor monomial by the following:

$$H\_{\mu}(t, a) = \frac{(t - a)^{\overline{\mu}}}{\Gamma(\mu + 1)} \prime$$

provided that the right-hand side exists. Observe that *Hµ*(*a*, *a*) = 0 and *Hµ*(*t*, *a*) = 0 for all *µ* ∈ {. . . , −2, −1} and *t* ∈ N*a*.

Let *u* : N*a* → R and *N* ∈ N1. The first order backward (nabla) difference of *u* is defined by the following:

$$\left(\nabla u\right)(t) = u(t) - u(t-1), \quad t \in \mathbb{N}\_{a+1\nu}$$

and the *N*-th order nabla difference of *u* is defined recursively by

$$\left(\nabla^N u\right)(t) = \left(\nabla\left(\nabla^{N-1} u\right)\right)(t), \quad t \in \mathbb{N}\_{a+N}.$$

Let *u* : N*a*+1 → R and *N* ∈ N1. The *N*-th order nabla sum of *u* based at *a* is given by the following:

$$\left(\nabla\_a^{-N} u\right)(t) = \sum\_{s=a+1}^t H\_{N-1}(t, \rho(s)) u(s), \quad t \in \mathbb{N}\_{a\nu}$$

where, by convention, ∇−*N a u* (*a*) = 0.

We define ∇−0 *a u* (*t*) = *u*(*t*) for all *t* ∈ N*a*+1.

**Definition 1.** *Let u* : N*a*+1 → R *and ν* > 0*. The ν-th order nabla sum of u based at a is given by the following [4]:*

$$\left(\nabla\_a^{-\nu}\mu\right)(t) = \sum\_{s=a+1}^t H\_{\nu-1}(t,\rho(s))\mu(s), \quad t \in \mathbb{N}\_{a\nu}$$

*where, by convention,* ∇−*ν a u* (*a*) = 0*.*

**Definition 2.** *Let u* : N*a*+1 → R*, ν* > 0 *and choose N* ∈ N1 *such that N* − 1 < *ν* ≤ *N. The ν-th order Riemann–Liouville nabla difference of u is given by the following [4]:*

$$\left(\nabla\_a^{\nu}u\right)(t) = \left(\nabla^N \left(\nabla\_a^{-(N-\nu)}u\right)\right)(t), \quad t \in \mathbb{N}\_{a+N}.$$

In [21,22], Jonnalagadda obtained the following properties of the Green's function *G*0(*t*,*s*).

**Theorem 1.** *Assume that the following condition holds [22]:*

*(A0) α, β, γ, δ* ≥ 0*, α* 2 + *β* 2 > 0*, γ* 2 + *δ* 2 > 0 *and β* ≥ *α.*

*Then,*

*1. G*0(*t*,*s*) ≥ 0 *for all* (*t*,*s*) ∈ N*b a* × N*b a*+1 *;*

$$\begin{aligned} \Lambda &= \frac{1}{\xi} \Big[ a\gamma H\_{\nu-1}(b,a)H\_{\nu-1}(b,a) + a\delta H\_{\nu-1}(b,a) \\ &\quad + (\beta - a)\gamma H\_{\nu-1}(b,a) + (\beta - a)\delta \Big]. \end{aligned}$$

**Theorem 2.** *Assume that the condition (A0) holds [21]. Then,*

$$\sum\_{s=a+1}^{b} G\_0(t, s) \le \Omega\_\star$$

*for all* (*t*,*s*) ∈ N*b a* × N*b a*+1 *, where*

$$\begin{split} \Omega = \frac{1}{\xi} \Big[ a\gamma H\_{2\nu-1}(b, a^\*+1) + a\delta H\_{2\nu-2}(b, a^\*+1) \\ &+ (\beta - a)\gamma H\_{\nu}(b, a) + (\beta - a)\delta H\_{\nu-1}(b, a) \Big]. \end{split}$$

We mention the following classical result that will be used in the next section.

**Lemma 1.** *Let X be a Banach space, A* : *X* → *X be a linear operator with the operator norm* k*A*k *[23] (page 795). Then, if* k*A*k < 1*, we have that* (*I* − *A*) −1 *exists and*

$$(I - A)^{-1} = \sum\_{n=0}^{\infty} A^n.$$

*Here, I is the identity operator.*

#### **3. Green's Function and Its Properties**

In this section, we construct the Green's function related to problem (4), and deduce some significant properties.

We denote by *X* the set of all maps from N*b a* into R. Clearly, *X* is a Banach space endowed with the maximum norm k · k. We assume the following condition throughout the paper.

(A1) There exists *g*¯ > 0 such that

$$|\lg(t)| \le \mathfrak{g} < \frac{1}{\Omega'} \quad t \in \mathbb{N}\_a^b.$$

We define *G* : N*b a* × N*b a*+1 → R by the following:

$$\mathcal{G}(t,s) = \sum\_{n=0}^{\infty} (-1)^n \mathcal{G}\_{\mathcal{U}}(t,s),\tag{5}$$

where *G*0(*t*,*s*) is given by (2), and set (see Figures 2–4).

$$\mathcal{G}\_n(t,s) = \sum\_{\tau=a+1}^b \mathcal{G}\_0(t,\tau)\mathcal{G}\_{n-1}(\tau,s)\mathcal{g}(\tau), \quad n \in \mathbb{N}\_1. \tag{6}$$

Then, we have the following result.

**Theorem 3.** *Assume that conditions* (*A*0) *and* (*A*1) *are fulfilled, then function G*(*t*,*s*)*, defined in* (5) *as a series of functions, is convergent for* (*t*,*s*) ∈ N*b a* × N*b a*+1 *. Moreover, G*(*t*,*s*) *is the Green's function for the boundary value problem* (4)*.*

**Figure 2.** Graphic of *G*1(*t*, 20) for *α* = *β* = *γ* = 1, *δ* = 0 (Dirichlet case), *µ* = 3/2, *a* = 5, *b* = 40 and *g* ≡ 1/100.

**Figure 3.** Graphic of *G*2(*t*, 20) for *α* = *β* = *γ* = 1, *δ* = 0 (Dirichlet case), *µ* = 3/2, *a* = 5, *b* = 40 and *g* ≡ 1/100.

**Figure 4.** Graphic of the first three iterates of *G*(*t*, 20) for *α* = *β* = *γ* = 1, *δ* = 0 (Dirichlet case), *µ* = 3/2, *a* = 5, *b* = 40 and *g* ≡ 1/100.

**Proof.** For any *h* ∈ *X* and *t* ∈ N*b a* , consider the following linear boundary value problem:

$$\begin{array}{rcl}-\left(\nabla\_{a}^{\nu-1}(\nabla u)\right)(t) + \mathbf{g}(t)u(t) &=& h(t), \quad t \in \mathbb{N}\_{a+2\prime}^{b} \\ uu(a+1) - \boldsymbol{\beta}(\nabla u)(a+1) &=& 0, \\ \gamma u(b) + \boldsymbol{\delta}(\nabla u)(b) &=& 0. \end{array} \tag{7}$$

By definition of the Green's function *G*0, the solutions *u* of this problem satisfy the following identity:

$$\mu(t) = \sum\_{s=a+1}^{b} G\_0(t,s) [h(s) - g(s)\mu(s)]\_{\prime\prime}$$

which is the same to

$$u(t) + \sum\_{s=a+1}^{b} \mathcal{G}\_0(t,s)\mathcal{g}(s)u(s) = \sum\_{s=a+1}^{b} \mathcal{G}\_0(t,s)h(s). \tag{8}$$

Now, define the operators *T*1 : *X* → *X* and *T*2 : *X* → *X* by the following:

$$\begin{aligned} (T\_1 h)(t) &= \sum\_{s=a+1}^b \mathcal{G}\_0(t, s) h(s), \quad t \in \mathbb{N}\_a^b, \\ (T\_2 u)(t) &= \sum\_{s=a+1}^b \mathcal{G}\_0(t, s) \mathcal{g}(s) u(s), \quad t \in \mathbb{N}\_a^b. \end{aligned}$$

Then, (8) can be expressed as the following:

$$(I + T\_2)\mu = T\_1 h.$$

Using condition (A1) and Theorem 1 the following is true:

$$\begin{split} \|\|T\_{2}\|\| = \max\_{\|u\|\|=1} \|\|T\_{2}u\|\| &= \max\_{\|u\|\|=1} \left[ \max\_{t \in \mathbb{N}\_{a}^{b}} |(T\_{2}u)(t)| \right] \\ &= \max\_{\|u\|=1} \left[ \max\_{t \in \mathbb{N}\_{a}^{b}} \left| \sum\_{s=a+1}^{b} G\_{0}(t,s)g(s)u(s) \right| \right] \\ &\leq \max\_{\|u\|=1} \left[ \max\_{t \in \mathbb{N}\_{a}^{b}} \sum\_{s=a+1}^{b} G\_{0}(t,s) |g(s)| |u(s)| \right] \\ &\leq \max\_{\|u\|=1} \left[ \bar{g} \|\|u\| \max\_{t \in \mathbb{N}\_{a}^{b}} \sum\_{s=a+1}^{b} G\_{0}(t,s) \right] \\ &< \max\_{\|u\|=1} \left[ \bar{g} \|u\| \Omega \right] = \bar{g} \Omega < 1. \end{split}$$

Then, by Lemma 1, we have the following:

$$
\mu = (I + T\_2)^{-1} T\_1 h = \sum\_{n=0}^{\infty} (-T\_2)^n T\_1 h. \tag{9}
$$

Arguing in a similar manner than in [20], we can deduce the following:

$$\left( (-T\_2)^n T\_1 h \right)(t) = \sum\_{s=a+1}^b (-1)^n \mathcal{G}\_n(t, s) h(s), \quad t \in \mathbb{N}\_{a\prime}^b \quad n = 0, 1, 2, \dots \tag{10}$$

Let us see now that the following inequality is fulfilled:

$$|(-1)^{\mathfrak{n}}\mathcal{G}\_{\mathfrak{n}}(t,s)| < \Lambda \left(\mathfrak{g}\Omega\right)^{\mathfrak{n}}, \quad n = 0, 1, 2, \dots \tag{11}$$

From Theorem 1, we have that (11) holds for *n* = 0. Assume now that (11) is true for some *n* = *k*. Then, the following is true:

$$\begin{split} \left| (-1)^{k+1} \mathcal{G}\_{k+1}(t,s) \right| &= \left| (-1)^{k+1} \sum\_{\tau=a+1}^{b} \mathcal{G}\_{0}(t,\tau) \mathcal{G}\_{k}(\tau,s) \mathcal{g}(\tau) \right| \\ &= \left| - \sum\_{\tau=a+1}^{b} \mathcal{G}\_{0}(t,\tau) (-1)^{k} \mathcal{G}\_{k}(\tau,s) \mathcal{g}(\tau) \right| \\ &\leq \sum\_{\tau=a+1}^{b} \mathcal{G}\_{0}(t,\tau) \left| (-1)^{k} \mathcal{G}\_{k}(\tau,s) \right| \left| \mathcal{g}(\tau) \right| \\ &< \Lambda \left( \mathcal{g} \Omega \right)^{k} \mathcal{g} \sum\_{\tau=a+1}^{b} \mathcal{G}\_{0}(t,\tau) \\ &< \Lambda \left( \mathcal{g} \Omega \right)^{k} \mathcal{g} \Omega = \Lambda \left( \mathcal{g} \Omega \right)^{k+1} . \end{split}$$

Thus, (11) holds for *n* = *k* + 1. By mathematical induction, (11) holds for any *n* = 0, 1, 2, . . ..

As a direct consequence of previous inequality and condition (A1), we deduce that for all (*t*,*s*) ∈ N*b a* × N*b a*+1 the following property is fulfilled:

$$\begin{aligned} |\mathcal{G}(t,s)| = \left| \sum\_{n=0}^{\infty} (-1)^n \mathcal{G}\_{\mathbb{N}}(t,s) \right| &\leq \sum\_{n=0}^{\infty} |(-1)^n \mathcal{G}\_{\mathbb{N}}(t,s)| \\ &< \Lambda \sum\_{n=0}^{\infty} (\underline{\mathbb{g}} \Omega)^n = \frac{\Lambda}{1 - \underline{\mathbb{g}} \Omega} < \infty, \end{aligned}$$

and, a a consequence, *G*(*t*,*s*) converges on N*b a* × N*b a*+1

Finally, expressions (5), (9) and (10) imply that for all *t* ∈ N*b a* the following equality is fulfilled:

$$\begin{split} u(t) = \sum\_{n=0}^{\infty} \left[ \sum\_{s=a+1}^{b} (-1)^{n} \mathbf{G}\_{n}(t,s) h(s) \right] &= \sum\_{s=a+1}^{b} \left[ \sum\_{n=0}^{\infty} (-1)^{n} \mathbf{G}\_{n}(t,s) \right] h(s) \\ &= \sum\_{s=a+1}^{b} \mathbf{G}(t,s) h(s). \end{split} \tag{12}$$

.

It is not difficult to verify that any function defined by (12) is a solution of the boundary value problem (7). So we conclude that problem (7) has a unique solution and, as a consequence, *G* is its related Green's function.

**Lemma 2.** *Assume conditions* (*A*0) *and* (*A*1)*. Let G be defined by* (5) *and the following:*

$$\mathcal{G}(s) = \frac{\mathcal{G}\_0(\rho(s), s)}{1 - \mathcal{g}\Omega}, \quad s \in \mathbb{N}\_{a+1}^b. \tag{13}$$

*Then,*

$$|\mathcal{G}(t,s)| \le \mathcal{G}(s), \quad (t,s) \in \mathbb{N}\_a^b \times \mathbb{N}\_{a+1}^b.$$

**Proof.** First, we prove the following:

$$|(-1)^{n}G\_{\mathfrak{n}}(t,s)| < G(s-1,s)(\mathfrak{g}\Omega)^{\mathfrak{n}}, \quad s \in \mathbb{N}\_{\mathfrak{a}+1\prime}^{b} \quad n = 0,1,2,\dots \tag{14}$$

Theorem 1 implies that inequality (14) is true for *n* = 0.

Assume now that (14) holds for some *n* = *k*. We will show that (14) holds for *n* = *k* + 1. Consider the following:

$$\begin{split} \left| (-1)^{k+1} G\_{k+1}(t,s) \right| &= \left| (-1)^{k+1} \sum\_{\tau=a+1}^{b} G\_0(t,\tau) G\_k(\tau,s) g(\tau) \right| \\ &= \left| - \sum\_{\tau=a+1}^{b} G\_0(t,\tau) (-1)^k G\_k(\tau,s) g(\tau) \right| \\ &\leq \sum\_{\tau=a+1}^{b} G\_0(t,\tau) \left| (-1)^k G\_k(\tau,s) \right| \left| g(\tau) \right| \\ &< G(s-1,s) (\tilde{g}\Omega)^k \mathfrak{g} \sum\_{\tau=a+1}^{b} G\_0(t,\tau) \\ &< G(s-1,s) (\tilde{g}\Omega)^k \mathfrak{g} \Omega = G(s-1,s) (\tilde{g}\Omega)^{k+1} \end{split}$$

Thus, (14) holds for *n* = *k* + 1 and the inequalities are deduced from mathematical induction.

.

Now, from (5), (13) and (14), for *s* ∈ N*b a*+1 , we obtain the following:

$$\begin{aligned} |G(t,s)| = \left| \sum\_{n=0}^{\infty} (-1)^n G\_n(t,s) \right| &\leq \sum\_{n=0}^{\infty} |(-1)^n G\_n(t,s)| \\ &< G(s-1,s) \sum\_{n=0}^{\infty} (\mathfrak{g}\Omega)^n \\ &= \frac{G\_0(\rho(s),s)}{1-\mathfrak{g}\Omega} = \mathfrak{G}(s), \end{aligned}$$

and the proof is complete.

From the previous result, we deduce the following consequence for *g* ≤ 0.

**Corollary 1.** *Assume that condition (A) is fulfilled and*

$$-\mathfrak{F} < \mathfrak{g}(t) \le \mathbf{0}, \quad t \in \mathbb{N}\_a^b.$$

*Then, G*(*t*,*s*) ≥ 0 *for each* (*t*,*s*) ∈ N*b a* × N*b a*+1 *.*

**Proof.** From Theorem 1, we know that *G*0(*t*,*s*) ≥ 0 for each (*t*,*s*) ∈ N*b a* × N*b a*+1 . The result follows immediately from (5) and (6).

#### **4. Existence of Solutions**

In this section, we derive two existence results for the nonlinear problem (1). Define the operator *T* : *X* → *X* (*X* defined in previous section) by the following:

$$f(Tu)(t) = \sum\_{s=a+1}^{b} G(t,s) \, f(s,u(s)), \quad t \in \mathbb{N}\_a^b. \tag{15}$$

In view of (12), it is clear that *u* is a fixed point of *T* if and only if *u* is a solution of (1). For any *R* > 0 given, we define the following set:

$$\mathcal{K}\_{\mathbb{R}} = \{ \mu \in \mathbb{X} : \|\mu\| < \mathbb{R} \}.$$

Clearly, K*R* is a non-empty open subset of *X*, 0 ∈ K*R* and *T* : K*R* → *X*. Now, denoting by

$$\max\_{t \in \mathbb{N}\_a^b} |f(t, 0)| = M \quad \text{and} \quad \sum\_{s=a+1}^b \tilde{G}(s) = K \text{ (} > 0 \text{)}.$$

we enunciate the following list of assumptions:

$$|f(t, \mu) - f(t, v)| \le L|\mu - v|.$$

(A4) There exists a continuous function *σ* : N*b a* → R+ and a continuous nondecreasing function *ψ* : R+ → R+ such that

$$|f(t, \mathfrak{u})| \le \sigma(t)\psi(|\mathfrak{u}|), \quad (t, \mathfrak{u}) \in \mathbb{N}\_a^b \times \mathbb{R}.$$

(A5) 0 < *LK* < 1.

First, we present a nonlinear alternative of Leray–Schauder for contractive maps.

**Theorem 4.** *(Theorem 3.2) Suppose U is an open subset of a Banach space X,* 0 ∈ *U and F* : *U*¯ → *X a contraction with F*(*U*¯ ) *bounded [24]. Then, either of the following is true:*

*1. F has a fixed point in U.* ¯

*2. There exist λ* ∈ (0, 1) *and u* ∈ *∂U with u* = *λFu, holds.*

Now, we establish sufficient conditions on existence of solutions for(1) using Theorem 4.

**Theorem 5.** *Assume (A0)–(A3), (A5) hold. If we choose R such that*

$$R \geq \frac{MK}{1 - LK'} \tag{16}$$

*then the boundary value problem* (1) *has a solution in* K*R.*

**Proof.** First, we show that *T* is a contraction. To see this, let *u*, *v* ∈ K*R*, *t* ∈ N*b a* , and consider the following:

$$\begin{aligned} \left| \left( \left( T u \right) (t) - \left( T v \right) (t) \right) \right| &= \left| \sum\_{s=a+1}^{b} \mathcal{G}(t, s) \left[ f(s, u(s)) - f(s, v(s)) \right] \right| \\ &\le \sum\_{s=a+1}^{b} | \mathcal{G}(t, s) || f(s, u(s)) - f(s, v(s)) | \\ &\le L \sum\_{s=a+1}^{b} \mathcal{G}(s) |u(s) - v(s)| \\ &\le LK || u - v ||\_{\prime} \end{aligned}$$

implying that

$$||Tu - Tv|| \le LK||u - v||.$$

Since

$$0 < LK < 1\_r$$

it follows that *T* is a contraction.

Next, we prove that *T*(K*R*) is bounded.

To see this, let *u* ∈ K*R* (k*u*k ≤ *R*), *t* ∈ N*b a* , and consider the following:

$$\begin{aligned} \left| \left( Tu \right) (t) \right| &= \left| \sum\_{s=a+1}^{b} G(t,s) \, f(s,u(s)) \right| \\ &\le \sum\_{s=a+1}^{b} |G(t,s)| |f(s,v(s))| \\ &= \sum\_{s=a+1}^{b} |G(t,s)| |f(s,u(s)) - f(s,0) + f(s,0)| \\ &\le \sum\_{s=a+1}^{b} \left| \bar{G}(s) |f(s,u(s)) - f(s,0)| + \sum\_{s=a+1}^{b} \bar{G}(s) |f(s,0)| \right| \\ &\le L \sum\_{s=a+1}^{b} \bar{G}(s) |u(s)| + M \sum\_{s=a+1}^{b} \bar{G}(s) \\ &\le Lk ||u|| + MK \le [LR + M]K, \end{aligned}$$

implying the following:

$$\|Tu\| \le [LR+M]K.$$

Thus, *T*(K*R*) bounded.

Now, suppose there exist *v* ∈ *∂*K*R* (k*v*k = *R*) and *λ* ∈ (0, 1) such that

$$
v = \lambda T v.\tag{17}$$

Using the definition of *T* in (17) and arguing as before, we obtain the following:

$$|v(t)| = |\lambda(Tv)(t)| \le [LR + M]\lambda K < [LR + M]K, \quad t \in \mathbb{N}\_{a\prime}^b$$

which implies the following:

$$\mathcal{R} = \|v\| < \left[LR + M\right]K$$

or, which is the same,

$$R < \frac{MK}{1 - LK'} $$

in contradiction with (16).

Hence, by Theorem 4, we deduce that operator *T* has a fixed point in K*R* and the proof is complete.

**Remark 1.** *We note that in the previous result, if M* = 0 *then we have that u* ≡ 0 *on* [0, 1] *is a solution of problem* (1)*. On the contrary, if M* > 0 *the obtained function is non trivial on* [0, 1]

Next, we enunciate a nonlinear alternative of Leray–Schauder for continuous and compact maps.

**Theorem 6.** *Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and* 0 ∈ *U [24] (Theorem 6.6). Suppose that F* : *U*¯ → *C is a continuous, compact map. Then, either of the following is true:*

Now, we establish sufficient conditions on existence of solutions for(1) using Theorem 6.

**Theorem 7.** *Assume that conditions (A0)–(A2), and (A4) hold. If we choose R such that*

$$\frac{R}{K\|\sigma\|\|\psi(R)} \ge 1,\tag{18}$$

*then the boundary value problem* (1) *has a solution in* K*R.*

**Proof.**  Now, suppose that there exist *v* ∈ *∂*K*R* (k*v*k = *R*) and *λ* ∈ (0, 1) such that (17) holds. Using the definition of *T* in (17), we obtain the following:

$$\begin{aligned} |v(t)| &= |\lambda(Tv)(t)| \\ &= \left| \lambda \sum\_{s=a+1}^{b} G(t,s) f(s,v(s)) \right| \\ &\le \lambda \sum\_{s=a+1}^{b} |G(t,s)| |f(s,v(s))| \\ &\le \lambda \sum\_{s=a+1}^{b} |G(t,s)| \sigma(s) \psi(|v(s)|) \\ &\le \lambda ||\sigma|| \psi(||v||) \sum\_{s=a+1}^{b} \tilde{G}(s) \\ &< K ||\sigma|| \psi(||v||). \end{aligned}$$

So, we deduce the following:

*R* = k*v*k < *K*k*σ*k*ψ*(k*v*k).

Thus,

$$\frac{R}{K||\sigma||\psi(R)} < 1.$$

This is a contradiction to (18).

Hence, by Theorem 6, the boundary value problem (1) has a solution in K*R*. The proof is complete.

**Remark 2.** *Note that since we have that*

$$\max\_{t \in \mathbb{N}\_a^b} G\_0(t, s) < \Lambda$$

*we can set the following:*

$$\overline{K} = \frac{\Lambda}{1 - \overline{\mathfrak{g}}\Omega}(b - a) > \sum\_{s = a + 1}^{b} \overline{\mathcal{G}}(s) = K.$$

*Thus, we can use K instead of K everywhere and we do not need to calculate the Green's function at all.*

*Indeed, in (A5), if we have* 0 < *LK* < 1, *this implies that* 0 < *L K* < 1.

*In Theorem 4.2, if we choose R* ≥ *MK* 1−*LK* , *then we will also have that R* ≥ *MK* 1−*LK since MK* ≥ *MK* 1−*LK* .

1−*LK Finally, in Theorem 4.4, if we choose R K*k*σ*k*ψ*(*R*) ≥ 1, *then we will also have that R K*k*σ*k*ψ*(*R*) ≥ 1 *since R K*k*σ*k*ψ*(*R*) ≥ *R K*k*σ*k*ψ*(*R*) .

#### **5. Examples**

In the section, we present some examples to illustrate the applicability of our main results.

**Problem 1.** *Consider the following nabla fractional boundary value problem:*

$$\begin{array}{rcl}-\left(\nabla\_{0}^{1/2}(\nabla u)\right)(t) + \frac{e^{-t}}{10}u(t) &=& \frac{1}{200}\sin(u(t) + t), \quad t \in \mathbb{N}\_{2}^{6} \\ u(0) = u(6) &=& 0. \end{array} \tag{19}$$

*Here, α* = 1*, β* = 1*, γ* = 1 *and δ* = 0*, a* = 0*, b* = 6 *and ν* = 3/2*.*

*In addition, g*(*t*) = *e* −*t*/10 *and f*(*t*, *u*) = (sin(*u* + *t*))/200*. Clearly, g* : N6 0 → R*; f* : N6 0 × R → R *is continuous and satisfies Lipschitz condition with respect to u on* N6 0 × R *with Lipschitz constant L* = 0.005*.*

*We have ξ* = *H*0.5(6, 0) ≈ 2.7071*,* Λ = *H*0.5(6, 0) ≈ 2.7071*,* Ω = *H*2(6, 1) = 3 *and g*¯ = 0.1 *so that* |*g*(*t*)| ≤ *g*¯ < 1 Ω *. Further,*

$$\overline{K} = \frac{\Lambda}{1 - \overline{\mathfrak{F}}\Omega}(b - a) \approx 23.2037.$$

*Observe that* 0 < *LK*¯ ≈ 0.116 < 1*. Additionally,*

$$M = \max\_{t \in \mathbb{N}\_0^6} |f(t, 0)| = \max\_{t \in \mathbb{N}\_0^6} \left| \frac{\sin t}{200} \right| \approx 0.00479462137.$$

*If we choose*

$$R \ge \frac{M\overline{K}}{1 - L\overline{K}} \approx 0.12585176\ldots$$

*then by Theorem 5 and Remark 2, the boundary value problem* (19) *has a solution in* K*R.*

**Problem 2.** *Consider the following nabla fractional boundary value problem:*

$$\begin{array}{rcl}-\left(\nabla\_{0}^{1/2}(\nabla u)\right)(t) + \frac{1}{20(t+1)}u(t) &=& \frac{u^{2}(t)}{10(t^{2}+10)}, \quad t \in \mathbb{N}\_{2'}\\u(0) + u(1) &=& 0, \\u(8) + u(9) &=& 0. \end{array} \tag{20}$$

*Here, α* = 2*, β* = 1*, γ* = 2 *and δ* = −1 *such that α* 2 + *β* 2 > 0 *and γ* 2 + *δ* 2 > 0*, a* = 0*, b* = 9 *and ν* = 3/2*.* 2

*In addition, g*(*t*) = 1 20(*t*+1) *and f*(*t*, *u*) = *u* (*t*) 10(*t* 2+10) *. Clearly, g* : N9 0 → R *and f* : N9 0 × R → R *are continuous.*

*We have ξ* ≈ 10.9616*,* Λ ≈ 2.9403*,* Ω = 15.8396 *and g*¯ ≈ 0.05 *so that* |*g*(*t*)| ≤ *g*¯ < 1 Ω *. Further,*

$$\overline{K} = \frac{\Lambda}{1 - \overline{g}\Omega}(b - a) \approx 127.2245.1$$

*In addition,*

$$|f(t,\mathfrak{u})| \le \sigma(t)\psi(|\mathfrak{u}|), \quad (t,\mathfrak{u}) \in \mathbb{N}\_0^9 \times \mathbb{R},$$

*where σ*(*t*) = 1 10(*t* 2+10) *and ψ*(*x*) = *x* 2 *. Observe that σ* : N9 0 → R+ *is continuous and ψ* : R+ → R+ *is continuous non-decreasing with*

$$||\sigma|| = \max\_{t \in \mathbb{N}\_0^9} |\sigma(t)| = 0.001.$$

*If we choose*

*R K*k*σ*k*ψ*(*R*) ≥ 1,

*that is, R* ≤ 7.8616*, then by Theorem 7 and Remark 2, the boundary value problem* (19) *has a solution in* K*R.*

**Author Contributions:** Conceptualization, A.C., N.D.D. and J.M.J.; methodology, A.C., N.D.D. and J.M.J.; software, A.C.; validation, A.C., N.D.D. and J.M.J.; formal analysis, A.C., N.D.D. and J.M.J.; investigation, A.C., N.D.D. and J.M.J.; resources, A.C., N.D.D. and J.M.J.; data curation, A.C.; writing—original draft preparation, J.M.J.; writing—review and editing, A.C., N.D.D. and J.M.J.; visualization, A.C., N.D.D. and J.M.J.; supervision, A.C., N.D.D. and J.M.J.; project administration, A.C., N.D.D. and J.M.J.; funding acquisition, A.C and N.D.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** The first author is partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER. The second author is supported by the Bulgarian National Science Fund under Project DN 12/4 "Advanced analytical and numerical methods for nonlinear differential equations with applications in finance and environmental pollution", 2017.

**Acknowledgments:** The authors thanks to the anonymous referees for their useful comments that have contributed to improve this paper.

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

#### **References**

## *Article* **Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces**

**Maria Alessandra Ragusa 1,2,\* and Fan Wu 3**

**Abstract:** In this paper, we investigate the regularity of weak solutions to the 3D incompressible MHD equations. We provide a regularity criterion for weak solutions involving any two groups functions (*∂*1*u*1 , *∂*1*b*1), (*∂*2*u*2, *∂*2*b*2) and (*∂*3*u*3, *∂*3*b*3) in anisotropic Lorentz space.

**Keywords:** MHD equations; weak solution; regularity criteria; anisotropic Lorentz space

**MSC:** 76W05; 35Q30; 35B65

#### **1. Introduction**

In this paper, we are concerned with regularity criteria for the weak solutions to the incompressible magneto-hydrodynamic (MHD) equations in R3 [1,2]:

**Citation:** Ragusa, M.A.; Wu, F. Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces. *Symmetry* **2021**, *13*, 625. https:// doi.org/10.3390/sym13040625

Academic Editor: Stephen Anco

Received: 26 February 2021 Accepted: 3 April 2021 Published: 8 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

$$\begin{cases} \partial\_t u + (\boldsymbol{u} \cdot \nabla) \boldsymbol{u} - \Delta \boldsymbol{u} + \nabla p = (\boldsymbol{b} \cdot \nabla) b, \\\partial\_t b + (\boldsymbol{u} \cdot \nabla) b - \Delta b = (\boldsymbol{b} \cdot \nabla) \boldsymbol{u}, \\\nabla \cdot \boldsymbol{u} = \nabla \cdot \boldsymbol{b} = 0, \\\ u(\boldsymbol{x}, 0) = u\_0(\boldsymbol{x}), b(\boldsymbol{x}, 0) = b\_0(\boldsymbol{x}), \end{cases} \tag{1}$$

where *u* = (*u*1, *u*2, *u*3) is the fluid velocity field, *b* = (*b*1, *b*2, *b*3) is the magnetic field, *p* is a scalar pressure, and *u*0, *b*0 is the prescribed initial data satisfying the compatibility condition ∇ · *u*0 = ∇ · *b*0 = 0 in the distributional sense. Physically, Equation (1) govern the dynamics of the velocity and magnetic fields in electrically conducting fluids, such as plasmas, liquid metals, and salt water.

Besides its physical applications, the MHD equations (1) have also mathematically significant. Duvaut and Lions [1] developed a global weak solution to (1) for initial data withfinite energy, that is,

$$\mu, b \in L^{\infty}\left(0, T; L^2(\mathbb{R}^3)\right) \cap L^2\left(0, T; H^1\left(\mathbb{R}^3\right)\right) \quad \text{for any} \quad T > 0.$$

It is well known that the issue of regularity for weak solutions to the 3D incompressible Navier- Stokes equations has been one of the most challenging open problem in mathematical fluid mechanics [3], as well as that for the 3D incompressible magneto-hydrodynamics (MHD) equations (see Sermange and Temam [2]). Many sufficient conditions (see e.g., [4–14] and the references therein) were derived to guarantee the regularity of the weak solution. He and Xin [15] first extended the classical Prodi-Serrin conditions of Navier-Stokes equations to the MHD equations, they obtained regularity criteria involving only on velocity *u*, i.e.,

$$u \in L^{q}\left(0, T; L^{p}\left(\mathbb{R}^{3}\right)\right) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} \le 1 \quad \text{and} \quad 3 < p \le \infty \tag{2}$$

or

$$\nabla u \in L^{q}\left(0, T; L^{p}\left(\mathbb{R}^{3}\right)\right) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} \le 2 \quad \text{and} \quad \frac{3}{2} < p \le \infty. \tag{3}$$

Later, He and Wang [16] showed that a weak solution (*u*, *b*) is regular, provided only ∇*ω*+ = (*u* + *b*) or ∇*ω*− = (*u* − *b*) belongs to Beirao da Veiga's class, that is,

$$
\nabla \omega^{+} \quad \text{or} \quad \nabla \omega^{-} \in L^{q} \left( 0, T; L^{p, \infty} \Big( \mathbb{R}^{3} \Big) \right) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} = 2 \quad \text{and} \quad 3 \le p \le \infty. \tag{4}
$$

Ni et al. [17] showed that one of the following conditions hold

( ∇*hu* ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞, *∂*3*b* ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞. (5) *u*3 ∈ *L q* 0, *T*; *L p* (R3 ) with 2 *q* + 3 *p* ≤ 1 and 3 < *p* ≤ ∞, *∂*3*u* ∈ *L p* 0, *T*; *L q* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞, *b*3 ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 1 and 3 < *p* ≤ ∞, *∂*3*b* ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞, (6) ( ∇*hu* ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞, ∇*hb* ∈ *L q* 0, *T*; *L p* R3 with 2 *q* + 3 *p* ≤ 2 and 3 2 < *p* ≤ ∞, (7)

then the weak solution (*u*, *b*) is regular on (0, *T*], where ∇*h* = (*∂*1, *∂*2). Recently, Jia [18] showed that condition (7) can be replaced by

$$\begin{cases} \nabla\_h \vec{u} \in L^q\left(0, T; L^p\left(\mathbb{R}^3\right)\right) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} \le 2 \quad \text{and} \quad \frac{3}{2} < p \le \infty, \\\nabla\_h \vec{b} \in L^q\left(0, T; L^p\left(\mathbb{R}^3\right)\right) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} \le 2 \quad \text{and} \quad \frac{3}{2} < p \le \infty, \end{cases} \tag{8}$$

where ˜ *f* = (*f*1, *f*2). Regularity condition (8) was further improved by Xu et al. [19], more precisely, they proved that if any two quantities of

$$\begin{cases} A\_i^{q,p}(T) := \partial\_i u\_i \in L^q(0, T; L^p(\mathbb{R}^3)) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} = 2 \quad \text{and} \quad \frac{3}{2} < p \le \infty, \\\ B\_i^{q,p}(T) := \partial\_i b\_i \in L^q(0, T; L^p(\mathbb{R}^3)) \quad \text{with} \quad \frac{2}{q} + \frac{3}{p} = 2 \quad \text{and} \quad \frac{3}{2} < p \le \infty, \end{cases} \tag{9}$$

where *i* = 1, 2, 3, then the solution is smooth on interval (0,T]. For readers interested in this topic for partial components, please refer to [20–26] for recent progresses.

Motivated by papers cited above, the aim of this article is to study the regularity of weak solutions for the 3D MHD equations (1) in term of the two partial derivative of the velocity components and magnetic components on framework of the anisotropic Lorentz space. Before stating our main Theorem, we shall first recall the definitions of some function spaces [27].

#### *Lorentz Spaces*

Given a measurable function *f* : R*n* → R define the distribution function of *f* by

$$d\_f(\mathfrak{a}) = \mathfrak{p}(\{\mathfrak{x} : |f(\mathfrak{x})| > \mathfrak{a}\}),$$

where *µ*(*A*) (or |*A*|) denotes the Lebesgue measure of a set *A*. We now define its decreasing rearrangement *f* ∗ : [0, ∞) → [0, ∞] as

$$f^\*(t) = \inf\left\{\alpha : d\_f(\alpha) \le t\right\},$$

with the convention that inf ∅ = ∞. The point of this definition is that *f* and *f* ∗ have the same distribution function,

$$d\_{f^\*}(\mathfrak{a}) = d\_f(\mathfrak{a})\_\*$$

but *f* ∗ is a positive non-increasing scalar function.

**Definition 1.** *Let* (*p*, *q*) ∈ [1, ∞] 2 *, the Lorentz space L p*,*q* (R3 ) *consists of all measurable functions f for which the quantity*

$$\|f\|\_{L^{p\mathcal{A}}} := \begin{cases} \left(\int\_0^\infty [t^{\frac{1}{p}}f^\*(t)]^q \frac{\mathrm{d}t}{\underline{t}}\right)^{\frac{1}{q}} & q < \infty, \\\sup\_{0 < t < \infty} t^{\frac{1}{p}} f^\*(t) & q = \infty, \end{cases}$$

*is finite.*

In order to give the following definition involving anisotropic Lorentz space, we denote *f* = *f*(*x*1, *x*2, *x*3) be a measurable function defined on R3 , *f* ∗ (*t*) = *f* ∗1 ,∗2,∗3 (*t*1, *t*2, *t*3). Here *f* ∗1 ,∗2,∗3 (*t*1, *t*2, *t*3) is the multivariate decreasing rearrangement of *f*(*x*1, *x*2, *x*3) obtained by applying decreasing rearrangement *f* ∗1 (*t*1, *x*2, *x*3) of *f*(*x*1, *x*2, *x*3) relating to the first variable *x*1, under fixed the second, the third variables *x*2, *x*3, and then applying decreasing rearrangement *f* ∗1 ,∗2 (*t*1, *t*2, *x*3) of *f* ∗1 (*t*1, *x*2, *x*3) with respect to the second variable *x*2 under fixed the first variable *t*1 of *f* ∗1 (*t*1, *x*2, *x*3) and variable *x*3, finally for variable *x*3, by the same trick, we obtain the multivariate decreasing rearrangement *f* ∗1 ,∗2,∗3 (*t*1, *t*2, *t*3).

Recently, many works have been done for mixed-norm spaces. Stefanov-Torres [28] obtained the boundedness of Calderón-Zygmund operators on mixed-norm Lebesgue spaces. Georgiadis et al. [29] obtained various properties of anisotropic Triebel-Lizorkin spaces with mixed norms. In [30], Chen-Sun introduced the iterated weak and weak mixed-norm spaces and given some applications to geometric inequalities.

**Definition 2.** *Let multi indexes p* = (*p*1, *p*2, *p*3), *q* = (*q*1, *q*2, *q*3) *be such that if* 0 < *pi* < ∞*, then* 0 < *qi* ≤ ∞*, and if pi* = ∞*, then qi* = ∞ *for every i* = 1, 2, 3 *[31]. An anisotropic Lorentz space L p*1 ,*q*1 (R*x*1 ; *L p*2,*q*2 (R*x*2 ; *L p*3,*q*3 (R*x*3 ))) *is the set of functions for which the following norm is finite:*

$$\left\| \left\| \left\| \|f\|\|\_{L^{p\_1\otimes 1}\_{x\_1}} \right\|\_{L^{p\_2\otimes 2}\_{x\_2}} \right\|\_{L^{p\_3\otimes 3}\_{x\_3}} \right\|\_{L^{p\_3\otimes 3}\_{x\_3}} \coloneqq \left( \int\_0^\infty \left( \int\_0^\infty \left[ t\_1^{\frac{1}{p\_1}} t\_2^{\frac{1}{p\_2}} t\_3^{\frac{1}{p\_3}} f^{\*\_1,\*\_2,\nu\_3}(t\_1,t\_2,t\_3) \right]^{q\_1} \frac{dt\_1}{t\_1} \right)^{\frac{q\_2}{q\_3}} \frac{dt\_2}{t\_2} \right)^{\frac{q\_3}{q\_2}} \frac{dt\_3}{t\_3} \right)^{\frac{1}{q\_3}}$$

Now, our main result reads:

**Theorem 1.** *Suppose that* (*u*0, *b*0) ∈ *L* 2 (R3 ) ∩ *L* 4 (R3 ) *and* ∇ · *u*0 = ∇ · *b*0 = 0 *in distributional sense. Let* (*u*, *b*) *be the Leray-Hopf weak solution of* (1) *on* (0, *T*]*. If any two quantities*

$$\begin{cases} A\_i(T) := \int\_0^T \left\| \left\| \left\| \partial\_i u\_i(t) \right\|\right\|\_{L^{p,\infty}\_{x\_1}} \right\|\_{L^{q,\infty}\_{x\_2}} \left\| \begin{array}{l} \frac{2}{2 - \left(\frac{1}{p} + \frac{1}{q} + \frac{1}{p}\right)} \\\\ \left\| \left\| \left\| \partial\_i b\_i(t) \right\|\right\|\_{L^{p,\infty}\_{x\_1}} \right\|\_{L^{q,\infty}\_{x\_2}} \end{array} \right\|\_{L^{q,\infty}\_{x\_3}} \end{cases} \tag{10}$$

.

*are finite, where i* = 1, 2, 3 *with* 2 < *p*, *q*,*r* ≤ ∞ *and* 1 − 1 *p* + 1 *q* + 1 *r* ≥ 0*, then the weak solution* (*u*, *b*) *is actually smooth on interval* (0, *T*]*.*

**Remark 1.** *While L p* (R3 ) ,→ *L p*,∞(R3 )*, clearly L p*,∞ *is a larger space than L p . Therefore, from this point of view, condition* (10) *can be regarded as an extension of* (7)*–*(9)*. In addition, our regularity criteria only depends on any two groups functions of* (*∂*1*u*1, *∂*1*b*1)*,* (*∂*2*u*2, *∂*2*b*2) *and* (*∂*3*u*3, *∂*3*b*3)*. Hence,* (10) *can be as a significant improvement of condition* (7) *and* (8)*. In addition, when b* = 0*, it is note that Theorem 1 is also new to the incompressible Navier-Stokes equations.*

**Remark 2.** *According to embedding relation L p* (R3 ) ,→ *L p*,∞(R3 )*, we can obtain the following regularity criteria on framework of anisotropic Lebesgue space,*

$$\begin{cases} A\_i(T) := \int\_0^T \left\| \left\| \| \partial\_i u\_i(t) \|\_{L^p\_{x\_1}} \right\|\_{L^q\_{x\_2}} \right\|\_{L^r\_{x\_3}}^2 dt < \infty, \\ B\_i(T) := \int\_0^T \left\| \left\| \| \partial\_i b\_i(t) \|\_{L^p\_{x\_1}} \right\|\_{L^q\_{x\_2}} \right\|\_{L^r\_{x\_3}}^2 dt < \infty, \end{cases} \tag{11}$$

*where we should point out that for Equation* (1)*, the regularity criterion* (11) *still new.*

**Remark 3.** *Notice that when fix p* = *q* = *r in condition* (11)*, the conditions* (9) *naturally turn out as stated in [19]. Furthermore, if let p* = *q* = *r in condition* (10)*, it is not difficult to find that our result improves the condition* (4) *significantly. Hence, regularity criteria* (10) *or* (11) *is much better. In other words, Theorem 1 can be regarded as a generalization of [16,18,19,23].*

Before ending this section, we state the following lemmas, which will be used in the proof of our main result.

**Lemma 1.** *(Young's Inequality for Lorentz Spaces [32,33]) Let 1* < *p* < ∞, 1 ≤ *q* ≤ ∞ *and* 1 *p* 0 + 1 *p* = 1, 1 *q* 0 + 1 *q* 0 = 1. *Suppose as well that* 1 < *p*1 < *p* 0 *and q* 0 ≤ *q* ≤ ∞. *If* 1 *p*2 + 1 = 1 *p* + 1 *p*1 *and* 1 *q*2 = 1 *q* + 1 *q*1 , *then the convolution operator,*

$$\ast : L^{p,q}(\mathbb{R}^n) \times L^{p\_1,q\_1}(\mathbb{R}^n) \mapsto L^{p\_2,q\_2}(\mathbb{R}^n)$$

*is a bounded bilinear operator.*

**Lemma 2.** *(Hölder's inequality in Lorentz spaces [33]) If* 1 ≤ *p*1, *p*2, *q*1, *q*2 ≤ ∞*, then for any f* ∈ *L p*1 ,*q*1 (R*n* ), *g* ∈ *L p*2,*q*2 (R*n* )*,*

$$\|\|f\g\|\|\_{L^{p,q}(\mathbb{R}^n)} \le \mathcal{C} \|\|f\|\|\_{L^{p\_1,q\_1}(\mathbb{R}^n)} \|\|\mathcal{g}\|\|\_{L^{p\_2,q\_2}(\mathbb{R}^n)}\star$$

*where* 1 *p* = 1 *p*1 + 1 *p*2 *and* 1 *q* = 1 *q*1 + 1 *q*2 *.*

For any *s* ≥ 0, even if *s* not an integer, we can define the homogeneous Sobolev space *H*˙ *s* (R*n* ):

$$\dot{H}^{\mathfrak{s}}(\mathbb{R}^n) = \{ f \in \mathcal{S}' : f \in L\_{loc}^1(\mathbb{R}^n) \text{ and } \int\_{\mathbb{R}^n} |\mathring{\xi}|^{2s} |\widehat{f(\xi)}|^2 d\xi < \infty \} $$

with the natural norm

$$\|f\|\_{\dot{H}^s} = \left(\int\_{\mathbb{R}^n} |\mathfrak{f}|^{2s} |\widehat{f(\xi)}|^2 d\xi\right)^{\frac{1}{2}}.$$

where S 0 denotes the space of the tempered distributions on R*n* .

**Lemma 3.** *For* 2 < *p* < ∞*, there exists a constant C* = *C*(*p*) *such that f* ∈ *H*˙ 1 *p* (R)*, then f* ∈ *L* 2*p p*−2 ,2(R) *and*

$$\left\|f\right\|\_{L^{\frac{2p}{p-2},2}} \le \mathbb{C} \left\|f\right\|\_{H^{\frac{1}{p}}}.\tag{12}$$

**Proof.** We first make the pointwise definition, *γ*(*ξ*) = |*ξ*| 1 *p* ˆ *f*(*ξ*) ; since *f* ∈ *H*˙ 1 *p* (R), *γ* ∈ *L* 2 (R). If we set *g* = F −1*γ*, then *g* ∈ *L* 2 (R) and k*g*k*L* 2 = k*γ*k*L* 2 = k *f* k *H*˙ 1 *p* . Now,

$$\hat{f}(\xi) = \frac{|\xi|^{\frac{1}{p}}\hat{f}(\xi)}{|\xi|^{\frac{1}{p}}} = \hat{\mathfrak{z}}(\xi)|\xi|^{-\frac{1}{p}}.$$

Combining the fact that if *Pα*(*x*) = |*x*| −*α* , then *P*[*α*(*ξ*) = *CαP*1−*α*(*ξ*). Thus we obtain *f* = *g* ∗ *C* −1 1− 1 *p P*1− 1 *p* . The function *P*1− 1 *p* = |*x*| − *p*−1 *p* ∈ *L p p*−1 ,∞ (R) but not in *L p p*−1 (R).  Let Λ *p* 1 be the Fourier multiplier defined as

$$\mathcal{F}\_1(\Lambda\_1^p f)(\xi\_1, \mathbf{x}\_2, \mathbf{x}\_3) = |\xi\_1|^p \mathcal{F}\_1 f(\xi\_1, \mathbf{x}\_2, \mathbf{x}\_3)$$

with

$$\mathcal{F}\_1 f(\mathfrak{f}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) = \int\_{\mathbb{R}} e^{-i\mathfrak{f}\_1 \mathfrak{x}\_1} f(\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) \mathrm{d}\mathfrak{x}\_1.$$

Λ *p* 2 and Λ *p* 3 can be defined analogously.  (15)

Combining the Fourier-Plancherel formula and the Hölder's inequality, we have

$$\begin{split} &C\left\lVert\frac{1}{\hbar^{\frac{1}{2}}}\frac{1}{2}\frac{1}{2}\frac{1}{1}f\right\rVert\_{L^{2}} \leq C\Big(\int\_{\mathbb{R}^{3}}|\xi\_{1}|^{\frac{2}{7}}|\xi\_{2}|^{\frac{2}{7}}|\xi\_{3}|^{\frac{2}{7}}|\mathcal{F}f(\xi\_{1},\xi\_{2},\xi\_{3})|^{\frac{2}{7}}\mathrm{d}\xi\_{1}\mathrm{d}\xi\_{2}^{\ast}\mathrm{d}\xi\_{3}\Big)^{\frac{1}{2}} \\ &= C\Big(\int\_{\mathbb{R}^{3}}|\xi\_{1}|^{\frac{2}{7}}|\mathcal{F}f(\xi)|^{\frac{2}{7}}|\xi\_{2}|^{\frac{2}{7}}|\mathcal{F}f(\xi)|^{\frac{2}{7}}|\xi\_{3}|^{\frac{2}{7}}|\mathcal{F}f(\xi)|^{\frac{2}{7}}|\mathcal{F}f(\xi)|^{2-\frac{2}{7}+\frac{2}{7}+\frac{2}{7}}\mathrm{d}\xi\_{1}\mathrm{d}\xi\_{2}\mathrm{d}\xi\_{3}\Big)^{\frac{1}{2}} \\ &\leq C\|\mathcal{F}f\|\_{L^{2}}^{1-\frac{1}{p}-\frac{1}{q}-\frac{1}{r}}\left(\int\_{\mathbb{R}^{3}}|\xi\_{1}|^{2}|\mathcal{F}f|^{2}\mathrm{d}\xi\right)^{\frac{1}{2p}}\left(\int\_{\mathbb{R}^{3}}|\xi\_{2}|^{2}|\mathcal{F}f|^{2}\mathrm{d}\xi\right)^{\frac{1}{2p}}\left(\int\_{\mathbb{R}^{3}}|\xi\_{3}|^{2}|\mathcal{F}f|^{2}\mathrm{d}\xi\right)^{\frac{1}{2p}} \\ &\leq C\|\partial\_{\widetilde{t}}f\|\_{L^{2}}^{\frac{1}{$$

**Remark 4.** *In fact, since L* 2*p p*−2 ,2 ,→ *L* 2*p p*−2 , 2*p p*−2 *for* 2 < *p* < ∞*, we have similar result for estimate* (14) *in anisotropic Lebesgue space (for more details refer to [34]). However, we should point out that Lemma 4 holds in Lorentz space mainly depends on the Sobolev's embedding in Lemma 3.*

#### **2. Proof of Theorem 1**

This section is devoted to the proof of Theorem 1. The proof is based on the establishment of a priori estimates under condition (10).

Firstly, we note that, by the energy inequality, for weak solution (*u*, *b*), we have

$$\|\|u\|\|\_{L^2}^2 + \|\|b\|\|\_{L^2}^2 + 2\int\_0^T \|\nabla u\|\|\_{L^2}^2 + \|\nabla b\|\_{L^2}^2 dt \le \|u\_0\|\_{L^2}^2 + \|b\_0\|\_{L^2}^2. \tag{17}$$

Next, let us convert (1) into a symmetric form. Writing

$$
\boldsymbol{\omega}^{\pm} = \boldsymbol{u} \pm \boldsymbol{b}\_{\prime}
$$

we find by adding and subtracting (1)1 with (1)2 ,

$$\begin{cases} \partial\_t \omega^+ + (\omega^- \cdot \nabla) \omega^+ - \Delta \omega^+ + \nabla p = 0, \\\partial\_t \omega^- + (\omega^+ \cdot \nabla) \omega^- - \Delta \omega^- + \nabla p = 0, \\\nabla \cdot \omega^+ = \nabla \cdot \omega^- = 0, \\\omega^+(0) = \omega\_0^+ \equiv u\_0 + b\_0, \quad \omega^-(0) = \omega\_0^- \equiv u\_0 - b\_0. \end{cases} \tag{18}$$

Taking the inner product of the *i*-th equation of (18)1 with |*ω* + *i* | 2*ωi* and (18)2 with |*ω* − *i* | 2*ωi* (for *i* = 1, 2, 3) and integrating by parts in R3 to get

$$\begin{split} &\frac{1}{4}\frac{\mathbf{d}}{\mathbf{d}t} \Big( \left\| \boldsymbol{\omega}\_{i}^{+}\right\|\_{L^{4}}^{4} + \left\| \boldsymbol{\omega}\_{i}^{-}\right\|\_{L^{q}}^{q} \Big) + \frac{1}{2} \Big( \left\| \nabla \left| \boldsymbol{\omega}\_{i}^{+}\right|^{2} \right\|\_{L^{2}}^{2} + \left\| \nabla \left| \boldsymbol{\omega}\_{i}^{-}\right|^{2} \right\|\_{L^{2}}^{2} \Big) \\ &+ \left\| \left| \boldsymbol{\omega}\_{i}^{+}\right| \cdot \left| \nabla \boldsymbol{\omega}\_{i}^{+}\right| \right\|\_{L^{2}}^{2} + \left\| \left| \boldsymbol{\omega}\_{i}^{-}\right| \cdot \left| \nabla \boldsymbol{\omega}\_{i}^{-}\right| \right\|\_{L^{2}}^{2} \\ &= -\int\_{\mathbb{R}^{3}} \partial\_{i} p \left| \boldsymbol{\omega}\_{i}^{+}\right|^{2} \boldsymbol{\omega}\_{i}^{+} \,\mathrm{d}\mathbf{x} - \int\_{\mathbb{R}^{3}} \partial\_{i} p \left| \boldsymbol{\omega}\_{i}^{-}\right|^{2} \boldsymbol{\omega}\_{i}^{-} \,\mathrm{d}\mathbf{x} \equiv I + I, \end{split} \tag{19}$$

we consider the (*u*, *b*) satisfying condition (10) with any two quantities of *Ai*(*T*) and *Bi*(*T*) for (*i* = 1, 2, 3):

$$\begin{cases} A\_i(T) := \int\_0^T \left\| \left\| \left\| \partial\_i u\_i(t) \right\|\_{L^{p,\infty}\_{x\_1}} \right\|\_{L^{q,\infty}\_{x\_2}} \right\|\_{L^{r,\infty}\_{x\_3}}^{\frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)}} dt < \infty, \\\ B\_i(T) := \int\_0^T \left\| \left\| \left\| \partial\_i b\_i(t) \right\|\_{L^{p,\infty}\_{x\_1}} \right\|\_{L^{q,\infty}\_{x\_2}} \right\|\_{L^{r,\infty}\_{x\_3}}^{\frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)}} dt < \infty. \end{cases}$$

In order to estimate the term *I* and *J* of (19), let us first establish an estimate between the *p* and the *ω*. Taking the divergence operator ∇· on both sides of the first equations of (18), it follows that

$$-\Delta p = \operatorname{div} \left( w^- \cdot \nabla w^+ \right) = \operatorname{div} \operatorname{div} \left( w^- \otimes w^+ \right).$$

Similarly, taking ∇div operator on both sides of the first equation of (18) to obtain

$$-\Delta(\nabla p) = \nabla \operatorname{div} \left( w^- \cdot \nabla w^+ \right) = \nabla \operatorname{div} \left( w^+ \cdot \nabla w^- \right).$$

By using the boundedness of Riesz transformations in *L p* (1 < *p* < ∞) space, so we have

$$\begin{cases} \|p\|\_{L^p} \le \mathbb{C} \|w^+\|\_{L^{2p}} \|w^-\|\_{L^{2p}}\\ \|\nabla p\|\_{L^p} \le \mathbb{C} \|w^+ \cdot \nabla w^-\|\_{L^{p\prime}}\\ \|\nabla p\|\_{L^p} \le \mathbb{C} \|w^- \cdot \nabla w^+\|\_{L^p} \end{cases} \tag{20}$$

Using the Hölder's inequality, Young's inequality, Lemma 4 and (20), we can deduce that

*I* = − Z R3 *∂i p*|*ω* + *i* | 2*ω* + *i dx* ≤ *C* Z R3 *p*|*ω* + *i* | 2 *∂iω* + *i* d*x* ≤ *∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 *L r*,∞ *x*3 k*p*k *L* 2*p p*−2 ,2 *x*1 *L* 2*q q*−2 ,2 *x*2 *L* 2*r r*−2 ,2 *x*3 |*ω* + *i* | 2 *L* 2 ≤*C ∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 *L r*,∞ *x*3 k*∂*1 *p*k 1 *p L* 2 k*∂*2 *p*k 1 *q L* 2 k*∂*3 *p*k 1 *r L* 2 k*p*k 1− 1 *p* + 1 *q* + 1 *r L* 2 k|*ω* + *i* | 2 k*L* 2 ≤*C ∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 *L r*,∞ *x*3 k∇*p*k 1 *p* + 1 *q* + 1 *r L* 2 k*p*k 1− 1 *p* + 1 *q* + 1 *r L* 2 k|*ω* + *i* | 2 k*L* 2 ≤*C ∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 *L r*,∞ *x*3 k∇*p*k 1 *p* + 1 *q* + 1 *r L* 2 k*ω* +k 3− 1 *p* + 1 *q* + 1 *r L* 4 k*ω* −k 1− 1 *p* + 1 *q* + 1 *r L* 4 ≤*e w* + · ∇*w* − 2 *L* 2 + *w* − · ∇*w* + 2 *L* 2 + *C ∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 k*ω* +k 4 *L* 4 + k*ω* −k 4 *L* 4 . (21)

Similarly, for *J*, we have

$$\begin{split} \mathcal{J} = &-\int\_{\mathbb{R}^3} \partial\_i p |\omega\_i^-|^2 \omega\_i^+ \, \mathrm{d}x \leq \mathcal{C} \Big| \int\_{\mathbb{R}^3} p |\omega\_i^-|^2 \partial\_l \omega\_i^- \, \mathrm{d}x \Big| \\ \leq & \epsilon \Big( \|w^+ \cdot \nabla w^-\|\_{L^2}^2 + \|w^- \cdot \nabla w^+\|\_{L^2}^2 \Big) + \mathcal{C} \Big{ \Big\| \, \left\| \left\| \partial\_l \omega\_i^- \right\|\_{L^{p\alpha}\_1} \right\|\_{L^{p\alpha}\_\infty} \Big\|\_{L^{p\alpha}\_\infty} \Big\|\_{L^{p\alpha}\_\infty}^{\frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{q}\right)}} \Big( \|\omega^+\|\_{L^4}^4 + \|\omega^-\|\_{L^4}^4 \Big). \end{split} \tag{22}$$

Inserting (21) and (22) into (19) and summing up with respect to the index *i* from 1 to 3, we get

1 4 *ω* + 4 *L* 4 + *ω* − 4 *L* 4 + 1 2 Z *t* 0 ∇ *ω* + 2 2 *L* 2 + ∇ *ω* − 2 2 *L* 2 *ds* + Z *t* 0 |*ω* +| · |∇*ω* +| 2 *L* 2 + |*ω* −| · |∇*ω* −| 2 *L* 2 *ds* ≤*C* Z *t* 0 3 ∑ *i*=1 *∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 + *∂iω* − *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 · k*ω* +k 4 *L* 4 + k*ω* −k 4 *L* 4 *ds* + *e* Z *t* 0 *w* + · ∇*w* − 2 *L* 2 + *w* − · ∇*w* + 2 *L* 2 *ds* + *C ω* + 0 4 *L* 4 + *ω* − 0 4 *L* 4 , (23)

where we have used that for any *p* ≥ 1 and some constant *Cγ*,*p* > 0,

$$\|\mathcal{C}^{-1}\_{\gamma,p}\|\mu\|\_{L^p}^{\gamma} \le \sum\_{i=1}^3 \|\mu\_i\|\_{L^p}^{\gamma} \lesssim \mathcal{C}\_{\gamma,p} \|\mu\|\_{L^p}^{\gamma}.$$

Due to the fact

$$\left|\nabla |w^+|^2\right| \le 2|w^+| \left|\nabla w^+\right| $$

and the inequality

$$\|\mu(t)\|\_{L^4} \le \frac{1}{2} (\left\|w^+(t)\right\|\_{L^4} + \left\|w^-(t)\right\|\_{L^4}),$$

$$\|\left|b(t)\right\|\|\_{L^4} \le \frac{1}{2} (\left\|w^+(t)\right\|\_{L^4} + \left\|w^-(t)\right\|\_{L^4}).$$

We rewrite inequality (23) as follows

1 4 k*u*(*t*)k 4 *L* 4 + k*b*(*t*)k 4 *L* 4 + 1 4 Z *t* 0 ∇|*u*| 2 2 *L* 2 + ∇|*b*| 2 2 *L* 2 *ds* + Z *t* 0 k|*u*| · |∇*u*|k 2 *L* 2 + k|*u*| · |∇*b*|k 2 *L* 2 + k|*b*| · |∇*u*|k 2 *L* 2 + k|*b*| · |∇*b*|k 2 *L* 2 *ds* ≤*C* Z *t* 0 3 ∑ *i*=1 *∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 + *∂iω* − *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 · k*u*k 4 *L* 4 + k*b*k 4 *L* 4 *ds* + *e* Z *t* 0 k*u* · ∇*u*k 2 *L* 2 + k*b* · ∇*u*k 2 *L* 2 + k*u* · ∇*b*k 2 *L* 2 + k*b* · ∇*b*k 2 *L* 2 *ds* + *C ω* + 0 4 *L* 4 + *ω* − 0 4 *L* 4 , (24)

and hence we get

1 4 k*u*(*t*)k 4 *L* 4 + k*b*(*t*)k 4 *L* 4 + 1 4 Z *t* 0 ∇|*u*| 2 2 *L* 2 + ∇|*b*| 2 2 *L* 2 *ds* + 1 4 Z *t* 0 k|*u*| · |∇*u*|k 2 *L* 2 + k|*u*| · |∇*b*|k 2 *L* 2 + k|*b*| · |∇*u*|k 2 *L* 2 + k|*b*| · |∇*b*|k 2 *L* 2 *ds* ≤*C* Z *t* 0 3 ∑ *i*=1 *∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 + *∂iω* − *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 · k*u*k 4 *L* 4 + k*b*k 4 *L* 4 *ds* + *C* k*u*0k 4 *L* 4 + k*b*0k 4 *L* 4 . (25)

Applying the Gronwall's inequality to obtain

sup 0≤*t*≤*T* k*u*(*t*)k 4 *L* 4 + k*b*(*t*)k 4 *L* 4 ≤*C* exp *C* Z *T* 0 3 ∑ *i*=1 *∂iω* + *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 + *∂iω* − *i L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 *dt* ≤*C* exp *C* Z *T* 0 3 ∑ *i*=1 k*∂iui*k*L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 + k*∂ibi*k*L p*,∞ *x*1 *L q*,∞ *x*2 2 2−( 1 *p* + 1 *q* + 1 *r* ) *L r*,∞ *x*3 *dt* (26)

<∞.

#### **3. Conclusions**

This paper studies the MHD equations, and obtains the a regularity criterion only involving the partial components of the ∇*u* and ∇*b*. In addition, the anisotropic Lorentz space used in this article is broader than the general Lebesgue and Lorentz spaces. It seems that a slightly modified the technique in Theorem 1 can be applied to other incompressible fluid equations such as micropolar equations and the magneto-micropolar equations.

**Author Contributions:** Both authors contributed equally to this work. Both authors have read and agreed to the published version of the manuscript.

**Data Availability Statement:** All data generated or analysed during this study are included in this published article.

**Acknowledgments:** The first author is partially supported by I.N.D.A.M-G.N.A.M.P.A. 2019 and the "RUDN University Program 5-100".

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

#### **References**

## *Article* **Aboodh Transform Iterative Method for Solving Fractional Partial Differential Equation with Mittag–Leffler Kernel**

**Michael A. Awuya \* and Dervis Subasi**

Department of Mathematics, Eastern Mediterranean University, Via Mersin 10, Famagusta TR99628, Turkey<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>