Prolongation of solutions and Lyapunov stability for Stieltjes dynamical systems
Lamiae Maia Noha El Khattabi Marlène Frigon


§ ABSTRACT 
In this article, we introduce Lyapunov-type results to investigate the stability of the trivial solution of a Stieltjes dynamical system. We utilize prolongation results to establish the global existence of the maximal solution. Using Lyapunov's second method, we establish results of (uniform) stability and (uniform) asymptotic stability by employing a Lyapunov function. Additionally, we present examples and real-life applications to study asymptotic stability of equilibria in two population dynamics models.


Retrieving stellar parameters and dynamics of AGB stars 
 
   with Gaia parallax measurements and CO^5BOLD RHD simulations
    E. Béguin
          1
          A. Chiavassa1
          A. Ahmad2
          B. Freytag2
          S. Uttenthaler3
          
          
          
    Received April 4, 2024; accepted July 16, 2024
======================================================================================================================================================






§ INTRODUCTION

In recent years, the field of differential equations has witnessed a notable surge in interest surrounding Stieltjes differential equations. This renewed focus is largely driven by the pursuit of results that not only unify existing findings but also extend those related to classical derivatives <cit.> through the  Stieltjes derivative.

The distinction between classical derivatives and their Stieltjes counterparts lies in the nature of the differentiation process. While classical derivatives are based on limits involving small increments of function values, the Stieltjes derivative operates with respect to a derivator g:ℝ→ℝ. This derivator is typically assumed to be left-continuous and nondecreasing, characteristics that allow for a broader range of applications, particularly in contexts where certain processes may exhibit discontinuities and/or stationary periods <cit.>. Such scenarios are common in various fields, including population dynamics, and physics, where classical differentiation has limitations in capturing the complexities of real-world phenomena.


Typically, investigations into first-order Stieltjes differential equations and systems focus on solutions defined on bounded intervals. However, Larivière in <cit.> turned his attention to Stieltjes differential equations on the positive real half-line. In doing so, he provided results related to the prolongation of solutions and the existence of the maximal solution. The motivation behind exploring these equations on the positive real half-line lies in the observation that many natural processes evolve over time without any inherent time limit, while some phenomena can exhibit finite-time blow-up, leading to abrupt changes or singularities. By considering the Stieltjes derivative, we aim to capture these nuanced behaviors and enhance our understanding of more complex dynamical systems from a common perspective, facilitated by the unification that this derivative provides <cit.>.

In the study of dynamical systems, the stability of equilibria holds significant importance. Here, the term equilibrium refers to a state that does not change dynamically, in the sense that if a system starts at an equilibrium point, it will stay in that state indefinitely. The interaction between species within many ecosystems often relies on feedback loops, which makes stability crucial for their functioning, and resilience. Although stability is typically desirable, there are some scenarios where stable equilibrium at zero can be critical and raise concerns for several raisons. For instance, in ecological systems, this concern arises from the vulnerability to perturbations of certain species, which may increase the risk of their extinction.

Within the realm of dynamical systems theory, Lyapunov's Second Method <cit.> stands as a fundamental approach for assessing the stability properties of a system near an equilibrium.  This stability analysis provides insights into whether small perturbations around an equilibrium lead to convergence (stability) or divergence (instability) of solutions. The core of this method lies in the concept of the Lyapunov function. This function was the subject of numerous works in the classical literature starting from the works <cit.> and references therein.

In this paper, we extend  Lyapunov stability results from the classical literature to the Stieltjes case by means of the Stieltjes derivative. In doing so, we address first the prolongation of solutions considering the Stieltjes dynamical system:

    𝐱_g'(t)    = 𝐟(t,𝐱(t))   for g-almost all t≥ t_0≥ 0,
    𝐱(t_0)    = 𝐱_0∈ℝ^n;

where 𝐟=(f_1,…,f_n):[0,+∞) ×ℝ^n →ℝ^n. We start by deriving corollary results using compact sets to characterize the maximal solution of (<ref>). Then, based on a generalized version of the Grönwall lemma <cit.> for the Stieltjes derivative, we establish the existence of global solutions over the whole positive real half-line. The prolongation of solutions will be essentially used later on to establish Lyapunov-like stability results for the system (<ref>), particulary when studying the asymptotic behaviour of solutions around an equilibrium. Our stability study is inspired by results from classical theory and works such as <cit.>, which address dynamic equations on time scales and impulsive differential equations. To the best of our knowledge, this is the first work to introduce Lyapunov's method adapted to Stieltjes differential equations.

This  paper is organized as follows: we present the theoretical framework and some preliminaries in Section 2.  In Section 3, we focus on prolongation of solutions and the characterization of the maximal solution to deduce the existence of a global solution. Section 4 is devoted to Lyapunov-like stability results using Lyapunov's second method. We start by defining the stability notions: (uniform) stability and (uniform) asymptotic stability. Afterward, we present stability results for each type of stability inspired from the works <cit.>, to extend some classical results from <cit.>. In the last section of this paper, we suggest two applications to dynamics of population to study the asymptotic stability of some critical equilibria: in the first application, we  model the dynamics of a population with Allee's effect <cit.> negatively impacted by train vibrations, and another application related to the dynamics of a population of Cyanobacteria in a cultured environment, keeping track of ammonia levels in the process.


§ PRELIMINARIES

For [a,b]⊂ℝ, and 𝐮:[a,b]→ℝ^n a regulated function,  the symbols 𝐮(t^+) and 𝐮(t^-)  will be used to denote

    𝐮(t^+)=lim_ϵ→ 0^+𝐮(t+ϵ),     for all t∈[a,b), 
    𝐮(t^-)=lim_ϵ→ 0^-𝐮(t-ϵ),     for all t∈ (a,b].


Throughout this work, we will consider g:ℝ→ℝ a monotone, nondecreasing and left-continuous function, also known as a derivator. We denote the set of discontinuity points of g by

    D_g={t∈ℝ: g(t^+ )- g(t)>0}.

In addition, we denote

    C_g ={s∈ℝ: g  is constant on  (s-ϵ, s+ϵ)  for some ϵ>0}.

The set C_g is an open of the usual topology and it can be written as a countable union of disjoint intervals

    C_g =⋃_n∈Λ(a_n ,b_n),

with Λ⊂ℕ, a_n,b_n ∈ℝ. We set N_g={a_n,b_n: n∈Λ}∖ D_g.

The function g defines a Lebesgue-Stieltjes measure μ_g:ℒ𝒮_g→ [0,∞] over ℒ𝒮_g the σ-algebra of subsets of ℝ containing all Borel sets <cit.>. We refer to the measurability with respect to the σ-algebra ℒ𝒮_g as g-measurability. For any interval [a,b) ∈ℒ𝒮_g, we have μ_g([a,b))=g(b)-g(a), and μ_g({t})=g(t^+)-g(t) for all t∈ℝ. Moreover, μ_g(C_g)=μ_g(N_g)=0, the reader is referred to <cit.> for more details.
We denote by  ℒ_g^1([a,b),ℝ) the quotient space of the set of μ_g-integrable functions on [a,b)⊂ℝ under the equivalence relation:
 
    f∼ h : f(t)=h(t)  μ_g-almost everywhere.

We define the norm ·_ℒ_g^1([a,b),ℝ) on the space ℒ_g^1([a,b),ℝ) by

    f_ℒ_g^1([a,b),ℝ):=∫_[a,b) |f(t)|  dμ_g(t),    for every  f∈ℒ_g^1([a,b),ℝ).

In the sequel, given I⊂ℝ and a functional space Z(I,ℝ), we set Z(I,ℝ^n):=∏_i=1^nZ(I,ℝ), and we denote Z_loc(I,ℝ^n) the space of functions 𝐮:I →ℝ^n satisfying 𝐮|_[a,b]∈ Z([a,b],ℝ^n) for every interval [a,b] ⊂ I.

The derivator g defines a pseudometric ρ:ℝ×ℝ→ℝ^+ given by

    ρ(s,t)=|g(s)-g(t)|,    for every  s,t ∈ℝ.

We denote τ_g the topology induced by the pseudometric ρ. The topology τ_g is not necessarily Hausdorff as mentioned in <cit.>.
 A set A ⊂ℝ is called g-open if, for every t ∈ A, there exists r > 0 such that

    {s ∈ℝ : ρ(t,s)< r}⊂ A.

 
If t ∈ D_g, then the interval (t-ϵ,t] is a g-open set. The reader is referred to <cit.> for more properties of the topology τ_g.


In the following definition, we define the g-derivative of a real-valued function.

Let u:[a,b] →ℝ be a function. The derivative of u with respect to g at a point t_0 ∈ [a,b]∖ C_g, is defined as:

    u_g'(t)=
    lim_s→ tu(s)-u(t)/g(s)-g(t)     t ∉ D_g,
    u(t^+)-u(t)/g(t^+)-g(t)    t ∈ D_g,

provided that the limit exists, in this case u is said to be g-differentiable at t.


In the next proposition, we appeal to the g-derivative of the composition of two functions  established in <cit.>. Another version of this formula can be found in <cit.>.

Let t∈ℝ∖ C_g, f:ℝ→ℝ and h a real function defined on a neighborhood of f(t). We assume that there exist h'(f(t)), f_g'(t) and that the function h is continuous at f(t^+).
Then, the composition h∘ f is g-differentiable in t_0, and we have the formulae

    (h∘ f)_g'(t)=
    
    h'(f(t))f_g'(t)      t ∉ D_g,  
    h(f(t^+))-h(f(t))/f(t^+)-f(t)f_g'(t)     t ∈ D_g.



Throughout this paper, let · denotes the maximum norm in ℝ^n defined by

    𝐱=max{|x_1|,…,|x_n|}   for 𝐱=(x_1,…,x_n)∈ℝ^n.


Now, we recall the g-continuity notion  first introduced in <cit.>.

Let 𝐮 : [a,b]→ℝ^n. We say that 𝐮 is g-continuous at t∈ [a,b] if, for every ϵ >0, there exists δ>0 such that
    
    ∀ s∈ [a,b],      |g(s)-g(t)|< δ𝐮(s)-𝐮(t)<ϵ.


We denote 𝒞_g([a,b],ℝ^n) the set of g-continuous functions 𝐮:[a,b]→ℝ^n on the interval [a,b]. The following proposition relates the regularity of f and g, the reader is referred to <cit.>.

  If 𝐮:[a,b] →ℝ^n is g-continuous on [a,b], then the following statements hold:

    
  * 𝐮 is left-continuous at every t ∈ (a,b].
    
  * If g is continuous at t ∈ [a,b), then so is 𝐮.
    
  * If g is constant on some [c,d] ⊂ [a,b], then so is 𝐮.


g-continuous functions presents interesting measurability properties, see <cit.>. Let ℬ𝒞_g([a,b],ℝ^n) denotes the Banach space of the bounded functions of 𝒞_g([a,b],ℝ^n)  with respect to the supremum norm.

In the next theorem, we focus on the particular case when the derivator g:ℝ→ℝ is increasing, and continuous on an interval [a,b]⊂ℝ, to derive a generalized version of Rolle's theorem and the Mean Value theorem for real-valued functions in the context of Stieltjes differentiation.


Let g:ℝ→ℝ be a left-continuous and nondecreasing function, continuous and increasing on an interval [a,b]⊂ℝ. Let f:[a,b]→ℝ be g-continuous on [a,b] and g-differentiable on (a,b) satisfying f(a)=f(b). Then, there exists c∈ (a,b) such that f_g'(c)=0.


 As f is g-continuous on [a,b], it results from Proposition <ref> that f is continuous on [a,b].  We set

    m:=min_t∈[a,b]f(t),   M:=max_t∈[a,b]f(t).

If m=M, then f is constant on [a,b] and for all t∈ (a,b), f_g'(t)=0. Otherwise if m<M, as f(a)=f(b) we have

    f(a)≠ m, and  f(b)≠ M.

Without loss of generality, we can assume that

    f(a)≠ M, and  f(b)≠ M,

if it is not the case, we consider m instead. Thus, there exists c∈ (a,b) such that f(c)=M. Therefore, there exists δ>0 such that (c-δ,c+δ)⊂(a,b) and f(s)≤ f(c)=M for all s∈(c-δ,c+δ). As s↗ c, g(s)≤ g(c), and

    lim_s→ c^-f(s)-f(c)/g(s)- g(c)≥ 0.

While, as s↘ c, g(s)≥ g(c), and

    lim_s→ c^+f(s)-f(c)/g(s)- g(c)≤ 0.

Since f is g-differentiable at c, we deduce that f_g'(c)=0.

As a corollary of Theorem <ref>, we state a version of the Mean Value theorem involving the Stieltjes derivative.

Let g:ℝ→ℝ be a left-continuous and nondecreasing function, continuous and increasing on an interval [a,b]⊂ℝ. Let f:[a,b]→ℝ be g-continuous on [a,b] and g-differentiable on (a,b). Then, there exists c∈ (a,b) such that f_g'(c)=f(b)-f(a)/g(b)-g(a).


  Let us consider the function F:[a,b]→ℝ defined by
  
    F(t)=f(t)-(f(b)-f(a)/g(b)-g(a)(g(t)-g(a))+f(a)),   for all t∈ [a,b].

Clearly F is g-continuous on [a,b] and g-differentiable on (a,b), satisfying F(a)=F(b).  Moreover, for all t∈ (a,b), we have that

    F_g'(t)=f_g'(t)-f(b)-f(a)/g(b)-g(a).

Applying Theorem <ref>, there exists c∈(a,b) such that F_g'(c)=0. Hence, there exists c∈(a,b) such that f_g'(c)=f(b)-f(a)/g(b)-g(a).


Now, we present the notion of g-absolute continuity.

A map F:[a,b]→ℝ is g-absolutely continuous, if, for every ε > 0, there exists δ > 0 such that, for any family
{(a_i , b_i)}_i=1^i=n of pairwise disjoint open subintervals of [a,b],

    ∑_i=1^n g(b_i)-g(a_i) <δ⇒∑_i=1^n |F(b_i)-F(a_i)| < ε.



We denote by 𝒜𝒞_g ([a,b],ℝ) the vector space of g-absolutely continuous functions F: [a,b]→ℝ on the interval [a,b].
In <cit.>, a Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals was introduced.

Let a,b ∈ℝ be such that a<b, and let F:[a,b] →ℝ. The following assumptions are equivalent.


  (1) The function F is g-absolutely continuous, i.e.  to each ε>0, there is some δ>0 such that, for any family {(a_j,b_j)}_j=1^m of pairwise disjoint open subintervals of [a,b],

    ∑_j=1^m(g(b_j)-g(a_j))<δ    ∑_j=1^m|F(b_j)-F(a_j)|<ε.


  (2) The function F satisfies the following conditions:


  (a) there exists F'_g(t) for g-almost all t∈ [a,b);

  (b) F'_g ∈ℒ^1_g([a,b),ℝ);

  (c) for each t  ∈ [a,b], we have

    F(t)=F(a)+∫_[a,t) F'_g(s)   dμ_g(s).




We denote by 𝒜𝒞_g([a,b],ℝ) the set of  g-absolutely continuous functions. We set 𝐮=(u_1,…,u_n)∈𝒜𝒞_g([a,b],ℝ^n) if  u_i∈𝒜𝒞_g([a,b],ℝ) for i=1,…,n.

The following proposition provides conditions ensuring the g-absolute continuity of the composition of two functions, the proof is based on arguments as in <cit.>.

  Let 𝐮:[a,b]→ B⊂ℝ^n be a g-absolutely continuous function, and let v:B→ℝ be a Lipschitz continuous function on B. Then, the composition v∘𝐮∈𝒜𝒞_g([a,b],ℝ).


In <cit.>, an exponential function was introduced.


Let p ∈ℒ^1_g([a,b),ℝ) be such that

    1+p(t)(g(t^+)-g(t)) > 0 for every  t ∈ [a,b)∩ D_g.

Let us define the function e_p(·,a) : [a,b] → (0,∞) for every t∈ [a,b] by

    e_p(t,a) = e^∫_[a,t)p̃(s) dμ_g(s),

where

    p̃(t) = 
    p(t)    if t ∈ [a,b]∖ D_g,
    log(1 + p(t)(g(t^+) - g(t)))/g(t^+)-g(t)   if t ∈ [a,b) ∩ D_g.


In particular, given p ∈ℒ^1_g([a,b),ℝ) a function satisfying Condition (<ref>), then p̃∈ℒ^1_g([a,b),ℝ), and e_p(·, a)∈𝒜𝒞_g([a,b),ℝ), the reader is referred to <cit.> and improvements in <cit.>.

Now, we appeal to the generalization of the Grönwall Lemma to the Stieltjes derivative introduced by Larivière in <cit.>, and further generalized in <cit.>. This lemma will play a crucial role in establishing global solutions defined on the positive
real half-line as we shall prove in the following section.

Let u∈𝒜𝒞_g([a,b],ℝ). Assume that there exist functions k,p∈ℒ^1_g([a,b),ℝ), satisfying 1+p(t)μ_g({t})>0 for all t∈ [a,b)∩ D_g,  such that
  
    u_g'(t)≤ k(t)+ p(t) u(t)  for g-almost all t∈[a,b).

  Then,
  
    u(t)≤ e_p(t,a) (∫_[a,t)e_p^-1(s,a) k(s)/1+p(s)μ_g({s})  dμ_g(s)+u(a)),    t∈ [a,b].








§ PROLONGATION OF SOLUTIONS AND MAXIMAL INTERVAL OF EXISTENCE

Let O be a nonempty open set of ℝ^n with respect to the usual topology τ_ℝ^n, and I a g-open set of ℝ containing t_0≥ 0 with sup I > t_0. If we set Ω :=I× O, then Ω is an open set with respect to the topology τ̂_g on ℝ×ℝ^n, generated by the open sets U× V such that U∈τ_g and V∈τ_ℝ^n. Here and afterwards, B_ℝ^n(𝐱,δ) denotes the open ball of ℝ^n centered at 𝐱∈ℝ^n with radius δ>0.

 Let us consider the Stieltjes dynamical system:

    𝐱_g'(t)    = 𝐟(t,𝐱(t))    for g-almost all t≥ t_0, t∈ I,
    𝐱(t_0)   =𝐱_0 ∈ O,

where 𝐟=(f_1,…,f_n):Ω∩([t_0,∞)×ℝ^n) →ℝ^n and Ω defined above satisfying the following assumptions:

  
  (H_Ω) For every (t,𝐱) ∈Ω with t≥ t_0,
  
          
  (a)  one of the following conditions hold:
            
          
  (a1) there exists δ>0 such that (t-δ,t+δ)× B_ℝ^n(𝐱,δ) ⊂Ω;
          
  (a2) if for every δ>0, (t-δ,t+δ)× B_ℝ^n(𝐱,δ) ⊄Ω, then t∈ D_g and there exists ϵ>0 such that (t-ϵ,t]× B_ℝ^n(𝐱,δ)⊂Ω;
          
  (b) (t_0,𝐱_𝐟,t_0^+)∈Ω, and if (t,𝐱) ∈Ω∩ (D_g×ℝ^n) such that (t,𝐱_𝐟,t^+)∈Ω, then (t,𝐱_𝐟,t^+) satisfies Condition (H_Ω)(a)(1). Here and afterward, the notation 𝐱_𝐟,t^+ refers to
          
    𝐱_𝐟,t^+:=𝐱+μ_g({t})𝐟(t,𝐱);

        
  
  (H_𝐟)


  * for all 𝐱∈ O, 𝐟(·,𝐱) is g-measurable;

  * 𝐟(·,𝐱_0)∈ℒ^1_g,loc([t_0,∞),ℝ^n);

  * 𝐟 is g-integrally locally Lipschitz continuous, i.e.  for every r>0, there exists a function L_r∈ℒ^1_g,loc([t_0,∞),[0,∞)) such that

    𝐟(t,𝐱)-𝐟(t,𝐲)≤ L_r(t) 𝐱-𝐲,

for g-almost all t∈ I ∩ [t_0,∞) and all 𝐱,𝐲∈B_ℝ^n(𝐱_0,r)∩ O.


We recall the local existence result <cit.>.

Assume that the conditions in (H_𝐟) hold. Then there exists τ>0 such that the system (<ref>) has a unique solution 𝐱∈𝒜𝒞_g([t_0,t_0+τ],ℝ^n).

In the sequel, as shown in <cit.>, the solution given by Theorem <ref> can be extended to intervals larger than [t_0,t_0+τ] up to a maximal interval as long as the graph of the solution does not leave Ω and the right-hand side limit of the solution at t_0+τ belongs to O. It should be noted that the derivator g may not necessarily be continuous at t_0. For this purpose, Condition (H_Ω) is necessary to seek the maximal interval of existence of the solution. Indeed, notice that Condition (H_Ω) guarantees existence of a local solution on an interval [t_0,t_0+τ] even in the case where t_0∈ D_g, and insures extension of the solution 𝐱 on some larger interval [t_0,t_0+ϵ], ϵ>τ if t_0+τ∈ D_g  and 𝐱(t_0+τ^+) ∈ O.

Let us define the set

    𝒮(t_0,𝐱_0):={𝐱:I_𝐱 = J_𝐱∩[t_0,∞)→ℝ^n: 𝐱 is a solution of the system (<ref>)},

where J_𝐱 is a g-open interval containing t_0 with sup J_𝐱 > t_0. In the sequel, we adopt the notation t_sup:=sup I_𝐱.


Let 𝐱,𝐲∈𝒮(t_0,𝐱_0).

  
  * We say that 𝐱 is smaller than 𝐲 (and we denote 𝐱≺𝐲), if and only if

  
  * I_𝐱⊂ I_𝐲;
  
  * sup I_𝐲> sup I_𝐱;
  
  * 𝐲|_I_𝐱=𝐱.
 In this case, we say that 𝐱 is extendible to the right and 𝐲 is a prolongation to the right of 𝐱.
   
  * We write that 𝐱≼𝐲𝐱≺𝐲 or 𝐱=𝐲.
   


 It is worth mentioning that given a solution 𝐱:[t_0,t_1)→ℝ^n such that 𝐱(t_1^-):=lim_t→ t_1^-𝐱(t) ∈ O and t_1∈ D_g, to not increase the notation, we will replace 𝐱 by the function
 𝐱:[t_0,t_1]→ℝ^n defined by

    𝐱=𝐱(t)    t∈ [t_0,t_1), 
    𝐱(t_1^-),     t=t_1.




The next result asserts that two prolongations to the right of a solution are equal on the common interval of existence, the reader is referred to <cit.> for the proof.

Assume that (H_𝐟) holds. Let 𝐱,𝐲,𝐳∈𝒮(t_0,𝐱_0) be such that 𝐲 and 𝐳 are two prolongations of 𝐱 then 𝐲=𝐳 on I_𝐲∩ I_𝐳.


In <cit.>, extendible solutions to the right were characterized as follows.


 Assume that (H_𝐟) holds.   Let 𝐱∈𝒮(t_0,𝐱_0). The following assumptions are equivalent:
  
    
  * 𝐱 is extendible to the right;
    
  * 
           
  * Graph(𝐱):={(t,𝐱(t)): t∈ I_𝐱} is bounded;
           
  * A∪ A^+ ⊂Ω where
           
    A={(t_sup,𝐬)∈ [t_0,∞)×ℝ^n: ∃{t_n}_n⊂ I_𝐱,  t_n ↗ t_sup and 𝐬=lim_n→∞𝐱(t_n)},

           
    A^+={(t_sup,𝐬_𝐟,t_sup^+):(t_sup,𝐬)∈ A}.

         
  



Let 𝐱∈𝒮(t_0,𝐱_0). We say that 𝐱 is a maximal  solution of (<ref>) defined on an interval I_𝐱 if,  for every 𝐲∈𝒮(t_0,𝐱_0) satisfying 𝐱≼𝐲, we have 𝐱=𝐲. I_𝐱 is referred to as the maximal interval of existence.



As shown in <cit.>, the existence of the maximal solution holds as a consequence of using Zorn's lemma <cit.>. For this purpose, let us consider 𝐱∈𝒮(t_0,𝐱_0), and define the set 𝒳:

    𝒳={𝐲∈𝒮(t_0,𝐱_0): 𝐱≼𝐲}.

𝒳 is a nonempty partially ordered set since 𝐱∈𝒳. Now, following the same argument as in the proof of <cit.>, by using the partial order ≼ defined in Definition <ref> instead, we deduce that every chain of 𝒳 has the largest element. This yields the existence of a maximal solution defined on ⋃_𝐲∈𝒳I_𝐲, constructed by taking the union of all the prolongations to the right of 𝐱. In addition, Theorem <ref> guarantees the uniqueness of the maximal solution, and we obtain the following theorem.


Assume that (H_Ω) and (H_𝐟) hold. There exists a unique maximal solution 𝐱∈𝒮(t_0,𝐱_0) such that ω(t_0,𝐱_0):=sup I_𝐱≤∞.



The next theorem highlights three alternative cases that occur, the reader is referred to <cit.> for the proof.

Assume that (H_Ω) and (H_𝐟) hold. Let  𝐱∈𝒮(t_0,𝐱_0) be the maximal solution of (<ref>), then one of the alternatives holds:
  
 
  (A1) ω(t_0,𝐱_0)=∞;
 
  (A2) ω(t_0,𝐱_0)<∞, and for every {t_n}_n ⊂ I_𝐱 such that t_n ↗ω(t_0,𝐱_0),  {𝐱(t_n)}_n is not bounded;
 
  (A3) ω(t_0,𝐱_0)<∞, moreover, there exists {t_n}_n ⊂ I_𝐱 satisfying t_n ↗ω(t_0,𝐱_0) and {𝐱(t_n)}_n is a bounded sequence, such that for every  subsequence {t_n_k}_k verifying 𝐱(t_n_k) →𝐦, we have that
     
    {(ω(t_0,𝐱_0), 𝐦),(ω(t_0,𝐱_0),𝐦_𝐟,ω(t_0,𝐱_0)^+)}⊄Ω.

  


  It is worth mentioning that the maximal solution 𝐱 of (<ref>) shall be g-absolutely continuous on every interval [a,b]⊂ I_𝐱, which immediately  holds from the three alternatives of Theorem <ref>. Notice that if Alternative (A1) holds then I_𝐱=[t_0,∞), if Alternative (A2) or (A3) holds, then  I_𝐱=[t_0,ω(t_0,𝐱_0)) or I_𝐱=[t_0,ω(t_0,𝐱_0)].


 Thanks to Theorem <ref> established by Larivière, we obtain the following corollary, which ensures the global existence of the maximal solution over the whole interval [t_0,∞) in the case where [t_0,∞)⊂ I.


Assume that (H_Ω) and (H_𝐟) hold. Let  𝐱∈𝒮(t_0,𝐱_0) be the maximal solution of (<ref>). If

    Ω^+:={(t,𝐮) ∈Ω: 𝐮_𝐟,t^+∈Ω}=Ω,

and there exists a compact set D⊂ O such that 𝐱(t) ∈ D for every t∈ I_𝐱, then ω(t_0,𝐱_0)=∞.


Let us assume that ω(t_0,𝐱_0)<∞. Since D is compact, for all {t_n}_n ⊂ I_𝐱 such that t_n ↗ω(t_0,𝐱_0), {𝐱(t_n)}_n is bounded.  Thus, it results from Theorem <ref> that Alternative (A3) holds. Therefore, there exists {t_n}_n ⊂ I_𝐱 with t_n ↗ω(t_0,𝐱_0) and {(t_n ,𝐱(t_n))}_n is bounded, such that for a fixed subsequence {t_n_k}_k verifying 𝐱(t_n_k) →𝐦, we have that
  
    {(ω(t_0,𝐱_0), 𝐦),(ω(t_0,𝐱_0),𝐦_𝐟,ω(t_0,𝐱_0)^+)}⊄Ω.

   On the other hand,  D being a compact set yields that
  
    (t_n_k,𝐱(t_n_k)) → (ω(t_0,𝐱_0),𝐦) ∈ [t_0,ω(t_0,𝐱_0)] × D ⊂Ω.

 Now, since Ω^+=Ω, we deduce that (ω(t_0,𝐱_0),𝐦_𝐟,ω(t_0,𝐱_0)^+) ∈Ω, which contradicts
 
    {(ω(t_0,𝐱_0), 𝐦),(ω(t_0,𝐱_0),𝐦_𝐟,ω(t_0,𝐱_0)^+)}⊄Ω.

  Hence, ω(t_0,𝐱_0)=∞.


Now based on Theorems <ref> and <ref>, we provide a characterization of the maximal solution of the problem (<ref>).

 Assume that (H_Ω) and (H_𝐟) hold. Let 𝐱∈𝒮(t_0,𝐱_0). The following assumptions are equivalent:
  
    
  * 𝐱 is maximal;
    
  * for every compact set K⊂Ω, there exists t_K ∈ I_𝐱 such that
    
    {(t,𝐱(t)),(t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊄K,

 for all t≥ t_K, t∈ I_𝐱.
  



 Assume that  𝐱 is maximal. By Theorem <ref>, two cases may occur:

Case 1: if ω(t_0,𝐱_0)=sup I_𝐱=∞,  by contradiction assume that there exist a compact set K ⊂Ω and a sequence {t_n}_n ⊂ I_𝐱 such that

    t_n↗ω(t_0,𝐱_0),   and {(t_n,𝐱(t_n)),(t_n,𝐱(t_n)+μ_g({t_n})𝐟(t_n,𝐱(t_n)))}⊂ K  for all  n∈ℕ.

This implies that {(t_n,𝐱(t_n))}_n is bounded. Therefore, there exists a convergent subsequence {(t_n_k,𝐱(t_n_k))}_k  such that

    (t_n_k,𝐱(t_n_k)) → (τ,𝐮)∈ K ⊂ [t_0,∞)× O.

Particularly, t_n_k↗τ which contradicts t_n ↗ω(t_0,𝐱_0)=∞.

Case 2: if ω(t_0,𝐱_0)< ∞, then by Theorem <ref>, there are two subcases:

Subcase 1: if Alternative (A2) holds, then I_𝐱=[t_0,ω(t_0,𝐱_0)) and 𝐱(t)→∞ as t ↗ω(t_0,𝐱_0). Thus, for every M>0 there exists t^* ∈ I_𝐱 such that 𝐱(t)≥ M for all t≥ t^* with t∈ I_𝐱. Hence, for every compact K ⊂Ω there exists t_K ∈ I_𝐱 such that for all t≥ t_K with t∈ I_𝐱, we have (t,𝐱(t))∉ K, in particular {(t,𝐱(t)), (t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊄K.

Subcase 2: if Alternative (A3) holds, then we distinguish two cases. If I_𝐱=[t_0,ω(t_0,𝐱_0)), then  Graph(𝐱) approaches the boundary of Ω and 
    (ω(t_0,𝐱_0),𝐱(ω(t_0,𝐱_0)^-))∉Ω.

Thus, for every compact K ⊂Ω, there exists t_K ∈ [t_0,ω(t_0,𝐱_0)) such that, for all t≥ t_K with t∈ I_𝐱, (t,𝐱(t)) ∉ K, which yields

    {(t,𝐱(t)),(t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊄K.

Now, if  I_𝐱=[t_0,ω(t_0,𝐱_0)], then ω(t_0,𝐱_0) ∈ D_g, (ω(t_0,𝐱_0),𝐱(ω(t_0,𝐱_0)))∈Ω and
 
    (ω(t_0,𝐱_0),𝐱(ω(t_0,𝐱_0))+μ_g({ω(t_0,𝐱_0)})𝐟(ω(t_0,𝐱_0), 𝐱(ω(t_0,𝐱_0))))∉Ω.

 Thus, for every compact K ⊂Ω, there exists t_K ∈ [t_0,ω(t_0,𝐱_0)] such that, for all t≥ t_K with t∈ I_𝐱, (t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t))) ∉ K, which yields

    {(t,𝐱(t)),(t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊄K.


Conversely, by contradiction, let us assume that 𝐱:I_𝐱→ℝ^n is not maximal. Thus, t_sup=sup I_𝐱 < ∞ and  𝐱 is extendible to the right. From Theorem <ref>, it follows that Graph(𝐱) is bounded and A∪ A^+ ⊂Ω. Thus, [t_0,t_sup]×Graph(𝐱) is compact, 𝐱(t_sup^-)=𝐱(t_sup)=𝐬∈ O and 𝐬_𝐟,t_sup^+∈ O. Consequently,  by Theorem <ref>, there exists a prolongation to the right 𝐱̂:[t_0,t_sup +ϵ) →ℝ^n such that 𝐱̂|_I_𝐱=𝐱. Therefore, for the compact K=Graph(𝐱)∪{(t_sup,𝐬_𝐟,t_sup^+)}⊂Ω, we have that

    {(t,𝐱(t)),(t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊂ K  for all  t∈ I_𝐱,

which yields a contradiction. Hence, 𝐱 is maximal.


The logical negation of Theorem <ref> also provides an interesting characterization of extendible solutions.


Assume that (H_Ω) and (H_𝐟) hold. Let 𝐱∈𝒮(t_0,𝐱_0). The following assumptions are equivalent:


  * 𝐱 extendible to the right;

  * there exist a compact set K⊂Ω, and a sequence {t_n}_n⊂ I_𝐱 with t_n ↗ω(t_0,𝐱_0) such that
    
    {(t_n,𝐱(t_n)),(t_n,𝐱(t_n)+μ_g({t_n})𝐟(t_n,𝐱(t_n)))}⊂ K,  for all n∈ℕ.

  


Given the generalization of the Grönwall lemma, Lemma <ref>, we can state the next theorem which provides the global existence of the solution over [t_0,∞) and a priori bound of the solution when [t_0,∞) ⊂ I and O=ℝ^n.


Assume that (H_Ω) and (H_𝐟) hold. Let  [t_0,∞) ⊂ I and O=ℝ^n and assume that the conditions in (H_𝐟) hold for 𝐟:[t_0,∞)×ℝ^n→ℝ^n satisfying the linear growth condition:

  
  (H_LG) there exist functions
  k∈ℒ^1_g,loc([t_0,∞),ℝ), and p∈ℒ^1_g,loc([t_0,∞),[0,∞)) such that

    𝐟(t,𝐮)≤ k(t)+ p(t)𝐮,

for g-almost all t∈[t_0,∞) and all 𝐮∈ O.
  
 If 𝐱∈𝒮(t_0,𝐱_0) is the maximal solution of (<ref>),  then

    𝐱(t)≤ e_p(t,t_0) (∫_[t_0,t)e_p^-1(s,t_0) k(s)/1+p(s)μ_g({s})  dμ_g(s)+𝐱_0),

for all t∈ [t_0,T], T∈ I_𝐱. Moreover, we have I_𝐱=[t_0,∞).


Let 𝐱∈𝒮(t_0,𝐱_0) be the maximal solution of (<ref>) defined on the maximal interval I_𝐱 with ω(t_0,𝐱_0)=sup I_𝐱. By (H_LG), we have for T>t_0 with T∈ I_𝐱 that

    (𝐱)_g'(t) ≤𝐱_g'(t)=𝐟(t,𝐱(t))≤ k(t)+ p(t)𝐱(t)  for g-almost all  t∈ [t_0,T).

Observe that · is Lipschitz continuous on ℝ^n. Thus, by Lemma <ref>, 𝐱(·)∈𝒜𝒞_g([t_0,T],ℝ). Using the generalized version of the Grönwall Lemma, Lemma <ref>, we obtain

    𝐱(t)≤  e_p(t,t_0) (∫_[t_0,t)e_p^-1(s,t_0) k(s)/1+p(s)μ_g({s})  dμ_g(s)+𝐱_0)    for all  t∈ [t_0,T].

Assume by contradiction that ω(t_0,𝐱_0)<∞. Thus, for all  t∈ [t_0,T], we obtain a larger bound of 𝐱:

    𝐱(t)≤sup_t∈ [t_0,T] e_p(t,t_0) (∫_[t_0,t)e_p^-1(s,t_0) |k(s)|/1+p(s)μ_g({s})  dμ_g(s)+𝐱_0).

Since  p∈ℒ^1_g,loc([t_0,∞),[0,∞)), then  e_p(t,t_0)(1+p(t)μ_g({t}))≥ 1 for all t∈ [t_0,T]. Consequently,

    𝐱(t)   ≤sup_t∈ [t_0,T]e_p(t,t_0) (∫_[t_0,t) |k(s)|  dμ_g(s)+𝐱_0)  
       =  e_p(T,t_0) (k_ℒ^1_g([t_0,T),ℝ)+𝐱_0).

As T↗ω(t_0,𝐱_0) <∞, 𝐱(t) ∈ B_ℝ^n(0,M_1) for all t∈ I_𝐱 where

    M_1=e_p(ω(t_0,𝐱_0),t_0) (k_ℒ^1_g([t_0,ω(t_0,𝐱_0)),ℝ)+𝐱_0).

As 𝐱(ω(t_0,𝐱_0)^-)=𝐦∈ℝ^n, then 𝐦_𝐟,ω(t_0,𝐱_0)^+∈ℝ^n. Therefore, for the compact set K=[t_0, ω(t_0,𝐱_0)]× B_ℝ^n(0,M) with M=max{M_1, 𝐦_𝐟,ω(t_0,𝐱_0)^+}, we have that

    {(t,𝐱(t)),(t,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))}⊂ K  for all  t∈ I_𝐱.

By Corollary <ref>, we obtain that 𝐱 is extendible to the right which is a contradiction. Hence, ω(t_0,𝐱_0)=∞.




§ LYAPUNOV-LIKE STABILITY RESULTS


In this section, we introduce Lyapunov-type results in the context of Stieltjes dynamical systems, based on the classical Lyapunov's second method, considering Stieltjes dynamical systems of the form:

    𝐱_g'(t)    = 𝐟(t,𝐱(t))    for g-almost all t≥ t_0≥ 0, t∈ I,

where 𝐟=(f_1,…,f_n):ℝ^+× B_ℝ^n(0,r_0)→ℝ^n satisfying 𝐟(t,0)=0 for all t≥ 0. This study permits to draw conclusions about the local behavior of solutions of the dynamical system (<ref>) around the equilibrium 𝐱=0, which is called in the stability literature the trivial solution. Before stating the assumptions on 𝐟 to guarantee the existence and the uniqueness of solution of the system (<ref>) starting at some t_0≥ 0, notice that if t_0 ∈ D_g, and 𝐱_0 ∈ B_ℝ^n(0,r_0) such that 𝐱_0+μ_g({t_0})𝐟(t_0,𝐱_0) ∉ B_ℝ^n(0,r_0), then existence of a solution 𝐱 satisfying 𝐱(t_0)=𝐱_0 cannot be guaranteed. Thus, we assume

  
  (H_r)there exists r∈(0,r_0] such that
  for all (t,𝐮) ∈ (ℝ^+ ∩ D_g)× B_ℝ^n(0,r), we have
  
    𝐮+μ_g({t}) 𝐟(t,𝐮) ∈ B_ℝ^n(0,r_0),

in other terms, for all t∈ (ℝ^+ ∩ D_g),
  
    B_ℝ^n(0,r)⊂{𝐮∈ B_ℝ^n(0,r_0): 𝐮+μ_g({t}) 𝐟(t,𝐮) ∈ B_ℝ^n(0,r_0) }.


Now, let us assume that 𝐟 fulfills (H_𝐟), where Ω is an open set of the form Ω=I× B_ℝ^n(0,r) satisfying (H_Ω) as in  Section 5.1 with I a g-open set containing the whole ℝ^+.

Therefore, under hypotheses (H_Ω), (H_𝐟), and (H_r), it follows from Theorem <ref> that, for every (t_0,𝐱_0)∈ℝ^+× B_ℝ^n(0,r), there exists a unique maximal solution 𝐱 = 𝐱(·,t_0,𝐱_0)∈𝒮(t_0,𝐱_0) ∩𝒜𝒞_g,loc(I_t_0,𝐱_0, ℝ^n) of (<ref>) defined on a maximal interval of existence that we denote I_t_0,𝐱_0 since it depends on the initial data (t_0,𝐱_0). As before, we denote ω(t_0,𝐱_0)=sup I_t_0,𝐱_0≤∞.


 §.§ Lyapunov stability notions

In this subsection, we present stability concepts within the framework of Stieltjes' differentiation. Through illustrative examples, we highlight the influence of the sets C_g and D_g on the change of stability properties.

  The trivial solution 𝐱=0 of the system (<ref>) is said to be
  
    
  ∙ stable if, for all ϵ >0 and t_0∈ℝ^+, there exists δ=δ(ϵ,t_0)∈(0,r) such that
    
    𝐱_0<δ   implies that   𝐱(t,t_0,𝐱_0)< ϵ for all   t∈ I_t_0,𝐱_0;

    
  ∙ uniformly stable if, for all ϵ >0, there exists δ=δ(ϵ) ∈(0,r)  such that for all t_0∈ℝ^+,
    
    𝐱_0<δ   implies that   𝐱(t,t_0,𝐱_0)< ϵ for all  t∈ I_t_0,𝐱_0.

    
    


  In the case where the trivial solution 𝐱=0 of the system (<ref>) is stable, for every t_0≥0 fixed, observe that for ϵ=r, there exists δ=δ(ϵ,t_0)∈ (0,r) such that for all 𝐱_0 ∈ B_ℝ^n(0,δ),
  
    𝐱(t,t_0,𝐱_0)< ϵ for all   t∈ I_t_0,𝐱_0.

  Using Theorem <ref> and (H_Ω) and (H_r), we deduce that I_t_0,𝐱_0=[t_0,∞). Furthermore, observe that if the stability of the trivial solution 𝐱=0 is uniform, then δ do not depend on t_0.


The following definition is a notion of asymptotic stability. This concerns the behavior of solutions as t→∞.

The trivial solution 𝐱=0 of the system (<ref>) is said to be


  ∙ asymptotically stable if it is stable and, for every t_0∈ℝ^+, there exists δ=δ(t_0)∈(0,r)  such that, for all 𝐱_0 ∈ B_ℝ^n(0,δ) and ϵ>0, there exists σ=σ(t_0,𝐱_0,ϵ) >0 such that
    
    𝐱(t,t_0,𝐱_0)< ϵ for all  t∈ [t_0+σ,∞) ∩ I_t_0,𝐱_0;


    
  ∙ uniformly asymptotically stable if it is uniformly stable and  there exists δ∈(0,r)  such that, for every ϵ>0, there exists σ=σ(ϵ)>0 such that, for all t_0 ∈ℝ^+ and 𝐱_0 ∈ B_ℝ^n(0,δ),
    
    𝐱(t,t_0,𝐱_0)<ϵ for all  t∈ [t_0+σ,∞) ∩ I_t_0,𝐱_0.

  





In the following example, we compare the stability properties of the trivial solution of a linear Stieltjes dynamical system to the ones in the classical case, observing the change of the stability properties depending on the sets C_g and D_g. The resolution of linear Stieltjes differential equations has been studied in the literature, see for instance <cit.>.

  Let us consider the linear Stieltjes dynamical system

    x_g'(t)    = c x(t)  for g-almost all t≥ t_0≥0,
    
       x(t_0)   =x_0∈ℝ,

 for c∈ℝ. In the classical case of derivation where g≡id_ℝ, for c>0, the equilibrium x=0 is not stable given that the solutions have the form x_0 e^c(t-t_0) and they are not bounded on [t_0,∞).  Thus, for every ϵ>0 and t_0≥0, there is no δ>0 such that
    
    |x_0|<δ   implies that    |x_0e^c(t-t_0)|< ϵ for all  t≥ t_0.

However, for c<0, the equilibrium x=0 is asymptotically stable since stability holds and given that the solutions x_0 e^c(t-t_0)→ 0 for all x_0 ∈ℝ. In the case where c=0, every constant z ∈ℝ is a uniformly stable equilibrium.

Now, let us reconsider the dynamical system (<ref>), where g:ℝ→ℝ is defined by g(t)=t for t≤ 1 and g(t)=1 for t≥ 1. Thus, the solutions of the problem (<ref>) have the form x_0 e^c(g(t)-g(t_0)), t≥ t_0  for every x_0∈ℝ and t_0≥ 0. Now, we show that the stability properties of the trivial solution x=0 of the system (<ref>) differ from the classical case, for c>0 and c<0.  For c>0, we deduce the stability of the trivial solution x=0. Indeed, for all ϵ>0 and t_0≥0, there exists δ=ϵ/e^c(g(1)-g(t_0)) such that

    |x_0|<δ   implies that    |x_0 e^c(g(t)-g(t_0))|< ϵ for all  t≥ t_0.

Whereas, for c<0 the equilibrium x=0 is merely uniformly stable compared to the classical case, since for all ϵ>0, there exists δ=ϵ>0 such that

    |x_0|<δ   implies that    |x_0 e^c(g(t)-g(t_0))|< ϵ for all  t≥ t_0.

Observe that asymptotic stability does not hold  for any c∈ℝ, since

    x_0 e^c(g(t)-g(t_0))→ x_0 e^c(g(1)-g(t_0))↛ 0,

as t→∞ for all t_0≥0 and x_0 ∈ℝ^*.


Next, we change slightly the dynamical system (<ref>), to incorporate jumps. Let us consider the derivator g_1(t)=t+∑_n∈ℕχ_[n,∞)(t) for all t∈ℝ, and  reconsider the dynamical system with g_1 instead:
 
    x_g_1'(t)    = 
                    c x(t)     t≥ t_0≥0, with  t∉ D_g_1=ℕ, 
    ν x(t),    t≥ t_0≥0, with  t∈ D_g_1,

where c,ν∈ℝ with ν∈(-1,∞)∖{0}. Again the solution of (<ref>) satisfying x(t_0)=x_0 has the form  x(t)=x_0 e_c_*(t,t_0)=x_0e^∫_[t_0,t)c_*(s)  dμ_g_1(s), where

    c_*(t)=
                                        c      t∈[t_0,∞)∖ D_g_1, 
    log(1+ν)      t∈[t_0,∞)∩ D_g_1.

In Figure <ref>, we can observe different patterns depending on the values of c and ν. This implies that the presence of discontinuities can both destabilize and restore the stability properties of a dynamical system.




 §.§ Stability results based on Lyapunov's function

In the classical case where g≡id_ℝ, Lyapunov's method is used to study the behavior of a trajectory of the system in a neighborhood of the trivial solution 𝐱=0, by means of a function V depending on time and state; known as a Lyapunov  function. This function can be understood as an energy representation of the system (<ref>), since in numerous applications, the function considered is the total energy of the system (<ref>) through time, see for instance <cit.> for an example of an energy-based Lyapunov function for physical systems. In our context, the derivator g takes into account the relevance of each moment during the process by means of the changes of the slopes of g accordingly. Put differently, g amplifies an alternative measurement for time, which may differ from the linear timeline typically used in the classical case where g≡id_ℝ, see for instance the works <cit.> where g represents the life cycle of some populations, also we refer to <cit.> for more applications. Nevertheless, in the context of Stieltjes differentiation,   we still can rely on Lyapunov's function, particularly,  based on the g-derivative of its composition with maximal solutions of the system (<ref>) under consideration. This g-derivative of the composition will then permit a better  understanding of how the energy of the system (<ref>) changes, but with respect to this new observed time described by g. More precisely, it will describe how the energy of the system varies in response to the variation of this new "curved" scale of time.




Based on the definition of the Stieltjes derivative in Definition <ref>, we define the partial Stieltjes derivative as follows.

Given a function V:ℝ^+× B_ℝ^n(0,r_0)→ℝ and t, x_1,…,x_n its arguments. The partial g-derivative of V with respect to  the first argument at a point (t,𝐱)∈(ℝ^+∖ C_g)×  B_ℝ^n(0,r_0) is defined as:

    ∂ V/∂_g t(t,𝐱)=
    lim_s→ tV(s,𝐱)-V(t,𝐱)/g(s)-g(t),    if  t ∈ℝ^+ ∖ D_g,  
    V(t^+,𝐱)-V(t,𝐱)/g(t^+)-g(t),    if  t ∈ℝ^+ ∩ D_g,

provided that the limits exist.





Combining Proposition <ref> and Definition <ref>, we obtain the following technical proposition. The proof involves a formula related to the g-derivative of the composition involving a function with two variables. Formulae of this fashion were stated without proof in <cit.> for t∉ D_g. In the next proposition, we derive formulae in the case where D_g and N_g are discrete.

Let g: ℝ→ℝ be a left-continuous and nondecreasing function such that D_g and N_g are discrete. Given a function V:ℝ^+× B_ℝ^n(0,r_0) →ℝ satisfying the following assumptions:

  
  * V(·,𝐮) is g-differentiable on ℝ^+∖ (D_g ∪ C_g)  for all 𝐮∈ B_ℝ^n(0,r_0);
  
  * V(t,·) ∈ C^1(B_ℝ^n(0,r_0),ℝ) for all t≥ 0;
  
  * ∂ V/∂_g t(·,𝐮) is continuous on ℝ^+∖ (D_g ∪ C_g) for all 𝐮∈ B_ℝ^n(0,r_0);
  
  * for (t_0,𝐱_0)∈ℝ^+×  B_ℝ^n(0,r), V(·,𝐱(·))∈𝒜𝒞_g,loc(I_𝐱,ℝ) for every solution 𝐱 :I_𝐱→ B_ℝ^n(0,r_0) of the system (<ref>).

Then, for all [a,b]⊂ I_𝐱, we obtain for g-almost all t∈ [a,b]∖ (D_g ∪ C_g) that

    V_g'(t,𝐱(t))=
     ∂ V/∂_g t(t,𝐱(t))+ ∑_i=1^n∂ V/∂ x_i(t,𝐱(t)) f_i(t,𝐱(t)).

Moreover, if t∈ [a,b)∩ D_g, then
 
    V_g'(t,𝐱(t))=V(t^+,𝐱(t)+ μ_g({t})𝐟(t,𝐱(t)))-V(t,𝐱(t))/g(t^+)-g(t).



For (t_0,𝐱_0)∈ℝ^+ × B_ℝ^n(0,r), let us consider  a solution 𝐱 :I_𝐱→ B_ℝ^n(0,r_0) of the system (<ref>). Thus, the composition V(·,𝐱(·))∈𝒜𝒞_g(I_𝐱,ℝ). Let [a,b]⊂ I_𝐱. For g-almost every t ∈ [a,b] ∖ (D_g ∪ C_g):

    V_g'(t,𝐱(t))   = lim_s→ tV(s,𝐱(s))-V(t,𝐱(t))/g(s)-g(t)
        =  lim_s→ tV(s,𝐱(s))-V(t,𝐱(s))/g(s)-g(t)+V(t,𝐱(s))-V(t,𝐱(t))/g(s)-g(t)
        =  lim_s→ tV(s,𝐱(s))-V(t,𝐱(s))/g(s)-g(t)
         +∑_i=1^n( V(t,(x_1(t),…,x_i-1(t),x_i(s),…,x_n(s)))/g(s)-g(t)
             -V(t,(x_1(t),…,x_i(t),x_i+1(s),…,x_n(s)))/g(s)-g(t)).

For s sufficiently close to t, and since D_g and N_g are discrete, g is continuous and increasing on the interval with endpoint points s and t. By applying Corollary <ref> to the function V(·,𝐱(s)), we obtain that there exists c between s and t such that

    V(s,𝐱(s))-V(t,𝐱(s))/g(s)-g(t)= ∂ V/∂_g t(c,𝐱(s)).

As s→ t, c→ t, and using Condition (3) we obtain

    V(s,𝐱(s))-V(t,𝐱(s))/g(s)-g(t)= ∂ V/∂_g t(c,𝐱(s))→∂ V/∂_g t(t,𝐱(t)).

Therefore,

    V_g'(t,𝐱(t))     =  ∂ V/∂_g t(t,𝐱(t))+ ∑_i=1^n∂ V/∂ x_i(t,𝐱(t))(x_i)_g'(t)
       =  ∂ V/∂_g t(t,𝐱(t))+ ∑_i=1^n∂ V/∂ x_i(t,𝐱(t))f_i(t,𝐱(t)).

For t ∈[a,b)∩ D_g, we obtain immediately that

    V_g'(t,𝐱(t))   =V(t^+,𝐱(t^+))-V(t,𝐱(t))/g(t^+)-g(t)
       =V(t^+,𝐱(t)+μ_g({t})𝐟(t,𝐱(t)))-V(t,𝐱(t))/g(t^+)-g(t).




In order to establish sufficient conditions for different types of stability of the trivial solution 𝐱=0 of the system (<ref>) as defined in Subsection 4.1, we introduce specific sets of functions.

A function V:ℝ^+× B_ℝ^n(0,r_0) →ℝ is said to belong  to class 𝒱^g_1 if it satisfies the following conditions:


  * V(t,·) is continuous for all t≥ 0;

  * V(·,𝐱(·))∈𝒜𝒞_g,loc(I_t_0,𝐱_0,ℝ) for every function
𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) of 𝒜𝒞_g,loc(I_t_0,𝐱_0,ℝ^n) maximal solution of the system (<ref>);

  * V(t,0)=0 for all t≥0.



 A  function φ:ℝ^+→ℝ^+ belongs to the class 𝒦 if it fulfills the following assumptions:
  
   
  * φ is continuous;
   
  * φ(0)=0;
   
  * φ is increasing.
  

Now, we state the first stability result.

Assume that Conditions (H_Ω), (H_𝐟) and (H_r) hold.
 If there  exist functions V ∈𝒱_1^g and a ∈𝒦 such that
 

  (a) a(𝐮)≤ V(t,𝐮), for all (t,𝐮)∈ℝ^+× B_ℝ^n(0,r_0);

  (b) for every (t_0,𝐱_0) ∈ℝ^+× B_ℝ^n(0,r), if 𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) is a maximal solution of the system (<ref>), then V_g'(t,𝐱(t)) ≤ 0 for g-almost all t∈ I_t_0,𝐱_0.

Then, the trivial solution of the system (<ref>) is
 

  (i)stable.

  (ii)uniformly stable if there exists b ∈𝒦 such that
    
    V(t,𝐮)≤ b(𝐮),    for all  (t,𝐮)∈ℝ^+× B_ℝ^n(0,r_0).

    



(i)  Since V∈𝒱_1^g, then, for all ϵ>0 and t_0∈ℝ^+, there exists δ=δ(ϵ, t_0) ∈(0,r) such that
      
    sup_𝐮<δV(t_0,𝐮)<a(ϵ).

Let 𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) be a maximal solution of the system (<ref>) and 𝐱_0<δ. It follows from Conditions (a) and (b)  that

    a(𝐱(t))≤ V(t,𝐱(t)) ≤ V(t_0,𝐱_0) < a(ϵ).

Thus, for all t∈ I_t_0,𝐱_0, we have that

    𝐱(t)=𝐱(t,t_0,𝐱_0) < ϵ.

 Therefore, the trivial solution 𝐱=0 is stable.

(ii)  Arguing as in (i), we can choose a δ=δ(ϵ) ∈(0,r) independent of t_0 such that

    b(δ)<a(ϵ).

Thus, using (<ref>), we obtain for all t∈ I_t_0,𝐱_0,

    a(𝐱(t))≤ V(t,𝐱(t)) ≤ V(t_0,𝐱_0)≤ b(𝐱_0)< b(δ)<a(ϵ).

This yields that the trivial solution 𝐱=0 is uniformly stable.

In the next theorem, we impose additional assumptions which will permit to insure the asymptotical stability of the trivial solution 𝐱=0 to the system (<ref>).


Assume that Conditions (H_Ω), (H_𝐟) and (H_r) hold. Let V ∈𝒱_1^g, a, b ∈𝒦, ϕ : ℝ^+ →ℝ^+ continuous, and  a g-measurable function w:ℝ^+→ℝ^+ be such that
 

  (a) a(𝐮) ≤ V(t,𝐮) for every (t,𝐮)∈ℝ^+× B_ℝ^n(0,r_0);

  (b) ϕ(s)=0 if and only if s=0;

  (c) for every (t_0,𝐱_0) ∈ℝ^+× B_ℝ^n(0,r), the maximal solution 𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) of the system (<ref>) satisfies
 
    V_g'(t,𝐱(t)) ≤ -w(t)ϕ(𝐱(t))   for g-almost all  t∈ I_t_0,𝐱_0;

 
  (d) inf_t_0 ∈ℝ^+lim_t→ +∞∫_[t_0,t_0+t) w(s) dμ_g(s) =∞.
 
If the trivial solution 𝐱=0 of the system (<ref>) is uniformly stable, then 𝐱=0 is asymptotically stable.



(i) The stability of the trivial solution 𝐱=0 holds from uniform stability. Let us choose  δ_0 ∈(0, r) associated to  an ϵ_0 ≤ r given by the uniform stability.  Now, for a fixed t_0 ≥ 0, let ϵ>0. Again, by the uniform stability, there exists δ∈ (0,δ_0) such that, for all t̂∈ℝ^+ and every 𝐱̂_0 satisfying 𝐱̂_0 < δ, one has

    𝐱̂(t,t̂,𝐱̂_0)< ϵ for all  t∈ [t̂,∞) ∩ I_t̂,𝐱̂_0.

We denote

    M = inf_s ∈ [δ,r_0)|ϕ(s)|.

By Condition (b), observe that M>0.

Let 𝐱_0 ∈ B_ℝ^n(0,δ_0). Since

    lim_t→ +∞∫_[t_0,t_0+t) w(s) dμ_g(s) =∞,

we can choose σ > 0 such that

    ∫_[t_0,t_0+σ) w(s) dμ_g(s) > V(t_0,𝐱_0)/M.

Let 𝐱 : I_t_0,𝐱_0→ B_ℝ^n(0,r_0) be a maximal solution of (<ref>). Using Remark <ref>, notice that I_t_0,𝐱_0=[t_0,∞). Now, if there exists t̂∈ [t_0,t_0+σ] such that 𝐱(t̂) < δ, then, by the uniform stability
 
    𝐱̂(t) < ϵ  for all  t ∈ [t̂,∞)∩ I_t̂,𝐱(t̂),

 where 𝐱̂:  I_t̂,𝐱(t̂)→ B_ℝ^n(0,r_0) is the maximal solution of (<ref>) satisfying the initial condition 𝐱̂(t̂) = 𝐱(t̂). By the uniqueness of the maximal solution, one has
 
    ω(t_0,𝐱_0) = ω(t̂,𝐱(t̂))=∞  and  𝐱(t) = 𝐱̂(t)  for all  t∈ [t̂,∞).

 Hence,
 
    𝐱(t) < ϵ  for all  t ∈ [t_0+σ,∞)∩ I_t_0,𝐱_0.

 On the other hand, if 𝐱(t)≥δ for all t ∈ [t_0,t_0+σ], then using Condition (a), Theorem <ref>, and (<ref>), we obtain

    a(𝐱(t_0+σ))    ≤ V(t_0+σ,𝐱(t_0+σ))
     
       =V(t_0,𝐱(t_0,t_0,𝐱_0))+∫_[t_0,t_0+σ)V_g'(s,𝐱(s)) dμ_g(s)
     
       ≤ V(t_0,𝐱_0) -∫_[t_0,t_0+σ) w(s) ϕ(𝐱(s)) dμ_g(s)
     
       ≤ V(t_0,𝐱_0) - M∫_[t_0,t_0+σ) w(s) dμ_g(s)
     
       < 0.

 This is a contradiction. Therefore, 𝐱=0 is asymptotically stable.


In the example below, we provide an application of Theorem <ref>.

  Let us consider the Stieltjes dynamical system

    x_g'(t) = f(t,x(t))  for g-almost all t≥ t_0≥0,     
       x(t_0)=x_0∈ℝ,

with g:ℝ→ℝ  defined by g(t)=t for all t≤ 1, and g(t)=t+1 for t> 1, and where f:ℝ^+×ℝ→ℝ is a function defined by

    f(t,x)=
             -xt/1+t^2    t∈ℝ^+∖ D_g, 
    ν x     t∈ℝ^+∩ D_g,

for some ν∈ℝ∖{-1,0}. The function f satisfies conditions of Theorem <ref>, thus the problem (<ref>) has a maximal solution x: [t_0,+∞)→ℝ for every (t_0,x_0) ∈ℝ^+×ℝ. Observe that x=0 is an equilibrium of the dynamical system (<ref>).


Let us define the function V:ℝ^+×ℝ→ℝ for every (t,x) ∈ℝ^+×ℝ by

    V(t,x)=
    x^2      t∈ [0,1],
    x^2/(1+ν)^2    t>1.

Clearly V∈𝒱_g^1, and a(|u|)≤ V(t,u)≤ b(|u|) for all (t,u)∈ℝ^+×ℝ, where a,b∈𝒦 are given by a(s)=min{s^2,s^2/(1+ν)^2} and b(s)=max{s^2,s^2/(1+ν)^2} for all s∈ℝ^+. In addition, for all (t,x)∈ℝ^+×ℝ:

    ∂ V/∂_g t(t,x)   =
                                           0     t∈ (0,1)∪(1,∞),
    x^2/(1+ν)^2-x^2/g(1^+)-g(1)    t=1,
    
       =
                                           0     t∈ (0,1)∪(1,∞),
    1-(1+ν)^2/(1+ν)^2x^2     t=1.

Thus, by means of Proposition <ref>, for t∈ [t_0,∞)∖ D_g, we obtain

    V_g'(t,x(t))   = ∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) f(t,x(t))
       = 
     -2t/1+t^2 x(t)^2     t∈ [t_0,∞)∩ [0,1),
    
     -2t/(1+t^2)(1+ν)^2x(t)^2     t∈ [t_0,∞)∩ (1,∞).

For t∈ [t_0,∞)∩ D_g, if t_0 ≤ 1, then t=1 and we have that

    V_g'(1,x(1))   =V(1^+,x(1^+))-V(1,x(1))/g(1^+)-g(1)
       =V(1^+,x(1)+ μ_g({1})f(1,x(1)))-V(1,x(1))/g(1^+)-g(1)
       =(1+ν)^2x(1)^2/(1+ν)^2-x(1)^2/g(1^+)-g(1)
       = 0.

 This implies that

    V_g'(t,x(t))≤-ω(t)ϕ(|x(t)|),  for every  t≥ t_0,

with w:ℝ^+ →ℝ^+, defined by

    w(t)=2t/1+t^2    t∈[0,1), 
    
           0     t=1,
    2t/(1+t^2)(1+ν)^2    t∈(1,∞),

 and ϕ∈𝒦, defined by ϕ(y)=y^2 for all y∈ℝ^+. Moreover, for every t_0∈ℝ^+  and t>1, we have

    ∫_[t_0,t_0+t) w(s) dμ_g(s)      = ∫_[t_0,t_0+t)∩([0,1)∪{1}∪ (1,∞)) w(s) dμ_g(s)
       ≥1/(1+ν)^2∫_[t_0,t_0+t)∩ (1,∞)2s/1+s^2 ds 
        =  1/(1+ν)^2(log(1+(t_0+t)^2)-sup{log(2),log(1+t_0^2)}).

As, inf_t_0 ∈ℝ^+lim_t→ +∞log(1+(t_0+t)^2)-sup{log(2),log(1+t_0^2)} =+∞, we obtain that

    inf_t_0 ∈ℝ^+lim_t→ +∞∫_[t_0,t_0+t) w(s) dμ_g(s)=+∞.

Thus, Condition (d) holds. Therefore, by means of Theorem <ref>, we deduce that x=0 is an asymptotically stable equilibrium. Figure <ref> illustrates the asymptotic behavior of solutions of the dynamical system (<ref>).



 In the following example, we reconsider the system (<ref>) in the case where the set D_g is infinite.
 
  Let us consider the dynamical system

    x_g'(t) = f(t,x(t))  for g-almost all t≥ t_0≥0,     
       x(t_0)=x_0∈ℝ,

where g:ℝ→ℝ is a derivator such that D_g={t_k}_k∈ℕ⊂(0,+∞), defined by

    g(t)=t+∑_k∈ℕχ_[t_k,+∞)(t)  for all t∈ℝ,

 and f:ℝ^+×ℝ→ℝ is a function defined by

    f(t,x)=
             -xt/1+t^2    t∈ℝ^+∖ D_g, 
    ν x     t∈ℝ^+∩ D_g,

for some ν∈[-2,-1). The function f satisfies conditions of Theorem <ref>, thus the problem (<ref>) has a maximal solution x: [t_0,+∞)→ℝ for every (t_0,x_0) ∈ℝ^+×ℝ. Observe that x=0 is an equilibrium of the dynamical system (<ref>).

Let us define the function V:ℝ^+×ℝ→ℝ for all (t,x) ∈ℝ^+×ℝ by

    V(t,x)=x^2

Clearly V∈𝒱_g^1, and a(|u|)≤ V(t,u)≤ b(|u|) for all (t,u)∈ℝ^+×ℝ, where a∈𝒦 is given by a(s)=s^2 and b(s)=s^2/(1+ν)^2 for all s∈ℝ^+. In addition, for all t∈ [t_0,∞)

    ∂ V/∂_g t(t,x)=0.

Thus, by means of Proposition <ref>, for g-almost every t∈ [t_0,∞)∖ D_g, we obtain

    V_g'(t,x(t))   = ∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) f(t,x(t))
       =  -2t/1+t^2 x(t)^2.

For t_k∈ [t_0,∞)∩ D_g, we have that

    V_g'(t_k,x(t_k))   =V(t_k^+,x(t_k^+))-V(t_k,x(t_k))/g(t_k^+)-g(t_k)
       =V(t_k^+,x(t_k)+ μ_g({t_k})f(t_k,x(t_k)))-V(t_k,x(t_k))/g(t_k^+)-g(t_k)
       = ((1+ν)x(t_k))^2-x(t_k)^2/g(t_k^+)-g(t_k)
       =ν(2+ν)x(t_k)^2.

 This implies that

    V_g'(t,x(t))≤-ω(t)ϕ(|x(t)|),  for g-almost every  t≥ t_0,

with w:ℝ^+ →ℝ^+, defined by

    w(t)=2t/1+t^2    t∈[0,t_1)∪(t_k,t_k+1),  k∈ℕ,
    
           -ν(2+ν)     t∈{t_k}_k∈ℕ,

and ϕ∈𝒦 defined by ϕ(y)=y^2 for all y∈ℝ^+. For t_0,t ∈ℝ^+, observe that w satisfies:

    ∫_[t_0,t_0+t)w(s) dμ_g(s)   =  ∑_s∈ [t_0,t_0+t)∩ D_g -ν(2+ν) μ_g({s}) + ∫_t_0^t_0+t2s/1+s^2 ds
       = ∑_s∈ [t_0,t_0+t)∩ D_g -ν(2+ν) μ_g({s}) + log(1+(t_0+t)^2/1+t_0^2).

Thus,

    inf_t_0 ∈ℝ^+lim_t→ +∞∫_[t_0,t_0+t)w(s) dμ_g(s)=+∞.

Consequently, Condition (d) holds. Therefore, by means of Theorem <ref>, we deduce that x=0 is an asymptotically stable equilibrium. Figure <ref> illustrates the asymptotic behavior of solutions of the dynamical system (<ref>).








Assume that Conditions (H_Ω), (H_𝐟) and (H_r) hold. If there  exist V ∈𝒱_1^g, a, b ∈𝒦, ϕ : ℝ^+ →ℝ^+ continuous, and  a g-measurable function w:ℝ^+→ℝ^+ such that


  (a) a(𝐮) ≤ V(t,𝐮) ≤ b(𝐮) for every (t,𝐮)∈ℝ^+× B_ℝ^n(0,r_0);

  (b) ϕ(s)=0 if and only if s=0;

  (c) for every (t_0,𝐱_0) ∈ℝ^+× B_ℝ^n(0,r), the maximal solution 𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) of the system (<ref>) satisfies
 
    V_g'(t,𝐱(t)) ≤ -w(t)ϕ(𝐱(t))   for g-almost all  t∈ I_t_0,𝐱_0.

 
  (d) lim_t→ +∞inf_t_0 ∈ℝ^+∫_[t_0,t_0+t) w(s) dμ_g(s) =+∞.
 
Then the trivial solution 𝐱=0 of the system (<ref>) is uniformly asymptotically stable.


By (ii) of Theorem <ref>, the trivial solution 𝐱=0 is uniformly stable. Thus, let us choose  δ_0 ∈(0, r) associated to  an ϵ_0 ≤ r. Let ϵ > 0. Again, by uniform stability, there exists   δ∈(0,r) such that, for all t̂∈ℝ^+ and every 𝐱̂_0 such that 𝐱̂_0 < δ, one has
 
    𝐱̂(t,t̂,𝐱̂_0)< ϵ for all  t∈ [t̂,∞) ∩ I_t̂,𝐱̂_0.

 Let M be as defined in (<ref>).  Since
 
    lim_t→ +∞inf_t_0 ∈ℝ^+∫_[t_0,t_0+t) w(s) dμ_g(s) =+∞,

 we can choose σ > 0 such that
 
    ∫_[t_0,t_0+σ) w(s) dμ_g(s) > b(δ_0)/M  for all  t_0 ∈ℝ^+.

 Let (t_0,𝐱_0) ∈ℝ^+× B_ℝ^n(0,δ_0) and 𝐱 : I_t_0,𝐱_0→ B_ℝ^n(0,r_0) a maximal solution of (<ref>).

 If there exists t̂∈ [t_0,t_0+σ] ∩ I_t_0,𝐱_0 such that 𝐱(t̂) < δ, then
 
    𝐱̂(t) < ϵ  for all  t ∈ [t̂,∞)∩ I_t̂,𝐱(t̂),

 where 𝐱̂:  I_t̂,𝐱(t̂)→ B_ℝ^n(0,r_0) is the maximal solution of (<ref>) satisfying the initial condition 𝐱̂(t̂) = 𝐱(t̂). By the uniqueness of the maximal solution, one has
 
    ω(t_0,𝐱_0) = ω(t̂,𝐱(t̂))   and  𝐱(t) = 𝐱̂(t)  for all  t∈ [t̂,∞)∩ I_t_0,𝐱_0.

 Hence,
 
    𝐱(t) < ϵ  for all  t ∈ [t_0+σ,∞)∩ I_t_0,𝐱_0.

 On the other hand, if 𝐱(t)≥δ for all t ∈ [t_0,t_0+σ], then using Condition (a), Theorem <ref>, and (<ref>), we obtain

    a(𝐱(t_0+σ))    ≤ V(t_0+σ,𝐱(t_0+σ))
     
       =V(t_0,𝐱(t_0,t_0,𝐱_0))+∫_[t_0,t_0+σ)V_g'(s,𝐱(s)) dμ_g(s)
     
       ≤ V(t_0,𝐱_0) -∫_[t_0,t_0+σ) w(s) ϕ(𝐱(s)) dμ_g(s)
       
       ≤ V(t_0,𝐱_0) - M∫_[t_0,t_0+σ) w(s) dμ_g(s)
       
        <b(𝐱_0) -b(𝐱_0)
       = 0.

 This is a contradiction. Hence, we conclude that 𝐱=0 is uniformly asymptotically stable.


 In the next example, we provide an application of Theorem <ref> for a system subject to impulses where the trivial solution 𝐱=0 is uniformly asymptotically stable.

 Let us consider the dynamical system

    x_g'(t) = f(t,x(t))  for g-almost all t≥ t_0≥0,     
       x(t_0)=x_0∈ℝ,

where g:ℝ→ℝ is a derivator such that D_g={t_k}_k∈ℕ⊂(0,+∞), defined by

    g(t)=t+∑_k∈ℕχ_[t_k,+∞)(t)  for all t∈ℝ,

 and f:ℝ^+×ℝ→ℝ is a function defined by

    f(t,x)=
             -t arctan(x)     t∈ℝ^+∖ D_g, 
    ν_k x     t=t_k,  k∈ℕ

where  {ν_k}_k⊂ℝ_+^* is a sequence satisfying

    lim_k→∞1/∏_i=1^k(1+ν_i)^2=a_0>0.

 The map f satisfies conditions of Theorem <ref>, thus the problem (<ref>) has a maximal solution x: [t_0,+∞)→ℝ for every (t_0,x_0) ∈ℝ^+×ℝ. Observe that x=0 is an equilibrium of the Stieltjes dynamical system (<ref>).

Let us define the function V:ℝ^+×ℝ→ℝ for all (t,x) ∈ℝ^+×ℝ by

    V(t,x)=
    x^2      t∈ [0,t_1],
    x^2/∏_i=1^k(1+ν_i)^2    t∈ (t_k,t_k+1],  k∈ℕ.

Clearly V∈𝒱_g^1, and a(|x|)≤ V(t,x)≤ b(|x|) for all (t,x)∈ℝ^+×ℝ, where a,b∈𝒦 are functions defined by a(s)=a_0s^2 and b(s)=s^2 for all s∈ℝ^+. In addition, for all (t,x)∈ℝ^+×ℝ:

    ∂ V/∂_g t(t,x)   =
                                           0     t∈ [0,t_1]∪ (t_k,t_k+1),  k∈ℕ,
    x^2/∏_i=1^k(1+ν_i)^2-x^2/∏_i=1^k-1(1+ν_i)^2/g(t_k^+)-g(t_k)   t=t_k,  k∈ℕ,
                                         
       =
    0      t∈ [0,t_1)∪ (t_k,t_k+1),  k∈ℕ,
    1-(1+ν_k)^2/∏_i=1^k(1+ν_i)^2x^2
        t=t_k,  k∈ℕ.

For g-almost every t∈ [t_0,∞)∖ D_g, we have that t∈[0,t_1) or there exists k∈ℕ such that t∈(t_k,t_k+1). Thus, by means of Proposition <ref>, we obtain if t∈[0,t_1):

    V_g'(t,x(t))   = ∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) f(t,x(t))
       = -2x(t)tarctan(x(t)) 
       ≤     -2t x(t)^2/1+x(t)^2,

where the last inequality follows from the Mean Value Theorem. While if there exists k∈ℕ such that t∈(t_k,t_k+1), then

    V_g'(t,x(t))   = ∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) f(t,x(t))
       =-2x(t)/∏_i=1^k(1+ν_i)^2tarctan(x(t))
       ≤ -2t/∏_i=1^k(1+ν_i)^2x(t)^2/1+x(t)^2.

For t_k∈ [t_0,∞)∩ D_g, we have that

    V_g'(t_k,x(t_k))   =V(t_k^+,x(t_k^+))-V(t_k,x(t_k))/g(t_k^+)-g(t_k)
       =V(t_k^+,x(t_k)+ μ_g({t_k})f(t_k,x(t_k)))-V(t_k,x(t_k))/g(t_k^+)-g(t_k)
       =(1+ν_k)^2x(t_k)^2/∏_i=1^k(1+ν_i)^2- x(t_k)^2/∏_i=1^k-1(1+ν_i)^2/g(t_k^+)-g(t_k)
       = 0 .

 Therefore, we conclude that

    V_g'(t,x(t))≤-ω(t)ϕ(|x(t)|),  forg-almost all  t≥ t_0,

where w:ℝ^+ →ℝ^+ is the function defined for every t∈ℝ^+ by

    w(t)=
          2t     t∈[0,t_1), 
    
           0    t=t_k,  k∈ℕ,
    2t/∏_i=1^k(1+ν_i)^2    t∈(t_k,t_k+1),  k∈ℕ,

and ϕ∈𝒦 the function given by ϕ(y)=y^2/1+y^2 for all y∈ℝ^+.

Observe that the function w satisfies

    lim_t→ +∞inf_t_0 ∈ℝ^+∫_[t_0,t_0+t)w(s) dμ_g(s)   ≥lim_t→ +∞inf_t_0 ∈ℝ^+∫_t_0^t_0+ta_02s ds
       =a_0lim_t→ +∞inf_t_0 ∈ℝ^+ 2tt_0+t^2
       =+∞.

By means of Theorem <ref>, we deduce that x=0 is uniformly asymptotically stable equilibrium. Figure <ref> illustrates the asymptotic behavior of solutions of the dynamical system (<ref>) with D_g=ℕ.



  Observe in Example <ref> that the function w satisfies

    lim_t→ +∞inf_t_0 ∈ℝ^+∫_[t_0,t_0+t) w(s) dμ_g(s) =0 ≠∞.

Thus, Condition (d) of Theorem <ref> does not hold. Consequently, uniform asymptotic stability cannot be deduced.


In the classical case where g≡id_ℝ, corollary results <cit.> and <cit.> are well-known when Conditions (b) of Theorem <ref> is replaced by V_g'(t,𝐱(t)) being negative definite along each maximal solution 𝐱 for every t≥ t_0. However, to present an analogous statement, we require an additional assumption to avoid the case when  lim_t→∞ g(t)=l<∞, and in particular,  when there exists T≥0 such that (T,∞)⊂ C_g, Example <ref> provides an interesting illustration of attractivity lack for asymptotic stability.

Assume that  lim_t→∞ g(t)=∞.  If there  exist V:ℝ^+× B_ℝ^n(0,r_0) →ℝ; V∈𝒱_1^g and a,b,ϕ∈𝒦 such that
 

  (a) a(𝐮) ≤ V(t,𝐮)≤ b(𝐮), for every (t,𝐮)∈ℝ^+× B_ℝ^n(0,r_0);

  (b)  for every (t_0,𝐱_0) ∈ℝ^+× B_ℝ^n(0,r), the maximal solution 𝐱:I_t_0,𝐱_0→ B_ℝ^n(0,r_0) of the system (<ref>) satisfies

    V_g'(t,𝐱(t))≤ -ϕ(𝐱(t)),  for g-almost all  t≥ t_0.

     
then, the trivial solution 𝐱=0 of the system (<ref>) is  asymptotically stable. Furthermore, if lim_t→ +∞inf_t_0 ∈ℝ^+ g(t+t_0)-g(t_0)=∞, then, the trivial solution 𝐱=0 of the system (<ref>) is uniformly asymptotically stable.

  Observe that Conditions of Theorem <ref> hold for ω≡ 1 as

    inf_t_0 ∈ℝ^+lim_t→ +∞∫_[t_0,t+t_0)ω(s) dμ_g(s) = inf_t_0 ∈ℝ^+lim_t→ +∞g(t+t_0)-g(t_0)=+∞.

Hence, the trivial solution is 𝐱=0 of the system (<ref>) is  asymptotically stable. Moreover, if lim_t→ +∞inf_t_0 ∈ℝ^+ g(t+t_0)-g(t_0)=∞, then, Theorem <ref> ensures that 𝐱=0  is uniformly asymptotically stable.



§ APPLICATIONS TO DYNAMICS OF POPULATION



 §.§ Stable equilibrium of a population subject to train vibrations

In this subsection, by means of a system of Stieltjes differential equations, we study the long-term impact of high-speed train vibrations and  noise pollution on a population of animals living near railways. Depending on the species and their sensitivity to vibrations, various implications can be observed, we cite for instance:

  
  ∙ Hearing damage resulting in from the significant noise and vibrations that can potentially harm animals with sensitive hearing such as certain small mammals, birds, and bats which rely heavily on their hearing for communication, navigation, detection of predators, and finding food, thus, prolonged exposure to train vibrations may lead to hearing impairment or damage, disrupting their normal behaviors and increasing their death rate.
  
  ∙ Increased stress levels since some animals may be startled by the vibrations. This can affect their feeding patterns, reduce their reproduction rate, or lead to emigration resulting in a loss of suitable habitat and altering the composition and diversity of the local ecosystem
  
  ∙ Ecological interactions disruption which can affect pollination  for instance if the vibrations deter insects that are important pollinators, disturb ground-dwelling organisms (insects, reptiles, and small mammals…) which can implicitly impact other species that rely on them as a food source.

To these aims, an Allee effect can be observed in this regard, especially when the survival of the population depends on a minimum threshold size M>0. In the follows, we denote x(t) as the number of individuals of a population living in a region near a railway with a carrying capacity K>M.
Let us assume that a certain number m>0 of trains pass through the area every day. We refer to {τ_i}_i=1^i=m as the moments when trains pass in a single day.
Once a train pass by the area, its impact is significant for a proportion of individuals living near the railway. They may experience vibrations or direct injuries. In the following analysis, we use a Stieltjes differential equation to model the dynamics of this population affected by train vibrations, and we study the asymptotic behavior of its solutions. In doing so,  we require a derivator g:ℝ→ℝ presenting discontinuities for t∈{τ_i}_i=1^i=m+24ℤ such that μ_g({t})  quantifies the rate at which the risk of damage varies. Depending on the specific τ_i, this rate can either increase or decrease, reflecting the varying impact of vibrations during daylight and nighttime hours. For simplicity,  we can take for instance:

    g(t)=t+∑_k∈ℕ∑_i=1^mχ_[τ_i+24k,∞)(t),   for all  t∈ℝ,

with μ_g({·})≡1 on D_g= {τ_i}_i=1^i=m+24ℤ.

In the sequel, we suggest to analyse the asymptotic behaviour of the dynamics of this population, through the study of asymptotic stability of the zero equilibrium of the Stieltjes dynamical system:

    x_g'(t) =f(t,x(t)), for g-almost every t≥ t_0≥0,     x(t_0) =x_0,

where f: ℝ^+×ℝ→ℝ is defined by

    f(t,x) =ρ x(1-x/K)(x/M-1),     t∉{τ_i+24k}_i=1^m,  k=0,1,2…
    
                      -dx,     t∈{τ_i+24k}_i=1^m,  k=0,1,2…

The parameters of the model can be understood as:


  K>M the carrying capacity of the environment.

  ρ>0 intrinsic rate of reproduction of the population.

  d∈(0,1) Constant related to the impact induced by trains, either a migration rate immediately following the passage of trains or a mortality rate for certain populations that live in close proximity to the railway.


To simplify the analysis, we make the assumption that a train passes every hour over a 24-hour period. Thus, g(t)= t+∑_i∈ℤχ_[i+1,∞)(t) for all t∈ℝ, D_g=ℤ  and μ_g({t})=1 for all t∈ D_g.

The dynamics present several equilibria. However, we will focus on the zero equilibrium, to study its local asymptotic stability within a region (-r_0,r_0); r_0>0, that will be determined to enhance the impact of environmental factors threatening this population.

Since the trivial solution x=0 is an equilibrium of the dynamical system (<ref>),  let us consider r_0≤ M. For all (t_0,x_0) ∈(ℝ^+∩ D_g)× (-r_0,r_0), if x is a solution of (<ref>), then x(t_0^+)=x(t_0)+μ_g({t_0})f(t_0,x(t_0))=(1-d)x(t_0) ∈(-r_0,r_0). Thus, f satisfies (H_r) for r=r_0. Combining this with Theorem <ref>, yields existence of a maximal solution x: I_t_0,x_0→ℝ for every (t_0,x_0) ∈ℝ^+×(-r_0,r_0). Let ω(t_0,x_0):=sup I_t_0,x_0. Let us construct this solution through local existence to show that ω(t_0,x_0)=∞. First of all, let τ∈(t_0,ω(t_0,x_0)) such that x:[t_0,τ]→ (-r_0,r_0) is a solution of (<ref>). Observe that if x(τ)=0, by uniqueness of the solution, we deduce that x≡ 0 which lies in (-r_0,r_0). Thus, two other interesting cases occur when x(τ) ≠ 0:

Case 1: if τ∈ D_g, then x(τ^+)=x(τ)+μ_g({τ})f(τ,x(τ))=(1-d)x(τ)∈ (-r_0,r_0).

Case 2: if τ∉ D_g, then we distinguish two subcases:

  Subcase 1: if x(τ)∈(0,r_0) then
x_g'(τ)=f(τ,x(τ))<0. Thus, by g-continuity of x at τ there exists τ_1 ∈(τ,ω(t_0,x_0)) such that x:[t_0,τ_1]→ (0,r_0).

  Subcase 2: if x(τ)∈(-r_0,0) then x_g'(τ)=f(τ,x(τ))>0. Thus, there exists  τ_2 >0 such that x:[τ,τ_2]→ (-r_0,0).

Repeating the same argument for each subinterval of I_t_0,x_0, we deduce that the solution x(t)∈[-λ_x_0,λ_x_0]⊂(-r_0,r_0) for all t∈ I_t_0,x_0, where

    λ_x_0=sup_t∈[t_0,τ]|x(t,t_0,x_0)|.

By Corollary <ref>, we deduce that ω(t_0,x_0)=∞. Hence, x:[t_0,∞)→ (-r_0,r_0) is a maximal solution of (<ref>).

Now, we define the function V:ℝ^+× (-r_0,r_0)→ℝ by V(t,x)=x^2 for all (t,x) ∈ℝ^+×(-r_0,r_0). Clearly V∈𝒱_g^1. For all (t_0,x_0) ∈ℝ^+× (-r_0,r_0), let x:[t_0,∞)→ (-r_0,r_0) be the maximal solution of (<ref>). Thus, using Proposition <ref>, we obtain for g-almost all  t ∈ [t_0,∞)∖ D_g:

    V_g'(t,x(t))   =
     ∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) x_g'(t)
       =∂ V/∂_g t(t,x(t))+∂ V/∂ x(t,x(t)) f(t,x(t))
       = 2ρ x(t)^2(1-x(t)/K)(x(t)/M-1).

While for t ∈ [t_0,∞)∩ D_g, we obtain

    V_g'(t,x(t))   =V(t^+,x(t^+))-V(t,x(t))/g(t^+)-g(t)
       =V(t^+,x(t)+ μ_g({t})f(t,x(t)))-V(t,x(t))/g(t^+)-g(t)
       =(1-d)^2x(t)^2-x(t)^2
       =(-2d+d^2)x(t)^2.

As (-2d+d^2)<0, it follows that V_g'(t,x(t)) is negative definite. Since a(|x|)≤ V(t,x)≤ b(|x|) for all (t,x)∈ℝ^+×(-r_0,r_0) with a,b∈𝒦 defined by a(s)=b(s)=s^2 for all y∈ℝ^+,  Corollary <ref> ensures that the trivial solution x=0 is  uniformly  asymptotically stable. The asymptotic behavior of solutions is illustrated in Figure <ref>.


 §.§ Comments

Figure <ref> illustrates the long-term effect of the high-speed trains vibrations. As shown in the figure,  if the vibrations lead to an Allee effect, the population will decline significantly over time.  This outcome raises alarm about the overall stability and the persistence of the whole ecosystem in this region. Specifically, it indicates a high likelihood of population extinction  when there is no estimation showing that the initial population exceeds M.




 §.§ Stable equilibrium of Bacteria-Ammonia dynamics

Cyanobacteria, similar to plants, participate in oxygenic photosynthesis as they are photosynthetic bacteria. Photosynthesis  is the biochemical process through which organisms convert light energy into chemical energy in the form of glucose or other organic compounds resulting oxygen release. The energy captured is then used to fuel the synthesis of organic molecules such as glucose, serving as an energy source for the bacteria.

Cyanobacteria  are found in diverse habitats, including freshwater, marine environments, and terrestrial ecosystems. In this section, we consider a species of cyanobacteria that has also the ability to fix nitrogen such as Anabaena cyanobacteria. Through nitrogen fixation, atmospheric nitrogen gas (N_2) is converted into a form that can be used by plants and other organisms by means the enzyme nitrogenase. This enzyme catalyzes the conversion of nitrogen gas (N_2) into ammonium ions (NH_4^+) based on a considerable amount of energy obtained from photosynthesis, these ammonium ions are assimilated into amino acids and proteins, which are essential for the growth and survival of Cyanobacteria. As a by product of nitrogen fixation, ammonia gas (NH_3) can be  released. Some of the assimilated ammonium ions (NH_4^+) are released back into the environment providing neighboring vegetative cells with a source of nitrogen.

To avoid losing valuable nitrogen nutrients and optimizing nitrogen utilization efficiency, this population has mechanisms allowing ammonium ions (NH_4^+) reabsorption, and ammonia gas (NH_3) assimilation through converting ammonia gas (NH_3) into ammonium ions (NH_4^+). In what follows, the term "ammonia" refers to both the protonated and unprotonated forms, which are denoted as (NH_3) and (NH_4^+) respectively <cit.>. It is worth mentioning that ammonia is commonly used in cleaning products, fertilizers. Beyond that, ammonia's cooling properties make it an essential refrigerant in air conditioning systems and refrigerators.

To optimize resource utilization and adapt to the varying environmental conditions, the population undergoes a day-night cycling of nitrogen fixation and carbon consumption <cit.>. This is due to the sensitivity of the nitrogenase enzyme responsible for nitrogen fixation to oxygen. Thus, nitrogen fixation is carried out during daylight hours when the photosynthesis can provide the necessary energy and oxygen levels are relatively low. During the nights, since oxygen levels within the cells would be higher, this population of cyanobacteria reduces their metabolic activity and growth and relies on stored carbon compounds to fulfill their energy needs. Our objective in the sequel is to observe the dynamics of this population which thrives in the presence of ammonia in a culture room without exposure to artificial light,  tracking the levels of the ammonia during this the process. Here, we assume that  the carbon dioxide (CO_2), the nitrogen gas (N_2) and nutrients supply are maintained steady as well as the PH level.  Since the population undergoes dormancy phases during the nights, we identify the days with intervals of the form [2k,2k+1], k=0,1,2,…, and the night with intervals [2k+1,2k+2], k=0,1,2,… In our example, we differentiate with respect to a derivator g whose variation describes the intensity of light, which is necessary for the photosynthesis process. We require that g presents smaller slops at the beginning and at the end of the daylight hours, with maximal slops at middays where t=2k+1/2, and remains constant during the dormancy phases in the night [2k+1,2k+2], k=0,1,2,… For instance, we consider g:ℝ→ℝ defined by

    g(t)=sin(π(t-1/2))+1/2    t∈[0,1] 
    
           1     t∈(1,2],

and g(t)=g(1)+g(t-2) for t≥ 2.

We denote N(t) the Biomass of cyanobacteria (grams per liter), and A(t) the ammonia concentration in the environment at time t≥ 0.  Since the population thrives in the presence of ammonia, we can assume that the growth rate is proportional to the level of ammonia. We denote ρ the maximal intrinsic coefficient of reproduction that the population can reach in the presence of one unit of ammonia with maximal sunlight intensity. Thus, the dynamics can be modeled  using the autonomous system of Stieltjes differential equations:

    𝐮_g'(t) = 𝐅(𝐮(t)), for g-almost every t≥ t_0≥0,
       𝐮(0) =𝐮_0 =(N_0,A_0),

where 𝐮=(N,A) and 𝐅=(F_1,F_2) : ℝ^2→ℝ^2 is defined by

    F_1(N,A)     =  ρ AN(1-N/K)
    
    F_2(N,A)     = α N-β AN.

where the parameters of the model can be understood as:


  K>0 the carrying capacity of the culture room, which forms a spacial constraint for growth.

  α>0 Constant related to the production of ammonia through nitrogen fixation.

  β>0 Constant related to the proportion of the reabsorption of ammonia by the population depending on the level of ammonia in the environment.

Observe that 𝐮^*=(K,α/β) is an equilibrium of the system (<ref>) among other equilibria. Its asymptotic stability would guarantee the persistence of the population with nonzero ammonia production. Therefore, we study local asymptotic stability in a neighborhood B_ℝ^2(𝐮^*,r_0); r_0>0, of this equilibrium. To this aim, we transfer our study in a neighborhood of 𝐱=0=(0,0) ∈ℝ^2. If we adopt the variable change 𝐱=(x_1,x_2):=𝐮-𝐮^*, we obtain the system

    𝐱_g'(t) = 𝐟(𝐱(t)), for g-almost every t≥ t_0≥0,
       𝐱(0) =𝐱_0 :=𝐮_0-𝐮^*,

where 𝐟=(f_1,f_2) : ℝ^2→ℝ^2 is defined by

    f_1(x_1,x_2)     =  -ρx_1/K(x_2+α/β)(x_1+K)
    
    f_2(x_1,x_2)     =  -β x_2(x_1+K).

𝐱=0=(0,0) ∈ℝ^2 is an equilibrium of the dynamical system (<ref>). Next, we prove that 𝐱=0 is  asymptotically stable, implying that 𝐮^*=(K,α/β) is an asymptotically stable equilibrium of the system (<ref>).

Let us consider r_0=min{K,α/β}, and let (t_0,𝐱_0) ∈ℝ^+× B_ℝ^2(0,r_0). Arguing as in the previous subsection, since the function 𝐟 satisfies conditions of Theorem <ref>, we prove existence of the maximal solution 𝐱=(x_1,x_2):[t_0,∞)→ B_ℝ^2(0,r_0) of the system (<ref>). Initially, let 𝐱: I_t_0,𝐱_0→ℝ^2  be the maximal solution of (<ref>) with ω(t_0,𝐱_0):=sup I_t_0,𝐱_0. Let τ∈(0,ω(t_0,𝐱_0))∖ C_g such that 𝐱=(x_1,x_2):[t_0,τ]→ B_ℝ^2(0,r_0) is a solution of (<ref>). In what follows, we analyze the possible cases:

Case 1: if x_1(τ),x_2(τ)∈(0,r_0) (resp. x_1(τ),x_2(τ)∈(-r_0,0)),  then, for i=1,2, we have
(x_i)_g'(τ)=f_i(𝐱(τ))<0 (resp. (x_i)_g'(τ)=f_i(𝐱(τ))>0). Therefore, by g-continuity of x at τ, there exists τ_1 ∈(τ,ω(t_0,𝐱_0)) (τ_1 can be chosen such that τ_1∉ C_g) such that x_i:[t_0,τ_1]→ (0,r_0) (resp. x_i:[t_0,τ_1]→ (-r_0,0)).


Case 2: if x_1(τ)∈(0,r_0) and x_2(τ)∈(-r_0,0), then
(x_1)_g'(τ)=f_1(𝐱(τ))<0 and (x_2)_g'(τ)=f_2(𝐱(τ))>0.  Therefore, there exists τ_2 ∈(τ,ω(t_0,𝐱_0))∖ C_g such that x_1:[τ,τ_2]→ (0,r_0), and x_2:[τ,τ_2]→ (-r_0,0).

Case 3: if x_1(τ)∈(-r_0,0) and x_2(τ)∈(0,r_0), then similarly to Case 2, we deduce that there exists τ_3 ∈(τ,ω(t_0,𝐱_0)) such that x_1:[τ,τ_3]→(-r_0,0), and x_2:[τ,τ_3]→ (0,r_0).

Repeating the same argument for each subinterval of I_t_0,𝐱_0, we deduce that the solution 𝐱(t)∈ [-λ_𝐱_0,1,λ_𝐱_0,1]× [-λ_𝐱_0,2,λ_𝐱_0,2]⊂ B_ℝ^2(0,r_0) for all t∈ I_t_0,𝐱_0, where

    λ_𝐱_0,i=sup_t∈[t_0,τ] |x_i(t,t_0,𝐱_0)|,    for  i=1,2 and 𝐱_0=(𝐱_0,1𝐱_0,2).

Using Corollary <ref>, we obtain that ω(t_0,𝐱_0)=∞. Hence, x:[t_0,∞)→ B_ℝ^2(0,r_0) is the maximal solution of (<ref>).


Now, let us consider the function V:ℝ^+× B_ℝ^2(0,r_0) →ℝ defined by V(t,𝐱)=x_1^2+x_2^2 for all t≥ 0 and 𝐱=(x_1,x_2)∈ℝ^2. It is clear that V∈𝒱_1^g.
Let (t_0,𝐱_0)∈ℝ^+× B_ℝ^2(0,r_0), and 𝐱:[t_0,∞)→ B_ℝ^2(0,r_0) be the maximal solution of (<ref>). By  means of Proposition <ref>, for g-almost all  t ∈ [t_0,∞), we obtain

    V_g'(t,𝐱(t))   =
     ∂ V/∂_g t(t,𝐱(t))+ ∑_i=1^2∂ V/∂ x_i(t,𝐱(t)) (x_i)_g'(t)
       =∂ V/∂_g t(t,𝐱(t))+ ∑_i=1^2∂ V/∂ x_i(t,𝐱(t)) f_i(t,𝐱(t))
       = -2ρx_1(t)^2/K(x_2(t)+α/β)(x_1(t)+K)-2β x_2(t)^2(x_1(t)+K).

Thus, V_g'(t,𝐱(t)) is negative definite. Since a(𝐳)≤ V(t,𝐳)≤ b(𝐳) for all (t,𝐳)∈ℝ^+× B_ℝ^2(0,r_0) where a,b∈𝒦 are defined by a(y)=y^2 and b(y)=2y^2 for all y∈ℝ^+, it follows from Corollary <ref> that 𝐱=0 is  uniformly asymptotically stable. Hence, 𝐮^*=(K,α/β) is a uniformly asymptotically stable equilibrium of the system (<ref>). The graph of asymptotic behavior of solutions near the equilibrium 𝐮^*=(K,α/β) is given in Figure <ref>.





§ FUNDING

Lamiae Maia was partially supported by the National Center of Scientific and Technical Research (CNRST) under Grant No. 60UM5R2021, Morocco.


§ ACKNOWLEDGMENT

Lamiae Maia would like to express her sincere gratitude towards Professor Marlène Frigon and the Département de mathématiques et de statistique of the Université de Montréal, for their warm hospitality and for funding her research stay at the aforementioned department when this article was finalized.

































































































































































































































































































































































































































































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