\documentclass[10pt]{studiamnew} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \sloppy \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{solution}[theorem]{Solution} \newtheorem{notation}[theorem]{Notation} \renewcommand{\theequation}{\thesection.\arabic{equation}} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{ex}{Example} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%% \def\N{\Bbb N} \def\Z{\Bbb Z} \def\R{\Bbb R} \def\C{\Bbb C} %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \newcommand{\Rr}{{\mathbb{R} }} \newcommand{\Nn}{{\mathbb{N} }} \newcommand{\Nc}{{\mathcal{N} }} \newcommand{\Kc}{{\mathcal{K} }} %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} % \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{4} \renewcommand{\currentvolume}{??} \renewcommand{\currentyear}{??} \renewcommand{\currentissue}{??} % \title[Fractional order impulsive coupled systems on the half line]{Existence results for some fractional order coupled systems with impulses and nonlocal conditions on the half line} \author{Khadidja Nisse} \address{Laboratory of Operators Theory and PDEs: Foundations and Applications,\\ Department of Mathematics, Faculty of Exact Sciences,\\ University of El Oued, Algeria } \email{nisse-khadidja@univ-eloued.dz} % %\author{John Smith} %\address{``Babe\c{s}-Bolyai'' University, \\ Faculty of Mathematics and Computer Sciences\\ %1, Kog\u{a}lniceanu Street,\\ %400084 Cluj-Napoca,\\ %Romania} %\email{Your e-mail address} % %\author{John Smith} %\address{``Babe\c{s}-Bolyai'' University, \\ Faculty of Mathematics and Computer Sciences\\ %1, Kog\u{a}lniceanu Street,\\ %400084 Cluj-Napoca,\\ %Romania} %\email{your e-mail address} % \subjclass{26A33, 93C23, 35E15, 47H10} \keywords{Fractional differential equation, coupled systems, generalized spaces in Perov's sens, nonlocal initial conditions, impulses} \begin{abstract} In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov's type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. \end{abstract} \maketitle \section{Introduction}\label{sec1}%%% Popov Recently, an intensive interest has been given to the investigation of differential equations of fractional order. This is motivated by the natural introduction of fractional operators in the modeling of several phenomena whose nonlocal dynamics involving long-term effects are taken into account. These models have been applied successfully in many fields such as in mechanics, bio-chemistry, electrical engineering, control, porous media, medicine, etc (see \cite{Diethelm,Kilbas}). %Systems of this type arise from mathematical modelling of many processes from a variety of disciplines, including physics, biology, chemistry, engineering and other sciences. Thus, initial value problems and boundary value problems for nonlinear competitive or cooperative differential systems from mathematical biology [4] and mathematical economics [3] can be put in the operator form (1). On the other hand, differential equations involving impulse effects appear as an appropriate model for some evolutionary problems. It is the case of many real-world processes that are subject of abrupt of changes in certain moments of times and arising in a variety of disciplines, including biology, population dynamics, electric technology, control theory, engineering, etc. For more details on this subject, we reefer to the monographs \cite{Ref2,Ref3}. % The classical Banach contraction principle was extended for contractive maps on spaces endowed with a vector-valued metric by Perov in 1964 [11] and Perov and Kibenko [12]. Up to now, there have been a number of attempts to generalize the Perov fixed point theorem in several directions and also there have been a number of applications in various fields of nonlinear analysis, for systems of ordinary differential and semilinear differential equations. Recently Precup [13] established the role of vector-valued metric convergence in the study of semilinear operator systems. In recent years, many authors studied the existence of solutions for systems of differential equations by using the vector version fixed point theorem; see [3, 10, 14, 8, 9] and the references therein. Banach's contractive principle is one of the most useful tools in nonlinear functional analysis that ensures the existence and uniqueness of a fixed point on complete metric spaces. One of the extensions of this principle for contractive mappings on spaces endowed with vector valued metrics, was done by Perov in \cite{Perov1964} and Perov and Kibento in \cite{Perov1966}. Many other generalizations in this direction have been investigated. In \cite{Precup2014}, Precup established the extension in Perov's sens of some fixed point theorem in spaces endowed with a family of pseudo-metrics. Many authors applied the vector version's fixed point theorems in the study of the existence of solutions for systems of differential and integral equations, see for example \cite{ Belbali2021,BERREZOUG2017,J.Nieto2020,Kadari2020,Wang2014} and the references therein. % Further extensions have tried to relax the metrical structure of the space, its completeness, or the contraction condition itself. % The classical Banach contraction principle was extended for contractive maps on spaces endowed with vector-valued metrics by Perov in 1964 [40] and Perov and Kibenko [41]. %% Many generalizations of this principle have been given in recent years. We can observe two main directions in this line of research, one is by finding a more general contraction inequality and the other is by modifying the structure of space. %%%%%%%%%%% %of mappings plays a crucial role in nonlinear functional analysis and ensures the existence and uniqueness of a fixed point on complete metric spaces. In this line of research, we consider in this work, the following nonlinear coupled system of fractional differential equations: % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation}\label{Sys} \begin{cases} ^{C}\!D_{0^+}^{\alpha } u( t) =f(t, u( t),v( t)), & t\in \mathcal{I}_i=\left]t_i,t_{i+1}\right],\, i\in \mathbb{N} \\%[5mm] ^{C}\!D_{0^+}^{\beta } v( t) =g(t, u( t),v( t)), & t\in \mathcal{I}_i=\left]t_i,t_{i+1}\right],\, i\in \mathbb{N} %\\ \end{cases} \end{equation} with coupled nonlocal initial conditions: \begin{equation}\label{Initial.con.Sys} \begin{cases} % u(0)=\varphi(u,v), & \\ v(0)=\psi (u,v), & \end{cases} \end{equation} and subject to coupled impulsive conditions: %%%%%%%%%%%\eqref{Sys}- \eqref{Impulsive.con.Sys} \begin{equation}\label{Impulsive.con.Sys} \begin{cases} % %u(0)=\varphi(u,v), & t\leq 0\\ \Delta u (t_{i}) = I_{i}( u( t_i),v(t_i)),& \,\, i\in \mathbb{N}^*={\mathbb{N}}\backslash {\lbrace 0 \rbrace} \\ \Delta v (t_{i}) = J_{i}( u( t_i),v(t_i)),& \,\, i\in \mathbb{N}^*={\mathbb{N}}\backslash{\lbrace 0 \rbrace} \end{cases} \end{equation} %%%%%%%%%%% where $^{C}\!D_{0^+}^{\alpha }$ and $^{C}\!D_{0^+}^{\beta }$ denote the Caputo fractional derivative operators with the fixed lower limit equals zero, of order $\alpha$ and $\beta $ in $\left] 0,1\right[$ respectively, %$a$, $b$ and $\tau_{i},\,\, i=1,2$ are real continuous functions defined on $\Rr _+ = [0 ,\, %+\infty [$ such that $ \, 0 \leq a(t) \leq b(t) \leq t \,$, $ f,g :\, {\mathbb R}_{+} \times \mathbb{R}^{2} \longrightarrow {\mathbb R}$ are nonlinear continuous functions, $ \Delta u (t_{i}) = u( t_{i}^{+} ) - u( t_{i}^{-} ) $, where $u( t_{i}^{+} ) $ and $u( t_{i}^{-} ) $ represent the right and left limits of $ u$ at $ t=t_{i}$ and $\lbrace t_i \rbrace_{i\in \mathbb{N}^*}$ is a sequence of points in $\mathbb{R}_+$ such that $t_i0$. It should be noted that the coupled nonlocal initial conditions \eqref{Initial.con.Sys} generalizes many other types of initial conditions considered in the literature, such as: classical initial conditions, multi-point conditions and integral conditions. After converting \eqref{Sys}- \eqref{Impulsive.con.Sys} into an equivalent fixed point problem in generalized gauge space, we apply some fixed point theorem of Perov's type, established in \cite{Precup2014}. Using this approach, we obtain a global existence-uniqueness results for \eqref{Sys}- \eqref{Impulsive.con.Sys} under weak conditions allowing the nonlinearity to increase indefinitely with time, which is not the case in many earlier results in the literature (see Remark \ref{Rq1}). This study allows us also, to improve and generalize some other existence results in the literature for systems of fractional differential equations without impulses (see Remark \ref{Rq_impvov}). The rest of the paper is organized as follows. In Section \ref{Sec:2} we recall some definitions from fractional calculus. We introduce also the fixed point theorem in generalized gauge spaces, on which our result is based, as well as some related concepts. The main result concerning the global existence-uniqueness result for \eqref{Sys}- \eqref{Impulsive.con.Sys} is established in Section \ref{Sec:3}. Finally, in Section \ref{Sec:4}, we provide an illustrative example. \section{Preliminaries} \label{Sec:2} % Let us recall the notion of the fractional derivatives. % % % For further details on some essential related properties, we refer to \cite{Diethelm,Kilbas}.\\ Let $n$ be a positive integer, $\alpha$ the positive real such that $n-1<\alpha\leq n$ and ${d^n}/{dt^n} $ the classical derivative operator of order $n$. \begin{defn} \label{def_int_fra} The Riemann-Liouville fractional integral, and the Riemann-Liouville fractional derivative, of a real function $u$ defined on $\Rr _+$ of order $\alpha$, are defined respectively by \begin{equation*} \begin{matrix} I_{0^+}^{\alpha}u\left( t\right) :=\dfrac{1}{\Gamma\left( \alpha\right) }\displaystyle{\int _{0}^{t}}( t-s) ^{\alpha-1}u\left( s\right) ds,\text{\ \ \ \ }t>0, \\ %%%%%%%%%%%%%%%%%%%%%%%% D_{0^+}^{\alpha}u\left( t\right) :=\dfrac{d^n}{dt^n}I_{0^+}^{n-\alpha}u\left( t\right) :=\dfrac{1}{\Gamma\left( n-\alpha\right) }\dfrac{d^n}{dt^n}\displaystyle{\int_{0}^{t}}( t-s) ^{n-\alpha-1}u\left( s\right) ds,\text{\ \ \ \ }t>0, \end{matrix} \end{equation*} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % %, % where $\Gamma\left( .\right) $ is the Gamma function, provided that the right hand sides exist point wise.%% \end{defn} \begin{defn} \label{def_Caputo der} The Caputo fractional derivative of a real function $u$ defined on $\Rr _+$ of order $\alpha$, noted by $^{C}D_{0^+}^{\alpha}$, is defined by %%%%%%%%%%%%%%%%%%%%%%%%%% % $$ ^{C}D_{0^+}^{\alpha}u\left( t\right) :=\left( D_{0^+}^{\alpha}\left[u- \displaystyle{\sum_{k=0}^{n-1} \frac{u^{(k)}(0)}{k!} (.)^k} \right] \right)\left(t\right),\text{\ \ \ \ }t>0, $$ provided that the right hand side exists point wise. % \end{defn} % %%%%%%%%%%%%%%% % Let us recall first some related concepts and properties. We denote by $\mathcal{M}_{n}(\mathbb{R}_{+})$, the set of all square matrices of order $n$ with positive real elements, $I$ the identity matrix of order $n$ and by $O$ the zero matrix of order $n$. \begin{defn} \cite{R.S.Varga1999} A square real matrix $M$ of order $n$, is said to be convergent to zero, if $ M^{k} \longrightarrow O, \mathrm{as}\ k \longrightarrow \infty $. \end{defn} \begin{defn} \cite{R.S.Varga1999} % R. S. Varga 1999 Let $M\in \mathcal{M}_{n}(\mathbb{R}_{+})$ with eigenvalues $\lambda_i,\, 1\leq i \leq n$, that is $\lambda_i \in \mathbb{R}$ such that $ det (M-\lambda_i I)=O$. Then $$ \rho \left( M\right)= \max\limits_{1\leq i \leq n}\, \left\vert \lambda_i \right\vert $$ is called the spectral radius of $M$. \end{defn} %%%%%%%%%%%%%%%%%%%%%%%%%%%note that using the property of convergent matrices given in Theorem \ref{Prop.Matrix} %\begin{thm}\cite[Theorem 3.15]{R.S.Varga1999}\label{Prop.Matrix} %If $M\in \mathcal{M}_{ n}(\mathbb{R}_{+})$ with $\rho \left( M\right)<1$, then $I-M$ %is non singular, and %$$ %(I-M)^{-1}=I+M+M^{2}+...+M^{n}+..., %$$ %the series on the right is converging. Conversely, if the series on the right %converges, then $\rho \left( M\right)<1$. %\end{thm} %%%%%%%%%%%%%%%%%%%%% \begin{lem}\cite{R.S.Varga1999}\label{Prop.Matrix} Let $M\in \mathcal{M}_{n}(\mathbb{R}_{+})$. The following assumptions are equivalent. \begin{itemize} \item[(i)] $M$ is convergent to zero. \item[(ii)] The matrix $I-M$ is non singular, and $$ (I-M)^{-1}=I+M+M^{2}+...+M^{n}+..., $$ \item[(iii)] $\rho \left( M\right) < 1$. %for every $\lambda in \mathbb{C}$ with $ det (M-\lambda I)=O$ \end{itemize} \end{lem} As it is pointed out in \cite{Nica2012}, the following lemma follows immediately from the characterization $(iii)$ in Lemma \ref{Prop.Matrix}. % Nica2012 \begin{lem}\label{Lemma.Nica}\cite{Nica2012} If $A$ is a square matrix that converges to zero and the elements of another matrix $B$ are small enough, then $A+B$ also converges to zero. \end{lem} We state now the extension of Gheorghiu's theorem for generalized contractions on complete generalized gauge spaces established in \cite{Precup2014}. % To this end, we start by some related concepts and properties Let $X$ be a generalized gauge space endowed with a complete gauge structure $ \mathfrak{D} = \left\{ D_{\nu} \right\}_{\nu \in \mathcal{N}}$, where $\mathcal{N}$ is an index set. For further details on gauge spaces and generalized gauge spaces we reefer to \cite{J.Dugundji1966,Precup2014}.% set be a family of vector valued pseudo-metrics on $X$ %%%%%%%%%% $j^n(K)=j(j^{n-1}(K))$ stands for the $n^{th}$ iterate of the mapping $j$, where $j^{0}(K)=K$. %%%%%%%%%%%%%% $j : \mathcal{K} \longrightarrow \mathcal{K}$ \begin{defn}\label{def.Gener.contr.}\cite{Precup2014} (Generalized contraction) Let $\left(X, \mathfrak{D} \right)$ be a generalized gauge space with $ \mathfrak{D} = \left\{ D_{\nu} \right\}_{\nu \in \mathcal{N}}$. A map $T:D(T)\subset X \longrightarrow X$ is called a generalized contraction, if there exists a function $w : \mathcal{N} \longrightarrow \mathcal{N}$ and $M\in \mathcal{M}_{n}(\mathbb{R}_{+})^{\mathcal{N}}, \, M= \left\{M_{\nu}\right\}_{\nu\in \mathcal{N}}$ such that \begin{equation}\label{Generalized contraction} D_{\nu}(T(u),T(v))\leq M_{\nu}D_{w(\nu)}(u,v), \,\,\, \forall u,v \in D(T),\, \forall \nu \in \mathcal{N} \end{equation} and % M_{w^{2}(\nu)} \begin{equation}\label{Condition series} \sum\limits_{k=1}^{\infty} M_{\nu}M_{w(\nu)}...M_{w^{k-1}(\nu)}D_{w^{k}(\nu)}(u,v)<\infty, \,\,\, \forall u,v \in D(T),\, \forall \nu \in \mathcal{N} \end{equation} \end{defn} %%%%%%%%%%%%%%%%%%%%%%%%% %% \begin{thm}\cite[Theorem 2.1]{Precup2014}\label{fixed pt th} Let $\left(X, \mathfrak{D} \right)$ be a complete generalized gauge space and let $T: X \longrightarrow X$ be a generalized contraction. Then, $T$ has a unique fixed point in $X$, which can be obtained by successive approximations starting from any element of $X$. % \end{thm} % \subsection{Equivalent system of integral equations} In the fractional case, there are two different approaches defining the concept of solutions for impulsive differential equations, which can be briefly described as follows (see \cite{V_12,A_12}): {\em Fractional derivatives with a fixed lower limit at the initial time.} This approach (denoted respectively by $V_2$ in \cite{V_12} and by $A_1$ in \cite{A_12}) considers that the lower limit of the fractional derivative is kept equal to the initial time on any interval between two consecutive impulses, with only modified initial conditions. {\em Fractional derivatives with varying lower limits.} This approach (denoted respectively by $V_1$ in \cite{V_12} and by $A_2$ in \cite{A_12}) neglects the lower limit of the fractional derivative at the initial time and moves it to each impulsive time. In this work, we will adopt the case of fixed lower limit. For any interval $\mathcal{I}$ of $\mathbb{R_+}$ (which may be unbounded), we denote by $\mathcal{C}(\mathcal{I})$ the set of all real continuous functions on $\mathcal{I}$ and by $u_i$ the restriction of $u\in \mathcal{C}(\mathbb{R_+})$ to $\mathcal{I}_i=\left]t_i,t_{i+1}\right],\,\,(i\in \mathbb{N})$. %%%%%%%%%%%%%%%% % %We consider in what follows, $X=\mathcal{PC}(\mathbb{R}_+)\times \mathcal{PC}(\mathbb{R}_+)$, Let $\mathcal{PC}(\mathbb{R}_+)$ be the set of all real valued piece-wise continuous functions on $\mathbb{R}_+$: %\begin{align*} %\mathcal{PC}(\mathbb{R}_+) %=\Big\{& u:\mathbb{R}_+\to \mathbb{R}: u_i \in \mathcal{C}\left(\mathcal{I}_i %\right)\\ %\text{ and for every}\,\,i\in \mathbb{N}^*:\\ %&%u\mid_{J_k} \in \mathcal{C}\left(J_k \right) %\text{ and }u( t_{i}^{+} ) \text{ exist for every\,} i\in \mathbb{N} %\Big\}. %\end{align*} % \begin{equation}\label{Space PC} \mathcal{PC}(\mathbb{R}_+)=\lbrace u:\mathbb{R}_+\to \mathbb{R}: u_i \in \mathcal{C}\left(\mathcal{I}_i \right)\text{\,and\,} u( t_{i}^{+} ) \text{\,exist for every\,} i\in \mathbb{N} \rbrace \end{equation} % endowed with the saturated family $ \{ d_\nu \,:\, \nu \in \mathcal{N} \}$ of pseudo-metrics, generating its topology, defined by % % \begin{equation}\label{con_lambda} d_{\nu}\left(u,v\right) =\max\limits_{t\in \nu} \left\{ e^{-\lambda t}\left\vert u\left( t\right)-v\left( t\right) \right\vert \right\}, \, \forall u,v \in \mathcal{PC}(\mathbb{R}_+), \end{equation} where $\nu$ runs over the set of all compact subsets of ${\mathbb R_+}$ denoted by $\mathcal{N}$, and $\lambda$ is a positive real number to be specified later. In what follows, we consider $X=\mathcal{PC}(\mathbb{R}_+)\times \mathcal{PC}(\mathbb{R}_+)$, endowed with the generalized complete gauge structure $ \mathfrak{D} = \left\{ D_{\nu} \right\}_{\nu \in \mathcal{N}}$ defined for $W_1=(u_1, v_1), W_2=(u_2, v_2)\in X$ by: \begin{equation} D_{\nu}\left(W_1,W_2\right)= \left(\begin{array}{cc} d_{\nu}(u_1,u_2) \\ [5mm] d_{\nu}(v_1,v_2)\end{array}\right), \end{equation} where $d_{\nu}$ is the pseudo-metric on $\mathcal{PC}\left(\mathbb{R}_+ \right)$ given in \eqref{con_lambda}. % Reproducing the proof of {\cite[Lemma 1]{my}}, in addition of {\cite[Lemma 2.6]{Lemme2.6_Fechan}} with a slight adaptation, we get the system of integral equations equivalent to (\ref{Sys})-\eqref{Initial.con.Sys} given by the following lemma. % % % % \begin{lem}\label{lem_eq int eq} Let $f,g,I_i,J_i\,(i\in \mathbb{N}^*)$ be continuous functions and $\varphi,\psi$ continuous functionals such that: \begin{equation}\label{C.compatibility with App A_1} \begin{array}{cc} \forall (u, v),(\tilde{u},\tilde{v}) \in X\,\, \text{if\,\,}u=\tilde{u}\text{\,\,and \,\,}v=\tilde{v}\text{\,\,on \,\,} [0,t_1[,\text{\,\,then \,\,} \\ \varphi\left(u,v\right) =\varphi\left(\tilde{u}, \tilde{v}\right) \text{\,\,\,\,and\,\,\,\,}\psi\left(u,v\right) =\psi\left(\tilde{u}, \tilde{v}\right) \end{array} \end{equation} Then, $(u,v)\in X$ is a solution of (\ref{Sys})-\eqref{Impulsive.con.Sys} if and only if $(u,v)$ is a solution of the following system of integral equations %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%% \scriptstyle{ %%%%%%%%%% \displaystyle{ \begin{equation}\label{integral Sys} %\hspace{-0.4cm} \begin{cases} u(t) = \left\lbrace \begin{array}{ll} \varphi(u,v) + \displaystyle{\int_{0}^{t}} \scriptstyle{\frac{( t-s ) ^{\alpha -1}}{\Gamma ( \alpha ) } f(s,u( s ),v( s )) }ds, & t \in \mathcal{I}_0 \\%[5mm] %%%%%%%% \varphi(u,v) + \displaystyle{\int_{0}^{t}} \scriptstyle{\frac{( t-s ) ^{\alpha -1}}{\Gamma ( \alpha ) }f(s,u( s ),v( s )) ds + \displaystyle{ \sum_{j=1}^{i}} \scriptstyle{ I_{j}( u( t_{j} ),v( t_{j} ))} }, & t \in \mathcal{I}_i \end{array} \right.\\[5mm] %%%%%%%%%%%%%%%%%%%%%%%% v(t) = \left\lbrace \begin{array}{ll} \psi(u,v) + \displaystyle{\int_{0}^{t}}\scriptstyle{\frac{( t-s ) ^{\beta -1}}{\Gamma(\beta )} g(s,u( s ),v( s))} ds, & t \in \mathcal{I}_0 \\%[5mm] % \psi(u,v) + \displaystyle{\int_{0}^{t}} \scriptstyle{\frac{( t-s ) ^{\beta -1}}{\Gamma(\beta )} g(s,u( s ),v( s)) ds + \displaystyle{ \sum_{j=1}^{i} \scriptstyle{ J_{j}( u( t_{j} ),v( t_{j} )) }}}, & t \in \mathcal{I}_i \end{array} \right. \end{cases} \end{equation} \end{lem} %%%%%%%%%%%%%%%%%%%%$X=\mathcal{C}(\mathbb{R})\times \mathcal{C}(\mathbb{R})$ %%%%%%%%%%%%% For $i=1,2$, let $T_i: X \rightarrow \mathcal{PC}(\mathbb{R}_+)$ be the operators defined for every $W:=(u,v)\in X$ by \begin{equation}\label{T_1} % %\hspace{-0.3cm} T_1(W)(t) =%\hspace{-0.1cm} \varphi(u,v) + \displaystyle{\int_{0}^{t}}\scriptstyle{\frac{( t-s ) ^{\alpha -1}}{\Gamma ( \alpha ) }f(s,u( s ),v( s ))} ds + \displaystyle{ \sum_{t_j< t} \scriptstyle{ I_{j}( u( t_{j} ),v( t_{j} )) }} \end{equation} %%%%%%%%%%%%%%%%%%% \begin{equation}\label{T_2} % %\hspace{-0.3cm} T_2(W)(t) =%\hspace{-0.1cm} \psi(u,v) + \displaystyle{\int_{0}^{t}}\scriptstyle{\frac{( t-s ) ^{\beta -1}}{\Gamma ( \beta ) }g(s,u( s ),v( s ))} ds + \displaystyle{ \sum_{t_j< t} \scriptstyle{ J_{j}( u( t_{j} ),v( t_{j} )) }} \end{equation} %%%%%%%%%%%%%%%%%%%% % Let us consider the operator: $T: X \rightarrow X$ defined by \begin{equation}\label{Operator T} T(u,v)=(T_1(u,v),T_2(u,v)), \,\,\,\,\,\,\forall (u,v) \in X , \end{equation} where $T_1$ and $T_2$ are given respectively by \eqref{T_1} and \eqref{T_2}. Thus, according to Lemma \ref{lem_eq int eq}, the solutions of \eqref{Sys}-\eqref{Impulsive.con.Sys} can be regarded as fixed points of $T$. % % ------------------------------------------------------------------------ \section{Main Results } \label{Sec:3} In this section, we will prove a global existence-uniqueness result for \eqref{Sys}-\eqref{Impulsive.con.Sys}, to this end, we consider the following assumptions: %To prove our main results, we need the following further assumptions. \begin{itemize} %\textcolor{red}{ % %%%%%%%%%%%%%%%%%%%%%%%%\sum\limits_{i=1}^{N} % \item[$\left( H_{1}\right) $] There exist continuous positive real valued functions $A_i,B_i:\,i=1,2$ defined on $\mathbb{R}_+$, and satisfying \begin{itemize} \item[ (i) ] $\left\vert f\left(t, \xi_1, \eta_1 \right) -f\left(t, \xi_2, \eta_2 \right) \right\vert \leq A_{1}\left(t\right)\left\vert \xi_1-\xi_2\right\vert + A_{2}\left(t\right)\left\vert \eta_{1}-\eta_{2}\right\vert\, $\\ $\left\vert g\left(t, \xi_1, \eta_1 \right) -g\left(t, \xi_2, \eta_2 \right) \right\vert \leq B_{1}\left(t\right)\left\vert \xi_1-\xi_2\right\vert + B_{2}\left(t\right)\left\vert \eta_{1}-\eta_{2}\right\vert\ ,$ \\ whenever the left hand sides are defined. \item [(ii) ] For $\lambda >0,\mu >1,\,q:=1+{1}/{\alpha}\,$ and $\,\,\tilde{q}:=1+{1}/{\beta}\,$, we have\\ %% %% S_{\lambda,\mu}, \tilde{S}_{\lambda,\mu}, R_{\lambda,\mu}, \tilde{R}_{\lambda,\mu} %% $$S_{\lambda,\mu}:=\displaystyle{ \int_{0}^{+\infty} A_{1}^{q}\left(s\right)e^{\frac{-{q}\lambda s}{\mu}} ds}<\infty \textrm{ and }\, \tilde{S}_{\lambda,\mu}:=\displaystyle{ \int_{0}^{+\infty} A_{2}^{q}\left(s\right)e^{\frac{-{q}\lambda s}{\mu}} ds}<\infty $$ %%%%%%%%%%%%%%% $$R_{\lambda,\mu}:=\displaystyle{ \int_{0}^{+\infty} B_{1}^{\tilde{q}}\left(s\right)e^{\frac{-\tilde{q}\lambda s}{\mu}} ds}<\infty \textrm{ and }\, \tilde{R}_{\lambda,\mu}:=\displaystyle{ \int_{0}^{+\infty} B_{2}^{\tilde{q}}\left(s\right)e^{\frac{-\tilde{q}\lambda s}{\mu}} ds}<\infty $$ \end{itemize} %%%%%%%%%%%%% \item[$( H_{2}) $] % % There exist fixed compacts $K_i,\tilde{K}_j$ and non negative real numbers $L_i,\tilde{L}_i,M_j,\tilde{M}_j ,(1\leq i\leq l,\, 1\leq j\leq m)$, satisfying what follows for every $(u_1, v_1),(u_2, v_2) \in X$: \begin{equation*} \begin{array}{ll} \left\vert \varphi\left(u_1, v_1\right) -\varphi\left(u_2, v_2\right) \right\vert \leq \sum\limits_{i=1}^{l}\left( L_i d_{K_i} (u_1-u_2)+ \tilde{L}_i d_{K_i} (v_1-v_2) \right)\\ %%%%%%%%%%%%%%%%%%%%% \left\vert \psi\left(u_1, v_1\right) -\psi\left(u_2, v_2\right) \right\vert \leq \sum\limits_{j=1}^{m}\left( M_j d_{\tilde{K}_j} (u_1-u_2)+ \tilde{M}_j d_{\tilde{K}_j} (v_1-v_2) \right) \end{array} \end{equation*} % % %%%%%%%%%%% \item[$( H_{3}) $] % There exist positive real sequences $\lbrace h_i \rbrace$,$\lbrace \tilde{h}_i \rbrace$,$\lbrace k_i \rbrace$ and $\lbrace \tilde{k}_i \rbrace$ that converge to $H,\tilde{H},K$ and $\tilde{K}$ respectively and satisfying for every $ \xi_1,\xi_2, \eta_1, \eta_2 \in \Rr$ and $i \in \mathbb{N}^*$, the following estimations: \begin{equation*} \begin{array}{ll} \left\vert I_i\left( \xi_1, \eta_1 \right) -I_i\left( \xi_2, \eta_2 \right) \right\vert \leq h_{i}\left\vert \xi_1-\xi_2\right\vert + \tilde{h}_{i}\left\vert \eta_{1}-\eta_{2}\right\vert\ \\ %%%%%%%%%%%%%%%%% \left\vert j_i\left( \xi_1, \eta_1 \right) -j_i\left( \xi_2, \eta_2 \right) \right\vert \leq k_{i}\left\vert \xi_1-\xi_2\right\vert + \tilde{k}_{i}\left\vert \eta_{1}-\eta_{2}\right\vert\ \end{array} \end{equation*} % %% %% %% % %% %% %\end{itemize}%%%%%%%%%%%%%%%% % \end{itemize} %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% {Rq1}{Rq_impvov} %%%%%%%%%%%%%%%%%%%\eqref{Sys}-\eqref{Impulsive.con.Sys} \begin{rem}\label{Rq1} % It is not hard to see that hypothesis $( H_{1}) $ includes as special cases the Lipschitz condition with constant or integrable arguments, widely used in the literature (see for example \cite{J.Nieto2020,Wang2014,Wang2018,BERREZOUG2017}). This being said, we emphasize here that hypothesis $( H_{1}.(ii)) $ allows the nonlinearity to increase indefinitely with time, which can not be covered by the previous special cases (that is when $A_i,B_i$ are constants or $A_i,B_i \in L^1(\mathbb{R}_+)$). Therefore, our work generalizes and complements many existing results in the literature. %We give as an example, the following functions $f$ and $g$: %$$ %f(t,\xi , \eta)= a_1 e^{\lambda_1 \frac{\alpha +1}{\mu_1 \alpha}t}\xi + b_1 %e^{\lambda_2 \frac{\alpha +1}{\mu_2 \alpha}t}\eta, \,\,\, %g(t,\xi , \eta)= a_2 e^{\lambda_3 \frac{\beta +1}{\mu_3 \beta}t}\xi + b_2 %e^{\lambda_4 \frac{\beta +1}{\mu_4 \beta}t}\eta, %$$ %where: $\,a_i, b_i \in \mathbb{R},\,(i=1,2)$, $\lambda_j>0,\, %\mu_j>1,\,(j=1,...,4).$ % %Hypothesis $( H_{2}.ii) $ covers a wide class of nonlinearties of \eqref{Sys}-\eqref{Impulsive.con.Sys}, including those satisfying Lipschitz condition with constant arguments, widely used in the literature. This being said, we emphasize that Hypothesis $( H_{2}.ii) $ allows the nonlinearity to increase indefinitely with time. We give as an example, the following functions $f$ and $g$: % \end{rem} %%%%%%%%%%%%%%%%%%%% % % % %%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%% % % %%%%%%%%%%%%%%%%%%%% %\newpage For $\lambda > 0$ and $\mu > 1$, let $M_{\alpha, \beta}\left(\lambda,\mu \right)$ be the square matrix defined by: \begin{equation}\label{Matrix M} M_{\alpha, \beta}\left(\lambda,\mu \right) :=\left(\begin{array}{cc} \sum\limits_{i=1}^{l}L_i+ \Lambda_ {\lambda,\mu}^\alpha+H \ \ \ \sum\limits_{i=1}^{l}\tilde{L}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\alpha+\tilde{H} \\ [5mm] \sum\limits_{i=1}^{m}M_i+\Lambda_ {\lambda,\mu}^\beta+K \ \ \ \ \sum\limits_{i=1}^{m}\tilde{M}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\beta+\tilde{K} \end{array} \right) \end{equation} Where %$\Lambda_ {\lambda,\mu}^\alpha, \tilde{\Lambda}_ {\lambda,\mu}^\alpha, %\Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ \begin{equation}\label{Coefficient of M} \begin{array}{cc} \Lambda_ {\lambda,\mu}^\alpha=\frac{1}{\Gamma(\alpha)\lambda^\alpha} \left( \frac{1}{(\alpha+1)^{\alpha^2}} \Gamma(\alpha^2) \right)^{\frac{1}{1+\alpha}} \left( S_{\lambda,\mu} \right) ^{\frac{\alpha}{1+\alpha}} \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \tilde{\Lambda}_ {\lambda,\mu}^\alpha=\frac{1}{\Gamma(\alpha)\lambda^\alpha} \left( \frac{1}{(\alpha+1)^{\alpha^2}} \Gamma(\alpha^2) \right)^{\frac{1}{1+\alpha}} \left( \tilde{S}_{\lambda,\mu} \right) ^{\frac{\alpha}{1+\alpha}} \\ %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \Lambda_ {\lambda,\mu}^\beta=\frac{1}{\Gamma(\beta)\lambda^\beta} \left( \frac{1}{(\beta+1)^{\beta^2}} \Gamma(\beta^2) \right)^{\frac{1}{1+\beta}} \left( R_{\lambda,\mu} \right) ^{\frac{\beta}{1+\beta}} \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \tilde{\Lambda}_ {\lambda,\mu}^\beta=\frac{1}{\Gamma(\beta)\lambda^\beta} \left( \frac{1}{(\beta+1)^{\beta^2}} \Gamma(\beta^2) \right)^{\frac{1}{1+\beta}} \left( \tilde{R}_{\lambda,\mu} \right) ^{\frac{\beta}{1+\beta}} \end{array} \end{equation} %%%%%%%%%%%%%%%%%%%%% \begin{thm}\label{Th1} % % Let $\left( H_{1}\right)-\left( H_{3}\right)$ and \eqref{C.compatibility with App A_1} hold true. Then, the system \eqref{Sys}-\eqref{Impulsive.con.Sys} admits a unique global solution in $X$ provided that: there exist $ \lambda > 0$ and $\mu > 1$ such that %the matrix $M_{\alpha, \beta}\left(\lambda,\mu \right)$, given in \eqref{Matrix M}, converges to zero. %%%%%%%%%%%%%%%% \begin{equation}\label{Condition in Th} % $\lambda > 0$ and $\mu > 1$ %\text{There exist\,\,} \lambda > 0 \text{and\,\,} \mu > 1 \text{The matrix\,\,}M_{\alpha, \beta}\left(\lambda,\mu \right) \text{given in \eqref{Matrix M}, converges to zero.} \end{equation} % % \end{thm} %%%%%%%%%%%%%%%%%%%%% \begin{proof} %% % Recall that the solutions of \eqref{Sys}-\eqref{Impulsive.con.Sys} are the fixed points of the operator $T$ defined in \eqref{Operator T}. We shall prove that $T$ is a generalized contraction in the sens of Definition \ref{def.Gener.contr.}, to deduce the result from Theorem \ref{fixed pt th}. To this end, let us define a mapping $w\,:\, \mathcal{N} \longrightarrow \mathcal{N} $ as follows: \begin{equation}\label{func w} %\quad w(\nu) = \left[0 ,\,\, \max\limits_{ 1\leq i\leq l , \\ 1\leq j\leq n }\lbrace \nu^{m},\, K_{i}^m,\, \tilde{K}_{j}^m \rbrace \right] , \end{equation} where $\nu^m$ denotes $\max \nu$ and $ K_{i},\, \tilde{K}_{j}$ are the compacts given by $\left( H_{2}\right)$. Note that according to \eqref{func w}, it follows that \begin{equation}\label{w carré =w} \text{For every\,\,\,} \nu \in \mathcal{N}: w^n(\nu) =w(\nu), \quad \forall n\geq 2 \end{equation} Let $\nu\in \mathcal{N}$ and $t\in \nu$. Using $\left( H_{1}(i)\right),\left( H_{2}\right),\left( H_{3}\right)$, we get: $$ \left\vert T_1(u_1, v_1)\left( t\right)- T_1(u_2, v_2)\left( t\right) \right\vert \leq , $$ $$ % \displaystyle{\int_{0}^{t}}\frac{\left( t-s\right) ^{\alpha-1}} {\Gamma\left( \alpha\right) } \left\{ A_1\left( s\right) \left\vert u_1\left( s \right) - u_2\left( s \right) \right\vert +A_2\left( s\right) \left\vert v_1\left( s \right) - v_2\left( s \right) \right\vert \right\} ds % $$ % $$ + \displaystyle{ \sum_{t_i 1$. Note that according to \eqref{func w}, the compacts $ \lbrack 0,t]$ and $K_i\,(1\leq i \leq l)$ are included in $w(\nu )$. Hence $$ \left\vert T_1(u_1, v_1)\left( t\right)- T_1(u_2, v_2)\left( t\right) \right\vert \leq \left\{ \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } A_1( s) e^{\lambda s \frac{\mu -1}{\mu}} ds \right\} d_{w(\nu )} ( u_1 - u_2) $$ %%%%%%%%%%% $$%\displaystyle{ \begin{array}{lll} %%%%%%%%%%%%%%% \,+\, \left\{ \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } A_2( s) e^{\lambda s \frac{\mu -1}{\mu}} ds \right\} d_{w(\nu )} ( v_1 - v_2) % \\ \\ % & & \\ %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% % \,+\, \sum\limits_{i=1}^{l}\left\{ L_i d_{w(\nu )} (u_1-u_2)+ \tilde{L}_i d_{w(\nu )} (v_1-v_2) \right\}\\ \\ % \,+\, H e^{\lambda t} d_{w(\nu )} (u_1-u_2) +\tilde{H} e^{\lambda t} d_{w(\nu )} (v_1-v_2) \end{array} $$ Now, multiplying the above inequality by $e^{-\lambda t}$, we get: %%%%%%%%%%%%%%%%%%% % \begin{equation}\label{first estimation} \begin{array}{cc} e^{-\lambda t} \left\vert T_1(u_1, v_1)\left( t\right)- T_1(u_2, v_2)\left( t\right) \right\vert \leq \\ \\ \left\{ \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s \frac{\mu -1}{\mu})} A_1( s) ds \right\} d_{w(\nu )} ( u_1 - u_2) \\ \\ %%%%%%%%%%%%%%%%%%%\left\{ \right\} + \left\{ \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s \frac{\mu -1}{\mu})} A_2( s) ds \right\} d_{w(\nu )} ( v_1 - v_2)\\ \\ + \sum\limits_{i=1}^{l}\left\{ L_i d_{w(\nu )} (u_1-u_2)+ \tilde{L}_i d_{w(\nu )} (v_1-v_2) \right\}\\ \\ + H d_{w(\nu )} (u_1-u_2) +\tilde{H} d_{w(\nu )} (v_1-v_2) \end{array} \end{equation} % Let us find estimates for the integrals in \eqref{first estimation}: %%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$ I:=\displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s \frac{\mu -1}{\mu})} A_1( s) ds= %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s)} A_1( s) e^{ \frac{-\lambda s}{\mu}} ds $$ Performing the change of variable $X=\lambda(t-s)$, we get: $$ I=\frac{1}{\Gamma( \alpha) \lambda^\alpha} \displaystyle{\int_{0}^{\lambda t }} X^{\alpha -1}e^{-X} A_1\left(t-\frac{X}{\lambda}\right)e^{-\frac{\lambda}{\mu}(t-\frac{X}{\lambda})}dX $$ In view of $\left( H_{1}.(ii)\right)$, H\"older's inequality gives: $$ \begin{array}{cc} I\leq & \dfrac{1}{\Gamma( \alpha) \lambda^\alpha} {\left\{ \displaystyle{\int_{0}^{\lambda t }} {\left( X^{\alpha -1}e^{-X}\right)}^{1+\alpha}dX \right\}}^{\frac{1}{1+\alpha}} \times \\ \\ & {\left\{ \displaystyle {\int_{0}^{\lambda t }} { \left(A_1\left(t-\frac{X}{\lambda}\right)e^{-\frac{\lambda}{\mu}(t-\frac{X}{\lambda})}\right) }^{1+\frac{1}{\alpha}} dX\right\} } ^{\frac{\alpha}{1+\alpha}} \end{array} % %%%%%%%%%%%% %%%%%%%%%%%%%% \left\{ \right\} %%$\Lambda_ {\lambda,\mu}^\alpha, \tilde{\Lambda}_ {\lambda,\mu}^\alpha, %\Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ $$ Consequently: \begin{equation}\label{est.A1} \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s \frac{\mu -1}{\mu})} A_1( s) ds\leq \Lambda_ {\lambda,\mu}^\alpha \end{equation} In the same way, we can prove that \begin{equation}\label{est.A2} \displaystyle{\int_{0}^{t}}\frac{( t-s) ^{\alpha-1}} {\Gamma( \alpha) } e^{-\lambda(t-s \frac{\mu -1}{\mu})} A_2( s) ds\leq \tilde{\Lambda}_ {\lambda,\mu}^\alpha \end{equation} %%%%%%%%%%%%%% In view of \eqref{est.A1} and \eqref{est.A2}, and after taking the maximum on $\nu$, the estimation \eqref{first estimation} can be rewritten as: \begin{equation}\label{d T_1} \hspace{0.0cm} \begin{array}{cc} d_{\nu}\left(T_1(u_1, v_1), T_1(u_2, v_2)\right) \leq & \hspace{-0.5cm} \Lambda_ {\lambda,\mu}^\alpha\, d_{w(\nu )} ( u_1 - u_2) + \tilde{\Lambda}_ {\lambda,\mu}^\alpha\, d_{w(\nu )} ( v_1 - v_2)\\ \\ % &\hspace{-0.5cm} + \sum\limits_{i=1}^{l}\left\{ L_i d_{w(\nu )} (u_1-u_2)+ \tilde{L}_i d_{w(\nu )} (v_1-v_2) \right\}\\ \\ &\hspace{-0.5cm} +\ H d_{w(\nu )} (u_1-u_2) +\tilde{H} d_{w(\nu )} (v_1-v_2) \end{array} \end{equation} %%%%%%%%%%%% % \Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ Similarly, we prove that the following inequality holds true for every $(u_1,v_1), (u_2,v_2) \in X$ and every $\nu\in \mathcal{N}$: \begin{equation}\label{d T_2} \hspace{0.0cm} \begin{array}{cc} d_{\nu}\left(T_2(u_1, v_1), T_2(u_2, v_2)\right) \leq & \hspace{-0.5cm} \Lambda_ {\lambda,\mu}^\beta\, d_{w(\nu )} ( u_1 - u_2) + \tilde{\Lambda}_ {\lambda,\mu}^\beta\, d_{w(\nu )} ( v_1 - v_2)\\ \\ % &\hspace{-0.5cm} + \sum\limits_{i=1}^{m}\left\{ M_i d_{w(\nu )} (u_1-u_2)+ \tilde{M}_i d_{w(\nu )} (v_1-v_2) \right\}\\ \\ &\hspace{-0.5cm} +\ H d_{w(\nu )} (u_1-u_2) +\tilde{K} d_{w(\nu )} (v_1-v_2) \end{array} \end{equation} %%%%%%%%%%%%%%%%%%%%%% Now, \eqref{d T_1} together with \eqref{d T_2} lead to what follows for every $(u_1,v_1), (u_2,v_2) \in X$ and every $\nu\in \mathcal{N}$: \begin{equation}\label{Contraction de T} D_{\nu}\left(T(u_1, v_1), T(u_2, v_2)\right) \leq M_{\alpha, \beta}\left(\lambda,\mu \right) D_{w(\nu)}\left(\left(u_1, u_2\right),\left(v_1, v_2\right) \right) \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % That is \eqref{Generalized contraction} holds true with $M_\nu =M_{\alpha, \beta}\left(\lambda,\mu \right)$, which is independent of $\nu$. Consequently the series (\ref{Condition series}) turns in our case into % \begin{equation}\label{C_series} % \sum\limits_{n=0}^{\infty} M^{n+1}_{\alpha, \beta}\left(\lambda,\mu \right) D_{w^n(\nu )}\left(u,v \right) \end{equation} According to \eqref{w carré =w}, we have: \begin{equation*}\label{w-bounded} sup\left\lbrace D_{w^n(\nu )}\left(u,v \right): n = 0, 1,2, \dots \right\rbrace = sup\left\lbrace D_{\nu }\left(u,v \right), D_{w(\nu )}\left(u,v \right) \right\rbrace< \infty. %% \end{equation*} Since, moreover $M_{\alpha, \beta}\left(\lambda,\mu \right)$ is convergent to zero, then the series in \eqref{C_series} converges too. That is $T$ is a generalized contraction and the result follows so, from Theorem \ref{fixed pt th} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % \end{proof} %%%%%%%%%%%%%%%% \begin{rem}\label{Equiv C. for conv tozero} In view of Lemme \ref{Prop.Matrix}, the following condition is equivalent to \eqref{Condition in Th} % %%%%% $\Lambda_ {\lambda,\mu}^\alpha, \tilde{\Lambda}_ {\lambda,\mu}^\alpha, %\Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ \begin{equation}\label{Rq.conv matrix M} \begin{array}{cc} \sqrt{\scriptstyle{\left( \sum\limits_{i=1}^{l}L_i+ \Lambda_ {\lambda,\mu}^\alpha+H -\sum\limits_{i=1}^{m}\tilde{M}_i- \tilde{\Lambda}_ {\lambda,\mu}^\beta-\tilde{K}\right)^2 +4 \left(\sum\limits_{i=1}^{l}\tilde{L}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\alpha+\tilde{H} \right ) %%%%%%%% \left(\sum\limits_{i=1}^{m}M_i+ \Lambda_ {\lambda,\mu}^\beta+K \right) }} \\ + \sum\limits_{i=1}^{l}L_i+ \Lambda_ {\lambda,\mu}^\alpha+H + \sum\limits_{i=1}^{m}\tilde{M}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\beta+\tilde{K} < 2 \end{array} \end{equation} % \end{rem} %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% The following Corollary provides a global existence-uniqueness result for a particular class of \eqref{Sys}-\eqref{Impulsive.con.Sys}. \begin{cor}\label{Cor 1} Assume that in addition of $\left( H_{1}\right)-\left( H_{3}\right)$ and \eqref{C.compatibility with App A_1}, the following hypothesis holds true: \begin{equation}\label{Hypothesis H1 iii} \forall \epsilon > 0 ,\, \exists \lambda > 0,\, \mu > 1, \text{\,such that:\,\,\,} S_{\lambda,\mu}, \, \tilde{S}_{\lambda,\mu},\, R_{\lambda,\mu},\, \tilde{R}_{\lambda,\mu}< \epsilon. \end{equation} Then, \eqref{Sys}-\eqref{Impulsive.con.Sys} admits a unique global solution in $X$ provided that: \begin{equation}\label{Matrix Q} %\text{The matrix\,\,} Q :=\left(\begin{array}{cc} \sum\limits_{i=1}^{l}L_i+H \ \ \ \sum\limits_{i=1}^{l}\tilde{L}_i+\tilde{H} \\ [5mm] \sum\limits_{i=1}^{m}M_i+K \ \ \ \ \sum\limits_{i=1}^{m}\tilde{M}_i+\tilde{K} \end{array} \right), \text{\,converges to zero} \end{equation} %\begin{equation}\label{C.conv matrix. Cor} % % \sqrt{\scriptstyle{ \left(\sum\limits_{i=1}^{l}L_i+H %-\sum\limits_{i=1}^{m}\tilde{M}_i-\tilde{K}\right)^2 %+4 \left(\sum\limits_{i=1}^{l}\tilde{L}_i+\tilde{H} \right ) %%%%%%%% %\left(\sum\limits_{i=1}^{m}M_i+K \right) %}} % \\ %+ \sum\limits_{i=1}^{l}L_i+H + %\sum\limits_{i=1}^{m}\tilde{M}_i+\tilde{K} < 2 % %\end{equation} \end{cor} %%%%%%%%%%%%%%%% \begin{proof} Note first that $M_{\alpha, \beta}\left(\lambda,\mu \right)=P_{\alpha, \beta}\left(\lambda,\mu \right) + Q$, where \begin{equation*}%\label{Matrix M} P_{\alpha, \beta}\left(\lambda,\mu \right) :=\left(\begin{array}{cc} \Lambda_ {\lambda,\mu}^\alpha \ \ \ \tilde{\Lambda}_ {\lambda,\mu}^\alpha \\ [5mm] \Lambda_ {\lambda,\mu}^\beta \ \ \ \ \tilde{\Lambda}_ {\lambda,\mu}^\beta \end{array} \right) \,% \end{equation*} % %By computing the eigenvalues of $Q$ and according to Lemma \ref{Prop.Matrix}, the condition \eqref{C.conv matrix. Cor} implies that $Q$ converges to zero. It is not hard to see that under hypothesis \eqref{Hypothesis H1 iii}, the elements of $P_{\alpha, \beta}\left(\lambda,\mu \right)$ are small enough. Hence, in view of \eqref{Matrix Q} together with Lemma \ref{Lemma.Nica}, $M_{\alpha, \beta}\left(\lambda,\mu \right)$ is convergent to zero and the result follows so from Theorem \ref{Th1}. %Where %$\Lambda_ {\lambda,\mu}^\alpha, \tilde{\Lambda}_ {\lambda,\mu}^\alpha, %\Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ %\begin{equation} \end{proof} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%{Rq_impvov2}{Rq_impvov} % When $I_i=J_i=0$ for every $i \in \mathbb{N}^*$, that is by omitting the impulsive condition \eqref{Impulsive.con.Sys}, then the problem \eqref{Sys}-\eqref{Impulsive.con.Sys} is reduced to: \begin{equation}\label{Sys without impulses} \begin{cases} ^{C}\!D_{0^+}^{\alpha } u( t) =f(t,u( t),v( t)), & t>0 \\%[5mm] ^{C}\!D_{0^+}^{\beta } v( t) =g(t,u( t),v( t)), & t>0 \\ u(0)=\varphi(u,v), & \\ v(0)=\psi (u,v), \end{cases} \end{equation} In this particular case, we have: $$ \Delta u (t_{i}):= u( t_{i}^{+} ) - u( t_{i}^{-} ) = I_{i}( u( t_i),v(t_i))=0 $$ and $$ \Delta v (t_{i}):= v( t_{i}^{+} ) - v( t_{i}^{-} ) = J_{i}( u( t_i),v(t_i))=0 $$ Which means that the space $X=\mathcal{PC}(\mathbb{R}_+)\times \mathcal{PC}(\mathbb{R}_+)$, where $\mathcal{PC}(\mathbb{R}_+)$ is defined by \eqref{Space PC}, becomes $\mathcal{C}(\mathbb{R}_+)\times \mathcal{C}(\mathbb{R}_+)$. So, as particular cases of Theorem \ref{Th1} and Corollary \ref{Cor 1}, we have the following Corollary. \begin{cor}\label{Cor 2} Under hypotheses $\left( H_{1}\right)-\left( H_{2}\right)$, the system \eqref{Sys without impulses} admits a unique global solution in $\mathcal{C}(\mathbb{R}_+)\times \mathcal{C}(\mathbb{R}_+)$ provided that: there exist $ \lambda > 0$ and $\mu > 1$ such that: \begin{equation}\label{M tilde} %\text{The matrix\,\,} \tilde{M}_{\alpha, \beta}\left(\lambda,\mu \right) =\left(\begin{array}{cc} \sum\limits_{i=1}^{l}L_i+ \Lambda_ {\lambda,\mu}^\alpha \ \ \ \sum\limits_{i=1}^{l}\tilde{L}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\alpha \\ [5mm] \sum\limits_{i=1}^{m}M_i+\Lambda_ {\lambda,\mu}^\beta \ \ \ \sum\limits_{i=1}^{m}\tilde{M}_i+ \tilde{\Lambda}_ {\lambda,\mu}^\beta \end{array} \right) \text{converges to zero} \end{equation} where $\Lambda_ {\lambda,\mu}^\alpha, \tilde{\Lambda}_ {\lambda,\mu}^\alpha, \Lambda_ {\lambda,\mu}^\beta, \tilde{\Lambda}_ {\lambda,\mu}^\beta$ are given by \eqref{Coefficient of M}. If in addition, \eqref{Hypothesis H1 iii} holds true, then \eqref{M tilde} is weakened to \begin{equation}\label{Matrix Q tilde} %\text{The matrix\,\,} \tilde{Q} :=\left(\begin{array}{cc} \sum\limits_{i=1}^{l}L_i \ \ \ \sum\limits_{i=1}^{l}\tilde{L}_i \\ [5mm] \sum\limits_{i=1}^{m}M_i \ \ \ \ \sum\limits_{i=1}^{m}\tilde{M}_i \end{array} \right) \text{\,converges to zero.} \end{equation} \end{cor} %%%%%%%%%%%%%%%% Wang2018 BERREZOUG2017 %%%%%%%%%%%%%%%%% \begin{rem}\label{Rq_impvov} Note that \eqref{Hypothesis H1 iii} includes the Lipschitz condition with constant and integrable arguments. In this case, the matrices $Q$ in \eqref{Matrix Q} and $\tilde{Q}$ in \eqref{Matrix Q tilde} are independent of $A_i, B_i\, (i=1,2)$. Moreover, with the classical initial conditions, $\sum\limits_{i=1}^{l}L_i$, $\sum\limits_{i=1}^{l}\tilde{L}_i$, $\sum\limits_{i=1}^{m} M_i$ and $\sum\limits_{i=1}^{m}\tilde{M}_i$ vanish. All this, allows us to see clearly that Corollary \ref{Cor 1} and Corollary \ref{Cor 2} provide significant improvements and generalizations of many recent results in the literature, such as {\cite[Theorem 15]{J.Nieto2020}}, {\cite[Theorem 3.3]{Wang2018}}, {\cite[Theorem 3.1]{Wang2014}}, {\cite[Theorem 3.2]{Wang2014}} and {\cite[Theorem 3.2]{BERREZOUG2017}}. \end{rem} %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% \section{Example}\label{Sec:4} Let us consider the following system: \begin{equation}\label{Sys.exemple} \begin{cases} ^{C}\!D_{0^+}^{\frac{15}{20} } u( t) =\displaystyle{\dfrac{1}{10}e^{\frac{t}{80}} \left(2u( t)+v( t)\right)}, & t>0,\,t\neq t_i=10^i,\, i\in \mathbb{N}^* \\[2mm] ^{C}\!D_{0^+}^{\frac{13}{20} } v( t) =\dfrac{1}{10}e^{\frac{t}{80}} \left(u( t)+v( t)\right), & t>0,\,t\neq t_i= 10^i,\, i\in \mathbb{N}^* \\%[5mm] %%%%%%%%%%%%%%%% \Delta u (t_i)\, = \frac{7}{ 25i(i+1)\left( 1 + \vert u\left(t_i\right)\vert \right)}+\frac{5}{ 25i(i+1)\left( 1 + \vert v\left(t_i\right)\vert \right)},& i\in \mathbb{N}^* \\[2mm] %%%%%%%%%%%%%%%% \Delta v (t_i)\, = \frac{6}{ 25\times 2^i\left( 1 + \vert u\left(t_i\right)\vert \right)}+\frac{9}{ 25\times 2^i\left( 1 + \vert v\left(t_i\right)\vert \right)},& i\in \mathbb{N}^*\\[2mm] %%%%%%%%%%%%%%%% u(0)=\frac{1}{10}\sup\limits_{\!t\in \lbrack 0 ,\, 1]}\, u\left( t\right)+\frac{1}{5}\sup\limits_{\!t\in \lbrack 0 ,\, \frac{1}{2}]}\, v\left( t\right) \\%[5mm] %%%%%%%%%%%%%%%% v(0)=\frac{1}{5}sin\left( u(\frac{1}{6})+v(\frac{1}{3}) \right) \end{cases} \end{equation} % %%%%%%%%%%%%%%%%%%{Sys} {Impulsive.con.Sys} The problem \eqref{Sys.exemple} is identified to \eqref{Sys}-\eqref{Impulsive.con.Sys}, with: \begin{equation*} \begin{matrix} \alpha=\frac{15}{20}, f\left(t, \xi ,\eta\right) =\dfrac{1}{10}e^{\frac{t}{80}}\left(2\xi+\eta\right),\\ \\ \,I_i\left( \xi ,\eta\right)=\dfrac{7}{ 25i(i+1)\left( 1 + \vert \xi\vert \right)}+\dfrac{5}{ 25i(i+1)\left( 1 + \vert \eta\vert \right)}\\ \\ %%%%%%%%%%%%%%%%% \beta=\frac{13}{20}, g\left(t, \xi ,\eta\right) =\dfrac{1}{10}e^{\frac{t}{80}}\left(\xi+\eta\right), \\ \\ J_i\left( \xi ,\eta\right)=\dfrac{6}{ 25\times 2^i\left( 1 + \vert \xi\vert \right)}+\dfrac{9}{ 25\times 2^i\left( 1 + \vert \eta\vert \right)}\\\\ %%%%%%%%%%%%%% \varphi(u,v)=\frac{1}{10}\sup\limits_{\!t\in \lbrack 0 ,\, 1]}\, u\left( t\right)+\frac{1}{5}\sup\limits_{\!t\in \lbrack 0 ,\, \frac{1}{2}]}\, v\left( t\right),\,\,\,\psi(u,v)=\frac{1}{5}sin\left( u(\frac{1}{6})+v(\frac{1}{3}) \right) \end{matrix} \end{equation*} It is not hard to see that $\left( H_{1}.(i)\right) $ is satisfied with: $$ A_1(t)=\dfrac{1}{5}e^{\frac{t}{80}},\,\,\, A_2(t)=B_1(t)=B_2(t)=\dfrac{1}{10}e^{\frac{t}{80}} $$ A straightforward computation leads to: $$ S_{\lambda,\mu}=\frac{7 \lambda}{30 \mu},\, \tilde{S}_{\lambda,\mu}=\frac{ \lambda}{10 \mu},\,R_{\lambda,\mu}=\tilde{R}_{\lambda,\mu}=\frac{23 \lambda}{260 \mu} $$ Which means that $\left( H_{1}.\left(ii\right)\right) $ is satisfied too. It can be easily seen that $\left( H_{2}\right) $ is satisfied with: \begin{equation*} \begin{matrix} l=2, L_1= \dfrac{1}{10}e^{\lambda}, L_2=0,\, \tilde{L}_1=0,\,\tilde{L}_2= \dfrac{1}{5}e^{\frac{\lambda}{2}},\, K_1=[ 0, 1],\, K_2=[ 0, \frac{1}{2}] \\ \\ %%%%%%%%%%%%%%%%%%%%%%%% m=2, M_1= \dfrac{1}{10}e^{\frac{\lambda}{6}}, M_2=0,\, \tilde{M}_1=0,\,\tilde{M}_2= \dfrac{1}{5}e^{\frac{\lambda}{3}},\, \tilde{K}_1=[ 0, \frac{1}{6}],\, \,\tilde{K}_2=[ 0, \frac{1}{3}] \end{matrix} \end{equation*} For all $i \in \mathbb{N}^*$ we have: \begin{equation*} \begin{matrix} \vert I_i\left( \xi_1 ,\eta_1\right) - I_i\left( \xi_2 ,\eta_2\right) \vert \leq \dfrac{7}{ 25i(i+1)} \vert \xi_1-\xi_2 \vert + \dfrac{5}{ 25i(i+1)} \vert \eta_1-\eta_2 \vert \\ \\ %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% \vert J_i\left( \xi_1 ,\eta_1\right) - J_i\left( \xi_2 ,\eta_2\right) \vert \leq \dfrac{6}{ 25\times 2^i} \vert \xi_1-\xi_2 \vert + \dfrac{9}{ 25\times 2^i} \vert \eta_1-\eta_2 \vert \end{matrix} \end{equation*} That is, $\left( H_{3}\right) $ is satisfied with: $$ \begin{matrix} \lbrace h_i \rbrace=\left \lbrace \frac{7}{ 25i(i+1)} \right\rbrace, \lbrace \tilde{h}_i \rbrace= \left\lbrace \frac{5}{ 25i(i+1)} \right\rbrace,\, \lbrace k_i \rbrace=\left\lbrace \frac{6}{ 25\times 2^i} \right\rbrace,\, \lbrace \tilde{k}_i \rbrace=\left\lbrace \frac{9}{ 25\times 2^i}\right \rbrace\\ \\ %%%%%%%%%%% H=\frac{7}{ 25}, \,\,\, \tilde{H}=\frac{5}{ 25}, \,\,\,K=\frac{6}{ 25}, \,\tilde{K}=\frac{9}{ 25} \end{matrix} $$ If we choose $\lambda=\frac{1}{2}$ and $\mu=20$, the matrix $M_{\alpha, \beta}\left(\lambda,\mu \right)$ given in \eqref{Matrix M}, becomes in this case: \begin{equation*} M_{\alpha, \beta}\left(\lambda,\mu \right) :=\left(\begin{array}{cc} 0.486788 \ \ \ 0.498721 \\ [5mm] 0.474757 \ \ \ \ 0.495513 \end{array} \right), \end{equation*} which admits the following eigenvalues: $\lambda_1=0.977761<1$ and $\lambda_2=0.00453945<1$ and consequently $M_{\alpha, \beta}\left(\lambda,\mu \right)$ converges to zero. 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