3.1 Introduction
**Fractional** di¤erential **equations** is a generalization **of** ordinary di¤erential **equations** **and** integration to arbitrary non-integer orders. **The** origin **of** **fractional** calculus goes back to Newton **and** Leibniz in **the** seventeenth century. **Fractional** di¤erential equa- tions appear naturally in a number **of** …elds such as physics, engineering, biophysics, blood ‡ow phenomena, aerodynamics, electron-analytical chemistry, biology, control **the**- ory, etc. An excellent account **of** **the** **study** **of** **fractional** di¤erential **equations** can be found in [2, 3, 19, 34, 39, 44, 45, 48, 49, 52, 54, 61] **and** **the** references therein. Boundary value problems for **fractional** di¤erential **equations** have been discussed in [5, 6, 15, 43, 56, 57, 58, 60, 61, 63, 64]. By contrast, **the** development **of** **stability** for solutions **of** **fractional** di¤erential **equations** is a bit slow. El-Sayed, Gaafar **and** Hama- dalla [22] discuss **the** **existence**, **uniqueness** **and** **stability** **of** solutions for **the** non-local non-autonomous system **of** **fractional** order di¤erential **equations** with delays

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2 UMAB University **of** Mostaganem
LPAM, Faculty SEI, UMAB University **of** Mostaganem, Algeria
Abstract In this paper, we **study** a nonlinear coupled system **of** n−**fractional** dif- ferential **equations**. Applying Banach contraction principle **and** Schaefer’s fixed point theorem, new **existence** **and** **uniqueness** results are established. We also give some concrete examples to illustrate **the** possible application **of** **the** established analytical results.

[29] S. M. Mechee, K. Tabatabaei **and** E. Celik, **The** numerical method for solving **differential** **equations** **of** Lane-Emden type by pade approximation, Discrete Dyn. Nat. Soc. 2011 (2011), 1–9.
[30] S. A. Okunuga, J. O. Ehigie **and** S. A. Sofoluwe, Treatment **of** Lane-Emden type **equations** via sec- ond derivative backward differentiation formula using boundary value technique, in: Proceedings **Of** **The** World Congress On Engineering, July 4 - 6, London, U.K., 2012, p. 224.

In **the** present paper, we provide a generic treatment **of** **the** limiting macroscopic equa- tion ( 1.1 ) **and** we allow for (infinite activity) jumps in Z. For instance, Hawkes processes can be recovered by setting c = 0 **and** ν = δ 1 . **The** strategy we adopt is based on approximations
using stochastic Volterra **equations** with L 2 kernels, whose **existence** **and** **uniqueness** theory is now well–established, see Abi Jaber et al. ( 2019a , b ) **and** **the** references therein. By doing so, we avoid **the** infinite-dimensional analysis used for super-processes, we also circumvent **the** need to **study** scaling limits **of** Hawkes processes, allowing for more generality in **the** choice **of** kernels K **and** input functions G 0 . Along **the** way, we derive a general **stability**

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Chapter 2
**Existence** **and** **Uniqueness** Results for Impulsive Evolution **Equations** **The** problem **of** **existence** **and** **uniqueness** **of** **the** solution **of** impulsive evolution **equations** is similar to that **of** **the** corresponding ordinary evolution **equations**. **The** linear impulsive evolution **equations** in a Banach space have, for **the** …rst time, been considered by D. D. Bainov [12 15]. **The** **existence** **of** solutions, classical **and** mild, are established by Hernandez [54; 55], J. H. Liu, [78], Y. V. Rogovchenko [94], W. Zhang, R. P. Agarwal, E. Akin-Bohner [116], Chen Fangqi, chen Yushu [43], Benchohra [20; 21; 24] **and** Lakshmikanthan, [67] for linear **and** nonlinear cases. In this chapter, we present some basic properties **of** **the** impulsive problem in a Banach space. For more details one may refer to Hernandez [54; 55] **and** J. H. Liu [78] : We construct a new impulsive evolution operator corresponding to **the** impulsive evolution system **and** introduce a suitable de…nition **of** a PC-mild solution. **The** impulsive evolution operator can be used to reduce **the** **existence** **of** PC-mild solution for nonhomogeneous linear impulsive system to **the** **existence** **of** …xed points for some operator equation.

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HIGH DIMENSIONAL **FRACTIONAL** COUPLED SYSTEMS: NEW **EXISTENCE** **AND** **UNIQUENESS** RESULTS
LOUIZA TABHARIT 1 **AND** ZOUBIR DAHMANI 2
Abstract. In this paper, we **study** a class **of** high dimensional coupled **fractional** **differential** systems using Caputo approach. We investigate **the** **existence** **of** solutions using Schaefer fixed point theorem. Moreover, new **existence** **and** **uniqueness** results are obtained by using **the** contraction mapping principle. Finally, Some examples are presented to illustrate our main results.

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We introduce some definitions **and** some auxiliairy results that will be used in **the** paper. We begin by **the** following definition:
Definition 2.1 [13] **The** Hadamard **fractional** integral **of** order α > 0 **of** a function f ∈ C([a, b]),0 ≤ a ≤ b ≤ ∞, is defined as

Abstract
This paper studies **the** **existence** **of** solutions for a coupled system **of** nonlinear **fractional** integro-**differential** **equations** involving Riemann-Liouville integrals with several continuous functions. New **existence** **and** **uniqueness** results are established using Banach fixed point theorem, **and** other **existence** results are obtained using Schaefer fixed point theorem. Some illustrative examples are also presented.

Abstract
In this paper, **the** Banach contraction principle **and** Schaefer theorem are applied to establish new results for **the** **existence** **and** **uniqueness** **of** solutions for some Caputo **fractional** **differential** **equations**. Some examples are also discussed to illustrate **the** main results.

Keywords: Caputo derivative, fixed point, **differential** equation, **existence**, **uniqueness**. 2010 MSC: 34A34, 34B10.
1. INTRODUCTION
In **the** last few decades, there has been an explosion **of** research activities on **the** application **of** **fractional** calculus to very diverse scientific fields ranging from **the** physics **of** diffusion **and** advection phenomena, to control systems, finance **and** eco- nomics. For more details, we refer **the** reader to [5, 6, 8, 13, 14, 15, 17, 21, 22] **and** **the** reference therein. Moreover, **the** **study** **of** **fractional** order **differential** **equations** is also important in various problems **of** applied sciences, **and** has attracted **the** atten- tion **of** many authors. Considerable work has been done in this field **of** research, for instance, see [1, 2, 3, 4, 7, 9, 10, 11, 12, 16, 18, 19, 20, 23].

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In this paper, we **study** a class **of** boundary value problems **of** nonlin- ear **fractional** **differential** **equations** with integral boundary conditions. Some new **existence** **and** **uniqueness** results are obtained by using Ba- nach fixed point theorem. Other **existence** results are also presented by using Krasnoselskii theorem.

iii) For each r > 0, there exists h r ∈ L 1 ([0, T ] , R) such that, for almost all t ∈ [0, T ]
**and** for all z with |z| < r, we have |f (t, z)| ≤ h r (t).
2.2 Fixed point theorems
**The** fixed point theorem, generally known as **the** Banach contraction principle, appeared in explicit form in Banach’s thesis in 1922 where it was used to establish **the** **existence** **of** a solution for an integral equation. Since then, because **of** its simplicity **and** usefulness, it has become a very popular tool in solving **existence** problems in many branches **of** mathematical analysis **and** forms an attractive tool which facilitates **the** **study** **of** **stability** for **the** **differential** **equations** with or without delay.

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1. Introduction **and** Preliminaries
**The** physical laws **of** dynamics are not always described by ordinary order dif- ferential **equations**. In some cases, their behavior is governed by **fractional** or- der diﬀerential **equations**. For more details, we refer **the** reader to **the** books **of** Hilfer in [14] **and** Podlubny in [28]. Other research works can be found in [2, 4, 7, 9, 20, 24, 25, 26, 27, 29, 31, 32]. Further, many authors have established **the** ex- istence **and** **uniqueness** **of** solutions for some **fractional** systems. **The** reader may re- fer to **the** following research papers [1, 3, 5, 6, 8, 10, 11, 12, 15, 16, 17, 18, 19, 21, 33, 34]. These papers treat problems with only one nonlinear term depending on two un- known functions. Other cases, where we have more than one nonlinearity de- pending on some unknown functions, are more complicated **and** have not been discussed in **the** above cited works.

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with functional delays by using Lyapunov functionals, many di¢ culties arise if **the** delay is unbounded or if **the** di¤erential equation in question has unbounded terms, (see [31], [32], [38], [64], [102]). In recent years, several investigators have tried **stability** by using a new technique. Particularly, Burton, Furumochi, Zhang **and** others began a **study** in which they noticed that some **of** these di¢ culties vanish or might be overcome by means **of** …xed point theory (see [3]–[22], [26]–[44], [66]–[68], [83]–[93], [94]–[97], [56], [113], [114]). **The** most striking object is that **the** …xed point method does not only solve **the** problem but has a signi…cant advantage over Liapunov’s direct method. **The** conditions **of** former are always average while those **of** **the** latter are pointwise. Further, while it remains an art to construct a Liapunov’s functional when it exists, a …xed point method, in one step, yields **existence** (sometimes **uniqueness**) **and** **stability**. All we need, to use **the** …xed point method, is a complete metric space, a suitable …xed point theorem **and** an elementary integral methods to solve problems that have frustrated investigators for decades.

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J. Ineq. Pure **and** Appl. Math. 5(2) Art. 40, 2004 http://jipam.vu.edu.au **differential** **equations**. **The** main purposes **of** this paper are to investigate **the**
hyper-exponent **of** convergence **of** distinct zeros **and** **the** hyper-order **of** infinite order solutions for **the** above equation. We will prove **the** following two theo- rems:

Zinelˆ aabidine Latreuch **and** Benharrat Bela¨ıdi
Abstract. **The** main purpose **of** this paper is to **study** **the** growth **of** **certain** combinations **of** entire solutions **of** higher order complex linear diﬀerential **equations**.
2010 Mathematics Subject Classiﬁcation. Primary 34M10; Secondary 30D35.