Haut PDF Study of the existence, uniqueness and stability of certain fractional differential equations

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

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

The study of stability for this kind of systems focuses a great interest in the research community. We can cite in this domain the works of Matignon [13] and Bonnet and Partington [2] for the stability of linear fractional systems, those of Khusainov [10], Bonnet and Partington [3], Chen and Moore [5] and Deng et al. [7] for fractional systems with time delay and Ladaci et al. [11] for fractional adaptive control systems. Ahn et al. have proposed robust stability test methods for fractional order systems [1], [6]. Recently Lazarevi´c [12] has studied the finite time stability of a fractional order controller for robotic time delay systems.

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.

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.

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].

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.

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.

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.

The present paper is related to another framework of influence originally introduced in Hoede and Bakker (1982) and later refined in our several works. In the original one-step model, agents have to make their acceptance-rejection decision on a certain issue. Each agent has an inclination to say either YES or NO, but due to a possible influence of the other agents, his final decision (YES or NO) may be different from his initial inclination. Our first results concerning this model are presented in Grabisch and Rusinowska (2010a) where we investigate several tools to analyze the influence. In particular, we define the influence indices to measure the influence of a coalition on an agent, introduce several influence functions and study their properties, investigate the concept of a follower of a coalition. In Grabisch and Rusinowska (2010b) we generalize the YES-NO model of influence to a framework in which each agent has an ordered set of possible actions, and in Grabisch and Rusinowska (2010c) we assume a continuum of actions. Our results presented in Grabisch and Rusinowska (2009) concern a comparison of the influence model with the framework of command games (Hu and Shapley (2003b,a)). We show that our framework of influence is more general than the framework of the command games. In particular, we define several influence functions which capture the command structure. For some influence functions we define the equivalent command games. In Grabisch and Rusinowska (Forthcoming 2011) we establish exact relations between the key concepts of the influence model and the framework of command games. More precisely, we study the relations between: influence functions and follower functions, command games and command functions, and between command games and influence functions.

The idea is to find critical points of the functional J defined as 1.2 . Indeed, we shall obtain the existence of a minimizer on a subset of almost periodic functions taking their values into S and using the hypothesis 2.2 we show that this minimizer is a critical point of the functional J on a space of functions with values into the whole space H. We conclude finally that the minimizer that we found is a Besicovitch

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:

of Theorem A. Let d 0 , d 1 , d 2 be complex constants that are not all equal to zero.If f (z) 6≡ 0 is any meromorphic solution of equation (1.2), then: (i) f, f 0 , f 00 all have infinitely many fixed points and satisfy λ (f − z) = λ  f 0 − z  = λ  f 00 − z  = ∞, (ii) the differential polynomial

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.