Sur l’existence et la ULAM-stabilité de certaines classes de systèmes dynamiques à dérivées arbitraires

Abstract

Purpose – In this paper, we investigate the existence of solutions for a class of Caputo fractional differential inclusions with two integral boundary conditions. The convex and nonconvex cases are separately considered. For the first case, an existence result is obtained by applying the Bohnenblust–Karlin’s fixed point theorem. For the second case, a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler is used. Further, an illustrating example is presented. Design/methodology/approach – In Section 2, we recall some basic concepts of the fractional calculus and the theory of multi-valued maps. Some well-known existence results are also recalled. In Section 3, we prove the existence result for the problem (1.1)–(1.3) when, in one case, the right-hand side is convex valued, and in the other case, nonconvex valued. The first result relies on the Bohnenblust–Karlin theorem, while the other is based upon a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler [37]. In Section 4, we propose an example to illustrate our results. Findings – This paper is concerned with the existence of solutions for a certain class of fractional differential inclusion with integral boundary conditions. Thanks to the Bohnenblust–Karlin’s fixed point theorem, an existence result is obtained. Originality/value – Considering the particular case of single-valued second member, this result is then used to derive an existence result for a certain type of singular boundary value problems.