A Bilocal Problem Associated to a Fractional Differential Inclusion of Caputo-Fabrizio Type

Abstract

A fractional differential inclusion defined by Caputo-Fabrizio fractional derivative with bilocal boundary conditions is studied. A nonlinear alternative of Leray-Schauder type, Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and Covitz-Nadler set-valued contraction principle are employed in order to obtain the existence of solutions when the set-valued map that define the problem has convex or non convex values.

Several Variational Inclusions for a Fractional Differential Inclusion of Caputo-Fabrizio Type

“…The next result ( [5]) is an extension of Filippov's theorem concerning the existence of solutions to a Lipschitzian differential inclusion to fractional differential inclusions of the form (1.1).…”

“…This new definition is able to describe better heterogeneousness, systems with different scales with memory effects, the wave movement on surface of shallow water, the heat transfer model, mass-spring-damper model etc.. Another good property of this new definition is that using Laplace transform of the fractional derivative the fractional differential equation turns into a classical differential equation of integer order. Some properties of this definition have been studied in [1,4,8] etc.. Several papers are devoted to the development of this new fractonal derivative [6,7,8,9,10,11] etc.. In Control Theory, mainly, if we want to obtain necessary optimality conditions, it is essential to have several "differentiability" properties of solutions with respect to initial conditions.…”

“…For all these results and for a complete discussion on this topic we refer to [2]. The proofs of our results follows by an approach similar to the classical case of differential inclusions ( [2]) and use a recent result ( [5]) concerning the existence of solutions of problem (1.1).…”