Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions

Bashir Ahmad,Ravi P Agarwal,Ahmed Alsaedi

doi:10.1186/s13661-016-0533-7

Abstract

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.

Highlights

) to a fixed point problem, we define the operator W : D → D as t (t – s)q–
1 Introduction In this paper, we study a boundary value problem of nonlinear fractional differential equations with semiperiodic and three-point boundary conditions given by cDqx(t) + f t, x(t) =, t ∈ [, ], < q ≤, x( ) = x( ), ξ x ( ) – ηx ( ) = ζ x( / ), ( . )
Multipoint nonlocal boundary value problems of ordinary, integro-differential, and partial differential equations have been extensively studied, and a variety of results can be found in the recent literature, for instance, see [ – ] and the references therein

Summary

Introduction

) to a fixed point problem, we define the operator W : D → D as t (t – s)q– Applies, and the operator W has a fixed point x ∈ P, which is a solution of problem 2.2 Examples Consider the three-point boundary value problem of nonlinear fractional differential equations cD / x(t) + f t, x(t) = , t ∈ [ , ], < q ≤ , x( ) = x( ),

Results

Conclusion