Existence theory for fractional differential equations with non-separated type nonlocal multi-point and multi-strip boundary conditions

Bashir Ahmad,Ahmed Alsaedi,Ravi P Agarwal,Sotiris K Ntouyas,Wafa Shammakh

doi:10.1186/s13662-018-1546-6

Abstract

We introduce a more general class of fractional-order boundary value problems involving non-separated type multi-point and multi-strip boundary conditions. Several existence and uniqueness results for the given problem are established by applying the tools of fixed-point theory. Some illustrative examples are also included. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

Highlights

In the last few decades, fractional-order differential equations equipped with a variety of boundary conditions have been studied
6 Conclusions We have studied a nonlinear fractional differential equation with nonlinearity depending on the unknown function together with its lower-order fractional derivative, equipped with a general type of non-separated boundary conditions involving finite many nonlocal points and sub-segments of the interval [0, 1]
Several existence and uniqueness results have been derived by applying different tools of the fixed point theory

Summary

Introduction

In the last few decades, fractional-order differential equations equipped with a variety of boundary conditions have been studied. 1 (1 – s)q–1 f s, x(s), cDβ x(s) and observe that problem (1.1)–(1.2) has solutions if and only if operator (2.9) has fixed points. Lemma 3.1 (Schauder’s fixed point theorem [25]) Let U be a closed, convex, and nonempty subset of a Banach space X.

Results

Conclusion