Existence results for Caputo type fractional differential equations with four-point nonlocal fractional integral boundary conditions

Bashir Ahmad,Ahmed Alsaedi,Afrah Assolami

doi:10.14232/ejqtde.2012.1.93

Abstract

This paper presents some existence and uniqueness results for a boundary value problem of fractional differential equations of order � 2 (1,2] with fourpoint nonlocal fractional integral boundary conditions. Our results are based on some standard tools of fixed point theory and nonlinear alternative of LeraySchauder type. Some illustrative examples are also discussed.

Highlights

A variety of problems involving differential equations of fractional order have been investigated by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions
It is found that the differential equations of arbitrary order provide an excellent instrument for the description of memory and hereditary properties of various materials and processes
In this paper, motivated by [22], we discuss the existence of solutions for a four-point nonlocal boundary value problem for Caputo type fractional differential equations of order α ∈ (1, 2] with fractional integral boundary conditions given by cDαx(t) = f (t, x(t)), 1 < α ≤ 2, t ∈ [0, 1]

Summary

Introduction

A variety of problems involving differential equations of fractional order have been investigated by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. It is found that the differential equations of arbitrary order provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. With these features, the fractional order models become more realistic and practical than the classical integer-order models. In this paper, motivated by [22], we discuss the existence of solutions for a four-point nonlocal boundary value problem for Caputo type fractional differential equations of order α ∈ (1, 2] with fractional integral boundary conditions given by cDαx(t) = f (t, x(t)), 1 < α ≤ 2, t ∈ [0, 1], x(0) = aIα−1x(η) = a η 0. 0 < η < σ < 1, where a and b are arbitrary real constants, cDα denotes the Caputo fractional derivative of order α and f : [0, 1] × R → R is a given continuous function

Preliminaries

Existence results