Existence of solutions for the delayed nonlinear fractional functional differential equations with three-point integral boundary value conditions

Kaihong Zhao,Kun Wang

doi:10.1186/s13662-016-1012-2

Abstract

This paper is concerned with the three-point integral boundary value problems of time-delay nonlinear fractional functional differential equations involving Caputo fractional derivatives of order $\alpha\in(2,3)$ . By employing the Schauder fixed point theorem, the Banach contraction principle, and a nonlinear alternative of Leray-Schauder type, some sufficient criteria are established to guarantee the existence of solutions. Our study improves and extends the previous results in the literature. As applications, some examples are provided to illustrate our main results.

Highlights

1 Introduction This paper is considered with the existence and uniqueness of solutions to the integral boundary value problems for the nonlinear fractional differential equations ( . )-( . ): cDα +u(t) + f t, ut, u (t), cDβ +u(t) =, t ∈ J = (, ], ( . )
Fractional differential equations have played a significant role in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electronanalytical chemistry, biology, control theory, fitting of experimental data, and so forth
Fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes

Summary

Introduction

This paper is considered with the existence and uniqueness of solutions to the integral boundary value problems (short for BVP) for the nonlinear fractional differential equations There are relatively scarce results dealing with the boundary value problems of fractional functional differential equations with time delays. The aim of this paper is to study the existence of solutions for triple-point boundary value problems of fractional functional differential equations with time delays and integer boundary value conditions. In [ ], Rehman et al studied the existence and uniqueness of solutions to nonlinear three-point boundary value problems for the following fractional differential equation:. By using standard fixed point theorems and Leray-Schauder degree theory, Ahmad et al [ ] investigated the existence and uniqueness of solutions of boundary value problem for the following nonlinear fractional differential equations:.

A C – Q t

Main results