Existence of solutions to nonlinear fractional differential equations with boundary conditions on an infinite interval in Banach spaces

Jingjing Tan,Caozong Cheng

doi:10.1186/s13661-015-0419-0

Jingjing Tan, Caozong Cheng

Open Access

https://doi.org/10.1186/s13661-015-0419-0

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Journal: Boundary Value Problems	Publication Date: Sep 3, 2015
Citations: 1	License type: CC BY 4.0

Affiliation: Beijing University of Technology

Abstract

In this paper, we consider a system of nonlinear differential equations in a Banach space with boundary conditions on an infinite interval and provide sufficient conditions for the existence of solutions of the system. Our method relies upon the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to illustrate the main results.

Highlights

Fractional differential equations are important mathematical models of some practical problems in many fields such as polymer rheology, chemistry physics, heat conduction, fluid flows, electrical networks, and many other branches of science
It should be noted that the theory of nonlinear fractional differential equation boundary value problems receives more and more attention
The main result of this paper is as follows

Summary

Introduction

Fractional differential equations are important mathematical models of some practical problems in many fields such as polymer rheology, chemistry physics, heat conduction, fluid flows, electrical networks, and many other branches of science (see [ – ]). For a Banach space, Salem [ ] solved the existence of solutions to the fractional boundary value problems by means of some standard tools of fixed point theory. By using Darbo’s fixed point theorem, Su [ ] obtained the existence of solutions to the following fractional differential equation: Dα +u(t) = f (t, u(t)), t ∈ J := [ , ∞), u( ) = , Dα +– u( ) = u∞, in a real Banach space. In Section , the existence results of solutions to BVP ( ) are discussed by using the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. (Kuratowski noncompactness measure) Let E be a real Banach space, S be a bounded subset of E. For a bounded subset D of the Banach space E, let α(D) denote the Kuratowski noncompactness measure of D.

Together with the definition

It is enough to verify that sup αE

Then the fractional differential equation boundary value problem

Note that

Findings

From this inequality and we can obtain that holds for