Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations

Badawi Hamza Elbadawi Ibrahim,Qixiang Dong,Zhengdi Zhang

doi:10.11648/j.ajam.20190706.13

Abstract

It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the purpose of this study, we concern ourselves with Captou type derivative of the order α∈(2, 3), with respect to boundary values that are nonzero. We establish sufficient conditions for the existence of solutions for boundary value problem of nonlinear variable coefficient of fractional order. On the other hand, the boundary value problem is formulated as follows: cDαu(t) + p(t)f(t, u(t)) + q(t) = 0, u(0) = a, u'(0) = b, u(1) = d. Where a, b, d ∈ R are constants. In this paper, we investigate the existence and uniqueness of solutions for a class of boundary value problem of the nonlinear variable coefficient of fractional differential equations. The existence of solutions involving Captuo fractional derivatives is discussed under the assumption that the bounded conditions are constants. By means of the Banach contraction mapping principle and Larry- Schauder alternative, the existence of solutions are obtained. Finally, some examples are discussed to illustrate the results, which are generalized to nonlinear fractional derivatives with variable coefficients.

Highlights

The theory of fractional differential equations has become an important area of study, see [1, 2, 3, 4]
In [13] the authors considered the existence of multiple positive solutions for the following fractional differential equations with a negatively perturbed term
In this paper we study the following boundary value problems for fractional differential equation (BVP in short) cDαu(t) + p(t)f (t, u(t)) + q(t) = 0, 0 < t < 1, (1)

Summary

Introduction

The theory of fractional differential equations has become an important area of study, see [1, 2, 3, 4]. Boundary value problems of fractional differential equations have applications in various fields of science such as physics, mechanics, chemistry, engineering, etc.[7, 6, 5, 8]. It has received great attention and a variety of results concerning the existence of solutions, based on different kinds of analytic techniques, which can be found in [11, 10, 9, 12].

Preliminaries

Existence Results

Example