A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions

Ahmed El-Sayed,Hari Srivastava,Fatma Gaafar

doi:10.3390/sym10100508

Abstract

In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the existence of a solution for the first-order nonlinear differential equation with m-point nonlocal BCs. The existence of solutions of our problems is investigated by applying the properties of the Riemann sum for continuous functions. Several examples are given in order to illustrate our results.

Highlights

IntroductionOur objective in this article is to investigate the existence of absolute continuous solutions of the nonlocal first-order boundary value problem (BVP) with the nonlinear function involving the Liouville-Caputo fractional derivative: dx
Our objective in this article is to investigate the existence of absolute continuous solutions of the nonlocal first-order boundary value problem (BVP) with the nonlinear function involving the Liouville-Caputo fractional derivative: dx= f t, D α x (t) dt a.e.(0 < t < 1; 0 < α 5 1), (1)together with either the Riemann-Stieltjes functional integral boundary condition given by Z 1x φ(s) dg(s) = x0Symmetry 2018, 10, 508; doi:10.3390/sym10100508 g : [0, 1] → [0, 1]; g(s) = 0 (2)www.mdpi.com/journal/symmetrySymmetry 2018, 10, 508 or the infinite-point boundary conditions given by
We have considered the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs

Summary

Introduction

Our objective in this article is to investigate the existence of absolute continuous solutions of the nonlocal first-order boundary value problem (BVP) with the nonlinear function involving the Liouville-Caputo fractional derivative: dx. BVPs with integral BCs arise naturally in semiconductor problems [10], thermal conduction problems [11], hydrodynamic problems [12], population dynamics model [13], and so on (see [14]) These BVPs were extensively studied by (among others) Akcan and Çetin [15], Boucherif [16], Benchohra et al [17], Chalishajar and Kumar [18], Dou et al [19], Li and Zhang [20], Liu et al [21], Song et al [22], Tokmagambetov and Torebek [23], Wang et al [24] and Yang and Qin [25] (see the references to the related earlier works which are cited in each of these investigations). If we have a way of getting the continuous solution of the m-point BVP, we can (in a simple way) get a solution to the BVP with the Riemann-Stieltjes integral or infinite points in the BCs

Preliminaries

Riemann-Stieltjes Functional Integral BCs

Infinite-Point Boundary Conditions

Further Illustrative Examples

Conclusions