Monotone Iterative Schemes for Positive Solutions of a Fractional  q -Difference Equation with Integral Boundary Conditions on the Half-Line

Min Jiang,Rengang Huang

doi:10.1155/2021/9384128

Monotone Iterative Schemes for Positive Solutions of a Fractional q -Difference Equation with Integral Boundary Conditions on the Half-Line

Min Jiang, Rengang Huang

Open Access

https://doi.org/10.1155/2021/9384128

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Journal: Journal of Mathematics	Publication Date: Feb 26, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Guizhou Minzu University

Abstract

In this paper, we study the boundary value problem of a fractional q -difference equation with nonlocal integral boundary conditions on the half-line. Using the properties of the Green function and monotone iterative method, the extremal solutions are obtained. Finally, an example is presented to illustrate our main results.

Highlights

We are concerned with the following fractional q-differential equation with integral boundary value problem on the half-line:
By applying the nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorems, the authors obtain some sufficient conditions for the existence of positive solutions to the considered problem
Our main purpose of this paper is by constructing a suitable Banach space, defining appropriate operators, and using the monotone iterative method, which is different from the method in [17] to obtain the existence and uniqueness of positive solutions

Summary

Introduction

We are concerned with the following fractional q-differential equation with integral boundary value problem on the half-line:. By applying the nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorems, the authors obtain some sufficient conditions for the existence of positive solutions to the considered problem. In [17], the authors investigate the existence of solutions for the following boundary value problem of nonlinear fractional q-difference equations on the half-line. To the best of our knowledge, there are few papers that consider the boundary value of nonlinear fractional q-difference equations with nonlocal conditions on the half-line, the study of such problems is very important. Our main purpose of this paper is by constructing a suitable Banach space, defining appropriate operators, and using the monotone iterative method, which is different from the method in [17] to obtain the existence and uniqueness of positive solutions

Preliminaries on q-Calculus and Lemmas

Main Results

Example

Conclusions