Uniqueness results for a mixed $ p $-Laplacian boundary value problem involving fractional derivatives and integrals with respect to a power function

Ahmed Alsaedi; Madeaha Alghanmi; Bashir Ahmad; Boshra Alharbi; Ahmed Alsaedi; Madeaha Alghanmi; Bashir Ahmad; Boshra Alharbi

doi:10.3934/era.2023018

Electronic Research Archive

2023, Volume 31, Issue 1: 367-385. doi: 10.3934/era.2023018

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Uniqueness results for a mixed $ p $-Laplacian boundary value problem involving fractional derivatives and integrals with respect to a power function

1.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, College of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia

Academic Editor: Arran Fernandez

Received: 17 August 2022 Revised: 24 October 2022 Accepted: 26 October 2022 Published: 02 November 2022

This paper is concerned with a mixed $ p $-Laplacian boundary value problem involving right-sided and left-sided fractional derivatives and left-sided integral operators with respect to a power function. We prove the uniqueness of positive solutions for the given problem for the cases $ 1 < p \le 2 $ and $ p > 2 $ by applying an efficient novel approach together with the Banach contraction mapping principle. Estimates for Green's functions appearing in the solution of the problem at hand are also presented. Examples are given to illustrate the obtained results.
- derivatives and integrals with respect to a power function,
- $ p $-Laplace operator,
- Green's function,
- existence,
- fixed point
Citation: Ahmed Alsaedi, Madeaha Alghanmi, Bashir Ahmad, Boshra Alharbi. Uniqueness results for a mixed $ p $-Laplacian boundary value problem involving fractional derivatives and integrals with respect to a power function[J]. Electronic Research Archive, 2023, 31(1): 367-385. doi: 10.3934/era.2023018

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Abstract

This paper is concerned with a mixed $ p $-Laplacian boundary value problem involving right-sided and left-sided fractional derivatives and left-sided integral operators with respect to a power function. We prove the uniqueness of positive solutions for the given problem for the cases $ 1 < p \le 2 $ and $ p > 2 $ by applying an efficient novel approach together with the Banach contraction mapping principle. Estimates for Green's functions appearing in the solution of the problem at hand are also presented. Examples are given to illustrate the obtained results.

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