Existence of solutions for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator

Tingting Xue,Tengfei Shen,Wenbin Liu

doi:10.1186/s13661-017-0900-z

Tingting Xue, Tengfei Shen + Show 1 more

Open Access

https://doi.org/10.1186/s13661-017-0900-z

Copy DOI

Abstract

This paper is concerned with the solvability for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator at resonance using Mawhin’s continuation theorem. Sufficient conditions for the existence of solutions have been acquired, and they would extend the existing results. Furthermore, an example is provided to illustrate the main result.

Highlights

1 Introduction The last two decades have witnessed a wide application of fractional differential equations in various fields of natural science and engineering technology
With some theoretical discussions conducted regarding boundary value problem (BVP for short) of differential equations so far, valuable results have been obtained for BVP of fractional differential equations
It is generally known that the p-Laplacian equations normally derive from nonlinear elastic mechanics and non-Newtonian fluid theory

Summary

Introduction

The last two decades have witnessed a wide application of fractional differential equations in various fields of natural science and engineering technology (see [ – ]). A lot of research regarding BVP of fractional differential equations with p(t)-Laplacian operator have been quite limited so far (see [ – ]). Shen and Liu [ ] studied the existence of solutions for the following BVP with p(t)-Laplacian operator at nonresonance and resonance by using Schaefer’s fixed point theorem and Mawhin’s continuation theorem:. Inspired by the above findings, this paper studies the BVP subjected to Sturm-Liouville type integral boundary conditions for fractional differential equations with p(t)-Laplacian operator:. If dim Ker L = codim Im L < +∞ and Im L is a closed subset in Y , the map L is a Fredholm operator with index zero If there exist such continuous projections as P : X → X and Q : Y → Y , which meet the conditions that Im P = Ker L and Ker Q = Im L, L|dom L∩KerP : dom L ∩ Ker P → Im L is reversible.

Main result

Conclusions