Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator

Qiaozhen Yuan,Wengui Yang

doi:10.1186/1029-242x-2014-481

Abstract

This paper investigates a class of four-point boundary value problems of fractional q-difference equations with p-Laplacian operator D β (ϕp(D α u(t))) = f (t, u(t)), t ∈ (0, 1), u(0) = 0, u(1) = au(ξ ), D α u(0) = 0, and D α u(1) = bD α u(η), where D α and D β are the fractional q-derivative of the Riemann-Liouville type, p-Laplacian operator is defined as ϕp(s )= |s| p–2 s, p >1 , andf (t, u) may be singular at t =0 , 1 oru = 0. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. Furthermore, two examples are presented to illustrate the main results. MSC: 39A13; 34B18; 34A08

Highlights

Fractional differential equations with p-Laplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, non-Newtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science
The existence results for the above boundary value problem were obtained by using the upper and lower solutions method associated with the Schauder fixed point theorem
By applying the upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at least one positive solution for the above fractional q-difference boundary value problem with p-Laplacian operator are established

Summary

Introduction

Fractional differential equations with p-Laplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, non-Newtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science. The theory of boundary value problems for nonlinear fractional q-difference equations has been addressed extensively by several researchers. Zhao et al [ ] showed some existence results of positive solutions to nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation using the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii fixed point theorem.

Results

Conclusion