Positive Solutions to Boundary-value Problems of P-laplacian Fractional Differential Equations with a Parameter in the Boundary

Abstract

In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D 0+(φp(D α 0+u(t))) + a(t)f(u) = 0, 0 < t < 1, u(0) = γu(ξ) + λ, φp(D α 0+u(0)) = (φp(D α 0+u(1))) ′ = (φp(D α 0+u(0))) ′′ = 0, where 0 < α 6 1, 2 < β 6 3 are real numbers, Dα 0+, D β 0+ are the standard Caputo fractional derivatives, φp(s) = |s|p−2s, p > 1, φ−1 p = φq , 1/p+1/q = 1, 0 6 γ < 1, 0 6 ξ 6 1, λ > 0 is a parameter, a : (0, 1) → [0,+∞) and f : [0,+∞)→ [0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameter λ are obtained. The uniqueness of positive solution on the parameter λ is also studied. Some examples are presented to illustrate the main results.

Cite this paper

@inproceedings{Han2012PositiveST, title={Positive Solutions to Boundary-value Problems of P-laplacian Fractional Differential Equations with a Parameter in the Boundary}, author={Zhenlai Han and Hongling Lu and Shurong Sun and Dianwu Yang}, year={2012} }