POSITIVE SOLUTIONS TO BOUNDARY-VALUE PROBLEMS OF P-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS WITH A PARAMETER IN THE BOUNDARY

In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D-0+(beta)(phi(p)(D(0+)(alpha)u(t))) vertical bar a(t)f(u) = 0, 0 < t < 1, u(0) = gamma u(xi) + lambda, phi(p)(D(0+)(alpha)u(0)) = (phi(p)(D(0+)(alpha)u(1)))' = (phi(p)(D(0+)(alpha)u(0)))'' = 0, where 0 < alpha <= 1, 2 < beta <= 3 are real numbers, D-0+(alpha), D-0+(beta) are the standard Caputo fractional derivatives, phi(p)(s) = vertical bar s vertical bar(p-2)s, p > 1, phi(-1)(p) = phi(q), 1/p+1/q = 1, 0 <= gamma < 1, 0 <= xi <= 1, lambda > 0 is a parameter, a : (0, 1) -> [0, +infinity) and f : [0, +infinity) -> [0, +infinity) 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 lambda are obtained. The uniqueness of positive solution on the parameter lambda is also studied. Some examples are presented to illustrate the main results.