Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials

Abstract

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))) + a(t)f(u) = 0, 0 < t < 1, u(0) = gamma u(h) + lambda, u'(0) = mu, phi(p)(D(0+)(alpha)u(0)) = (phi(p)(D-0+(alpha) u(0)) = (phi(p)(D(0+)(alpha)u(1)))' = (phi(p)(D(0+)(alpha)u(0)))'' = (phi(p)(D(0+)(alpha)u(0)))''' = 0, where 1 < alpha <= 2, 3 < beta <= 4 are real numbers 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 <= h <= 1, lambda, mu > 0 are parameters 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 parameters.. and.. are obtained. The uniqueness of positive solution on the parameters lambda and mu is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.