Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations

Abstract

In this paper, we investigate the problem of existence and nonexistence of positive solutions for the nonlinear boundary value problem of fractional order: Dαu(t)+λa(t)f(u(t)) = 0, 0<t<1, n−1<α⩽n, n⩾3, u(0) = u″(0) = u‴(0) = ̤̤… = u(n−1)(0) = 0, γu′(1)+βu″(1) = 0, where Dα is the Caputo’s fractional derivative and λ is a positive parameter. By using Krasnoeselskii’s fixed‐point theorem of cone preserving operators, we establish various results on the existence of positive solutions of the boundary value problem. Under various assumptions on a(t) and f(u(t)), we give the intervals of the parameter λ which yield the existence of the positive solutions. An example is also given to illustrate the main results.