Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems

Abstract

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problem D(0+)(alpha)u(t) + f(t, u(t)) = 0, 0 < t < 1, 2 < alpha <= 3, u(0) = u'(0) = 0, D-0(alpha-1),u(1) = beta u(xi), 0 < xi < 1, where D-0+(alpha) denotes Riemann-Liouville fractional derivative, beta is positive real number, beta xi(alpha-1) >= 2 Gamma(alpha), and f is continuous on [0, 1] x [0,infinity). As an application, one example is given to illustrate the main result.