Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems

Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function see, e.g., 1, 2 and to problems of analyticity in the complex domain 3 . Moreover, Delbosco and Rodino 4 considered the existence of a solution for the nonlinear fractional differential equation D 0 u f t, u , where 0 < α < 1 and f : 0, a × R → R, 0 < a ≤ ∞ is a given continuous function in 0, a × R. They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang 5 considered the existence of positive solution for equation D 0 u f t, u , where 0 < α < 1 and f : 0, 1 × 0,∞ → 0,∞ is a given continuous function by using the suband supersolution methods. In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation