Existence of positive solution for singular fractional differential equation
Introduction
Many papers and books on fractional calculus differential equation have appeared recently [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function and to problems of analyticity in the complex domain (see, for example [3], [7]). Moreover, Delbosco and Rodino [2] considered the existence of a solution for the nonlinear fractional differential equation , where , and is a given function, continuous in . They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Very recently, Zhang [9] considered the existence of positive solution for equation , where , and is a given continuous function, by using the sub- and super-solution method.
In this paper, we discuss the existence of a positive solution to boundary value problem of nonlinear fractional differential equation:where is a real number, is the Caputo’s differentiation, and (that is f is singular at ).
About the development of the existence theorems for fractional differential equation and also many references with real applications, we refer to the Survey paper by Kilbas and Trujillo [5], [6]. Concerning the definitions of a fractional integral and derivative and related basic properties used in the text, the readers can refer to Samko et al. [8] or Delbosco and Rodino [2].
In conclusion we obtain two results about this boundary value problem, by using Krasnoselskii’s fixed point theorem in a cone and nonlinear alternative of Leray–Schauder, respectively.
Section snippets
Background materials and preliminaries
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent literature. Definition 2.1 The Riemann–Liouville fractional integral of order of a function is given byprovided that the right side is pointwise defined on . Definition 2.2 The Caputo’s fractional derivative of order of a continuous function is given by
Main results
For our construction, we let with be Banach space.We seek solutions of (1.1) that lie in a cone P, defined byDefine operator Lemma 3.1 Let is continuous, and . Suppose that is continuous function on [0, 1]. Then the functionis continuous on [0, 1]. Proof By the continuity of and It is easily to know that . The proof is divided into
Acknowledgements
This work is supported by the National Nature Science Foundation of China (10371006) and the Mathematics Tianyuan Foundation of China (10626033).
References (11)
Positive solutions for boundary value problem of nonlinear fractional differential equation
J. Math. Anal. Appl.
(2005)- et al.
Existence and uniqueness for a nonlinear fractional differential equation
J. Math. Anal. Appl.
(1996) - et al.
A class of analytic function defined by fractive derivative
J. Math. Anal. Appl.
(1994) The existence of a positive solution for a nonlinear fractional differential equation
J. Math. Anal. Appl.
(2000)On the solution of some simple fractional differential equation
Int. J. Math. Sci.
(1990)
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