Abstract
In this paper, by means of the Avery-Peterson fixed point theorem, we establish the existence result of a multiple positive solution of the boundary value problem for a nonlinear differential equation with Riemann-Liouville fractional order derivative. An example illustrating our main result is given. Our results complement previous work in the area of boundary value problems of nonlinear fractional differential equations (Goodrich in Appl. Math. Lett. 23:1050-1055, 2010).
Highlights
1 Introduction In this paper, we consider the positive solution for the following boundary value problem of the differential equation involving the Riemann-Liouville fractional order derivative: Dα +u(t) + f t, u(t), Dβ +u(t) =, t ∈ (, ), ( . )
By means of some fixed point theorems on cone, existence and multiplicity results of positive solutions were obtained
Bai and Lü proved that the Green’s function of problem ( . ) did not satisfy a Harnack-like inequality, which is a crucial property when seeking the existence of positive solutions by means of cone theory
Summary
) may be applied to the description of the deformation of beam and has been studied with a variety of boundary conditions and nonlinearities; see, for example, [ , ] and references therein In these works, the existence results of positive solutions for FBVPs were all established under the assumption that the derivative of the unknown function u(t) was not involved in the nonlinear term explicitly. In [ ], Avery and Peterson gave a new triple fixed point theorem, which can be regarded as an extension of the Leggett-Williams fixed point theorem By using this method, many results concerning the existence of at least three positive solutions of boundary value problems of differential equation with integer order were established; see [ – ].
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