Abstract
In this paper, we study the existence and uniqueness of positive solutions to a class of multipoint boundary value problems for singular fractional differential equations with the p-Laplacian operator. Here, the nonlinear source term f permits singularity with respect to its time variable t. Some fixed-point theorems such as the Leray-Schauder nonlinear alternative, the Schauder fixed-point theorem, and the Banach contraction mapping principle and the properties of the Gauss hypergeometric function are used to prove our main results. And by employing the upper and lower solutions technique, we derive a new approach to obtain the maximal and minimal solutions to the given problem. Finally, we present some examples to demonstrate our existence and uniqueness results.
Highlights
In this paper, we consider the existence of positive solutions of the following m-point boundary value problems for singular nonlinear fractional differential equations >>>>>>>>>>>>>>>>: Dα0+xð0Þ 0, i=1 φpðDα0+xð1ÞÞ
A function x ∈ fuju ∈ C1⁄20, 1, Dα0+u ∈ C1⁄20, 1, Dβ0+ðφpðDα0+uÞÞ ∈ Cð0, 1Þg is called a solution of problem (1) if it satisfies the fractional differential equation and the boundary conditions of (1)
The Gauss hypergeometric function 2 F1ða, b ; c ; sÞ is defined in the unit disk as the sum of the hypergeometric series
Summary
We consider the existence of positive solutions of the following m-point boundary value problems for singular nonlinear fractional differential equations. Many researchers have derived some important results for solutions to boundary value problems of fractional differential equations with singularity with respect to the time variable (see [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]). Motivated by the above works, in this paper, we first apply the Leray-Schauder nonlinear alternative to establish the existence of solutions to our problem (1) and use the Schauder fixed-point theorem and upper and lower control functions to derive the upper and lower solutions method to obtain the maximal and minimal solutions. We prove the uniqueness of solutions to the given problem by using some useful properties of the Gauss hypergeometric function 2 F1ða, b, c, sÞ and the Banach contraction mapping principle
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