Abstract
We concentrate on a category of singular boundary value problems of fractional differential equations with integral boundary conditions, in which the nonlinear function f is singular at t=0, 1. We use Banach’s fixed-point theorem and Hölder’s inequality to verify the existence and uniqueness of a solution. Moreover, also we prove the existence of solutions by Krasnoselskii’s and Schaefer’s fixed point theorems.
Highlights
1 Introduction The current work concentrates on the existence and uniqueness of solutions for a category of singular nonlinear fractional differential equations (NFDEs) subject to integral boundary conditions (BCs)
There is a vast literature on this subject, where the basic concepts, properties, and applications of fractional-order operators are introduced [1,2,3,4,5,6], and the related initial and boundary value problems are studied [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]
⎧ ⎨Dα0+ y(t) = f (t, y(t)), 0 < t < 1, ⎩y(0) = y (1) = y (0) = 0, where 2 < α ≤ 3, Dα0+ stands for the Caputo derivative, and f : (0, 1] × [0, +∞) → [0, +∞) satisfies limt→0+ f (t, ·) = +∞. They hypothesized that tσ f (t, y(t)) is continuous on [0, 1] × [0, +∞) and employed nonlinear alternative and Krasnoselskii’s fixed point theorem to extract two positive solutions to this problem
Summary
The current work concentrates on the existence and uniqueness of solutions for a category of singular nonlinear fractional differential equations (NFDEs) subject to integral boundary conditions (BCs). ⎧ ⎨Dα0+ y(t) = f (t, y(t)), 0 < t < 1, ⎩y(0) = y (1) = y (0) = 0, where 2 < α ≤ 3, Dα0+ stands for the Caputo derivative, and f : (0, 1] × [0, +∞) → [0, +∞) satisfies limt→0+ f (t, ·) = +∞ They hypothesized that tσ f (t, y(t)) is continuous on [0, 1] × [0, +∞) and employed nonlinear alternative and Krasnoselskii’s fixed point theorem to extract two positive solutions to this problem. Several papers have dealt with problems for singular NFDEs containing integral boundary conditions [29,30,31,32,33] He [29] discussed the existence and multiplicity of positive solutions for NFDEs with integral BCs.
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