Jing Ren, Ling Bai, Chengbo Zhai, A decreasing operator method for a fractional initial value problem on infinite interval, Vol. 2023 (2023), Article ID 35, pp. 1-9

Full Text: PDF
DOI: 10.23952/jnfa.2023.35

Received August 1, 2023; Accepted October 13, 2023; Published Nobember 1, 2023

Abstract. This paper considers the uniqueness of positive solutions for a fractional initial value problem involving Caputo fractional derivative
$^{C}D_{0^+}^{\alpha}x(t)=[p(t)+q(t)(f(t,x(t))]^{-1},$ $0\leq t <\infty,$
$x(0)=b_0, x'(0)=b_1, x''(0)=b_2,$
where $2<\alpha\leq3,$ $f:[0,\infty)\times[0,\infty)\rightarrow [0,\infty)$ , and $p(t)$ and $q(t)$ are continuous functions. By imposing some suitable conditions on f, p, and q, we obtain the uniqueness of positive solutions for the problem, and we construct an iterative scheme to approximate the unique solution. Our approach is based on a fixed point theorem of decreasing operators on cones. In addition, two simply examples are presented to illustrate our main result.

How to Cite this Article:
J. Ren, L. Bai, C. Zhai, A decreasing operator method for a fractional initial value problem on infinite interval, J. Nonlinear Funct. Anal. 2023 (2023) 35.