Monotonicity and asymptotic behavior of solutions for Riemann-Liouville fractional differential equation

TL;DR

This paper studies the monotonicity and long-term behavior of solutions to Riemann-Liouville fractional differential equations, introducing new methods based on fractional integral functions and fixed point theorems.

Contribution

It presents novel results on the asymptotic behavior of solutions using monotonicity properties, which were not previously explored in this context.

Findings

Existence of at least one decreasing solution under certain conditions

Asymptotic behavior characterized through monotonicity analysis

Illustrative examples validating the theoretical results

Abstract

In this paper, we first investigate the monotonicity and limit problem of the fractional integral functions. By fixed point theorem and these new results of the fractional integral functions, we present that the Riemann-Liouville fractional differential equations has at least one decreasing solution in $C_{1-\beta}^{+}(0,+\infty)$. The asymptotic behavior of solutions is also discussed under some different conditions. The novelty in this paper is that we investigate the asymptotic behavior of Riemann-Liouville fractional differential equations by the monotonicity of functions. Finally, several examples are given to illustrate our main results.