Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional-Order Integral Boundary Value Problem

Abstract

We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system $-{{𝒟}_{\mathbf{t}}}^{\alpha }x(t)=f(t,x(t),x\text{'}(t),x”(t),\dots ,x^{(n-2)}(t))$, , $x(0)=x\text{'}(0)=\cdots =x^{(n-2)}(0)=0$, $x^{(n-2)}(1)={\int }_0^1{x}^{(n-2)}(s)dA(s)$, where , $n\in \mathbb{N}$ and $n\ge 2$, ${{𝒟}_{\mathbf{t}}}^{\alpha }$ is the standard Riemann-Liouville derivative, ${\int }_0^{1}x(s)dA(s)$ is linear functionals given by Riemann-Stieltjes integrals, $A$ is a function of bounded variation, and $dA$ can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder's fixed-point theorem and upper and lower solution method.