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

Min Jia,Xuemai Gu,Xin Liu

doi:10.1155/2012/294694

Abstract

We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system , 0 < t < 1, x(0) = x′(0) = ⋯ = x(n−2)(0) = 0, , where n − 1 < α ≤ n, n ∈ ℕ and n ≥ 2, is the standard Riemann‐Liouville derivative, 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.

Highlights

Fractional-order models have proved to be more accurate than integer order models, that is, there are more degrees of freedom in the fractional-order models.we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, and signal processing of wireless communication system
Hao et al 13 studied the existence of positive solutions for the BVP 1.5 with integer order n, and nonlinear term is replaced by a t f t, x t, where a can be singular at t 0, 1, f can be singular at x 0 and no singularity at t 0, 1
Motivated by the results mentioned above, in this paper, we study the existence, uniqueness, and asymptotic behavior of positive solutions for the BVP 1.4 where the nonlinear terms and boundary conditions all involve derivatives of unknown functions and with Riemann-Stieltjes integral boundary condition, f may be singular at xi 0 and t 0, 1

Summary

Introduction

Fractional-order models have proved to be more accurate than integer order models, that is, there are more degrees of freedom in the fractional-order models.we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, and signal processing of wireless communication system. Motivated by systems 1.1 – 1.3 and their application background in electromagnetic and signal processing of wireless communication system, in this paper, we consider the existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal fractional differential equation. Hao et al 13 studied the existence of positive solutions for the BVP 1.5 with integer order n, and nonlinear term is replaced by a t f t, x t , where a can be singular at t 0, 1, f can be singular at x 0 and no singularity at t 0, 1. Motivated by the results mentioned above, in this paper, we study the existence, uniqueness, and asymptotic behavior of positive solutions for the BVP 1.4 where the nonlinear terms and boundary conditions all involve derivatives of unknown functions and with Riemann-Stieltjes integral boundary condition, f may be singular at xi 0 and t 0, 1. Our main tool relies on Schauder’s fixed-point theorem and upper and lower solution method

Preliminaries and Lemmas

Main Results

Conclusion