Existence and Uniqueness of Positive Solution for a Nonlinear Multi-Order Fractional Differential Equations
Changyou Wang *
Key Laboratory of Industrial Internet of Things and Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China and Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Lu Gao
Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Qi Dou
Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
*Author to whom correspondence should be addressed.
Abstract
This paper is concerned with a class of nonlinear multi-order fractional differential equation. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and Schauder fixed point theorem, the existence and uniqueness of positive solution for the equation are investigated. Moreover, some properties concerning the maximal, minimal, and continuation of solutions are obtained.
Keywords: Multi-order fractional differential equation, Erdelyi-Kober operator, positive solution, upper and lower solutions, existence and uniqueness, fixed point theorem