Coupled Systems of Nonlinear Integer and Fractional Differential Equations with Multi-Point and Multi-Strip Boundary Conditions

Bin Di,Huihui Pang,Guo Chen

doi:10.3390/math8060935

Abstract

We first consider a second order coupled differential system with nonlinearities involved two unknown functions and their derivatives, subject to a new kinds of multi-point and multi-strip boundary value conditions. Since the coupled system contains two dependent variables and their derivatives, the classical method of upper and lower solutions on longer applies. So we adjust and redefine the forms of upper and lower solutions, to establish the existence results. Secondly, we study a Caputo fractional order coupled differential system with discrete multi-point and integral multi-strip boundary value conditions which are very popular recently, and can accurately describe a lot of practical dynamical phenomena, such as control theory, biological system, electroanalytical chemistry and so on. In this part the existence and uniqueness results are achieved via the Leray-Schauder’s alternative and the Banach’s contraction principle. Finally, an example is presented to illustrate the main results.

Highlights

We study the existence results to a coupled systems of nonlinear differential equations with multi-point and multi-strip boundary conditions
In Reference [13], by using the method of upper and lower solutions, authors established an existence results of solutions to a second order coupled differential systems integral boundary value problems in which the nonlinear terms of the system are only related to the unknown functions
Subject to the coupled system of second-order differential equations with nonlinearities depending on two unknown functions as well as their derivatives, by defining the appropriate upper and lower solutions, combining with Nagumo conditions, the truncation function is constructed successfully

Summary

Introduction

We study the existence results to a coupled systems of nonlinear differential equations with multi-point and multi-strip boundary conditions. In Reference [13], by using the method of upper and lower solutions, authors established an existence results of solutions to a second order coupled differential systems integral boundary value problems in which the nonlinear terms of the system are only related to the unknown functions. Because of the complexity of the form of the problem (3) and (4), we have encountered a lot of resistance in calculating the related Green’s functions and discussing their properties In this part, the existence results are obtained by applying Leray-Schauder’s alternative, while the uniqueness of solution is established via Banach’s contraction principle.

A Coupled System of Second-Order Differential Equations

A Coupled System of Fractional Differential Equations

Conclusions