Existence and uniqueness of solutions to distributional differential equations involving Henstock–Kurzweil–Stieltjes integrals

Wei Liu,Mingxia Zhang,Dafang Zhao,Guoju Ye

doi:10.33044/revuma.v60n2a11

Abstract

This paper is concerned with the existence and uniqueness of solutions to the second order distributional differential equation with Neumann boundary value problem via Henstock–Kurzweil–Stieltjes integrals.The existence of solutions is derived from Schauder’s fixed point theorem, and the uniqueness of solutions is established by Banach’s contraction principle. Finally, two examples are given to demonstrate the main results.

Highlights

In this paper we apply the Henstock–Kurzweil–Stieltjes integral to establish the existence and uniqueness of solutions to the second order distributional differential equation− D2x = f (t, x) + g(t, x)Du, t ∈ [0, 1], (1.1)subject to the Neumann boundary value conditionDx(0) = Dx(1) = 0, (1.2)where D, D2 stand for the first order and the second order distributional derivative, respectively, x and u are regulated functions such that both are left-continuous on (a, b] and right-continuous at a, the function f (·, x(·)) is Henstock–Kurzweil integrable, and g(·, x(·)) is a function of bounded variation.We know that regulated functions contain continuous functions and functions of bounded variation as special cases
Where D, D2 stand for the first order and the second order distributional derivative, respectively, x and u are regulated functions such that both are left-continuous on (a, b] and right-continuous at a, the function f (·, x(·)) is Henstock–Kurzweil integrable, and g(·, x(·)) is a function of bounded variation
When u is an absolutely continuous function, its distributional derivative is the usual derivative and we obtain the ordinary differential equation; when u is a function of bounded variation, Du can be identified with a Stieltjes measure, and (1.1) is called measure differential equation

Summary

Introduction

In this paper we apply the Henstock–Kurzweil–Stieltjes integral to establish the existence and uniqueness of solutions to the second order distributional differential equation (in short DDE). Some existence and uniqueness results and the multiplicity of positive solutions for the Neumann boundary value problem have been established. In [1], a Lyapunov-type inequality and Schauder’s fixed point theorem were used to obtain existence and uniqueness results for the Neumann boundary value problem as. In [13], by using a fixed point theorem in a cone, the authors established the existence and multiplicity of the positive solutions to the second order Neumann boundary value problem (1.3) with parameter. With the help of Henstock–Kurzweil–Stieltjes integrals, we establish the existence and uniqueness of solutions for the second order DDE with Neumann boundary value problem.

Preliminaries

Main results