Existence of solutions for second-order integral boundary value problems

Yujun Cui,Yumei Zou

doi:10.15388/na.2016.6.6

Abstract

In this paper, using a new comparison result and monotone iterative method, we consider the existence of solution of integral boundary value problem for second-order differential equation. To obtain corresponding results, we also discuss second order differential inequalities. The interesting point is that the one-sided Lipschitz constant is related to the first eigenvalues corresponding to the relevant operators.

Highlights

We will devote to considering the existence of solution of the following integral boundary value problem for second-order differential equation, using the method of upper and lower solutions and its associated monotone iterative technique
It is worth mentioning that integral boundary value problems for differential equations appear often in investigations connected with applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics [8, 9, 21]
One of the basic problems considered in the theory of integral boundary value problems for differential equations is to establish convenient conditions guaranteeing the existence of solutions of those equations

Summary

Introduction

We will devote to considering the existence of solution of the following integral boundary value problem for second-order differential equation, using the method of upper and lower solutions and its associated monotone iterative technique. One of the basic problems considered in the theory of integral boundary value problems for differential equations is to establish convenient conditions guaranteeing the existence of solutions of those equations. There is a vast literature devoted to the applications of this method to differential equations with different boundary conditions, for details, see [4, 7, 15, 16, 24, 29]. Motivated by [3], in this paper, we first present a new comparison theorem for the operator −u − λu with integral boundary value condition, and by using the monotone iterative technique, we investigate the extremal solutions of (1).

Preliminaries and lemmas

Main results

Example