Existence of solutions for fourth-order nonlinear boundary value problems

Mingzhu Huang

doi:10.1186/s13662-021-03354-4

Abstract

In this paper, we discuss the existence and approximation of solutions for a fourth-order nonlinear boundary value problem by using a quasilinearization technique. In the presence of a lower solution α and an upper solution β in the reverse order alpha geq beta , we show the existence of (extreme) solution.

Highlights

1 Introduction In this paper, we are concerned with the existence and approximation of solutions for the fourth-order nonlinear boundary value problem
The quasilinearization method is one of important tools to deal with nonlinear boundary value problems, see [1,2,3,4,5] and the references therein
For the case that a lower solution α is not greater than an upper solution β, we refer the reader to the papers [7,8,9,10]

Summary

Introduction

We are concerned with the existence and approximation of solutions for the fourth-order nonlinear boundary value problem. By using the quasilinearization technique, the author obtained the existence and approximation of solutions of (1.2) in the presence of a lower solution α and an upper solution β in the reverse order α ≥ β. There are a few papers which studied fourth-order boundary value problems with the help of the quasilinearization technique, see [11,12,13,14]. Inspired by [6, 14], in this paper, we study the existence of solution for (1.1) in the presence of a lower solution α and an upper solution β in the reverse order α ≥ β.

Comparison principle

Main results