Positive solutions of higher-order Sturm\u2013Liouville boundary value problems with fully nonlinear terms

Yongxiang Li,Qian Wen

doi:10.1186/s13662-018-1636-5

Abstract

In this paper we consider the existence of positive solutions of nth-order Sturm–Liouville boundary value problems with fully nonlinear terms, in which the nonlinear term f involves all of the derivatives u',ldots, u^{(n-1)} of the unknown function u. Such cases are seldom investigated in the literature. We present some inequality conditions guaranteeing the existence of positive solutions. Our inequality conditions allow that f(t, x_{0}, x_{1},ldots, x_{n-1}) is superlinear or sublinear growth on x_{0}, x_{1},ldots, x_{n-1}. Our discussion is based on the fixed point index theory in cones.

Highlights

1 Introduction In this paper, we consider the existence of positive solutions of the nth-order Sturm– Liouville boundary value problem (BVP)
For some of the simple cases that the nonlinearity f does not contain a derivative term, the existence of positive solutions has been researched by many authors; see [2,3,4,5,6,7,8,9,10,11]
We present some inequality conditions on the nonlinearity f (t, x0, x1, . . . , xn–1) when |(x0, x1, . . . , xn–1)| is small or large enough to guarantee the existence of positive solutions

Summary

Introduction

We consider the existence of positive solutions of the nth-order Sturm– Liouville boundary value problem (BVP). They converted BVP (1.5) to an equivalent (m – q)th-order Sturm– Liouville boundary value problem of integral-differential equations, and using Krasnoselskii’s fixed point theorem in cones they obtained existence results of one or more positive solutions. This method is not applicable to the more general BVP (1.1) owing to the presence of a derivative u(n–1) in the nonlinearity f.

Preliminaries

Main results