Building Infinitely Many Solutions for Some Model of Sublinear Multipoint Boundary Value Problems

Guy Aymard Degla

doi:10.1155/2015/732761

Abstract

We show that the sublinearity hypothesis of some well-known existence results on multipoint Boundary Value Problems (in short BVPs) may allow the existence of infinitely many solutions by using Tietze extension theorem. This is a qualitative result which is of concern in Applied Analysis and can motivate more research on the conditions that ascertain the existence of multiple solutions to sublinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differential equations with multiple solutions.

Highlights

BVPs occur in most of the branches of sciences, engineering, and technology, for example, boundary layer theory in fluid mechanics, heat power transmission theory, space technology, and control and optimization theory
The disconjugacy of the higher order differential linear operator L means that every nontrivial solution of the ordinary differential equation Lx = 0 has less than n zeros counting their multiplicities
Many authors have proved the existence of at least one nontrivial solution for sublinear Boundary Value Problems that can be transformed to the model problem

Summary

Introduction

BVPs occur in most of the branches of sciences, engineering, and technology, for example, boundary layer theory in fluid mechanics, heat power transmission theory, space technology, and control and optimization theory. Pn are given real-valued continuous functions on [a, b]; for instance, Lx = x(n). This means that L has a Polya factorization; that is, there exist n smooth positive functions Vi ∈ Cn−i+1([a, b]), 1 ≤ i ≤ n, such that

Results

Conclusion