Solvability ofNth Order Linear Boundary Value Problems

P Almenar,L Jódar

doi:10.1155/2015/230405

Abstract

This paper presents a method that provides necessary and sufficient conditions for the existence of solutions ofnth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.

Highlights

Let I be a compact interval in R and let us consider the differential operator L : Cn(I) → C(I) defined byLy = an (x) y(n) (x) + an−1 (x) y(n−1) (x) + ⋅ ⋅ ⋅ (1)+ a0 (x) y (x), x ∈ I, where ai(x) ∈ C(I), 0 ≤ i ≤ n
This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems
The purpose of this paper is to investigate the existence of solutions of the nth order boundary value problem μ

Summary

Introduction

Let I be a compact interval in R and let us consider the differential operator L : Cn(I) → C(I) defined by. In the books of Krasnosel’skii [16] and Deimling [17], one can find the grounds for this approach, which originated in Krasnosel’skii and Krein and Rutman’s works (see [16, 18]) and was later pursued by Gentry and Travis [19], Schmitt and Smith [20], Keener and Travis [21], Tomastik [22, 23], Kreith [24], Hankerson and Peterson [25], Hankerson and Henderson [26], Eloe [27,28,29,30,31,32], and Diaz [33] and more recently by Graef [34, 35], Zhang et al [36], Zhang [37], Sun et al [38], or Hao et al [39], among many others Most of these papers are focused on obtaining comparison theorems for different problems and different boundary conditions, whereas some of them deal with the existence of solutions for specific boundary value problems.

The Application of the Cone Theory to the Operator M

Application to a Broad Boundary Value Problem

Some Examples

Discussion