On a Singular Second-Order Multipoint Boundary Value Problem at Resonance

S A Iyase,O F Imaga

doi:10.1155/2017/8579065

Abstract

The aim of this paper is to derive existence results for a second-order singular multipoint boundary value problem at resonance using coincidence degree arguments.

Highlights

In this paper we derive existence results for the second-order singular multipoint boundary value problem of the form x󸀠󸀠 (t) = f (t, x (t), x󸀠 (t)) + g (t), 0 < t < 1, x󸀠 (0) = 0, (1)m−2 x (1) = ∑ aix, i=1 where f : [0, 1] × R2 → R is Caratheodory’s function (i.e., for each (x, y) ∈ R2 the function f(⋅, x, y) is measurable on [0, 1]; for a.e. t ∈ [0, 1], the function f(t, ⋅, ⋅) is continuous on R2)
[2] Ma and O’Regan derived existence results for the same equation when f and g have a singularity at t = 1 and ∑mi=−12 ai ≠ 1
One says that the linear operator L : dom L ⊂ X → Z is a Fredholm mapping of index zero if Ker L and Z/ Im L are of finite dimension, where Im L denotes the image of L

Summary

Introduction

In this paper we derive existence results for the second-order singular multipoint boundary value problem of the form x󸀠󸀠 (t) = f (t, x (t) , x󸀠 (t)) + g (t) , 0 < t < 1, x󸀠 (0) = 0, (1). In [1] Gupta et al studied the above equation when f and g have no singularity and ∑mi=−12 ai ≠ 1 They obtained existence of a C1[0, 1] solution by utilising the LeraySchauder continuation principle. In [2] Ma and O’Regan derived existence results for the same equation when f and g have a singularity at t = 1 and ∑mi=−12 ai ≠ 1. International Journal of Differential Equations (A2) g : [0, 1] → R is such that ∫01(1 − t)|g(t)| < ∞

Preliminaries

Main Result