Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

Yuji Liu

doi:10.1155/2012/352159

Abstract

This paper is concerned with the integral type boundary value problems of the second order differential equations with one‐dimensional p‐Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one‐dimensional p‐Laplacian term [ρ(t)Φ(x’(t))]’ involved with the function ρ, which makes the solutions un‐concave.

Highlights

The multipoint boundary-value problems for linear second order ordinary differential equations ODEs for short was initiated by Il’in and Moiseev 1
The nonempty convex closed subset P of X is called a cone in X if ax ∈ P for all x ∈ P and a ≥ 0 and x ∈ X and −x ∈ X imply x 0
H3 f t, c, 0 /≡ 0 on any finite subinterval of R for each c ∈ R, f : R3 → R is a Carathedory function, that is, i t → f t, x, 1/Φ−1 ρ t y is measurable for any x, y ∈ R2, ii x, y → f t, x, 1/Φ−1 ρ t y is continuous for a.e. t ∈ R, iii for each r > 0, there exists nonnegative function φr ∈ L1 R such that max{|u|, |v|} ≤ r implies f t, u, Φ−1

Summary

Introduction

The multipoint boundary-value problems for linear second order ordinary differential equations ODEs for short was initiated by Il’in and Moiseev 1. Let a, b, c, d, h > 0 be positive constants, α, ψ be two nonnegative continuous concave functionals on the cone P , γ, β, θ be three nonnegative continuous convex functionals on the cone P .

Results

Conclusion