Theory of Singular Boundary Value Problems

The following sections are included:

• PREFACE

• Acknowledgement

• CONTENTS

This book is intended as an introduction to the existence theory of regular and singular two-point boundary value problems for ordinary differential equations. Over the last twenty years or so major developments have arisen in this subject and we try to give an insight into some of these developments. As the book is aimed to reach a general audience this will mean that some of the results are not given in their most general formulation. We refer the interested reader to the reference list for further study…

This chapter presents fixed point theory of Leray-Schauder type. The theory developed here will play a major role in establishing the existence principles in chapter 3. If X is any normed linear space and f a function mapping X to X, then x₀ ∈ X is called a fixed point of f if f(x₀) = x₀…

The nonlinear alternative of chapter 2 is used in the chapter to obtain existence principles for singular differential equations with Sturm Liouville, Neumann, Periodic and Bohr boundary data. First we establish a result known as the Arzela-Ascoli theorem in the literature [3,19]. Let −∞ < a < b < ∞. A collection of real valued functions A = {f_i : [a, b] → R} is said to be uniformly bounded if there exists a constant M > 0 with |f_i(x)| ≤ M for all x ∈ [a,b] and for all f_i ∈ A. A is said to be equicontinuous if for every ε > 0 there exists δ = δ(ε) > 0 such that if x₀, x₁ ∈ [a, b] then |x₀ − x₁| < δ implies |f_i(x₀) − f_i(x₁)| < ε for all f_i ∈ A…

We begin by discussing the spectrum of a symmetric completely continuous linear operator. Rayleigh Ritz type minimization theorems, are then obtained for various regular and singular Sturm-Liouville problems.

Throughout this chapter let H be a Hilbert space, with inner product denoted by 〈.,.〉 and A a linear operator from D(A) into H. Here D(A), the domain of A, is a subspace dense in H…

This chapter examines singular second order differential equations subject to Sturm Liouville or Bohr boundary conditions. In particular we will discuss the method of upper and lower solutions…

This chapter is divided into two main sections. We begin by using the Rayleigh-Ritz minimization theorems of chapter 4 to obtain various existence results for two point boundary value problems

with y satisfying either the homogeneous Sturm Liouville data or the homogeneous mixed boundary data. The technique in using norms to establish existence results for singular and nonsingular problems has been used extensively in the literature of boundary value problems…

This chapter discusses existence results for “nonresonant” and “resonant” singular boundary value problems. In particular we will examine the second order equation

where λ_m−1 < μ < λ_m (nonresonant case) or μ = λ_m (resonant case), m = 1,2,… Here λ₀ = −∞ and the , i = 1,… are the eigenvalues of an appropriate linear problem to be described shortly…

This chapter is devoted to the study of boundary value problems on the semiinfinite interval. These problems have applications in many areas, for example the search for positive classical solutions (ground states) of the Poisson equation in Rⁿ(n ≥ 2)

reduces to the search of radial solutions to …

In the study of nonlinear phenomena, for example in fluid theory [5] and boundary layer theory [4], many mathematical models give rise to singular boundary value problems of the form

with ø ∈ C[0, 1], ø ≥ 0 on [0, 1] and γ > 0. One is usually interested in the existence of a nonnegative solution…

A simple but frequently used mathematical model for steady-state diffusion with reaction in one dimension is

Here y is the concentration and a is usually a small positive integer; the sign on the right hand side of (10.1) determines whether the chemical species of concentration y is created or destroyed in the reaction. If the medium occupying 0 ≤ t ≤ 1 is homogeneous, then the ratio of reaction rate to diffusion rate, the Thiele modulus ø², is a constant. But more general models are also used, including fractional and negative values of α, see [1] for example; if the reaction is concentration dependent and if the medium is not homogeneous the model y″ = ±ø²f(t,y) is appropriate…

The following sections are included:

• Index