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1998 | Buch

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Existence Theory for Nonlinear Integral and Integrodifferential Equations

verfasst von: Donal O’Regan, Maria Meehan

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

The theory of integral and integrodifferential equations has ad vanced rapidly over the last twenty years. Of course the question of existence is an age-old problem of major importance. This mono graph is a collection of some of the most advanced results to date in this field. The book is organized as follows. It is divided into twelve chap ters. Each chapter surveys a major area of research. Specifically, some of the areas considered are Fredholm and Volterra integral and integrodifferential equations, resonant and nonresonant problems, in tegral inclusions, stochastic equations and periodic problems. We note that the selected topics reflect the particular interests of the authors. Donal 0 'Regan Maria Meehan CHAPTER 1 INTRODUCTION AND PRELIMINARIES 1.1. Introduction The aim of this book is firstly to provide a comprehensive existence the ory for integral and integrodifferential equations, and secondly to present some specialised topics in integral equations which we hope will inspire fur ther research in the area. To this end, the first part of the book deals with existence principles and results for nonlinear, Fredholm and Volterra inte gral and integrodifferential equations on compact and half-open intervals, while selected topics (which reflect the particular interests of the authors) such as nonresonance and resonance problems, equations in Banach spaces, inclusions, and stochastic equations are presented in the latter part.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Preliminaries

Abstract

The aim of this book is firstly to provide a comprehensive existence theory for integral and integrodifferential equations, and secondly to present some specialised topics in integral equations which we hope will inspire further research in the area. To this end, the first part of the book deals with existence principles and results for nonlinear, Fredholm and Volterra integral and integrodifferential equations on compact and half-open intervals, while selected topics (which reflect the particular interests of the authors) such as nonresonance and resonance problems, equations in Banach spaces, inclusions, and stochastic equations are presented in the latter part.

Donal O’Regan, Maria Meehan

Chapter 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations

Abstract

In this chapter, a Nonlinear Alternative of Leray-Schauder type will be used to establish an existence principle for the operator equation

$$ \left\{ {\begin{array}{*{20}c} {y'(t) = V{\rm{ y(t)}}} \\ {y(0) = y_{0'} } \\\end{array}} \right. $$

(2.1.1)

on both the compact interval [0, T], and the half-open interval [0, T]. Various cases of the operator V will be discussed. In particular we consider cases when V is composed of either Fredholm or Volterra integral operators, which when coupled with (2.1.1), provide us with existence principles for Fredholm and Volterra integrodifferential equations.

Donal O’Regan, Maria Meehan

Chapter 3. Solution Sets of Abstract Volterra Equations

Abstract

In this chapter we examine the solution set of the abstract Volterra equation

$$ \left\{ {\begin{array}{*{20}c} {x'(t) = x(t){\rm{ a}}{\rm{.e t}} \in {\rm{[0,T]}}} \\ {x(0) = x_{0;} } \\\end{array}} \right. $$

(3.1.1)

here V is the abstract Volterra operator. By placing mild conditions on the operator V we will show that the set of solutions of (3.1.1) is a R _δ set [6] (so in particular nonempty, compact and connected). Aronszajn in 1942 was the first to discuss the structure of the solution set of differential equations. However it was only in the late 1970’s that a general theory was established. In particular we refer the reader to the papers of Szufla (see [14]) and his coworkers. In this chapter we present a modern theory (adapted from [4, 13]) on the topological structure of the solution set of abstract Volterra equations. The technique used in this chapter involves applying a well known result of Szufla (see [6 page 161] for an elementary proof) together with a trick involving the Urysohn function. For convenience we recall the result of Szufla here.

Donal O’Regan, Maria Meehan

Chapter 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals

Abstract

In this chapter we present existence theory for the nonlinear Fredholm integral equation

$$ y(t) = h(t) + \smallint _0^T k(t,s)g(s,y(s))ds,$$

(4.1.1)

and the nonlinear Volterra integral equation

$$ y(t) = h(t) + \smallint _0^t k(t,s)g(s,y(s))ds,$$

(4.1.2)

when both are defined on the compact interval [0, T]. Naturally we first concern ourselves with existence principles for both equations.

Donal O’Regan, Maria Meehan

Chapter 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals

Abstract

In this chapter, we examine the Fredholm integral equation

$$ y(t) = \smallint _0^T k(t,s)g(s,y(s))ds, $$

(5.1.1)

and the Volterra integral equation

$$ y(t) = \smallint _0^t k(t,s)g(s,y(s))ds, $$

(5.1.2)

when both are defined on the half-open interval [0, T), with 0 ≤ T ≤ ∞. We present a comprehensive collection of existence principles for (5.1.1) and (5.1.2). In particular we establish the existence of a solution y ∈ L ^p[0, T) (1 ≤ p < ∞) of both equations, the existence of a solution y ∈ C _l[0, ∞) of both equations (with T = ∞), and the existence of a solution y ∈ BC[0, ∞) of (5.1.1) when T = ∞. Examining the Volterra integral equation independently, yields a further two existence principles providing us with conditions under which a solution y ∈ L _loc ^p [0, T) or y ∈ C[0, T) exists for (5.1.2). Using the existence principles established, several existence results are presented for both equations.

Donal O’Regan, Maria Meehan

Chapter 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations

Abstract

In this chapter we consider the nonresonant operator equations

$$ y(t) = {\rm{ (}}\gamma {\rm{ + }}\Upsilon (t))Ly(t) + Ny(t) $$

(6.1.1)

and

$$ y(t) = {\rm{ (}}\gamma {\rm{ + }}\Upsilon {\rm{(t))Ly(t) + }}\mu {\rm{Ny(t),}} $$

(6.1.2)

where both are defined on [0, T]. With X equal to either L ^p[0, T], p ≥ 2 or C[0, T], we assume throughout that L : L ²[0, T] → X is a linear, completely continuous, self-adjoint, nonnegative operator, and N : X → X is possibly nonlinear. The spectral theory of L is discussed in detail and plays an integral part in all the results. In particular the hypotheses on γ and τ rely on having knowledge of the eigenvalues of L. We present an existence principle which establishes the existence of a solution y ∈ X of (6.1.1), and from this we obtain an existence result for (6.1.2) when the nonlinear operator N satisfies a growth condition.

Donal O’Regan, Maria Meehan

Chapter 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations

Abstract

Having discussed nonresonant operator and integral equations in Chapter 6, we now turn our attention to the more difficult problem of providing an existence theory for resonant operator and integral equations.

Donal O’Regan, Maria Meehan

Chapter 8. Integral Inclusions

Abstract

This chapter studies integral inclusions in Banach spaces. In particular we discuss the Volterra integral inclusion

$$ y(t){\rm{ }} \in {\rm{ g(t) + }}\smallint _0^t k(t,s){\rm{ F(s,y(s))ds for t }} \in {\rm{ [0,T]}} $$

(8.1.1)

and the Hammerstein integral inclusion

$$ y(t){\rm{ }} \in {\rm{ g(t) + }}\smallint _0^T k(t,s){\rm{ F(s,y(s))ds for t }} \in {\rm{ [0,T]}} $$

(8.1.2)

Here F : [0, T] × E → E is a multivalued map with nonempty compact values; E is a real Banach space. In section 8.2 we present some existence results for (8.1.1) and (8.1.2) when F is a Carathéodory multifunction of u.s.c. or l.s.c. type satisfying some measure of noncompactness assumption. The theory of differential inclusions, usually when dim E < ∞, has received a lot of attention over the last twenty years or so. In this chapter a mixture of old and new ideas are presented so that a general existence theory for multivalued equations can be obtained. The ideas in this chapter were adapted from Deimling [4], Frigon [6] and O’Regan [13]. In particular the technique used in this chapter relies on Ky Fan’s or Schauder’s Fixed Point Theorem [16] together with a trick introduced in [9] and a result of Fitzpatrick and Petryshyn [5].

Donal O’Regan, Maria Meehan

Chapter 9. Approximation of Solutions of Operator Equations on the Half Line

Abstract

In this chapter we discuss existence and approximation for the nonlinear operator equation on the half line

$$ y(t){\rm{ = F y(t) on [0,}}\infty ). $$

(9.1.1)

Solutions will be sought in C([0, ∞]), R _k), k ∈ N ⁺ = {1, 2, …}. A particular example of (9.1.1) will be the nonlinear integral equation

$$ y(t){\rm{ = h(t) + }}\smallint _0^\infty {\rm{ k(t,s,y(s)) ds for t }} \in {\rm{ [0,}}\infty ). $$

(9.1.2)

Finite section approximations for (9.1.2) are given by

$$ y(t){\rm{ = h(t) + }}\smallint _0^n {\rm{ k(t,s,y(s)) ds for t }} \in {\rm{ [0,}}\infty ). $$

(901.3)

for n ∈ N ⁺. Note that (9.1.3)ⁿ, for fixed n ∈ N ⁺, determines y(t) for t > n in terms of y(x) for x ∈ [0, n] so in fact the finite section approximations reduce to integral equations on bounded intervals (we note as well that various discretization techniques, such as numerical integration, are available for the approximate solution of (9.1.3)ⁿ, n ∈ N ⁺ fixed). The technique which we present in this chapter to establish existence and approximation of solutions to (9.1.2) (or more generally (9.1.1)) involves using a new fixed point approach for equations on the half line (see [7, 10‐12]) together with the well known notion of strict convergence (see [2‐4]). The ideas presented were adapted from [2, 13].

Donal O’Regan, Maria Meehan

Chapter 10. Operator Equations in Banach Spaces Relative to the Weak Topology

Abstract

A lot has been written on differentia] and integral equations in a Banach space relative to the strong topology over the last twenty years or so; see [11, 12] and their references (and also Chapter 8). However only a few results have been obtained for equations in a Banach space relative to the weak topology. The first paper [19] appeared in 1971. There Szep discussed in detail the abstract Cauchy problem

$$ \left\{ {\begin{array}{*{20}c} {y'{\rm{ = f(t'y) on [0}}{\rm{.T]}}} \\ {y(0) = y_0 } \\\end{array}} \right. $$

(10.1.1)

with: [0, T] × B → B a weakly-weakly continuous function and B a reflexive Banach space. The nonreflexive case was examined by Cramer, Lakshmikantham and Mitchell [5] and more recently by Kubiaczyk and Szufla [10] and Szufla [20]. Motivated by the above studies, our goal in this chapter is to present an existence theory for the general operator equation in Banach spaces relative to the weak topology. Using a well known fixed point theorem we will obtain a variety of existence principles for (10.1.2). Our general theory will include as particular cases the Cauchy problem (10.1.1) and the Volterra integral equation the integral in (10.1.3) is understood to be the Pettis integral. The theory presented in this chapter was adapted from O’Regan [13‐16]. Also in this chapter we will discuss approximation of solutions to (10.1.2) using the notion of a set of collectively compact operators in a Banach space relative to the weak topology.

Donal O’Regan, Maria Meehan

Chapter 11. Stochastic Integral Equations

Abstract

In this chapter we first present some random fixed point theorems for random operators. These results rely on classical continuation methods; in particular on the idea of an essential map. In section 11.3 our fixed point theory will then be applied to obtain a general existence principle for stochastic integral equations of Volterra type. This principle will then be used to establish the existence of sample solutions to a class of stochastic integral equations. The ideas in this chapter were adapted from the papers of Deimling, Ladde and Lakshmikantham [2], Itoh [6] and O’Regan [12].

Donal O’Regan, Maria Meehan

Chapter 12. Periodic Solutions for Operator Equations

Abstract

In this chapter a variety of existence results will be presented for the periodic operator equation

$$ \left\{ {\begin{array}{*{20}c} {y'(t){\rm{ = N y(t) a}}{\rm{.e on [0,T]}}} \\ {y(0) = y(T).} \\\end{array}} \right. $$

(12.1.1)

Here N : C[0, T] → L ¹ [0, T] is a continuous operator and y takes values in R. By a solution to (12.1.1) we mean a function y ∈ AC[0, T] with y satisfying the equation in (12.1.1) almost everywhere and with y(0) = y(T). In this abstract setting very little is known concerning the existence of solutions to (12.1.1). In the particular case when (12.1.1) reduces to a first order differential equation, namely,

$$ \left\{ {\begin{array}{*{20}c} {y'(t){\rm{ = f(t,y(t)) a}}{\rm{.e on [0,T]}}} \\{y(0) = y(T)} \\\end{array}} \right. $$

(12.1.2)

there are many results in the literature (here: [0, T] × R → R is a L ¹-Carathéodory function); we refer the reader to [6, 9, 10, 11]. The ideas in this chapter were adapted from Meehan and O’Regan [7].

Donal O’Regan, Maria Meehan

Backmatter

Titel: Existence Theory for Nonlinear Integral and Integrodifferential Equations
verfasst von: Donal O’Regan
Maria Meehan
Verlag: Springer Netherlands
Electronic ISBN: 978-94-011-4992-1
Print ISBN: 978-94-010-6095-0
DOI: https://doi.org/10.1007/978-94-011-4992-1

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Preliminaries

Chapter 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations

Chapter 3. Solution Sets of Abstract Volterra Equations

Chapter 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals

Chapter 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals

Chapter 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations

Chapter 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations

Chapter 8. Integral Inclusions

Chapter 9. Approximation of Solutions of Operator Equations on the Half Line

Chapter 10. Operator Equations in Banach Spaces Relative to the Weak Topology

Chapter 11. Stochastic Integral Equations

Chapter 12. Periodic Solutions for Operator Equations

Backmatter