main-content

## Über dieses Buch

Methods in Nonlinear Integral Equations presents several extremely fruitful methods for the analysis of systems and nonlinear integral equations. They include: fixed point methods (the Schauder and Leray-Schauder principles), variational methods (direct variational methods and mountain pass theorems), and iterative methods (the discrete continuation principle, upper and lower solutions techniques, Newton's method and the generalized quasilinearization method). Many important applications for several classes of integral equations and, in particular, for initial and boundary value problems, are presented to complement the theory. Special attention is paid to the existence and localization of solutions in bounded domains such as balls and order intervals. The presentation is essentially self-contained and leads the reader from classical concepts to current ideas and methods of nonlinear analysis.

## Inhaltsverzeichnis

### Chapter 0. Overview

Abstract
As the title suggests, this book presents several methods of nonlinear analysis for the treatment of nonlinear integral equations. To this end the book is developed on two levels which interfere. The first level is devoted to the abstract results of nonlinear analysis: compactness criteria, fixed point theorems, critical point results, and general principles of iterative approximation. The second level is devoted to the applications for systems of nonlinear integral equations (nonlinear integral equations in R n ). Here we present existence, uniqueness, localization, and approximation results for Fredholm, Volterra, and Hammerstein integral equations, and an integral equation with delay.

### Chapter 1. Compactness in Metric Spaces

Abstract
In this chapter we first define the notions of a compact metric space and of a relatively compact subset of a metric space. Then we state and prove Hausdorff’s theorem of the characterization of the relatively compact subsets of a complete metric space in terms of finite and relatively compact ε--nets. Furthermore, we prove the Ascoli-Arzèla and Fréchet-Kolmogorov theorems of characterization of the relatively compact subsets of C (K; R n ) and L p (Ω; R n ), respectively. Here K is a compact metric space, Ω ⊂ R N is a bounded open set and 1 ≤ p < ∞.

### Chapter 2. Completely Continuous Operators on Banach Spaces

Abstract
In this chapter we define the notion of a completely continuous operator from a Banach space to another Banach space and we present some simple properties of the completely continuous operators. Next we prove Brouwer’s fixed point theorem. Finally, we prove the famous Schauder fixed point theorem which, like Banach’s contraction principle, represents a fundamental result in nonlinear analysis.

### Chapter 3. Continuous Solutions of Integral Equations via Schauder’s Theorem

Abstract
This chapter presents three examples of nonlinear integral operators which are completely continuous on some spaces of continuous functions: the Fredholm integral operator, the Volterra integral operator, and a particular integral operator with delay. Simultaneously, by means of Schauder’s fixed point theorem we prove existence theorems for continuous solutions of the integral equations associated to these operators.

### Chapter 4. The Leray-Schauder Principle and Applications

Abstract
In applications one of the drawbacks of Schauder’s fixed point theorem is the invariance condition T (D)D which has to be guaranteed for a bounded closed convex subset D of a Banach space. The Leray-Schauder principle [32] makes it possible to avoid such a condition and requires instead that a ‘boundary condition’ is satisfied. In this chapter we shall prove the Leray-Schauder principle and we shall apply it in order to obtain existence results for continuous solutions of integral equations. In particular, we give results on the existence of continuous solutions of initial value and two-point boundary value problems for nonlinear ordinary differential equations in R n . The results will be better than those established by means of Schauder’s theorem.

### Chapter 5. Existence Theory in L p Spaces

Abstract
In this chapter we present three examples of continuous operators acting in L p spaces, namely: the Nemytskii superposition operator; the Fredholm linear integral operator; and the Hammerstein nonlinear integral operator. As applications we shall prove via the Leray-Schauder principle several existence results in L p for Hammerstein and Volterra-Hammerstein integral equations in R n. We show that these results immediately yield existence theorems of weak solutions (in Sobolev spaces) to the initial value and two-point boundary value problems for ordinary differential equations in R n, under some more general conditions than the continuity. Notice the weak solutions are functions which satisfy the differential equations almost everywhere (a.e., that is, except a set of measure zero).

### Chapter 6. Positive Self-Adjoint Operators in Hilbert Spaces

Abstract
In this part we present some variational methods with applications to the existence of L p solutions of the Hammerstein integral equation in R n
$$u(x) = \int\limits_\Omega K \left( {x,y} \right)f\left( {y,u\left( y \right)} \right)dy\quad a.e\quad on\quad \Omega$$
(6.1)

### Chapter 7. The Fréchet Derivative and Critical Points of Extremum

Abstract
In this chapter we present the notion of Fréchet derivative of a functional and we illustrate it by some examples. Then we build a functional E : L 2 (Ω; R n ) → R whose Fréchet derivative is the operator
$$I - {H^ * }{N_f}H:{L^2}\left( {\Omega ;{R^n}} \right) \to {L^2}\left( {\Omega ;{R^n}} \right)$$
associated to (6.8). We prove the infinite-dimensional version of the classical Fermat’s theorem about the connection between extremum points and critical points, and we give sufficient conditions for that a functional admits minimizers. The abstract results are then applied to establish the existence of L p solutions for Hammerstein integral equations in R n .

### Chapter 8. The Mountain Pass Theorem and Critical Points of Saddle Type

Abstract
In Chapter 9 we shall continue the investigation of the L p solutions of the Hammerstein integral equations under the assumption that f (x, 0) = 0, that is, the null function is a solution. We are now interested in non-null solutions. The technique we use is based on the so called mountain pass theorem of Ambrosetti-Rabinowitz [3]. By this method one can establish the existence of a critical point u of the functional E which in general is not an extremum point of E, and has the property that in any neighborhood of u there are points v and w with E (v) < E (u) < E (w). Such a critical point is said to be a saddle point of E.

### Chapter 9. Nontrivial Solutions of Abstract Hammerstein Equations

Abstract
This chapter deals with nontrivial solvability in balls of abstract Hammer-stein equations and Hammerstein integral equations in R n by a variational approach. The variational method for treating Hammerstein integral equations goes back to the papers of Hammerstein [17] and Golomb [15]. For further contributions see the monographs by Krasnoselskii [19], Krasnoselskii-Zabreiko-Pustylnik-Sobolevskii [20] and Vainberg [43]. For more recent results see the papers by Moroz-Vignoli-Zabreiko [24] and Moroz-Zabreiko [25], [26]. The results presented in this chapter were adapted from Precup [31]–[34].

### Chapter 10. The Discrete Continuation Principle

Abstract
The Banach contraction principle was generalized by Perov (see Perov-Kibenko [40]) for contractive maps on spaces endowed with vector-valued metrics. Also, Granas [25] proved that the property of having a fixed point is invariant under homotopy for contractions on complete metric spaces. This result was completed in Precup [50] (see also O’Regan-Precup [38] and Precup [51], [52]) by an iterative procedure of discrete continuation along the fixed points curve. This chapter presents a variant for contractive maps on spaces with vector-valued metrics, first given in Precup [54].

### Chapter 11. Monotone Iterative Methods

Abstract
The basic notion in this chapter is that of an ordered Banach space. We try to localize solutions of an operator equation u = T (u) in an ordered interval [u 0, v 0] of an ordered Banach space X. In addition we look for solutions which are limits of increasing or decreasing sequences of elements of X. The basic property of the operator T is monotonicity. This combined with certain properties of the ordered Banach space X guarantees the convergence of monotone sequences. Thus we may say that this chapter explores the contribution of monotonicity to compactness.

### Chapter 12. Quadratically Convergent Methods

Abstract
The succesive approximation method as well as the monotone iterative methods described previously are not very fast. To explain this let us consider an operator T : X → X. All these methods give us convergent sequences (u k ) of the form
$${u_{k + 1}} = T({u_k}),k \in N$$
having as limit a fixed point u * of T. For the results in Chapter 10, we have
$$d({u_{k + 1}},{u^*}) \le Md({u_{k,}}{u^*})$$
.