Methods in Nonlinear Integral Equations
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 v
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Overview 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 Rn). Here we present existence, uniqueness, localization, and approximation results for Fredholm, Volterra, and Hammerstein integral equations, and an integral equation with delay. Although most of the methods are classical, in several cases new points of view, extensions and new applications are presented. Thus the selected topics include: a unified existence theory for continuous and integrable solutions of integral equations; Schechter's bounded mountain pass theorem; a nontrivial solvability theory for Hammerstein integral equations; a localization result for a superlinear elliptic problem; the discrete continuation principle for generalized contractions; new fixed point results for increasing and decreasing operators in ordered Banach spaces; monotone iterative techniques; and the quasilinearization method for a delay integral equation from biomathematics. Of course, the selective topics reflect the particular interests of the author. Let us now briefly describe the contents of this book without insisting on the exact meaning of the notation and notions which are used. Chapter 1 presents basic concepts and results of compactness in general metric spaces, and in particular, in the spaces C (K; Rn) and LP (0; Rn) , where K is a compact metric space, 0 eRN is bounded open and 1 R. Precup, Methods in Nonlinear Integral Equations © Springer Science+Business Media Dordrecht 2002
2
Chapter 0
1 :::; p < 00. We state and prove Hausdorff's theorem of characterization of the relatively compact subsets of a complete metric space, in terms of finite or relatively compact E-nets. Then we prove the Ascoli-Arzela compactness criterion in C (K; Rn), and the Frechet-Kolmogorov theorem of characterization of relatively compact subsets of V (0; Rn) (1:::; P < (0) . We conclude this chapter by the following unified compactness criterion for both C (0; Rn) and V (0; Rn).
Theorem 0.1 Let 0 C RN be bounded open and p E [1,00]. Let Y c V (0; Rn) for 1 :::; p < 00, and Y c C (0; Rn) for p = 00. Assume that there exists a function v E V (0) such that
lu (x)1
:::; v (x)
for almost every x E 0 and all u E Y. Then Y is relatively compact in V (0; Rn) if and only if sup ITh (u) - ulLP(o .Rn)
uEY
Here Oh
h,
= 0 n (0 - h) and
Th (u)(x)
-+
0
as h
-+
o.
= u (x + h) for all x
E 0h·
Chapter 2 is devoted to the concept of a completely continuous operator. We prove the theorem of representation of the completely continuous operators as uniform limits of sequences of continuous operators of finite rank. Also we pres
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