Geometric Properties of Banach Spaces and Nonlinear Iterations

Frontmatter

1. Some Geometric Properties of Banach Spaces

In the first part of this monograph (Chapters 1 to 5), we explore selected geometric properties of Banach spaces that will play crucial roles in our study of iterative algorithms for nonlinear operators in various Banach spaces.

In this chapter, we introduce the classes of uniformly convex and strictly convex spaces, and in Chapter 2, we shall introduce the class of smooth spaces. All the results presented in these two chapters are well-known and standard and can be found in several books on geometry of Banach spaces, for example, in Diestel [206], or in Lindenstrauss and Tzafriri [312]. Consequently, we shall skip some details and long proofs.

2. Smooth Spaces

In this chapter, we introduce the class of smooth spaces. We remark immediately that there is a duality relationship between uniform smoothness and uniform convexity. In the sequel, we shall examine this relationship. We begin with the following definition.

3. Duality Maps in Banach Spaces

In trying to develop analogue of the identity (1.1) in Banach spaces more general than Hilbert spaces, one has to find a suitable replacement for inner product, ⟨.,.⟩. In this chapter, we present the notion of duality mappings which will provide us with a pairing between elements of a normed space E and elements of its dual space E*, which we shall also denote by ⟨.,.⟩ and will serve as a suitable analogue of the inner product in Hilbert spaces.

4. Inequalities in Uniformly Convex Spaces

Among all Banach spaces, Hilbert spaces are generally regarded as the ones with the simplest geometric structures. The reason for this observation is that certain geometric properties which characterize Hilbert spaces (e.g., the existence of inner product; the para l lelogram law or equivalently the polariza¬tion identity; and the fact that the proximity map or nearest point mapping in Hilbert spaces is Lipschitz with constant 1) make certain problems posed in Hilbert spaces comparatively straightforward and relatively easy to solve. In several applications, however, many problems fall naturally in Banach spaces more general than Hilbert spaces. Therefore, to extend the techniques of so¬lutions of problems in Hilbert spaces to more general Banach spaces, one needs to establish identities or inequalities in general Banach spaces analo¬gous to the ones in Hilbert spaces. As shown by recent works, several authors have conducted worthwhile research in this direction (e.g., Al'ber ([3], [4], [9], Beauzamy [26], Bynum [61, 62], Clarkson [191], Lindenstrauss ([309], [310]), Hanner [247], Kay [276], Lim [306, 303], Lindenstrauss and Tzafriri [311], Prus and Smarzewski [387], Reich [408], Tribunov [491], Xu [509], Xu [523], Xu and Roach [525], and a host of other authors). In this chapter (and also in Chapter 5), we shall describe some of the results obtained primarily within the last thirty years or so. Applications of these results to iterative solutions of nonlinear equations in Banach spaces will be given in subsequent chapters.

5. Inequalities in Uniformly Smooth Spaces

In this chapter, we obtain analogues of the identities (1.1) and (1.2) in smooth spaces. We begin with the following definitions.

6. Iterative Method for Fixed Points of Nonexpansive Mappings

We begin this chapter with the following well known definition and theorem.

7. Hybrid Steepest Descent Method for Variational Inequalities

Let (E, ρ) be a metric space and K be a nonempty subset of E. For every x∊E, the distance between the point x and K is denoted by ρ(x, K) and is defined by the following minimum problem: \(\rho {\rm{(}}x{\rm{,}}K{\rm{) : = }}\mathop {{\rm{inf}}}\limits_{y \in K} {\rm{ }}\rho {\rm{(}}x{\rm{,}}y{\rm{)}}{\rm{.}}\) The metric projection operator (also called the nearest point mapping) Pk defined on E is a mapping from E to 2K such that \(P_K {\rm{(}}x{\rm{) : = \{ }}z{\rm{ }} \in {\rm{ }}K{\rm{ : }}\rho {\rm{(}}x{\rm{, }}z{\rm{) = }}\rho {\rm{(}}x{\rm{,}}K{\rm{)\} }}\forall {\rm{ }}x{\rm{ }} \in {\rm{ }}E{\rm{.}}\).

8. Iterative Methods for Zeros of Ф – Accretive-Type Operators

In this chapter, we continue to apply the Mann iteration method introduced in Chapter 6. Here, we use it to approximate the zeros of Ф-strongly accretive operators (and to approximate fixed points of Ф-strong pseudo-contractions).

9. Iteration Processes for Zeros of Generalized Ф —Accretive Mappings

10. An Example; Mann Iteration for Strictly Pseudo-contractive Mappings

We have seen (Chapter 6) that the Mann iteration method has been successfully employed in approximating fixed points (when they exist) of nonexpansive mappings. This success has not carried over to the more general class of pseudo-contractions. If K is a compact convex subset of a Hilbert space and T : K → K is Lipschitz, then, by Schauder fixed point theorem, T has a fixed point in K. All efforts to approximate such a fixed point by means of the Mann sequence when T is also assumed to be pseudo-contractive proved abortive. In 1974, Ishikawa introduced a new iteration scheme and proved the following theorem.

11. Approximation of Fixed Points of Lipschitz Pseudo-contractive Mappings

In Chapter 10, we stated the Ishikawa iteration theorem (Theorem 10.1) which converges strongly to a fixed point of a Lipschitz pseudo-contractive map T defined on a compact convex subset of a Hilbert space.

It is still an open question whether or not Theorem 10.1 can be extended to Banach spaces more general than Hilbert spaces. However, two other iteration methods have been introduced and have successfully been employed to approximate fixed points of Lipschitz pseudo-contractive mappings in certain Banach spaces more general than Hilbert spaces.

12. Generalized Lipschitz Accretive and Pseudo-contractive Mappings

We have seen in Chapter 11 that several convergence results have been proved on iterative methods for approximating zeros of Lipschitz accretive-type (or, equivalently, fixed points of Lipschitz pseudo-contractive-type) nonlin ear mappings. We have also seen (Chapter 9) that a natural generalization of the class of Lipschitz mappings and the class of mappings with bounded range is that of generalized Lipschitz mappings. In this chapter, by means of an iteration process introduced by Chidume and Ofoedu [152], we prove con vergence theorems for fixed points of generalized Lipschitz pseudo-contractive mappings in real Banach spaces.

13. Applications to Hammerstein Integral Equations

In this chapter, we shall examine iterative methods for approximating solu tions of important nonlinear integral equations involving accretive-type op erators. In particular, we examine iteration methods for solving nonlinear integral equations of Hammerstein type.

14. Iterative Methods for Some Generalizations of Nonexpansive Maps

Some generalizations of nonexpansive mappings which have been studied extensively include the (i) quasi-nonexpansive mappings; (ii) asymptotically nonexpansive mappings; (iii) asymptotically quasi-nonexpansive mappings.

For the past 30 years or so, iterative algorithms for approximating fixed points of operators belonging to subclasses of these classes of nonlinear mappings and defined in appropriate Banach spaces have been flourishing areas of research for many mathematicians. For the classes of mappings mentioned here in (i) to (iii), we show in this chapter that modifications of the Mann iteration algorithm and of the Halpern-type iteration process studied in chapter 6 can be used to approximate fixed points (when they exist).

15. Common Fixed Points for Finite Families of Nonexpansive Mappings

Markov ([320])(see also Kakutani [270]) showed that if a commuting family of bounded linear transformations T_α, α ϵ ▵ (▵ an arbitrary index set) of a normed linear space E into itself leaves some nonempty compact convex subset K of E invariant, then the family has at least one common fixed point. (The actual result of Markov is more general than this but this version is adequate for our purposes).

Motivated by this result, De Marr studied the problem of the existence of a common fixed point for a family of nonlinear maps, and proved the following theorem.

16. Common Fixed Points for Countable Families of Nonexpansive Mappings

Various authors have studied iterative schemes similar to that of Bauschke (Theorem BSK, Chapter 15) in more general Banach spaces on one hand and using various conditions on the sequence {λ_n} on the other hand (see, for example, Colao et al. [192], Yao [532], Takahashi and Takahashi [482], Plubtieng and Punpaeng [385], Ceng et al. [193], Chidume and Ali [125], Jung [264], Jung et al. [265], O'Hara et al. [361], Zhou et al. [559]). Most of the results in these references are proved for finite families of nonexpansive mappings defined in Hilbert spaces.

Convergence theorems have also been proved for common fixed points of countable infinite families of nonexpansive mappings. Before we proceed, we first state the following important theorem.

17. Common Fixed Points for Families of Commuting Nonexpansive Mappings

In this chapter, we present an iteration process which has been studied for approximating common fixed points for families of commuting nonexpansive mappings defined on a compact convex subset of a Banach space.

We first prove the following lemmas which are connected with real num bers.

18. Finite Families of Lipschitz Pseudo-contractive and Accretive Mappings

In this chapter, we study an iteration process for approximating a common fixed point (assuming existence) for a family of Lipschitz pseudo-contractive mappings in arbitrary real Banach spaces.

19. Generalized Lipschitz Pseudo-contractive and Accretive Mappings

In this chapter, we construct an iterative sequence for the approximation of common fixed points of finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive operators (assuming exis tence). These classes of mappings have been defined in Chapter 12. Fur thermore, the iteration scheme introduced here and the method of proof are of independent interest.

20. Finite Families of Non-self Asymptotically Nonexpansive Mappings

21. Families of Total Asymptotically Nonexpansive Maps

22. Common Fixed Points for One-parameter Nonexpansive Semigroup

Let K be a nonempty subset of a normed space E. A family of mappings {T(t) : t ≥ 0} is called a one-parameter strongly continuous semigroup of nonexpansive mappings (or, briefly a nonexpansive semigroup) on K if the following conditions hold.

23. Single-valued Accretive Operators; Applications; Some Open Questions

Set-valued accretive operators in Banach spaces have been extensively studied for several decades under various continuity assumptions. In the first part of this chapter we establish a recent incisive finding that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael's Selection Theorem is needed. In Section 23.3, this result is used to extend known theorems concerning the existence of zeros for such operators, as well as, showing existence of solutions for variational inclusions. In Section 23.4, we make some general comments on some fixed point theorems; the rest of the chapter is devoted to some examples of accretive operators; examples of nonexpansive retracts; open problems; and some suggestions for further reading.

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