Geometric Properties of Banach Spaces and Nonlinear Iterations

1. Front Matter

   Pages i-xvii

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2. Some Geometric Properties of Banach Spaces

   Pages 1-9

3. Smooth Spaces

   Pages 11-18

4. Duality Maps in Banach Spaces

   Pages 19-28

5. Inequalities in Uniformly Convex Spaces

   Pages 29-44

6. Inequalities in Uniformly Smooth Spaces

   Pages 45-55

7. Iterative Method for Fixed Points of Nonexpansive Mappings

   Pages 57-86

8. Hybrid Steepest Descent Method for Variational Inequalities

   Pages 87-111

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

   Pages 113-127

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

    Pages 129-140

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

    Pages 141-149

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

    Pages 151-160

13. Generalized Lipschitz Accretive and Pseudo-contractive Mappings

    Pages 161-167

14. Applications to Hammerstein Integral Equations

    Pages 169-191

15. Iterative Methods for Some Generalizations of Nonexpansive Maps

    Pages 193-204

16. Common Fixed Points for Finite Families of Nonexpansive Mappings

    Pages 205-214

17. Common Fixed Points for Countable Families of Nonexpansive Mappings

    Pages 215-229

18. Common Fixed Points for Families of Commuting Nonexpansive Mappings

    Pages 231-242

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

    Pages 243-250

20. Generalized Lipschitz Pseudo-contractive and Accretive Mappings

    Pages 251-256

The contents of this monograph fall within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: geometric properties of Banach spaces and nonlinear iterations, a topic of intensive research e?orts, especially within the past 30 years, or so. In this theory, some geometric properties of Banach spaces play a crucial role. In the ?rst part of the monograph, we expose these geometric properties most of which are well known. As is well known, among all in?nite dim- sional Banach spaces, Hilbert spaces have the nicest geometric properties. The availability of the inner product, the fact that the proximity map or nearest point map of a real Hilbert space H onto a closed convex subset K of H is Lipschitzian with constant 1, and the following two identities 2 2 2 ||x+y|| =||x|| +2 x,y +||y|| , (?) 2 2 2 2 ||?x+(1??)y|| = ?||x|| +(1??)||y|| ??(1??)||x?y|| , (??) which hold for all x,y? H, are some of the geometric properties that char- terize inner product spaces and also make certain problems posed in Hilbert spaces more manageable than those in general Banach spaces. However, as has been rightly observed by M. Hazewinkel, “... many, and probably most, mathematical objects and models do not naturally live in Hilbert spaces”. Consequently,toextendsomeoftheHilbertspacetechniquestomoregeneral Banach spaces, analogues of the identities (?) and (??) have to be developed.

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• Convexity
• Families of operators
• Hammerstein equations
• Iterative methods
• Nonexpansive mapping
• Smooth function
• Variational inequalities
• operator theory

From the reviews: “The aim of the present book is to give an introduction to this very active area of investigation. … the book is of great help for graduate and postgraduate students, as well as for researchers interested in fixed point theory, geometry of Banach spaces and numerical solution of various kinds of equations - operator differential equations, differential inclusions, variational inequalities.” (S. Cobzaş, Studia Universitatis Babeş-Bolyai. Mathematica, Vol. LIV (4), December, 2009) “The topic of this monograph falls within the area of nonlinear functional analysis. … The main purpose of this book is to expose in depth the most important results on iterative algorithms for approximation of fixed points or zeroes of the mappings mentioned above. … this book picks up the most important results in the area, its explanations are comprehensive and interesting and I think that this book will be useful for mathematicians interested in iterations for nonlinear operators defined in Banach spaces.” (Jesus Garcia-Falset, Mathematical Reviews, Issue 2010 f)

• Abdus Salam International Centre for Theoretical Physics, Mathematics Section, Trieste, Italy

  Charles Chidume

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