Geometric Properties of Banach Spaces and Nonlinear IterationsThe 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. |
Contents
Some Geometric Properties of Banach Spaces | 1 |
Smooth Spaces | 11 |
Duality Maps in Banach Spaces | 19 |
Finite Families of Nonself Asymptotically Nonexpansive | 20 |
Inequalities in Uniformly Convex Spaces | 29 |
ст | 44 |
6 | 54 |
11 | 113 |
Common Fixed Points for Finite Families of Nonexpansive | 205 |
Common Fixed Points for Countable Families | 215 |
Common Fixed Points for Families of Commuting | 230 |
Finite Families of Lipschitz Pseudocontractive | 243 |
Generalized Lipschitz Pseudocontractive and Accretive | 251 |
Families of Total Asymptotically Nonexpansive Maps | 271 |
Common Fixed Points for Oneparameter Nonexpansive | 283 |
23 | 299 |
19 | 119 |
An Example Mann Iteration for Strictly | 141 |
Approximation of Fixed Points of Lipschitz | 151 |
Generalized Lipschitz Accretive and Pseudocontractive | 161 |
Applications to Hammerstein Integral Equations 169 | 168 |
Iterative Methods for Some Generalizations | 193 |
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Geometric Properties of Banach Spaces and Nonlinear Iterations Charles Chidume No preview available - 2009 |
Common terms and phrases
accretive mappings accretive operators Appl arbitrary asymptotically nonexpansive mappings Banach limit bounded subset Chidume closed convex subset common fixed point Contraction Mapping converges strongly convex Banach space Corollary equations exists family of nonexpansive finite family fixed point property following conditions following theorem Gâteaux differentiable norm Hence Hilbert space implies integer Ishikawa iteration method iteration process j(xn Lemma Let T1 lim sup Lipschitz mappings Lipschitz pseudo-contractive map Lp spaces Mann iteration Math nonempty closed convex nonempty subset normed linear space prove the following q-uniformly quasi-nonexpansive mappings real Banach space real Hilbert space real sequence recursion formula reflexive Banach space satisfying the following sequence in 0,1 sequence xn smooth Banach space ẞn Strong convergence uniformly continuous uniformly convex Banach uniformly Gâteaux differentiable uniformly smooth Banach unique solution variational inequality Xn+1 Zegeye