# Characteristic Inequalities in Banach Spaces and Applications

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**Characteristic Inequalities in Banach Spaces and Applications.**

**TABLE OF CONTENT**

Dedication iii

Preface v

Acknowledgement vii

1 Preliminaries 1

1.1 Basic notions of functional analysis . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Differentiability in Banach spaces . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Duality mapping in Banach spaces . . . . . . . . . . . . . . . . . . . 5

1.1.3 The signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Convex functions and sub-differentials . . . . . . . . . . . . . . . . . 8

2 Characteristic Inequalities 11

2.1 Uniformly convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Strictly convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Inequalities in uniformly convex spaces . . . . . . . . . . . . . . . . . 14

2.2 Uniformly smooth spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Inequalities in uniformly smooth spaces . . . . . . . . . . . . . . . . . 19

2.2.2 Characterization of uniformly smooth spaces by the duality maps . . 21

3 Sunny Nonexpansive Retraction 23

3.1 Construction of sunny nonexpansive retraction in Banach spaces . . . . . . . 23

4 An Application of Sunny Nonexpansive Retraction 33

Bibliography 37

**INTRODUCTION**

Basic notions of functional analysis: In this chapter, we recall some definitions and results from linear functional analysis. Proposition 1.1.1 (The Parallelogram Law) Let X be an inner product space.

Then for arbitrary x, y ∈ X, kx + yk 2 + kx − yk 2 = 2 kxk 2 + kyk 2 . Theorem 1.1.1 (The Riesz Representation Theorem) Let H be a Hilbert space and let f be a bounded linear functional on H.

Then there exists a unique vector y0 ∈ H such that f(x) = hx, y0i for each x ∈ H and ky0k = kfk.

Theorem 1.1.2 Let X be a reflexive and strictly convex Banach space, K be a nonempty, closed, and convex subset of X. Then for any fixed x ∈ X there exists a unique m∗ ∈ K such that kx − m∗ k = inf k∈K kx − kk. Proof.

Let x ∈ X be fixed, and define Px : X → R ∪ {∞} by Px(k) = kx − kk, if k ∈ K, ∞, if k /∈ K. Clearly Px is convex. Indeed, let λ ∈ (0, 1) and k1, k2 ∈ X. If any of k1 or k2 is not in K, then Px(λk1 + (1 − λ)k2) ≤ λPx(k1) + (1 − λ)Px(k2) since the right handside is ∞.

Now suppose both elements are in K. Then Px(λk1 + (1 − λ)k2) = kλ(k1 − x) + (1 − λ)(k2 − xk ≤ kλ(k1 − x)k + k(1 − λ)(k2 − xk = λPx(k1) + (1 − λ)Px(k2). We next show that Px is lower semicontinuous.

By the continuity of the map k 7→ kx − kk, k ∈ K, we have Px is lower semicontinuous on K. We now show Px is lower semicontinuous on Kc .

Let x0 ∈ Kc and α ∈ R such that α < Px(x0). Since K is closed, Kc is an open neighbourhood of x0 and α < Px(y) ∀y ∈ Kc . Hence Px is lower semicontinuous on Kc and therefore on the whole X. Obviously Px is proper.

Next, we show that Px is coercive. Let y ∈ X. Then Px(y) ≥ kyk − kxk ≥ kyk − kyk 2 = kyk 2 provided kyk ≥ 2kxk. This implies that Px(y) → ∞ as kyk → ∞.

Thus, Px is lower semicontinuous, convex, proper, and coersive. Hence there exists m∗ ∈ X such that Px(m∗ ) ≤ Px(m) ∀m ∈ X. Since Px(y) = ∞ for all y ∈ Kc and K 6= ∅, we must have m∗ ∈ K. Furthermore kx−m∗k = Px(m∗ ) ≤ Px(k) = kx − kk, ∀k ∈ K.

This completes the proof. We now show that m∗ ∈ K is unique. Indeed, if x ∈ K then m∗ = x and hence it is unique.

Suppose x ∈ Kc and m 6= n such that kx − mk = kx − nk ≤ kx − kk ∀k ∈ K, then 1 kx − mk k 1 2 ((x − m) + (x − n))k < 1. This implies that kx − 1 2 (m + n)k < kx − mk and this contradict the fact that m is a minimizing vector in K.

Therefore m∗ ∈ K is unique. Corollary 1.1.1 Let X be a uniformly convex Banach space and K be any nonempty, closed and convex subset of X. Then for arbitrary x ∈ X there exists a unique k ∗ ∈ K such that kx − k ∗ k = inf k∈K kx − kk.

Remark If H is a real Hilbert space and M is any nonempty, closed, and convex subset of H then in view of the above corollary, then there exists a unique map PM : H → M defined by x 7→ PMx, where kx − PMxk = inf m∈M kx − mk. This map is called the projection map.

The following properties of projection map PM of H onto M are well known. (1) z = PMx ⇔ hx − z, m − zi ≤ 0 ∀m ∈ M. (2) kPMx − PMyk 2 ≤ hx − y, PMx − PMyi ∀x, y ∈ H, which implies that kPMx − PMyk ≤ kx − yk ∀x, y ∈ H, i.e., PM is nonexpansive. (3) PM(PMx + t(x − PMx)) = PMx ∀t ≥ 0.

BIBLIOGRAPHYArkady Aleyner and Simeon Reich; An Explicit Construction of Sunny Nonexpansive Retractions in Banach Spaces, Fixed Point Theory and Applications 2005:3(2005)295- 305

C.E. Chidume; Applicable functional analysis, International Centre for Theoretical Physics Trieste, Italy, July 2006.

C. E. Chidume ; Geometric Properties of Banach Spaces and Nonlinear Iterations, International Center for Theoretical Physics, Triesty, Italy, June 2008

C.E. Chidume., Zegeye, H. and Shahzad, N.; Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings, Fixed Point Theory Application. (2005), no. 2, 233-241.

Haim Brezis; Functional Analysis,Sobolev Spaces and Partial Differential Equations,Springer-Verlag New York Inc. 2010.

Joseph Diestel; Geometry of Banach Spaces, Springer-Verlag(1975).djvu

N. Shioji and W. Takahashi; Strong Convergence of Approximated Sequences for Nonexpansive Mappings in Banach Spaces, Proc. Amer. Math. Soc. 125(1997), no. 12, 3641-3645.

R. E. Bruck; Nonexpansive Projections on Subset of Banach Spaces, Pacific J. Math. 47(1973), 341-355.

Tomas Dominguez Benavides, Genaro Lopez Acedo and Hong-Kun Xu; Construction of Sunny Nonexpansive Retraction in Banach Spaces, Bulletin of the Australian Mathematical Society / Volume 66 / Issue 01 / August 2002, pp 9-16.

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