Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-contractive Maps in Hilbert Spaces

Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-contractive Maps in Hilbert Spaces

Chapter One

PREAMBLE TO THE STUDY

The contribution of this thesis falls under a branch of mathematics called Functional Analysis. Functional Analysis as an independent mathematical discipline started at the turn of the 19th century and was finally established in 1920’s and 1930’s, on one hand under the influence of the study of specific classes of linear operators-integral operators and integral equations connected with them-and on the other hand under the influence of the purely intrinsic development of modern mathematics with its desire to generalize and thus to clarify the true nature of some regular behaviour. Quantum Mechanics also had a great influence on the development of Functional Analysis, since its basic concepts, for example energy, turned out to be linear operators (which physicists at first rather loosely interpreted as infinite dimensional matrices) on infinite dimensional spaces.

Chapter Two

Preliminaries

In this Chapter we present mostly geometric conditions ensuring that convergence is strong. Most of the properties will be established in the frame work of Hilbert spaces. Although a lot of results can be extended to larger classes of spaces, we will only do so in some specific cases, since our aim is to emphasize unity in terms of tools and approach.

Definitions and Technical Results About Convergent Sequences of Real

Definition (Strong Convergence)

Let H be a Hilbert space. We say that {x_n} converges (strongly) to x if lim_n_→∞ ǁx_n −xǁ =

Thisis written lim_n_→∞ x_n = 0 or simply x_n → x.

Definition (Weak Convergence)

Let H be a Hilbert space. We say that the sequence {x_n}∞n=0 of elements of a Hilbert space H converges weakly to x ∈ H if there is an x ∈ H such that for every f ∈ H^∗, lim_n_→∞ f (x_n) = f (x). We call the point x a weak limit of the sequences {x_n}∞n=0 and write x_n ~ x.

Proposition 2.1.3 A sequence {x_n} in a real Hilbert space H converges weakly to a point

x ∈ H if and only if lim⟨x_n, z⟩ = ⟨x, z⟩ ∀ z ∈ H.

Proof:

Suppose x_n ~ x, then f (x_n) → f (x) ∀ f ∈ H^∗. Let z ∈ H be arbitrary (but fixed).

Then define g : H → f = ঩ or C by g(x) = ⟨x, z⟩. Then g ∈ H^∗, since

g(αx + βy) = ⟨αx + βy, z⟩ = ⟨αx, z⟩ + ⟨βy, z⟩ = α⟨x, z⟩ + β⟨y, z⟩ = αg(x) + βg(y) Also, | g(x) |=| ⟨x, z⟩ |≤ ǁxǁǁzǁ. This implies that g is bounded and ǁgǁ ≤ ǁzǁ.

Since g ∈ H^∗ and x_n ~ x then g(x_n) → g(x)

⇒ ⟨x_n, z⟩ → ⟨x, z⟩

⇐) Suppose ⟨x_n, z⟩ → ⟨x, z⟩ ∀ z ∈ H then we prove that x_n ~ x.

Let f ∈ H^∗ be arbitrary. Then by the Riez Representation Theorem, there exists a unique z ∈ H such that f (x) = ⟨x, z⟩ ∀ x ∈ H. Thus f (x_n) = ⟨x_n, z⟩ and since ⟨x_n, z⟩ → ⟨x, z⟩, we have that f (x_n) → f (x), which implies that x_n ~ x

Definition 2.1.4 A sequence {a_n} is said to be monotone increasing if a_n₊₁ ≥ a_n for all

n ≥ 1, and monotone decreasing if a_n₊₁ ≤ a_n for n ≥ 1. We say {a_n} is monotone (or monotonic) if it is of one of these two types.

We now state and prove the following important results concerning sequences of real numbers.

Lemma 2.1.1 [19] Let {a_n} and {b_n} be two sequences of a normed space X and {t_n} a sequence of real numbers. If the following conditions

0≤ t_n ≤ t < 1 and ^∞n=1 t_n = ∞,
a_n₊₁= (1 − t_n)a_n + t_nb_n for all n ≥ 1,

Chapter Three

Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces

Main Result

In Chapter 2, we presented mostly geometric, conditions ensuring that the convergence is strong.

In this Chapter, we introduce a modified Ishikawa iteration for Lipschitz pseudocontractive map in real Hilbert spaces.

Algorithm 3.1.1 Let H be a real Hilbert space and K be a closed convex subset of H.

Let T : K → K be a Lipschitz pseudo-contractive mapping such that F (T ) ∅. Let

{α_n}, {β_n} and {γ_n} be real sequences in (0, 1). For given x₁ ∈ K, let {x_n} be generated iteratively by

x_n₊₁ = P_k[(1 − α_n − γ_n)x_n + γ_nTy_n];

y_n = (1 − β_n)x_n + β_nTx_n, n ≥ 1 (3.0)

Theorem 3.1.2 Let H be a real Hilbert space and K be a closed convex subset of H. Let

T : K → K be a L-Lipschitz pseudo-contractve mapping such that F (T ) the sequences {α_n}, {γ_n}, {β_n} ∈ (0, 1) satisfying

(i) β_n(1 − α_n) > γ_n∀n ≥ 1 (ii) lim_n_→∞ α_n = 0 and Σα_n = ∞

∅. Assume

0< α ≤ γ_n ≤ β_n ≤ β < [

1 1+L2+1]

for all n ≥ 1. Then the sequence {x_n} generated

by (3.0) strongly converges to a fixed point of T .

Content Summary

Proof of Theorem 3.1.2

Since F (T ) /= ∅, we can take p ∈ F (T ). From (3.0) we have

ǁx_n₊₁ − pǁ = ǁP_k[(1 − α_n − γ_n)x_n + γ_nTy_n] − pǁ

≤ ǁ(1 − α_n − γ_n)x_n + γ_nTy_n − pǁ

= ǁ(1 − α_n − γ_n)(x_n − p) + γ_n(Ty_n − p) − α_npǁ

≤ ǁ(1 − α_n − γ_n)(x_n − p) + γ_n(Ty_n − p)ǁ

+α_nǁpǁ (3.1)

Now consider

ǁ(1 − α_n − γ_n)(x_n − p) + γ_n(Ty_n − p)ǁ = ǁ(1 − α_n)[(1 − γ_n)(x_n − p) + γ_n(Ty_n − p)]

+α_n[−γ_nx_n + γ_nTy_n]ǁ2

References

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Banach, Sur les Operations Dans Les Emsembles Abstraits et Leur Application Aux Equations Integrals. Fundamenta Mathematicae 3, 133-181 (1922).
H. Bauschke, J. Borwein, On Projection Algorithms for Solving Convex Feasibility Problems. SIAM Rev. 38, 367-426 (1996).
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Proof of Theorem 3.1.2

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