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CONVERGENCE IN NORM OF MODIFIED KRASNOSELSKII-MANN ITERATION FOR FIXED POINTS OF ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS

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Abstract

This  project  report  deals with  the  class of asymptotically  demicontractive mappings  in Hilbert spaces. We noted some historical aspects concerning the concept of asymptotically demicontractivity and  studied  a regularized  variant of the  Krasnoselskii-Mann  iteration scheme, which ensured the strong convergence of the generated sequence towards the least norm element of the set of fixed points of asymptotically  demicontractive mapping.

Chapter  1

Introduction

1.1     General Introduction

For the past  30 years or so, the study  of Krasnoselskii-Mann  iterative  procedure  for the approximation of fixed points  of nonexpansive  mappings  and  fixed points  of some gen- eralisations  of nonexpansive  mappings  have been flourishing areas  of research  for many Mathematicians.  Recently,  Mainge and  Maruster  [22], studied  the  convergence in norm of modified Krasnoselkii-Mann  iterations  for fixed points of demicontractive mapping.  In summary,  they considered the following algorithm  in a real Hilbert space H:

x1  ∈ H,  xn+1 = (1 − αn)(1 − βn)xn + αnT [(1 − βn)xn]    n ≥ 1            (1.1.1) The above equation  can also be expressed in the form:

x1  ∈ H,  vn    =   (1 − βn)xn

xn+1   =  (1 − αn)vn  + αnT vn     n ≥ 1                           (1.1.2)

They proved that  if T : H → H is demicontractive, that  is, ||T x − p||2 ≤ ||x − p||2 + k||x − T x||2   for some k ∈ [0, 1),  ∀ x ∈ H, ∀ p ∈ F (T ), where F (T ) (assumed  to be nonempty) is the set of fixed points of T , then  {xn} converges strongly  to the least norm element of F (T ) under some mild assumptions  on the iterative  parameters {αn}, {βn}  which are real sequences in (0, 1). We observe that  the algorithm  (1.1.2) will reduce to Mann iteration  if

βn = 0. That  is, if βn = 0, ∀ n ≥ 1, (1.1.2) reduces to

x1  ∈ H,  xn+1 = (1 − αn)xn + αnT xn     n ≥ 1                           (1.1.3) The important thing here is that  their algorithm yielded strong convergence, but the Mann

iterative algorithm yields only weak convergence. It was shown, however, that  for the Mann iterative  algorithm,kxn − T xnk →  0  as n →  ∞, that  is, the sequence {xn}n≥1 generated by Mann’s algorithm  is an approximate  fixed point sequence of nonexpansive  operator  T . Thus,  to obtain  strong  convergence with Mann algorithm,  additional  requirements  on T and/or its domain are usually required.

1.2     Aim and Scope of Study

The aim of this project is to propose an algorithm (analogous to (1.1.2)) above for asymp- totically  demicontractive mappings which will yield strong convergence; and in particular show that  the proposed algorithm  converges strongly to the least norm element of the set of fixed points of asymptotically  demicontractive mappings.

Definition 1.2.1   Let X  be a nonempty  set and  let T : X  →  X  be a self map.  A fixed point of T is an element x ∈ X  for which T x = x.  The set of fixed points of T is denoted by F (T ).  That  is, F (T ) = {x ∈ X : T x = x}.

Example 1.2.2   Let T : R →  R    be a real  valued function  defined by T (x)  = x2,  then

F (T ) = {x ∈ X : T x = x} = {0, 1}

Remark 1.2.3   We quickly remark  that not all functions  have fixed points.  For  example, Let f : R →  R,    be a real valued function  defined by f(x) = x + 1, then f has no fixed point since there is no x∗  ∈ R such that f(x∗) = x∗, that is, there is no x∗  ∈ R such that x∗  + 1 = x∗.

Definition 1.2.4   A vector  space,  or  linear  space,  over a scalar  field F  = R or  C is a nonempty set X, whose elements are called vectors, together with two operations,  addition and multiplication  by scalars,

+ : X × X → X    and    . : R × X → X

defined by                       (x, y) → x + y      and          (t, x) → t.x,

respectively, with the properties  that:

1.  (X, +) is a commutative group, that is,

(a)  x + y = y + x    ∀ x, y ∈ X  (commutative  property)

(b)  x + (y + z) = (x + y) + z    ∀ x, y, z ∈ X  (associative  property)

(c)  there is a vector 0 ∈ X  called zero element, such that x + 0 = 0 + x    ∀ x ∈ X

(d)  for every x ∈ X  there exists a vector −x ∈ X,  called the additive inverse of x

such that x + (−x) = (−x) + x = 0.

2.  ∀ x, y ∈ X and α,  β ∈ R, (a) α.(x + y)) = α.x + α.y

(b)  (α + β).x = α.x + β.y

(c)  (α.β).x = α.(β.x)

(d)  1.x = x for 1 ∈ R, x ∈ X

Definition 1.2.5   A normed space is a pair  (X, ||.||), where X  is a vector space over the scalar field F = (R or C) and ||.|| : X → R is a function satisfying the following conditions:

1.  kxk ≥ 0     ∀ x ∈ X,

2.  kxk = 0 if and only if x = 0 in X,

3.  kαxk = |α|kxk         ∀ x ∈ X,   ∀ α ∈ R,

4.  kx + yk ≤ kxk + kyk ∀ x, y ∈ X.

Remark 1.2.6   In normed  spaces, the norm  defines a metric,  that  is, if k.k : X × X  →

[0, ∞)  is a norm on X  then for all x, y ∈ X, the function k.k induces a metric  given by

ρ(x, y) = kx − yk.

Thus,  we have a notion of completeness in terms  of the norm.  We say that  (X, k.k) is a Banach  space if and  only if ρ(x, y) is a complete  metric  space, that  is, if every Cauchy sequence {xn} in (X, ρ) converges to some element, say x ∈ X.

Definition 1.2.7   Let X be a vector  space over the field of scalars  F  = (R  or  C).   An inner  product  h., .i : X × X  →  F  on X  is a function  with domain  X × X  and range  F such that the following conditions  are satisfied:

1.  hx, xi ≥ 0 ∀ x ∈ X,

2.  hx, xi  = 0 if and only if x = 0,

3.  hx, yi = hy, xi,

4.  hx, y + zi = hx, yi + hx, zi     ∀ x, y, z ∈ X,

5.  hαx, yi = α hx, yi     ∀ x, y ∈ X,     ∀ α ∈ F . The pair (X, h., .i) is called an inner  product space.

Remark 1.2.8 In number 3 of Definition 1.2.7, bar denotes complex conjugate.  Thus, if

X  is a real Vector space, then hx, yi = hy, xi.

Lemma 1.1

Let (X, h., .i) be an inner product  space.  The function k.k : X →  [0, ∞)  defined for each

x ∈ X by

||x|| = hx, xi2                                                                                                (1.2.1)

is a norm on X. Thus,  we have a notion  of completeness in terms  of the  norm.  We say that  X is a Hilbert  space if it is a complete inner product  space, that  is, if every Cauchy sequence {xn} ⊂ X converges to some element, say x ∈ X.

Example 1.2.9   Let X = R,    the set of real numbers with the usual addition  and scalar multiplication.  Define h., .i : R ×R → R    ∀ x, y ∈ R by hx, yi = xy,  then h., .i is an inner product on R and (R, h., .i) is a Hilbert space.

Example 1.2.10 Let X :=  l2(C)  = {z = {zj }∞

: zj  ∈ C,  j = 1, 2, 3, …}, P∞

|zj |2  <

∞}.  Define h., .i : X × X → C    ∀ z, w ∈ X by hz, wi = P∞

zj wj ,  then h., .i is an inner

product on X  and (X, h., .i) is a Hilbert space

We remark  that  not all inner product  spaces are Hilbert  spaces.  For example, take X = C[a, b] = {f : [a, b] →  R|f  is continuous} with the  usual pointwise  addition  and  scalar multiplication,

Define h., .i : X × X → R    ∀ f, g ∈ X by

Z b

hf, gi =

f(t)g(t)dt

a

then  h., .i is an inner product  on X  so that  the pair (X, h., .i) is an inner product  space that  is not a Hilbert space.

Definition 1.2.11 Let X be a vector space over the real numbers.  A subset K ⊆ X is said to be convex if for all x, y ∈ K  and all t in the interval [0, 1], the point (1 − t)x + ty ∈ K. In other words, every point on the line segment connecting  x and y is in K.

Definition 1.2.12 A mapping T : C → C is called demiclosed at zero if for any sequence

n=1  

{xn}∞

⊂  C  which converges weakly to an  x  ∈  C,  where C  is closed and  convex, the

strong convergence of the sequence {T xn}∞

to zero in C implies that T x = 0

Definition 1.2.13 Let X and Y  be Banach spaces.  A mapping T : X → Y  is called com- pletely continuous,  if it maps a weakly convergent sequence in X  to a strongly convergent sequence in Y . That  is, xn * x    implies    kT xn − T xkY  → 0, as n  → ∞.



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