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

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  → ∞.

₦5,000.00 – DOWNLOAD FULL PROJECT DOWNLOAD FULL PROJECT Added to cart

---

---

---

---

Related Project Topics :