STRONG CONVERGENCE OF  MODIFIED AVERAGING ITERATIVE ALGORITHM FOR ASYMPTOTICALLY NONEXPANSIVE MAPS

ABSTRACT

Let H  be a real Hilbert  space and  K  a nonempty,  closed and  convex subset  of H.  Let

T : K  →  K  be an asymptotically  nonexpansive  map with a nonempty  fixed points  set.

n=1

Let {αn}∞

and  {tn}∞

be real sequences in (0,1).  Let {xn} be a sequence generated

from an arbitrary x0  ∈ K by

n=1

      yn = PK [(1 − tn)xn],       n ≥ 0

n=1

xn+1 = (1 − αn)yn + αnT nyn,     n ≥ 0.

where PK    : H  →  K  is the  metric  projection.   Under  some appropriate mild conditions

n=1

on {αn}∞

and {tn}∞

, we prove that  {xn} converges strongly  to fixed point of T .  No

compactness assumption  is imposed on T and or K and no further requirement is imposed on the fixed point set F ix(T

Chapter  1

Introduction

1.1     General Introduction

The theory of fixed points of nonlinear operators  has found many powerful and important applications  in diverse fields such as Differential Equations,Topology, Economics, Biology, Chemistry,  Engineering,  Game Theory,  Physics,  Dynamics,  Optimal  Control,  and Func- tional  Analysis .  Iterative  algorithms  for approximating fixed points  of some nonlinear operators belonging to certain classes of mappings that  generalize nonexpansive mappings and defined in appropriate Banach spaces have been flourishing area of research for many mathematicians.The class of nonlinear  mappings  we studied  in this  work, is the  class of asymptotically  nonexpansive  mappings.  This class of asymptotically  nonexpansive  map- pings which has  engaged the  interest  of many  researchers(for  example  see [18] and  the references there in ) was first introduced  by Goebel and Kirk [10] in the year 1972.

Definition 1.1.1:  Let K  be a nonempty  subset of a normed linear space E.  A mapping

T : K → K is said to be nonexpansive if

kT x − T yk ≤ kx − yk, ∀x, y ∈ K.

Definition 1.1.2:  Let K  be a nonempty  subset of a normed linear space E.  A mapping

T : K  →  K  is called asymptotically nonexpansive,  if there  exists a sequence {kn},  kn  ∈

1

[1, ∞)  such that  limn→∞ kn = 1, and

kT nx − T nyk ≤ knkx − yk

holds for each x, y ∈ K  and for each integer  n ≥ 1.  It is clear that  every nonexpansive mapping  is asymptotically nonexpansive  with kn = 1 ∀n ≥ 1.

The  following example  reveals that  the  class of asymptotically nonexpansive  mappings

properly contains  the class of nonexpansive mappings.

Example 1 (Goebel and Kirk [10]).  Let B  denote  the  unit  ball in the  Hilbert  space l2

and let T be defined as follows:

1

T : (x1, x2, x3, …) → (0, x2, a2x2, a3x3, …).

i=2

where {ai} is a sequence of numbers such that  0 < ai < 1 and Q∞

1

ai = 2 .

Then,  T is Lipschitz and kT x − T yk ≤ 2kx − yk, ∀x, y ∈ B.

Moreover,

Thus,

i

kT ix − T iyk ≤ 2 Y aj kx − yk ∀i = 2, 3, …

j=2

i

lim ki = lim 2 Y aj = 1

But let,

i→∞

i→∞

2

j=2

x = (

3

, 0, 0, …)

and

1

y = (

2

, 0, 0, …),

clearly x, y are in B.

4     1        7      1

kT x − T yk = | 9 − 4 | = 36 > 6 = kx − yk.

therefore T is not nonexpasnsive.

2

1.2     Some Banach Spaces and their Properties.

In this section we give the definition of some special Banach spaces as well as some of their geometric properties.

2

Definition 1.2.1     A Banach space E is said to be uniformly convex if for any    ∈ (0, 2], there exists δ = δ( ) > 0 such that  for all x, y ∈ E with ||x|| ≤ 1, ||y|| ≤ 1 and ||x−y|| >  , then  || 1 (x + y)|| ≤ 1 − δ.  Geometrically,  a Banach  space is uniformly convex if the unit ball centred  at the origin is uniformly round.  Again, uniform convexity is a property  of the norm on E.

The modulus of convexity of E is defined by

0 <  ≤ 2.

δE ( ) = inf {1 − ||

x + y

2   || : ||x|| = 1 = ||y||, ||x − y|| ≥  },

Definition 1.2.2     A Banach  space E  is said to be smooth  if for all x ∈ E, x = 0 with

||x|| = 1, there exists f ∈ E∗   such that  hx, fi = 1. Smoothness is a property  of the norm. In fact, E is smooth if and only if ∀x, y ∈ E, x = 0

lim  ||x + ty|| − ||x||

(1.1)

t→0                t

exists.   The  limit,  when it  exits,  is of the  form fx(y)  with  fx  ∈  E∗    and  is called  the

Ga´teaux derivative  of the norm in E. Thus T is smooth if and only if the norm is Ga´teaux differentiable.

Definition 1.2.3    A Banach space is uniformly smooth if and only if the limit (1.1) exists uniformly on the set

U = {(x, y) ∈ E × E : ||x|| = ||y|| = 1}.

Definition 1.2.4     A Banach  space E  is said to be an Opial space (see for example [1],

n=1

[16]) if for each sequence {xn}∞

in E which converges weakly to a point x ∈ E

lim inf ||xn − x|| < lim inf ||xn − y||,

3

for all y ∈ E, y = x. It is known (see [17]) that  every Hilbert space and every lp(1 < p < ∞) space enjoy the property.  Also in [9], D Van showed that  any separable Banach space can be equivalently  re-normed  so that  it  satisfies Opial  condition.   Indeed,  for any  normed space E  the existence of a weakly sequentially  continuous  duality  map implies that  E  is

an Opial’s space, but the converse implication does not hold.  Notably, the Lebesgue space

Lp  is not an Opial space for p = 2.

∞

Definition 1.2.5  A function  Φ : [0, ∞)  →  [0, ∞)  is said to be a guage function  if Φ is continuous  and strictly  increasing with Φ(0) = 0 and  lim Φ(t) =    .

t→∞

Definition 1.2.6  Let E be a real Banach space.  Let E∗   denote the topological dual of E and 2E∗  be the collection of all subsets of E∗.  Let Φ : [0, ∞) → [0, ∞) be a gauge function. The mapping  JΦ:E → 2E∗   defined by

JΦx = {f ∈ E∗   : hx, fi = kxkkfk, kfk = Φ(kxk)}.

is called duality  map with gauge function  Φ, where h., .i denotes the generalized duality paring  between  E  and  E∗.   We note  that  if 1 < q < ∞,  then  Φ(t) = tq−1   is a gauge function.   The  duality  mapping  Jq   : E  →  2E∗   with  gauge Φ(t) = tq−1   defined for each x ∈ E by

JΦx = {f ∈ E∗   : hx, fi = kxkkfk, kfk = kxkq−1}.

is called the generalised duality mapping.  If q = 2, we obtain

∗

J2 := J : E → 2E  ,

defined for all x ∈ E by

J2x := J(x) = {f ∈ E∗   : hx, fi = kxk2 = kfk2}.

J2 is known as the normalised duality  map.  It is well known ( see for instance  [2, 8]) that for 1 < q < ∞,    Jq (x) = kxkq−2J(x), for x ∈ X, x = 0. The following theorem  has been proved for uniformly convex Banach space.

4

Theorem  1.2.7(Xu, [30]) Let p > 1 and r > 0 be two fixed real numbers.Then a Banach space X  is uniformly convex if and  only if there  exists a continuous,  strictly  increasing

and convex function

g : R+ → R+,  g(0) = 0,

such that  for all x, y ∈ Br and 0 ≤ λ ≤ 1,

p

p

kλx + (1 − λ)yk

≤ λkxk

+ (1 − λ)kykp

− Wp(λ)g(kx − yk).                  (1.3)

where Wp(λ) := λp(1 − λ) + λ(1 − λ)p   and Br := {x ∈ X : kxk ≤ r}.

The next result below establishes  the existence of a fixed point for asymptotically nonex- pansive mapping in a nonempty,  closed, convex and bounded subset of a uniformly convex Banach space.

Theorem 1.2.8([10]  Theorem  1) Let  K  be a nonemtpty, closed, convex and  bounded subset  of a uniformly  convex Banach  space X,  let  F  : K  →  K  be an  asymptotically nonexpansive  mapping,then F  has a fixed point.

Proof For each x ∈ K  and r > 0, Let S(x, r) denote the spherical ball centred  at x with radius r. Let y ∈ K be fixed, and let the set Ry  consist of those numbers ρ for which there

exists an integer k such that

i=k

K ∩ (∩∞

S(F iy, ρ)) = ∅.

If d is the diameter  of K  then  d ∈ Ry, so Ry  = ∅.  Let ρ0  = g.l.b.Ry, and for each    > 0,

k=1

(∩

define C    = ∪∞

∞

i=k

S(F iy, ρ +  )).  Thus  for each    > 0 the set C   ∩ K  are nonempty

and convex, so reflexivity of X implies that

C = ∩ >0(C¯   ∩ K) = ∅.

Note that  for x ∈ C and η > 0 there exists an integer N such that  if i ≥ N,  kx − F iyk ≤

ρ0 + η.

5

Now let x ∈  C  and  suppose the  sequence {F nx} does not  converge to  x (i.e.,  suppose

F x = x).  Then there exists    > 0 and a subsequence {F ni x} of {F nx} such that

kF ni x − xk ≥  , i = 1, 2, ….

For m > n,

kF nx − F mxk ≤ knkx − F m−nxk,

where  kn  is the  Lipschitz  constant  for F n  obtained  from the  definition  of asymptotic

ρ0+α

nonexpansiveness.  Assume ρ0  > 0 and choose α > 0 so that  (1 − δ(         ))(ρ0  + α) < ρ0.

Select n so that  kx − F nxk ≥    and also that  kn(ρ0 +

large, then  m > N implies

α

2 ) ≤ ρ0 + α.  If N ≥ n is sufficiently

and we have

α

2

kx − F m−nyk ≤ ρ0 +

kF nx − F myk ≤ knkx − F m−nyk ≤ ρ0 + α,

kx − F myk ≤ ρ0 + α.

Thus by uniform convexity of X, if m > n,

x + F nx        m

−

k

(               )     F

2

yk ≤ (1 − δ(

ρ

0

))(ρ0  + α) < ρ0,

+ α

and this contradicts the definition of ρ0. Hence we conclude ρ0  = 0 or F x = x. But ρ0  = 0 implies {F ny} is a Cauchy sequence yielding F ny → x = F x as n → ∞. Therefore the set consists of a single point which is a fixed point under F .

Kirk, Yanez and Shin [13] improved the result of Goebel and Kirk [10]. They proved that if a reflexive Banach space E has the property  that  each of its closed, bounded and convex subset  has  the  fixed point  property  for nonexpansive  maps,  then  it  will also have  the fixed point property for asymptotically  nonexpansive  mapping  which has a nonexpansive iterate.

Theorem 1.2.9  Let  H  be a real Hilbert  space and  C  a nonempty  closed and  convex subset  of H.  Let T : C →  C be an asymptotically  non expansive map.  Then  F ix(T ) =

{x ∈ C : T x = x} is closed and convex.

6

Proof: Convexity.

For any x, y ∈ F ix(T ) and α ∈ (0, 1). Let zα := (1 − α)x + αy.  Then,

kT nzα − zαk2   =   kT nzα − [(1 − α)x + αy]k2

=  k(1 − α)(T nzα − x) + α(T nzα − y)k2

k

k

=  (1 − α)kT nzα − x  2

+ αkT

nzα − y  2

− α(1 − α)kx − yk2

≤  (1 − α)k2 kzα − xk2 + αk2 kzα − yk2 − α(1 − α)kx − yk2

n                              n

k

2

=  (1 − α)k2 k(1 − α)x + αy − xk2 + αk2 k(1 − α)x + αy − y  2

− α(1 − α)kx − yk

n                                                      n

k

=  α2(1 − α)k2 kx − yk2 + αk2 k(1 − α)x + (1 − α)yk2  − α(1 − α)kx − y  2

n                            n

k

=  α2(1 − α)k2 kx − yk2 + α(1 − α)2k2 kx − yk2 − α(1 − α)kx − y  2

n                                            n

=  α(1 − α)[αk2  + (1 − α)k2  − 1]kx − yk2

n                      n

n

=  α(1 − α)(k2 − 1)kx − yk2 → 0.

Therefore,

kT nzα − zαk → 0.

Now,

0   ≤  kT zα − zαk

≤  kT zα − T nzαk + kT nzα − zαk

≤   k1kzα − T n−1zαk + kT nzα − zαk → 0.

Hence

T zα = zα.

We now show that  F ix(T ) is closed.  Let {xn} ⊆ F ix(T ) be arbitrary and let xn →  x as

n → ∞, we show that  x is in F ix(T ).

T x = T  lim xn = lim T xn = lim xn  = x.

n→∞

n→∞

7

n→∞

Two other  definitions of asymptotically  nonexpansive  maps has also appeared  in the lit- erature.   One of the  definitions which is weaker than  Definition 1.1.2 was introduced  by Kirk[12] and requires that

lim sup sup(kT nx − T nyk − kx − yk) ≤ 0.

n→∞

y∈K

for every x ∈ K and that  T N   be continuous  for some integer N > 1.

The other definition which has appeared  require that

lim sup(kT nx − T nyk − kx − yk) ≤ 0 ∀x, y ∈ K.

n→∞

This,  however,  has  been  shown  to  be  unsatisfactory from  the  point  of view of fixed point  theory.   Tingly [24] constructed an example of a closed convex set K  in a Hilbert

space and a continuous  map T : K  →  K  which actually  satisfies the following condition

limn→∞ kT

nx − T

nyk = 0 ∀x, y ∈ K but  has no fixed point.

1.3       Iterative Algorithms for  Asymptotically Nonex- pansive Mappings

1.3.1   Modified Mann Iterative Algorithm

The averaging iteration  process,

xn+1 = (1 − αn)xn + αnT nxn, n ≥ 1,

where T : K  →  K  is asymptotically  nonexpansive  in the  sense of definition  1.1.2, K  a closed, convex and bounded subset of a Hilbert space was introduced  by Schu[23].

In [21] Schu used the modified Mann iteration  method,

xn+1 = (1 − αn)xn + αnT nxn, n ≥ 1

and proved the following theorem.

8

Theorem(1.3.1):  Let H be a Hilbert  space, K  a nonempty  closed convex and bounded

subset of H.  Let T : K → K be a completely continuous asymptotically nonexpansive map

n=1

with sequence {kn}∞

with kn ∈ [1, ∞) for all n ≥ 1, limn→∞ kn = 1 and P∞

n=1

n

(k2

− 1) <

n=1

∞.   Let {αn}∞

be a sequence in [0,1] satifying the  condition    < αn < 1 −   for some

  > 0. Then the sequence {xn} generated  from an arbitrary x1  ∈ K by

xn+1 = (1 − αn) xn + αnT nxn, n ≥ 1,

converges strongly to a fixed point of T .

1.3.2   Iterative  method of Schu

In this  subsection,  consider  algorithm  for approximating fixed points  of asymptotically nonexpansive  mappings which deals with almost fixed points,

xn := µnT nxn

of an asymptotically  nonexpansive  mappings T . Schu [24] proved the convergence of this sequence {xn} to some fixed point of T under additional  assumption  that  T is uniformly asymptotically  regular  and (I − T ) is demiclosed.  These assumptions  had actually  been made  by Vijayaraju[27]  to  ensure  the  existence of a fixed point  of T  By strengthening the asymptotic  regularity  of T , Schu established  the convergence of an explicit iteration

scheme,

zn+1 := µnT nzn

to some fixed point of T .

1.3.3   Halpern-type process

One of the most useful results concerning algorithms for approximating fixed points of non- expansive mappings in real uniformly smooth Banach spaces is the celebrated convergence theorem of Riech[29] who proved that  the implicit sequence {xn} defined as,

1                1

xn = nu + (1 − n)T xn.

9

converges strongly  to a fixed point  of T .  Several authors  have tried  to  obtain  a result analogous to that  of Reich [19] for asymptotically  nonexpansive  mappings.   Suppose K is a nonempty bounded closed convex subset of a real uniformly smooth Banach space E and T : K →  K  is an asymptotically  nonexpansive  mapping  with sequence kn ≥ 1 for all

n ≥ 1. Fix u ∈ K and define, for each integer n ≥ 1, the contraction  mapping Sn : K → K

by, (see [6]),

n                         k

S  (x) = (1 − tn n

)u + tn

kn

T nx,

where {tn} ⊂ [0, 1) is any sequence such that  tn  →  1.Then  by the  Banach  Contraction

Mapping Principle,  there is a unique point xn fixed by Sn, i.e. there is xn such that

n                  k

x   = (1 − tn n

)u + tn

kn

T nxn.

For  some  existing  results  on  asymptotically   nonexpansive  maps,  an  interested   reader should see [4,7,14,18,20,22] and the references there in.

1.4     Organization of Thesis

We have introduced  in this chapter  (Chapter One), various iteration  methods  for asymp- totically  nonexpansive  maps and some existing results on them.  We also studied  some of the important Banach spaces which are encountered  in this work and their properties.

In Chapter  Two (the  Preliminaries),  we presented  most of the classical results on the se- quences of real numbers  encountered  in Operator  Theory.   We also looked at  projection maps and some other results vital to this work.

In Chapter  Three,  we study  certain  averaging iterative  algorithm  for approximating the fixed point of asymptotically  nonexpansive mappings introduced  by Goebel and Kirk [10].

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