File Type: MS Word (DOC) & PDF

File Size:363KB

Number of Pages:36

## ABSTRACT

In this thesis, we consider the problem of approximating solution of generalized equilibrium prob-

lems and common xed point of nite family of strict pseudocontractions. The result obtained is

applied in approximation of solution of generalized mixed equilibrium problems and common xed

point of nite family of strict pseudocontractions. Our theorems improve and unify some existing

results that were recently announced by several authors. Corollaries obtained and our method of

proof are of independent interest.

** **

## TABLE OF CONTENTS

Certication ii

Acknowledgement vi

Dedication viii

1 INTRODUCTION 1

1.1 Background of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Some Facts in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 LITERATURE REVIEW 9

3 SOME AUXILIARY RESULTS 14

4 MAIN RESULTS 16

4.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 CONCLUSION 25

ix

## CHAPTER ONE

INTRODUCTION

The content of this thesis falls within the area of nonlinear operator theory. This area has attracted

attention of several researchers due to its wide range of application in dierent areas of pure and

applied sciences. The research documented in this thesis concentrated on the following topic:

Approximation of solution of generalized equilibrium problems and common xed point of nite

family of strict pseudocontractions.

1.1 Background of Study

In sciences, engineering, economics and in some other areas where there is a quantitative analysis,

we are greatly interested in describing how systems evolve in time, that is, in describing system’s

dynamics. We will restrict ourselves to one dimensional case for the purpose of illustration. We will

always write u = u(t), which is the state of the system. We think of the dependent variable u as

the state variable of a system that is varying with time t, which is the independent variable. Thus,

knowing u is virtually the same as knowing what state the system is, at time t. For example, u(t)

could be the number of patience admitted in a hospital, the quantity of data processed by CPU,

the concentration of a chemical substance such as sugar in the body, the number of immigrants

into a country, the current in an electrical circuit, the speed of a spacecraft, or the monthly sales

of an advertised item. Knowledge of u(t) for a given system tells us how the system changes with

respect to time. Often, we relate the state u(t) to its rates of change, as expressed by its derivatives

u

0

(t), u

00

(t); ; and so on. It is important to note that some of the dynamical system can be

described by the following model,

du

dt

+ Au = f(t; u(t)): (1.1)

Where A is an operator dened on some appropriate spaces. Equation (1.1) is called nonhomoge-

neous rst order ordinary dierential equation if f(t; u(t)) 6= 0, otherwise it is homogeneous rst

order ordinary dierential equation. Assuming that u(t) is a solution to equation (1.1) and suppose

that t0 is the initial reference time that we want to start studying the above model, we can always

use u(t) to make comparative analysis of the behaviour of the dynamical system between the time

t0 and t. If f(t; u(t)) = 0, then equation (1.1) becomes

du

dt

+ Au = 0: (1.2)

If we put A 0 in equation (1.1), then, equation (1.1) reduces to

du

dt

= f(t; u(t)): (1.3)

1

Picard proved that under some certain assumptions on f, its domain and co-domain, that problem

(1.3) is equivalent to problem of nding xed point of an operator T dened by

(Tu)(t) = (t)

= u0 +

Z t

t0

f(s; u(s))ds; (1.4)

where T is a self map dened on some appropriate innite dimensional function space and u0 =

u(t0). Though equation (1.3) looks simple, it happens that most times, we do not have exact

solution of equation (1.3) rather the numerical solution. This numerical solution corresponds to

the approximated xed point of some nonlinear operators. Furthermore, it is well known that at

equilibrium state, du

dt = 0, hence at equilibrium state, equation (1.2) becomes

Au = 0: (1.5)

Consequently, equation (1.2) reduces to problem of nding zero (zeros) of A which corresponds

(correspond) to problem of nding xed point of some operator T by dening A I T.

We recall that if a function f is twice dierentiable at a point x i.e f00(x) exists and f00(x) 6= 0

and f0(x) = 0 then, x is an extremum point. This leads us to the following question: How do we

get the optimizer of a function whenever it exist without necessarily dierentiating f in the usual

sense? We have to note that some of the important operators involved in optimization problems

are not dierentiable in the usual sense. We give an example to illustrate our point. Consider the

map f : H ! R dened by

f(x) = kxk;

where H is a real Hilbert space. It is well known that f is not dierentiable at zero. However,

it is easy to see that zero is the minimizer. From the foregoing analysis, it is worthy to study

optimization problems.

Let us consider the problem of nding u 2 K such that

f(u; y) 0; 8 y 2 K; (1.6)

where K is a nonempty, closed and convex subset of real Hilbert space H and f : K K ! R; a

bifunction. We observe that it includes xed point problems and optimization problems as special

cases. Furthermore, if we consider a nonlinear operator A : K ! H and a problem of nding

x 2 K such that

f(u; y) = hAu; y ui 0; 8 y 2 K: (1.7)

We obtain another special case of equation (1.6)

If however, we consider the problem of nding u 2 K such that

f(u; y) + hAu; y ui 0; 8 y 2 K; (1.8)

then, we have a new problem which include problems (1.6) and (1.7) as special cases, we are going

to study problem (1.8) extensively in this thesis.

Problem (1.6) was introduced by Blum and Oettli (1994) and Noor and Oettli (1994). It has a

great impact and in uence in the development of several branches of Pure and Applied Sciences.

Motivated by the above example and forgoing analysis, We are interested in studying some iterative

algorithm for approximating the solution of equation (1.8) and common xed point of nite family

of strict pseudocontractions.

We present some preliminary results, denitions and some well known facts in Hilbert spaces,

understanding them plays a crucial role in comprehending the entire work. We shall therefore, im-

mediately turn to the preliminary section where most of the necessary denitions and explanation

of terms are displayed.

2

1.2 Preliminary

In this section, we give denitions of some crucial concepts that shall be needed in sequel.

Denition 1.1. Let T : D(T) H ! H be a map. then, T is said to be

(i) Asymptotically k-strictly pseudocontraction in the intermediate sense (Sahu, et ai., 2008)

with sequence f ng if there exists a constant k 2 [0; 1) and a sequence f ng [0;1) with

limn!1 n = 0 such that for all x; y 2 K and for all n 2 N;

lim sup

n!1

sup

x;y2K

(kTnx Tnyk (1 + n)kx yk2 kk(I Tn)x (I Tn)yk2) 0: (1.9)

(ii) k-Lipschitz if there exists k 0 such that for all x; y 2 D(T);

kTx Tyk kkx yk:

If k 2 [0; 1) in (ii); then T is called contraction and if k 2 [0; 1]; then the mapping T is called

nonexpansive.

(iii) k-strictly pseudocontractive mapping if there exists a constant k 2 [0; 1) such that for all

x; y 2 D(T):

kTx Tyk2 kx yk2 + kkx Tx (y Ty)k2:

(iv) rmly nonexpansive if for all x; y 2 D(T);

kTx Tyk2 hTx Ty; x yi :

(v) monotone if for all x; y 2 D(T); hTx Ty; x yi 0:

(vi) -inverse strongly monotone if there exists > 0 such that for all x; y 2 D(T);

hTx Ty; x yi kTx Tyk2:

Furthermore, a point x 2 D(T) is called xed of T if Tx = x.

Remark 1.2. (i) It has been shown by Marino and Xu (2007) that the class of strict pseu-

docontractions are Lipschitz with Lipschitz constant 1+k

1k . Therefore, the class of strict

Pseudocontractions is a subclass of uniformly continuous mappings, as well as a subclass of

Lipschitz pseudocontractive mappings.

(ii) It is easy to see that every nonexpansive map is 0-strictly pseudocontraction. Hence, the

class of strict pseudocontractions contains the class of nonexpansive maps. We, however,

emphasize that the converse is false. In fact, we have the following example.

Example 1.3. Let H be a real Hilbert space and let T : H ! H be dened by

T(x) = 2x

It is not dicult to see that T is not nonexpansive map. We argue as follow to show that T is

strictly pseudocontraction. First, we observe that for any x; y 2 H;

kTx Tyk2 = 4kx yk2 = (1 + 3)kx yk2

=

1 +

3

9

(9)

kx yk2

= kx yk2 +

3

9

k3(x y)k2

= kx yk2 +

1

3

k(1 + 2)x (1 + 2)y)k2

3

= kx yk2 +

1

3

k(1 (2))x (1 (2))y)k2

= kx yk2 +

1

3

k(I T)x (I T)yk2

kx yk2 + kk(I T)x (I T)yk2; 8 k 2

1

3

; 1

:

Denition 1.4. The generalized mixed equilibrium problems (abbreviated GMEP) for operators

f; ; B is a problem of nding u 2 K such that

f(u; y) + (y) (u) + hBu; y ui 0; 8 y 2 K; (1.10)

where K is nonempty, closed and convex subset of a real Hilbert space H, f is a real valued

bifunction with domain K K, is a proper extended real valued function with domain K, that

is, : K ! R [ f+1g and B an operator dened from K to H. The solution set of (1.10) is

denoted by

GMEP(f;;B) := fu 2 K : f(u; y) + (y) (u) + hBu; y ui 0; 8 y 2 K:

It is easy to see that u 2 GMEP(F;;B) implies that

u 2 D() := fu 2 H : (u) < +1g:

If 0 B in (1.10), then, inequality (1.10) reduces to the Classical equilibrium problem

(abbreviated EP(f)), that is, the problem of nding u 2 K such that

f(u; y) 0; 8 y 2 K: (1.11)

Solution set of (1.11) is denoted by

EP(f) := fu 2 K : f(u; y) 0; 8 y 2 Kg:

If 0 f in (1.10), then (1.10) reduces to the Classical variational inequality problem

GMEP(0; 0;B), that is, the problem of nding u 2 K such that

hBu; y ui 0; 8 y 2 K: (1.12)

Solution set of (1.12) is denoted by

V:I(B;K) = fu 2 K : hBu; y ui 0 ; 8 y 2 Kg:

If B 0 f in (1.10), then (1.10) reduces to the following minimization problem: nd u 2 K

such that

(y) (u); 8 y 2 K: (1.13)

Solution set of (1.13) is denoted by Argmin(), where

Argmin() := fu 2 K : (y) (u); 8 y 2 Kg:

If B 0 in (1.10), then (1.10) reduces to the mixed equilibrium problem (abbreviated MEP(f;; 0),

that is, the problem of nding u 2 K such that

f(u; y) + (y) (u)+ 0; 8 y 2 K: (1.14)

4

Solution set of (1.14) is denoted by

MEP(f; ) := fu 2 K : f(u; y) + (y) (u)+ 0; 8 y 2 Kg:

If 0 in (1.10), then (1.10) reduces to the Generalized equilibrium problem, that is,

the problem of nding u 2 K such that

f(u; y) + hBu; y ui 0; 8 y 2 K: (1.15)

Solution set of (1.15) is denoted by

GEP(f;B) := fu 2 K : f(u; y) + hBu; y ui 0; 8 y 2 Kg:

If f 0 in (1.10), then (1.10) reduces to the Generalized variational inequality problems,

that is, the problem of nding u 2 K such that

(u) (y) + hBu; y ui 0; 8 y 2 K: (1.16)

Solution set of (1.16) is denoted by

GV I(;B;K) := fu 2 K : (u) (y) + hBu; y ui 0; 8 y 2 Kg:

From the forgoing discussion so far, we observe that (1.10) solves three dierent types of prob-

lems simultaneously i.e., it solves problem of optimization, variational inequality and equilibrium

problems.

Throughout this thesis, we assume that our bifunction f, satises the following conditions,

namely:

A1 f(x; x) = 0; 8 x 2 K;

A2 f is monotone in the sense that

f(x; y) + f(y; x) 0; 8 x; y 2 K;

A3 f is hemi-continuous, that is,

lim sup

t!0+

f(tz + (1 t)x; y) f(x; y); 8 x; y; z 2 K;

A4 The function f(x; 🙂 is convex and lower semicontinous, 8 x 2 K. Though the following

denition is well known, we still present it here for clarity sake.

Denition 1.5. Let E be a real vector space.The map

1. k:k : E ! [0;1) satisfying the following conditions:

(i) kxk 0; 8 x 2 E and kxk = 0 if and if x = 0,

(ii) For any 2 R, kxk = jjkxk; 8 x 2 E,

(iii) kx + yk kxk + kyk; 8 x; y 2 E,

is called a norm on E and the pair (E; k:k) is called a normed vector space.

2. h:; :i : E E ! R satisfying the following conditions:

(i) hx; xi 0; 8 x; y 2 E and hx; xi = 0 if and only if x = 0,

(ii) symmetricity, that is, hx; yi = hx; yi ; 8 x; y 2 E,

5

(iii) bilinear, that is, linear in both rst and second argument.

is called real inner product on E and the pair (E; h:; :i) is called a real inner product

space.

Remark 1.6. If (E; h:; :i) is an inner product space and we consider the map k:k : E ! R dened

by kxk =

p

hx; xi. One can easily verify that k:k is a norm on E. It is called the norm induced by

the inner product.

From now onward, we will always assume that:

(i) H is a real Hilbert space.

(ii) K is nonempty, closed and convex subset of H.

(iii) h:; :i is an inner product associated with H.

(iv) k:k is the norm induced by the inner product.

(v) F(T) = fx 2 D(T) : Tx = xg:

Denition 1.7. Let fxng be a sequence in H. Then, fxng is said to converge to x 2 H

(i) strongly, if 8 > 0; 9 n 2 N such that 8 n n; kxn xk < ;

(ii) weakly, if 8 f 2 H, the sequence ff(xn)gn1 converges to f(x) in R with the usual topology.

Denition 1.8. A net (or generalized sequence ) in H indexed by A := [0; 1] is an operator from

A to H. It is denoted by fxg2A:

Denition 1.9. (i) Let fxg2A be a net in H, fxg2A converges to a vector x as ! 0 if

fxg2A lies eventually in every neighbourhood of x. i.e 8 V 2 Nbh(x); 9 b 2 A such that

b ) x 2 V:

(ii) A point x 2 H is a cluster point of the net fxg2A if fxg2A frequently lies in every

neighbourhood of x. i.e 8 V 2 Nbh(x); 8 b 2 A; 9 2 A such that b and x 2 V:

Denition 1.10. Let f : H ! R [ f1g and and x0 2 H, where H is a real Hilbert as we have

pointed out before. Then, f is lower semicontinuous at x0 if, for every net (x)2A H such that

x ! x0 as ! 0+, Then, f(x0) lim inf

!0+

f(x)

1.2.1 Some Facts in Hilbert Spaces

(i) Given a nonempty, closed and convex subset K of H, let PK : H ! K be the projection

operator. It is well known that for arbitrary vector x 2 H, z = PKx if and only if

hx z; y zi 0; 8 y 2 K: (1.17)

The following identities are also well known in Hilbert spaces:

(ii) for any t 2 [0; 1] and for any x; y 2 H;

ktx + (1 t)yk2 = tkxk2 + (1 t)kyk2 t(1 t)kx yk2: (1.18)

(iii) for any x; y 2 H

kx yk2 = kxk2 + kyk2 2 hx; yi : (1.19)

(iv) It is also well known that given any vector y 2 H; there exists fy 2 H such that

fy(x) = hx; yi ; 8 x 2 H: (1.20)

Where H denotes the dual space of H, i.e the set of all bounded linear operators from H to R.

Remark 1.11. It is easy to see using equation (1.20) that xn * x if and only if for any y 2 H;

hxn; yi ! hx; yi.

6

1.3 Statement of Problem

Several Authors have published articles on how to approximate the solution of generalized equi-

librium problems and common xed points of nite family of strict pseudocontractions

For example, Marino and Xu (2007) proved that: Given a self mapping T from a nonempty, closed

and convex subset K of a real Hilbert space H , the sequence fxng dened recursively by the

formula

xn+1 = nxn + (1 n)Txn; n 0; (1.21)

converges weakly to a xed point of T. Where the initial guess x0 2 K is arbitrary, and fng is

a real control sequence in the interval (0; 1): They proved the above result under the additional

hypothesis that

(i) T is k-strictly pseudocontraction that admits at least a xed point,

(ii) k < n < 1; for all n 1 and

1X

n=0

(n k)(1 n) = 1:

Hu and Cai (2011) proved the following theorem for class of asymptotically pseudocontractive

mapping in the intermediate sense:

Theorem 1.12. (Hu and Cai, 2011 ) Let C be a nonempty, closed and convex subset of a real

Hilbert space H and N 1 be an integer, f : C C ! R be a bifunction satisfying A1 – A4 and A

be an -inverse strongly monotone mapping of C into H. Let, for each 1 i N; Ti : C ! C be

a uniformly continuous ki -strictly asymptotically pseudocontractive mapping in the intermediate

sense for some 0 ki < 1 with sequences f ng [0;1) such that

1X

n=1

n;i < 1 and fcn;ig [0;1)

such that limn!1 cn;i = 0: Let k = maxfki : 1 i Ng; n = maxf n;i : 1 i Ng and

cn = maxfcn;i : 1 i Ng. Assume that F := \Ni

=1F(Ti) \ EP is nonempty. Let fxng and fung

be sequences generated initially by arbitrary element x1 2 C and then by

8><

>:

f(un; y) + hAxn; y uni + 1

rn

hy un; un xni 0; 8 y 2 K;

zn = (1 n)un + nTk(n)

i(n) un;

xn+1 = nun + (1 n)zn;

(1.22)

where fng; fng and frng satisfy the following conditions:

(i) 0 < a n 1; fng (0; 1);

(ii) 0 < n 1 k < 1; fng (0; 1);

(iii)

1X

n=1

ncn < 1;

(iv) 0 < b rn c 2.

Then, the sequences fxng and fung converge weakly to an element of F.

Huang and Ma (2014) proved the following theorem by slightly adjusting scheme (1.12) and con-

sidering the class of strict pseudocontractions. They obtained the following theorem:

7

Theorem 1.13. (Huang and Ma, 2014) Let K be a nonempty, closed and convex subset of a real

Hilbert space H. Let T : C ! H be a -inverse-strongly monotone mapping. Let F be a bifunction

from C C to R satisfying conditions (A1)(A4). Let S : C ! C be a k-strict pseudocontraction.

Assume that F := EP(F; T)

T

F(S) is not empty. Let fng; fng; f ng and fng be sequences

in (0; 1). Let frng be a sequence in (0; 2), and let feng be a bounded sequence in C. Let fxng be

a sequence generated in the following manner:

8><

>:

x 2 C;

F(un; u) + hTxn; u uni + 1

rn

hu un; un xni 0 8 u 2 K;

xn+1 = nxn + n(nun + (1 n)Sun) + nen n 1;

(1.23)

Assume that the sequences fng; fng; f ng nd fng; frng satisfy the following restrictions: 0 <

a n a

0

< 1, 0 k n b < 1, 0 < c rn d < 2 and

1X

n=1

n < 1: Then, the sequence

fxng converges weakly to some point x 2 F, where x = limn!1 PF xn:

The problem is that all their results concluded weak convergence which seems to be less useful in

applications compare to strong convergence. In this thesis, we studied the above problem and we

constructed iterative algorithm by modifying the operators used in scheme (1.12) as Huang and

Ma did, drop the error term introduced in scheme (1.13) and use a modied Halpern scheme which

seems better than Mann’s scheme in several ways to study the convergence analysis of the new

problem.

1.4 Motivations

Our motivation arises from application point of view, the work of Huang and Ma (2014) and that

of Hu and Cai (2011) precisely theorems (1.12) and (1.13), respectively.

1.5 Objectives

Our objectives are the following :

(i) to introduce a scheme that will have computational advantage over the existing ones.

(ii) to prove strong convergence theorem using our scheme which seems to be more useful in

application.

1.6 Limitations

We proved our result in Hilbert space setting, so we are faced with the challenge of whether our

result is valid in more general Banach spaces. It is dicult in practice to get operators that

are inverse strongly monotone.This calls for further research on how to relax the inverse strongly

monotone condition.