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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
1􀀀k . 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.

 

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