ABSTRACT
Let H be a real Hilbert space. Let K; F : H ! H be bounded,
continuous and monotone mappings. Let fung1 n=1 and fvng1 n=1 be
sequences in E dened iteratively from arbitrary u1; v1 2 H by
un+1 = un n(Fun vn) n(un u1);
vn+1 = vn n(Kvn + un) n(vn v1); n 1
where fng1 n=1 is a real sequence in (0; 1) such that
P1
n=1 2n
P < 1 and 1
n=1 n = +1. Then, the sequence fung1 n=1 converges strongly to
u.
Let H be a real Hilbert space. For each i = 1; 2; :::m, let Fi; Ki : H !
H be bounded and monotone mappings. Let fung1n
=1; fvi;ng1 n=1; i =
1; 2; : : : ;m be sequences in H dened iteratively from arbitrary u1; vi;1 2
H by
(
un+1 = un nn
un +
Pm
i=1Kivi;n
nn(un u1);
vi;n+1 = vi;n nn(Fiun vi;n) nn(vi;n vi;1); i = 1; 2; : : : ;m
where fng1n
=1; fng1 n=1 and fng1 n=1 are real sequences in (0; 1)
such that n = o(n); n = o(n) and
P1
i=1 nn = +1. Suppose
that u +
Pm
i=1KiFiu = 0 has a solution in H. Then, there exist real
constants “0; “1 > 0 and a set
W such that if n “0n and
n “1n; 8n n0, for some n0 2 N and w := (u; x1
; x2
; : : : ; x
m) 2
(where xi
= Fiu; i = 1; 2; : : : ;m), the sequence fung1 n=1 converges
strongly to u.
Let E be a re exive real Banach space with uniformly G^ateaux differentiable
norm. Let K be a nonempty closed convex subset of E
and fTng1 n=1 be a sequence of Ln-Lipschitzian mappings of K into
viii
Abstract ix
itself with Ln 1;
1P
n=1
(Ln 1) < 1. Let
1T
n=1
F(Tn) 6= ;. For
a xed 2 (0; 1) and each n 2 N, dene Sn : K ! K by Snx :=
(1 )x + Tnx; 8x 2 K: Let fng1 n=1 be a sequence in [0; 1] such
that lim
n!1
n = 0 and
1P
n=1
n = 1 . Let fxng1n
=1 be a sequence in K
dened by x1 = u 2 K and
xn+1 = nu + (1 n)Snxn;
for all n 2 N. Suppose that (Kmin)
1T
n=1
F(Tn) 6= ; and lim
n!1
jjTn+1xn
Tnxnjj = 0. Then, fxng1 n=1 converges strongly to some common xed
points of fTng1 n=1.
Let K be a nonempty, closed and convex subset of a real Hilbert space
H. Let F be a bi-function from K K satisfying (A1) (A4), a
-inverse-strongly monotone mapping of K into H, A an -inversestrongly
monotone mapping of K into H and M : H ! 2H a maximal
monotone mapping. Let T : H ! H be a nonexpansive mapping
such that
:= F(T) \ I(A;M) \ EP 6= ; and suppose f : H ! H
is a contraction map with constant 2 (0; 1). Suppose fxng1 n=1 and
fung1n
=1 are generated by x1 2 H,
(
F(un; y) + h xn; y uni + 1
rn
hy un; un xni 0 8y 2 K;
xn+1 = nxn + (1 n)T
h
nf(xn) + (1 n)JM;(un Aun)
i
;
for all n 1, where fng1 n=1 and fng1n
=1 are sequences in [0,1] and
frng1 n=1 (0;1) satisfying:
(i) 0 < c n d < 1;
(ii) lim
n!1
n = 0;
1P
n=1
n = 1;
(iii) 2 (0; 2],
(iv) 0 < a rn b < 2; lim
n!1
jrn+1 rnj = 0,
then fxng1n
=1 converges strongly to z, where z := P
f(z) and P
f(z)
is the metric projection of f(z) onto
.
Let K be a nonempty, closed and convex subset of a real Hilbert space
H. Let F be a bi-function from K K to R satisfying (A1)(A4),
a -inverse-strongly monotone mapping of K into H, A an -inversestrongly
monotone mapping of K into H and M : H ! 2H a maximal
monotone mapping. Let = := fT(u) : 0 u < 1g be a one-parameter
nonexpansive semigroup on H such that
:= F(=)\I(A;M)\EP 6= ;
and suppose f : H ! H is a contraction mapping with a constant 2
(0; 1). Let ftng (0;1) be a real sequence such that limn!1 tn = 1.
Abstract x
Suppose fxng1 n=1 and fung1 n=1 are generated by x1 2 H,
8><
>:
F(un; y) + h xn; y uni + 1
rn
hy un; un xni 0 8y 2 K
wn = JM;(un Aun)
xn+1 = nxn + (1 n)
1
tn
R tn
0 T(u)[nf(xn) + (1 n)wn]du
;
for all n 1, where fng1n
=1 and fng1 n=1 are sequences in (0,1) and
frng1 n=1 (0;1) satisfying:
(i) lim
n!1
n = 0;
P1
n=1 jn+1 nj < 1,
(ii) lim
n!1
n = 0;
1P
n=1
n = 1;
P1
n=1 jn+1 nj < 1,
(iii) 2 (0; 2],
(iv) 0 < a rn b < 2;
P1
n=1 jrn+1 rnj < 1,
(v) lim
n!1
jtntn1j
tn
1
n(1n) = 0,
then fxng1 n=1 converges strongly to z, where z := P
f(z) and P
f(z)
is the metric projection of f(z) onto
.
Let E be a uniformly convex real Banach space which is also uniformly
smooth. Let C be a nonempty, closed and convex subset of E. Let F
be a bifunction from C C satisfying (A1) (A4). Suppose fTng1 n=0
is a countable family of relatively nonexpansive mappings of C into E
such that
:= (\1 n=0F(Tn)) \EP(F) 6= ;. Let fng; fng and f ng
be sequences in (0; 1) such that n + n + n = 1. Suppose fxng1 n=0
is iteratively generated by u; u0 2 E,
xn = Trnun;
un+1 = J1(nJu + nJxn + nJTnxn); n 0;
with the conditions
(i) lim
n!1
n = 0;
1P
n=0
n = 1;
(ii) 0 < b n n 1;
(iii) frng1 n=0 (0;1) satisfying lim inf
n!1
rn > 0.
Then, fxng1 n=0 converges strongly to
u, where
u is the generalized
projection of u onto
.
Let E be a 2-uniformly convex real Banach space which is also uniformly
smooth. Let C be a nonempty, closed and convex subset of
E. Suppose B : C ! E is an operator satisfying (B1) (B3) and
fTng1n
=1 is an innite family of relatively-quasi nonexpansive mappings
of C into itself such that F := V I(C;B) \
\mk
=1 GMEP(Fk; ‘k)
\
\1 n=1 F(Tn)
6= ;. Let fxng1 n=0 be iteratively generated by x0 2
Abstract xi
C; C1 = C; x1 = C1x0;
8>>>>>>><
>>>>>>>:
n = CJ1(Jxn nBxn)
yn = J1(nJn + (1 n)JTnn)
zn = J1(nJx0 + (1 n)Jyn)
un = TGm
rm;nTGm1
rm1;n :::TG2
r2;nTG1
r1;nzn
Cn+1 = fw 2 Cn : (w; un) (w; xn) + n(jjx0jj2 + 2hw; Jxn Jx0i)g
xn+1 = Cn+1×0; n 1;
where J is the duality mapping on E. Suppose fng1n
=1; fng1 n=1
and f ng1n
=1 are sequences in (0; 1) such that lim inf
n!1
n(1 n) >
0; lim
n!1
n = 0 and fng1 n=1 [a; b] for some a; b with 0 < a <
b < c2
2 , where 1
c is the 2-uniformly convexity constant of E and
frk;ng1 n=1 (0;1); (k = 1; 2; :::;m) satisfying lim inf
n!1
rk;n > 0; (k =
1; 2; :::;m). Suppose that for each bounded subset D of C, the ordered
pair (fTng;D) satises either condition AKTT or condition AKTT.
Let T be the mapping from C into E dened by Tx := lim
n!1
Tnx for all
x 2 C and suppose that T is closed and F(T) = \1 n=1F(Tn). Then,
fxng1 n=0 converges strongly to F x0
TABLE OF CONTENTS
Acknowledgements vi
Abstract viii
1 General Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Iterative algorithms for Hammerstein equations . . . . . . . . 1
1.3 Algorithms for common xed points . . . . . . . . . . . . . . 10
1.4 Algorithm for common solutions of three problems . . . . . . 27
1.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
I Approximation of Solution of Equations of Hammer-
stein Type 46
2 Strong Convergence Theorem for Approximation of Solu-
tions of Equations of Hammerstein Type 47
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Approximation of Solutions of Generalized Equations of Ham-
merstein Type 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
II Iterative Algorithm for Common Fixed Points of a
xii
Family of Mappings 64
4 Strong Convergence Theorems for a Mann-Type Iterative
Scheme for a Family of Lipschitzian Mappings 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
III Algorithms for Common Solutions of Common Fixed
Point Problems for a Family of Nonlinear Maps; Varia-
tional Inequality Problems and Equilibrium Problems 72
5 An Iterative Method for Fixed Point Problems, Variational
Inclusions and Generalized Equilibrium Problems 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Application to optimization problem . . . . . . . . . . . . . . 86
6 An Iterative Method for Nonexpansive Semigroups, Varia-
tional Inclusions and Generalized Equilibrium Problems 88
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 A New Iterative Scheme for a Countable Family of Rela-
tively Nonexpansive Mappings and an Equilibrium Problem
in Banach Spaces 104
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Strong Convergence Theorems for Nonlinear Mappings, Vari-
ational Inequality Problems and System of Generalized Mixed
Equilibrium Problems 114
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9 Conclusions and Future Work 132
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.2 Suggestions For Future Work . . . . . . . . . . . . . . . . . . 133
CHAPTER ONE
General Introduction
1.1 Introduction
The contributions of this thesis fall within the general area of nonlinear functional
analysis, an area with vast amount of applicability in recent years, as
such becoming the object of an increasing amount of study. We devote our
attention to three important topics within the area.
1. Approximation of solution of nonlinear equations of Hammerstein type.
2. Iterative algorithms for common xed points of a family of mappings
and,
3. Algorithms for common solutions of common xed point problems for
a family of nonlinear maps; variational inequality problems; and equilibrium
problems.
1.2 Iterative algorithms for Hammerstein equa-
tions
A nonlinear integral equation of Hammerstein type (see, e.g., Hammerstein
[102]) is one of the form
u(x) +
Z
k(x; y)f(y; u(y))dy = h(x) (1.2.1)
where dy is a -nite measure on the measure space
; the real kernel k
is dened on
; f is a real-valued function dened on
R and is,
1
General Introduction 2
in general, nonlinear and h is a given function on
. If we now dene an
operator K by
Kv(x) =
Z
k(x; y)v(y)dy; x 2
;
and the so-called superposition or Nemytskii operator F by Fu(y) := f(y; u(y))
then, the integral equation (1.2.1) can be put in operator theoretic form as
follows:
u + KFu = 0; (1.2.2)
where, without loss of generality, we have taken h 0.
Interest in equation (1.2.2) stems mainly from the fact that several problems
that arise in dierential equations, for instance, elliptic boundary value
problems whose linear parts possess Green’s functions can, as a rule, be
transformed into the form (1.2.2). Among these, we mention the problem of
the forced oscillations of nite amplitude of a pendulum (see, e.g., Pascali
and Sburlan [152], Chapter IV).
Example 1.2.1 The amplitude of oscillation v(t) is a solution of the prob-
lem
d2v
dt2 + a2 sin v(t) = z(t); t 2 [0; 1]
v(0) = v(1) = 0;
(1.2.3)
where the driving force z(t) is periodical and odd. The constant a 6= 0
depends on the length of the pendulum and on gravity. Since the Green’s
function for the problem
v
00(t) = 0; v(0) = v(1) = 0;
is the triangular function
k(t; x) =
t(1 x); 0 t x;
x(1 t); x t 1;
problem (1.2.3) is equivalent to the nonlinear integral equation
v(t) =
Z 1
0
k(t; x)[z(x) a2 sin v(x)]dx: (1.2.4)
If Z 1
0
k(t; x)z(x)dx = g(t) and v(t) + g(t) = u(t);
then (1.2.4) can be written as the Hammerstein equation
u(t) +
Z 1
0
k(t; x)f(x; u(x))dx = 0;
where f(x; u(x)) = a2 sin[u(x) g(x)].
General Introduction 3
Equations of Hammerstein type play a crucial role in the theory of optimal
control systems and in automation and network theory (see, e.g., Dolezale
[90]). Several existence and uniqueness theorems have been proved for equations
of the Hammerstein type (see, e.g., Brezis and Browder [20, 21, 22],
Browder [29], Browder and De Figueiredo [33], Browder and Gupta [32],
Chepanovich [47], De Figueiredo [91]).
Let H be a separable real Hilbert space and C be a closed subspace of H.
For a given f 2 C, consider the Hammertein equation
(I + KF)u = h (1.2.5)
and its nth Galerkin approximation given by
(I + KnFn)un = Ph; (1.2.6)
where Kn = P
nKPn : H ! Cn and Fn = PnFP
n : Cn ! H, where the
symbols have their usual meanings (see [152]). Under this setting, Brezis
and Browder [22] proved the following approximation theorem.
Theorem 1.2.2 Let H be a separable real Hilbert space. Let K : H ! C
be a bounded continuous monotone operator and F : C ! H be an angle-
bounded and weakly compact mapping. Then, for each n 2 N, the Galerkin
approximation (1.2.6) admits a unique solution un in Cn and fung1 n=1 con-
verges strongly in H to the unique solution u 2 C of the equation (1.2.5).
Prior to 2001, the only known method for approximating solutions of nonlinear
Hammerstein equations, as far as we know, was the Galerkin method
(1.2.6) of Brezis and Browder. The diculties associated with using Galerkin
method are well known.
In [70], Chidume and Osilike proved the following convergence theorem.
Theorem 1.2.3 Let E be a real Banach space with a uniformly convex dual
E. and suppose that:
(i) F is a nonlinear set-valued accretive map of E into itself with open
domain D;
(ii) K is a linear single-valued accretive map of E into itself with domain
D(K) such that Im(F) D(K); K1 exists and satises
hK1x K1y; j(x y)i jjx yjj2
for all x; y 2 Im(K) and > 0. Suppose also that for each h 2 Im(K) the
equation h 2 x + KFx has a solution x 2 D. Dene the set-valued map S
with domain D by Sx = K1h K1x Fx + x; x 2 D. Let fcng be a
real sequence satisfying:
General Introduction 4
(iii) 0 cn < 18n 1;
(iv)
P1
i=1 cn = 1;
(v) cnb(cn) < 1:
Then there exist a neighbourhood B = Bd(x) D of x and a real number
N0 0 such that for any n N0, and any initial guess x1 2 B, the sequence
fxng generated from x1 by
xn+1 = (1 cn)xn + cnn; 8n 2 Sxn;
remains in D and converges strongly to x.
We remark here that the recurrence formula used in Theorem 1.2.3 involves
K1 which is also assumed to be strongly monotone and this, apart from
limiting the class of mappings to which such iterative scheme is applicable,
is also not convenient in applications. Furthermore, convergence is guaranteed
if the initial guess is chosen in a neighbourhood of the solution which
is not known precisely. This is similar to a result of Bruck [37].
Due to further research, Chidume and Zegeye [75] introduced an iterative
algorithm for solutions of nonlinear Hammerstein equations (1.2.2) which
involves Cartesian product of Banach spaces and proved the following strong
convergence theorem.
Theorem 1.2.4 (Chidume and Zegeye [75]) Let X be a real q-uniformly
smooth Banach space. Let F;K : X ! X with D(K) = F(X) = X be
bounded maps such that the following conditions hold:
(i) for each u1; u2 2 X, there exists a strictly increasing function 1 :
[0;1) ! [0;1); 1(0) = 0 such that
hFu1 Fu2; jq(u1 u2)i 1(jju1 u2jj)jju1 u2jjq1;
(ii) for each u1; u2 2 X, there exists a strictly increasing function 2 :
[0;1) ! [0;1); 2(0) = 0 such that
hKu1 Ku2; jq(u1 u2)i 2(jju1 u2jj)jju1 u2jjq1;
(iii) i(t) (d + ri)t for all t 2 [0;1) and i = 1; 2 for some ri > 0 and
d := q1(1 + dq c12q1).
Assume that 0 = u + KFu has solution u in X. Let E := X X be
with norm jjzjjq
E = jjujjq
X + jjvjjq
X for z = (u; v) 2 E and dene the map
T : E ! E by Tz := T(u; v) = (Fu v; u + Kv). Then there exists a
real number 0 > 0 such that, if the real sequence fng [0; 0] satises
the following conditions: (i) lim
n!1
n = 0; (ii)
P1
i=1 n = 1, then for
arbitrary z0 2 E, the sequence fzng dened by
zn+1 = zn nTzn; n 0;
converges strongly to z = [u; v] where v = Fu and u is the unique solution
of 0 = u + KFu:
General Introduction 5
We observe that the operators K and F in Theorem 1.2.4 need not be de-
ned on compact or angle-bounded subset of X and the iterative algorithm
does not involve K1. The initial guess does not have to be chosen in a
neighbourhood of the solution. Therefore, Theorem 1.2.4 improves the results
of Chidume and Osilike. However, the draw back of Theorem 1.2.4
is that the result cannot be used by non-specialists because the iterative
algorithm is in a Cartesian product.
In 2005, Chidume and Zegeye [77] obtained an auxiliary operator, dened
in a real Hilbert space in terms of K and F that is monotone whenever
K and F are, and constructed a coupled iterative algorithm that converges
strongly to the solution of equation (1.2.2) in the original Banach space. In
particular, they proved the following theorem for approximation of solution
of a nonlinear integral equation of Hammerstein type in a real Hilbert space.
Theorem 1.2.5 (Chidume and Zegeye, [77]) Let H be a real Hilbert space.
Let F;K : H ! H be bounded monotone mappings satisfying the range con-
dition. Suppose the equation 0 = u+KFu has a solution in H. Let fng1 n=1
and fng1n
=1 be real sequences in [0; 1) satisfying the following conditions:
(i) lim
n!1
n = 0;
(ii)
P
nn = 1; lim
n!1
n
n
= 0;
(iii) lim
n!1
n1
n
1
nn
= 0;
Let fung1n
=1 and fvng1 n=1 be sequences in H dened iteratively from arbitrary
u1; v1 2 H by
un+1 = un n(Fun vn) nn(un w);
vn+1 = vn n(Kvn + un) nn(vn w);
where w 2 H is arbitrary but xed. Then, there exists d > 0 such that if
n d and n
n
d2 for all n 1, then the sequences fung1 n=1 and fvng1 n=1
converge strongly to u and v respectively, in H, where u is the solution
of the equation 0 = u + KFu and v = Fu.
In 2009, Chidume and Djitte [61] studied a new iteration method introduced
by Chidume and Zegeye [75] which does not involve K1 and which converges
strongly to a solution of (1.2.2) when K and F are Lipschitz and
-strongly accretive. In particular, they proved the following theorem for
approximation of solution of (1.2.2) in a real q-uniformly smooth Banach
space.
Theorem 1.2.6 (Chidume and Djitte [61]) For q > 1, let X be a real q-
uniformly smooth Banach space and F;K : X ! X be maps with D(K) =
F(X) = X such that the following conditions hold:
General Introduction 6
(i) F is Lipschitzian with constant LF and there exists a positive constant
> 0 such that
hFu1 Fu2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(ii) K is Lipschitzian with constant LK and there exists a positive constant
> 0 such that
hKu1 Ku2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(iii) (1 + cq)(1 + dq) 2q; > q
q , where
q :=
[(1 + cq)(1 + dq) 2q]
1 + cq
;
L := max(LF ;LK); := min(; ); q := d2q
L + dqL + q
q0 dqL + dq and q0 is
the Holder conjugate of q. Let fung and fvng be sequences in X dened as
follows:
u0 2 X; un+1 = un (Fun vn); n 1;
v0 2 X; vn+1 = vn (Kvn + un); n 1;
with
0 < < min
n 1
q
;
q
0q
1
q1
o
;q :=
1
q (01)
0
q
;
where 0 is any number such that0 > q
q q
and the symbols have their
usual meanings. Assume that u + KFu = 0 has a solution u. Then, fung
converges to u in X, fvng converges to v in X and u is the unique solution
of u + KFu = 0 with v = Fu.
Furthermore, it is observed that in Lp spaces, 1 < p < 2, the condition
(1 + cq)(1 + dq) 2q does not necessarily hold, so, Chidume and Djitte [61]
used a dierent tool to obtain the conclusions of Theorem 1.2.6 in these
spaces. They proved the following theorem in Lp spaces, 1 < p < 2.
Theorem 1.2.7 (Chidume and Djitte [61]) Let X = Lp(1 +
p
1 < p
2) and F;K : X ! X be maps with D(K) = F(X) = X such that the
following conditions hold:
(i) F is Lipschitzian with constant LF and there exists a positive constant
> 0 such that
hFu1 Fu2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(ii) K is Lipschitzian with constant LK and there exists a positive constant
> 0 such that
hKu1 Ku2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
General Introduction 7
(iii) Assume that := min(; ) > p(2 p)2q; > q
q , where
q :=
[(1 + cq)(1 + dq) 2q]
1 + cq
;
L := max(LF ;LK); := min(; ); q := d2q
L + dqL + q
q0 dqL + dq and q0 is
the Holder conjugate of q. Let fung and fvng be sequences in X dened as
follows:
u0 2 X; un+1 = un (Fun vn); n 1;
v0 2 X; vn+1 = vn (Kvn + un); n 1;
with
0 < < min
n 1
p
;
p
0p
1
p1
o
;q :=
1
p (01)
0
p2(2 p)
;
where 0 is any number such that0 >
p(2p) and p is as in Theorem
1.2.6. Assume that u + KFu = 0 has a solution u. Then, fung converges
to u in X, fvng converges to v in X and u is the unique solution of
u + KFu = 0 with v = Fu.
Furthermore, Chidume and Djitte [60] extended Theorem 1.2.6 and Theorem
1.2.7 and proved that an explicit coupled iteration process recently introduced
by Chidume and Zegeye [75] which does not involve K1 converges
strongly to a solution of (1.2.2) when K and F are bounded and strongly
accretive. In particular, they proved the following theorems.
Theorem 1.2.8 (Chidume and Djitte, [60]) For q > 1, let X be a real q-
uniformly smooth Banach space and F;K : X ! X be maps with D(K) =
F(X) = X such that the following conditions hold:
(i) F is bounded and there exists a positive constant > 0 such that
hFu1 Fu2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(ii) K is bounded and there exists a positive constant > 0 such that
hKu1 Ku2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(iii) (1+cq)(1+dq) 2q; := min(; ) > q
q , where q := [(1+cq)(1+dq)2q]
1+cq
.
Let fung and fvng be sequences in X dened iteratively from arbitrary points
u1; v1 2 X as follows:
un+1 = un n(Fun vn); n 1;
vn+1 = vn n(Kvn + un); n 1;
General Introduction 8
where fng is a positive sequence satisfying lim
n!1
n = 0;
P1
n=1 n = +1
and
P1
n=1 qn
< 1. Then, there exists a constant d0 > 0 such that if
0 < n d0; fung converges to u in X, fvng converges to v in X and u
is the unique solution of u + KFu = 0 with v = Fu.
Theorem 1.2.9 (Chidume and Djitte [60]) Let X = Lp(1 +
p
1 < p
2) and F;K : X ! X be maps with D(K) = F(X) = X such that the
following conditions hold:
(i) F is bounded and there exists a positive constant > 0 such that
hFu1 Fu2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
(ii) K is bounded and there exists a positive constant > 0 such that
hKu1 Ku2; jq(u1 u2)i jju1 u2jj2
X 8u1; u2 2 X;
Let fung and fvng be sequences in X dened iteratively from arbitrary points
u1; v1 2 X as follows:
un+1 = un n(Fun vn); n 1;
vn+1 = vn n(Kvn + un); n 1;
where fng is a positive sequence satisfying lim
n!1
n = 0;
P1
n=1 n = +1
and
P1
n=1 qn
< 1. Then, there exists a constant d0 > 0 such that if
0 < n d0; fung converges to u in X, fvng converges to v in X and u
is the unique solution of u + KFu = 0 with v = Fu.
Chidume and Djitte [60, 61] remarked that Theorem 1.2.6 and Theorem
1.2.8 hold in X = Lp(2 p +
p
2 + 4) and Theorem 1.2.7 and Theorem
1.2.9 hold in X = Lp(1 +
p
1 < p 2). Therefore, Chidume and
Djitte [60, 61] included these interesting open questions.
Open question 1. Do Theorem 1.2.7 and Theorem 1.2.9 hold in Lp spaces
for all p such that 1 < p 2?
Open question 2. Do Theorem 1.2.6 and Theorem 1.2.8 hold in Lp spaces,
2 p < 1?
Recently, Chidume and Ofoedu [68] extended the results of Chidume and
Zegeye [77] (Theorem 1.2.5 above). They continued to study modications of
the coupled iterative algorithm introduced in [77]. They proved the following
strong convergence theorem for approximation of solution of a nonlinear
integral equation of Hammerstein type in 2-uniformly smooth real Banach
space.
General Introduction 9
Theorem 1.2.10 (Chidume and Ofoedu, [68]) Let E be a 2-uniformly smooth
real Banach space. Let F;K : E ! E be bounded and accretive mappings.
Let fung1 n=1 and fvng1 n=1 be sequences in E dened iteratively from arbitrary
u1; v1 2 E by
un+1 = un nn(Fun vn) nn(un u1);
vn+1 = vn nn(Kvn + un) nn(vn v1);
where fng1 n=1; fng1 n=1 and fng1n
=1 are real sequences in (0; 1) such that
n = o(n); n = o(n) and
P1
i=1 nn = +1. Suppose that u+KFu = 0
has a solution in E. Then, there exist real constants “0; “1 > 0 and a set
W = E E such that if n “0n and n “1n; 8n n0, for some
n0 2 N and w := (u; v) 2
(where v = Fu), the sequence fung1 n=1
converges strongly to u.
A prototype of the parameters in Theorems 1.2.5 and 1.2.10 is n = (n+1)a
and n = (n + 1)b with 0 < b < a and a + b < 1. We verify that these
choices satisfy, in particular, condition (iii) of Theorem 1.2.5. In fact, using
the fact that (1 + x)p 1 + px, for x > 1 and 0 < p < 1, we have
0
n1
n
1
nn
=
h
1 +
1
n
b
1
i
(n + 1)a+b
b
(n + 1)a+b
n
= b
n + 1
n
1
(n + 1)1(a+b)
! 0; as n ! 1:
Quite recently, Chidume and Djitte [64] made a signicant improvement on
the Galerkin method of Brezis and Browder [22] and proved the following
convergence theorem.
Theorem 1.2.11 (Chidume and Djitte, [64]) Let H be a real Hilbert space.
Let F;K : H ! H be bounded monotone mappings satisfying the range con-
dition. Suppose the equation 0 = u+KFu has a solution in H. Let fng1 n=1
and fng1 n=1 be real sequences in [0; 1) satisfying the following conditions:
(i) lim
n!1
n = 0;
(ii)
P
nn = 1; lim
n!1
n
n
= 0;
(iii) lim
n!1
n1
n
1
nn
= 0;
Let fung1n
=1 and fvng1 n=1 be sequences in H dened iteratively from arbitrary
u1; v1 2 H by
un+1 = un n(Fun vn) nn(un w);
vn+1 = vn n(Kvn + un) nn(vn w);
where w 2 H is arbitrary but xed. Then, there exists d0 > 0 such that
if n d0n for all n n0 for some n0 1, then the sequence fung1 n=1
converges strongly to u, a solution of the equation 0 = u + KFu.
General Introduction 10
In Chapter 2 of this thesis, following the ideas of Chidume and Zegeye [77]
and Chidume and Djitte [64], we construct a new coupled explicit iterative
procedure that converges strongly to the solution of equation (1.2.2). The
parameters of this new scheme studied here admit the canonical choice 1
n,
which is not the case in the theorems of Chidume and Zegeye [77], Chidume
and Djitte [64] and Chidume and Ofoedu [68]. Furthermore, the 3(three)
iteration parameters of Chidume and Zegeye [77] and 4(four) iteration parameters
of Chidume and Ofoedu [68] have been reduced to 1(one) parameter
satisfying the conditions:
P1
n=1 2n
< 1 and
P1
n=1 n = +1. This
improves signicantly the eciency of the algorithm studied in Chidume
and Zegeye [77], Chidume and Djitte [64] and Chidume and Ofoedu [68].
Our results improve on the results of Chidume and Djitte [60, 61, 64].
Furthermore, in Chapter 3, we extend the results of Chapter 2 to generalized
equations of Hammerstein type, i.e., equations of the type
u +
Xm
i=1
KiFiu = 0 (1.2.7)
in real Hilbert spaces. This equation is sometimes called equation of Urysohn
type (see, e.g., [152], p. 203). Here, we introduce a new explicit iteration
scheme which converges strongly to a solution of (1.2.7) when Ki; Fi; i =
1; 2; : : : ;m are bounded and monotone assuming existence.
1.3 Algorithms for common xed points
Denition 1.3.1 Let E be a real linear space. A mapping T : D(T) E !
E is said to be nonexpansive if
jjTx Tyjj jjx yjj 8x; y 2 D(T):
A point x 2 D(T) is called a xed point of T if Tx = x.
Markov [136] (see also Kakutani [111]) showed that if a commuting family
of bounded linear transformations T; 2 ( an arbitrary index set) of
a normed linear space E into itself leaves some nonempty compact convex
subset K of E invariant, then the family has at least one common xed point.
(The actual result of Markov is more general than this but this version is
adequate for our purposes). Motivated by this result, De Marr studied the
problem of the existence of a common xed point for a family of nonlinear
maps, and proved the following theorem.
Theorem 1.3.2 (De Marr [88], p.1139) Let E be a Banach space and K
be a nonempty compact convex subset of E. If F is a nonempty commuting
family of nonexpansive mappings of K into itself, then the family F has a
common xed point in K.
General Introduction 11
Browder proved the result of De Marr in a uniformly convex Banach space,
requiring that K be only nonempty closed bounded and convex.
Theorem 1.3.3 (Browder [23], Theorem 1) Let E be a uniformly convex
Banach space, K a nonempty closed convex and bounded subset of E, fTg
a commuting family of nonexpansive self-mappings of K. Then, the family
fTg has a common xed point in K.
Bauschke [9] was the rst to introduce a Halpern-type iterative process for
approximating a common xed point for a nite family of r nonexpansive
self-mappings. He proved the following theorem.
Theorem 1.3.4 (Bauschke [9], Theorem 3.1) Let K be a nonempty closed
convex subset of a Hilbert space H and T1; T2; : : : ; Tr be a nite family of
non- expansive mappings of K into itself with F := \ri
=1F(Ti) 6= ; and
F = F(TrTr1 : : : T1) = F(T1Tr : : : T2) = : : : = F(Tr1Tr2 : : : T1Tr). Let
fng be a real sequence in [0; 1] which satises C1 : lim
n!1
n = 0; C2 :
P1
n=0 n = +1 and C3 :
P1
n=1 jn n+rj < 1. Given points u; x0 2 K;
let fxng be generated by
xn+1 = n+1u + (1 n+1)Tn+1xn; n 0; (1.3.1)
where Tn = Tn mod r. Then, fxng converges strongly to PF u, where PF :
H ! F is the metric projection.
Takahashi et al. [196] extended this result to uniformly convex Banach
spaces. O’Hara et al. [149] proved a complementary result to that of
Bauschke, still in the framework of Hilbert spaces, replacing C3 with the following
new condition: C4 : lim
n!1
n
n+r
= 1, or equivalently, lim
n!1
nn+r
n+r
= 0.
Their main theorems are the following.
Theorem 1.3.5 (O’Hara et al. [149], Theorem 3.3) Let fng (0; 1) sat-
isfy lim
n!1
n = 0 and
P1
n=0 n = +1. Let K be a nonempty closed and
convex subset of a Hilbert space H and let Tn : K ! K; n = 1; 2; : : :
be nonexpansive mappings such that F := \1i
=1F(Ti) 6= ;. Assume that
V1; V2; : : : ; Vn : K ! K are nonexpansive mappings with the property:
for all k = 1; 2; : : : ;N and for any bounded subset C of K, there holds
limn!1 supx2C jjTnx Vk(Tnx)jj = 0. For x0; u 2 K, dene
xn+1 = n+1u + (1 n+1)Tn+1xn; n 0: (1.3.2)
Then, xn ! Pu, where P is the projection from H.
General Introduction 12
Theorem 1.3.6 (O’Hara et al. [149], Theorem 4.1) Let K be a nonempty
closed convex subset of a Hilbert space H and T1; T2; : : : ; TN be nonexpan-
sive self-mappings of K with F := \Ni
=1F(Ti) 6= ;. Assume that F =
F(TrTr1 : : : T1) = F(T1Tr : : : T2) = : : : = F(Tr1Tr2 : : : T1Tr). Let fng
(0; 1) satisfy the following conditions: (i) lim
n!1
n = 0; (ii)
P1
n=0 n =
+1 and (iii) lim
n!1
n
n+N
= 1. Given points x0; u 2 K, the sequence
fxng K is dened by
xn+1 = n+1u + (1 n+1)Tn+1xn; n 0: (1.3.3)
Then, xn ! PF u, where PF is the projection of u onto F.
Jung [109] extended the results of O’Hara et al. [149] to uniformly smooth
Banach spaces. Furthermore, Jung et al. [110] studied the iteration scheme
(1.3.1), where the iteration parameter fng satises the following conditions:
C1 : lim
n!1
n = 0; C2 :
P1
n=0 n = +1 and C5 : jn+N nj
o(n+N) + n, where
P1
n=1 n < 1.
They proved the following theorem.
Theorem 1.3.7 (Jung et al. [110], Theorem 3.1) Let E be a uniformly
smooth Banach space with a weakly sequentially continuous duality map-
ping J : E ! E and let K be a nonempty closed convex subset of E.
Let T1; T2; : : : ; TN be nonexpansive mappings from K into itself with F :=
\Ni
=1F(Ti) 6= ; and F = F(TNTN1 : : : T1) = F(T1TN : : : T2) = : : : =
F(TN1TN2 : : : T1TN). Let fng (0; 1) satisfy the following conditions:
(i) lim
n!1
n = 0; (ii)
P1
n=0 n = +1 and (iii) jn+N nj o(n+N)+
n, where
P1
n=1 n < 1. Then, the iterative sequence fxng dened by
(1.3.1) converges strongly to QF u, where QF is a sunny nonexpansive re-
traction of K onto F.
Zhou et al. [219] proved that the conditions: (C1) and (C2) are indeed
sucient to guarantee the strong convergence of the iteration sequence of
(1.3.1) in each of the following situations: (a) E is a Hilbert space; (b) E
is a Banach space with weakly sequentially continuous duality map and the
sequence fxng of (1.3.1) is weakly asymptotically regular; (c) E is a re exive
Banach space whose norm is uniformly G^ateaux dierentiable and in which
every weakly compact convex subset of E has the xed point property for
nonexpansive mappings, and the sequence fxng of (1.3.1) is asymptotically
regular. Their main results are the following theorems.
Theorem 1.3.8 (Zhou et al. [219], Theorem 6) Let E be a re exive Banach
space with a uniformly G^ateaux dierentiable norm and a weakly continuous
duality mapping J’ for some gauge function ‘. Let K be a nonempty closed
General Introduction 13
convex subset of E. Assume that every weakly compact convex subset of E
has the xed point property for nonexpansive mappings. Let T1; T2; : : : ; Tr
be nonexpansive mappings of K into itself such that F := \ri
=1F(Ti) 6=
;. Assume also that F = F(TrTr1 : : : T1) = F(T1Tr : : : T2) = : : : =
F(Tr1Tr2 : : : T1Tr). Let fng be a sequence in (0; 1) which satises (C1)
and (C2). Let fxng be the sequence dened by (1.3.1) and assume that fxng
is weakly asymptotically regular, then the sequence fxng converges strongly
to a common xed point of T1; T2; : : : ; Tr.
Theorem 1.3.9 (Zhou et al. [219], Theorem 10) Let E be a re exive Ba-
nach space with a uniformly G^ateaux dierentiable norm, and let K be
a nonempty closed convex subset of E. Assume that every weakly com-
pact convex subset of E has the xed point property for nonexpansive map-
pings. Let T1; T2; : : : ; Tr be nonexpansive mappings of K into itself such
that F := \ri
=1F(Ti) 6= ;. Assume also that F = F(TrTr1 : : : T1) =
F(T1Tr : : : T2) = : : : = F(Tr1Tr2 : : : T1Tr). Let fng be a sequence in
(0; 1) which satises (C1) and (C2). Let fxng be the sequence dened by
(1.3.1) and assume that fxng is weakly asymptotically regular, then the se-
quence fxng converges strongly to a common xed point of T1; T2; : : : ; Tr.
We remark here that in their proofs of Theorems 1.3.8 and 1.3.9, the authors
use the concept of Banach limits, proving in the process two results
involving these limits.
In all the above discussion, T1; T2; : : : ; TN remain self-mappings of a nonempty
subset of the Banach space E. If, however, the domain of T1; T2; : : : ; TN,
D(Ti) K; i = 1; 2; : : : ;N, is a proper subset of E and Ti maps K into E
for each i, then the recursion formula (1.3.1) may fail to be well dened.
To overcome this, an algorithm for non-self mappings was dened for the
scheme (1.3.1) by Chidume et al. [79]. Using this algorithm, Chidume et al.
[79] proved the following theorems.
Theorem 1.3.10 (Chidume et al., [79]) Let K be a nonempty closed convex
subset of a re exive Banach space E with a uniformly G^ateaux dierentiable
norm. Assume that K is a sunny nonexpansive retract of E with Q as sunny
nonexpansive retraction. Assume that every nonempty closed bounded con-
vex subset of K has the xed point property for nonexpansive mappings. Let
Ti : K ! E; i = 1; 2; : : : ; r be family of nonexpansive mappings which are
weakly inward with F := \ri
=1F(Ti) 6= ; and F = F(QTrQTr1 : : :QT1) =
F(QT1QTr : : :QT2) = : : : = F(QTr1QTr2 : : :QT1QTr). For a given u; x0 2
K, let fxng be generated by the algorithm
xn+1 = n+1u + (1 n+1)QTn+1xn; n 0;
where Tn = Tn mod r and fng is a real sequence in [0; 1] satisfying the
following conditions: (i) lim
n!1
n = 0; (ii)
P1
n=0 n = +1 and either
General Introduction 14
(iii)
P1
n=0 jn+r nj < 1 or (iii) lim
n!1
n+rn
n+r
= 0. Then, fxng
converges strongly to a common xed point of the family fT1; T2; : : : ; Trg.
Further, if Px0 = lim
n!1
xn for each x0 2 K, then P is sunny nonexpansive
retraction of K onto F.
Theorem 1.3.11 (Chidume et al., [79]) Let K be a nonempty closed convex
subset of a real strictly convex Banach space E with a uniformly G^ateaux dif-
ferentiable norm. Assume that K is a sunny nonexpansive retract of E with
Q as sunny nonexpansive retraction. Assume that every nonempty closed
bounded convex subset of K has the xed point property for nonexpansive
mappings. For each i = 1; 2; : : : ; r, let Ti : K ! E be family of nonexpan-
sive mappings which are weakly inward with F := \ri
=1F(Ti) 6= ;. Let Si :
K ! E be a family of mappings dened by Si := (1i)I+iTi; 0 < i < 1,
for each i = 1; 2; : : : ; r. For a given u; x0 2 K, let fxng be generated by the
algorithm
xn+1 = n+1u + (1 n+1)QSn+1xn; n 0;
where Sn = Sn mod r and fng is a real sequence in [0; 1] satisfying the
following conditions: (i) lim
n!1
n = 0; (ii)
P1
n=0 n = +1 and either
(iii)
P1
n=0 jn+r nj < 1 or (iii) lim
n!1
n+rn
n+r
= 0. Then, fxng
converges strongly to a common xed point of the family fT1; T2; : : : ; Trg.
Further, if Px0 = lim
n!1
xn for each x0 2 K, then P is sunny nonexpansive
retraction of K onto F.
We remark that the requirement that the underlying space E be a Hilbert
space, or satisfy Opial’s condition, or admit weak sequential continuous duality
map imposed in several theorems, in particular, in the theorems of
Bauschke [9], in Theorems 1.3.5, 1.3.6, 1.3.7 and 1.3.8 excludes the application
of any of these theorems in, for example, Lp spaces, 1 < p < 1; p 6= 2
because it is well known that these spaces do not admit weak sequentially
continuous duality mappings and do not satisfy Opial’s condition.
In 2008, Chidume and Ali [52] introduced a new iteration scheme with respect
to which these strong conditions on the underlying space are dispensed
with, and conditions C1 and C2 are sucient to guarantee the strong convergence
of the sequence generated by the recursion formula of the iterative
scheme to a common xed point of T1; T2; : : : ; Tr. In particular, they proved
the following theorem.
Theorem 1.3.12 (Chidume and Ali [52], Theorem 3.1) Let E be a re ex-
ive Banach space with a uniformly G^ateaux dierentiable norm. Let K
be a nonempty closed convex subset of E. Let T1; T2; : : : ; TN be a family
of nonexpansive self-mappings of K, with F := \Ni
=1F(Ti) 6= ; and F =
F(TNTN1 : : : T1) = F(T1TN : : : T2) = : : : = F(TN1TN2 : : : T1TN). Let
General Introduction 15
fng be a sequence in (0; 1) satisfying the following conditions: C1 : lim
n!1
n =
0; C2 :
P1
n=0 n = +1. For a xed 2 (0; 1), dene Sn : K ! K
by Snx := (1 )x + Tnx8x 2 K where Tn = Tn mod N. For arbi-
trary xed u; x0 2 K, let B := fx 2 K : TNTN1 : : : T1x = x + (1
)u; for some > 1g be bounded and let
xn+1 = n+1u + (1 n+1)Sn+1xn; for n 0:
Assume lim
n!1
jjTnxn Tn+1xnjj = 0. Then, fxng converges strongly to a
common xed point of the family T1; T2; : : : ; TN.
We remark that the iteration process of Chidume and Ali [52] can be used,
for example, in the cases when E is a re exive Banach space with uniformly
G^ateaux dierentiable norm, and in which every weakly compact convex
subset of E has the xed point property for nonexpansive mappings, and
Tn satises a mild condition. Moreover, the underlying space will not be
required to admit weak sequential continuous duality maps or to satisfy
Opial’s condition. In addition, the sequence fxng will not be assumed to be
asymptotically regular and the method of proof does not involve the use of
Banach limits.
Consequently, Chidume and Ali [52] proved the following convergence theorem
for non-self maps which complements the results of Chidume et al.
[79].
Theorem 1.3.13 (Chidume and Ali [52], Theorem 4.1) Let K be a nonempty
closed convex subset of a real re exive Banach space E which has a uniformly
G^ateaux dierentiable norm. Let Ti : K ! E; i = 1; 2; : : : ;N be a family
of nonexpansive mappings which are weakly inward with F := \Ni
=1F(Ti) =
F(QTNQTN1 : : :QT1) = F(QT1QTN : : :QT2) = : : :
= F(QTN1QTN2 : : :QT1QTN) 6= ;. For a xed 2 (0; 1), dene Sn :
K ! K by Snx := (1)x+QTnx 8x 2 E. For arbitrary xed u; x0 2 K,
let B := fx 2 K : TNTN1 : : : T1x = x + (1 )u; for some > 1g be
bounded and let the sequence fxng be generated iteratively by
xn+1 = n+1u + (1 n+1)Sn+1xn; for n 0;
where Tn = Tn mod N and fng is a real sequence which satises (C1)
and (C2). Assume lim
n!1
jjQTnxn QTn+1xnjj = 0. Then, fxng converges
strongly to a common xed point of the family T1; T2; : : : ; TN. Further, if
Pu = lim
n!1
xn for each u 2 K, then P is sunny nonexpansive retraction of
K onto F.
We observe that all the theorems proved in Chidume and Ali [52] hold, in
particular, in Lp spaces, 1 < p < 1.
General Introduction 16
Another important class of nonlinear mappings more general than the class
of nonexpansive mappings called asymptotically nonexpansive mapping was
introduced in 1972 by Goebel and Kirk [97]. Let K be a nonempty subset
of a normed linear space, a mapping T : K ! K is called asymptotically
nonexpansive if there exists a real sequence fkng [1;1) with lim
n!1
kn = 1
such that jjTnx Tnyjj knjjx yjj for all x; y 2 K and n = 1; 2; : : : :.
It was proved [97] that if K is a nonempty closed, convex and bounded
subset of a real uniformly convex Banach space and T has a xed point. It
is clear that every nonexpansive mapping is asymptotically nonexpansive.
The following example shows that the class of asymptotically nonexpansive
mappings properly contains the class of nonexpansive mappings.
Example 1.3.14 (Goebel and Kirk, [97]) Let B be a unit ball of the real
Hilbert space l2 and let T : B ! B be dened by T(fx1; x2; : : :g) = f0; x21
; a2x2; a3x3; : : :g
where fang is a sequence of numbers such that 0 < an < 1 and
Q1
n=2 an = 1
2 :
Then jjTxTyjj 2jjxyjj; for all x; y 2 B and moreover, jjTnxTnyjj
knjjx yjj, with kn := 2
Qn
i=2 ai. Observe that T is not nonexpansive and
that lim
n!1
kn = 1, so that T is asymptotically nonexpansive mapping.
The averaging iteration process xn+1 := (1 n)xn + nTxn where T :
K ! K is asymptotically nonexpansive, K is a closed convex and bounded
subset of a Hilbert space was introduced by Schu [172]. He considered the
following iteration scheme: E = H, a Hilbert space, K is a nonempty closed
convex and bounded subset of H, T : K ! K is completely continuous
and asymptotically nonexpansive, for each x0 2 K; xn+1 := (1 n)xn +
nTxn; n 0, where
P1
n=0(k2n
1) < 1 and fng is a real sequence satisfying
appropriate conditions. He proved that fxng converges strongly to a xed
point of T. This result has been extended to uniformly convex Banach
spaces in the following theorems.
Theorem 1.3.15 (Rhoades, [169]) Let E be uniformly convex and K be a
nonempty closed convex and bounded subset of E. Suppose T : K ! K is
completely continuous and asymptotically nonexpansive; for each x0 2 K,
let fxng be dened as follows:
xn+1 = (1 n)xn + nTxn; n 0
where
P1
n=0(krn
1) < 1 for some r > 1; 1 n 1 for all positive
integer n and some > 0. Then lim
n!1
jjxn Txnjj = 0.
Theorem 1.3.16 (Rhoades, [169]) Let E be uniformly convex and K be a
nonempty closed convex and bounded subset of E. Suppose T : K ! K is
completely continuous and asymptotically nonexpansive; for each x0 2 K,
let fxng be dened as follows:
xn+1 = (1 n)xn + nTxn; n 0
General Introduction 17
where
P1
n=0(krn
1) < 1; r := maxf2; pg and 1 n 1 for all
positive integer n and some > 0. Then, fxng converges strongly to some
xed point of T.
Let E be a real uniformly smooth Banach space, K be a nonempty bounded
closed convex subset of E, and T : K ! K be a nonexpansive mapping.
Then, for a xed u 2 K and each integer n 1, by the Banach Contraction
Mapping Principle, there exists a unique xn 2 K such that
xn =
1
n
u + (1
1
n
)Txn: (1.3.4)
It follows immediately from this equation that lim
n!1
jjxnTxnjj = 0. One of
the most useful results concerning algorithms for approximating xed points
of nonexpansive mappings in real uniformly smooth Banach spaces is the
celebrated convergence theorem of Reich [168] who proved that the implicit
sequence fxng dened in equation (1.3.4) actually converges strongly to a
xed point of T. Several authors have tried to obtain a result analogous
to that of Reich [168] 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 2 K and dene, for each integer
n 1, the contraction mapping Sn : K ! K by
S(x) =
1
tn
kn
u + tn
kn
Tnx; (1.3.5)
where ftng 2 [0; 1) is any sequence such that tn ! 1. Then, by the Banach
Contraction Mapping Principle, there exists a unique point xn xed by Sn,
i.e., there exists xn such that
xn =
1
tn
kn
u + tn
kn
Tnxn: (1.3.6)
The question now arises as to whether or not this sequence converges to a
xed point of T. A partial answer is given in the following theorem.
Theorem 1.3.17 (Lim and Xu, [124]) Let E be a uniformly smooth Banach
space, K be a nonempty closed convex and bounded subset of E; T : K ! K
be an asymptotically nonexpansive mapping with sequence kn 2 [1;1). Fix
u 2 K and let ftng [0; 1) be chosen such that lim
n!1
tn = 1 and lim
n!1
kn1
kntn
=
0. Then, (i) for each integer n 0, there is a unique xn 2 K such that
xn =
1
tn
kn
u + tn
kn
Tnxn;
suppose in addition that lim
n!1
jjxn Txnjj = 0, then, (ii) the sequence fxng
converges strongly to a xed point of T.
General Introduction 18
Chidume et al. [65] extended Theorem 1.3.17 to re exive Banach spaces with
uniformly G^ateaux dierentiable norms. As an application, they proved that
the explicit sequence fzng iteratively generated by
z1 2 K; zn+1 =
1
tn
kn
u + tn
kn
Tnzn; n 1;
converges strongly to a xed point of the asymptotically nonexpansive mapping
T.
We now have the following theorems.
Theorem 1.3.18 (Chidume et al., [65]) Let E be a real Banach space with
uniformly G^ateaux dierentiable norm possessing uniform normal structure,
K be a nonempty closed convex and bounded subset of E; T : K ! K be an
asymptotically nonexpansive mapping with sequence kn 2 [1;1). Let u 2 K
be xed and let ftng (0; 1) be such that lim
n!1
tn = 1 and lim
n!1
kn1
kntn
= 0.
Then, (i) for each integer n 0, there is a unique xn 2 K such that
xn =
1
tn
kn
u + tn
kn
Tnxn;
suppose in addition that lim
n!1
jjxn Txnjj = 0, then, (ii) the sequence fxng
converges strongly to a xed point of T.
Theorem 1.3.19 (Chidume et al., [65]) Let E be a real Banach space with
uniformly G^ateaux dierentiable norm possessing uniform normal structure,
K be a nonempty closed convex and bounded subset of E; T : K ! K be an
asymptotically nonexpansive mapping with sequence kn 2 [1;1). Let u 2 K
be xed and let ftng (0; 1) be such that lim
n!1
tn = 1 and lim
n!1
kn1
kntn
= 0.
Dene the sequence fzng iteratively generated by
z1 2 K; zn+1 =
1
tn
kn
u + tn
kn
Tnzn; n 1:
Then, (i) for each integer n 0, there is a unique xn 2 K such that
xn =
1
tn
kn
u + tn
kn
Tnxn;
suppose in addition that lim
n!1
jjxn Txnjj = 0; jjzn Tnznjj = o
1 tn
kn
,
then, (ii) the sequence fzng converges strongly to a xed point of T.
The concept of non-self asymptotically nonexpansive mappings was introduced
by Chidume et al. [71] as an important generalization of asymptotically
nonexpan- sive self-mappings.
General Introduction 19
Using an Ishikawa-like scheme, Takahashi and Tamura [195] proved strong
and weak convergence of a sequence dened by xn+1 = nS[nTxn + (1
n)xn] + (1 n)xn to a common fxed point of a pair of nonexpansive
mappings T and S. Wang [510] used a similar scheme and the denition of
Chidume et al. [71] to prove strong and weak convergence theorems for a
pair of non-self asymptotically nonexpansive mappings. More precisely he
proved the following theorems.
Theorem 1.3.20 (Wang, [202]) Let K be a nonempty closed convex subset
of uniformly convex Banach space E. Suppose T1; T2 : K ! E are two non-
self asymptotically nonexpansive mappings with sequences fkng and flng 2
[1;1) such that
P1
n=1(kn 1) < 1;
P1
n=1(ln 1) < 1; kn ! 1; ln ! 1 as
n ! 1, respectively. Let fxng be generated by
8<
:
x1 2 K;
xn+1 = P((1 n)xn + nT1(PT1)n1yn);
yn = P((1 n)xn + nT2(PT2)n1xn); n 1;
(1.3.7)
where fng and fng are sequences in [; 1 ] for some 2 (0; 1). If one
of T1 and T2 is completely continuous, and F(T1) \ F(T2) 6= ;, then fxng
converges strongly to a common xed point of T1 and T2.
Theorem 1.3.21 (Wang, [202]) Let K; T1; T2; fkng; flng and fxng be as in
Theorem 1.3.20. If one of T1 and T2 is semi-compact, and F(T1)\F(T2) 6= ;,
then, fxng converges strongly to a common xed point of T1 and T2.
Theorem 1.3.22 (Wang, [202]) Let K; T1; T2; fkng; flng and fxng be as
in Theorem 1.3.20. If E satises Opial’s condition, and F(T1)\F(T2) 6= ;,
then, fxng converges weakly to a common xed point of T1 and T2.
In 2007, Chidume and Ali [54] introduced an iteration process for approximating
common xed points for nite families of non-self asymptotically
nonexpansive mappings. For these families of operators, strong convergence
theorems are proved in uniformly convex Banach spaces and weak convergence
theorems are proved in real uniformly convex Banach spaces that
satisfy Opial’s condition, or have Frechet dierentiable norms, or whose
dual spaces have the Kadec-Klee property. In particular, they proved the
following theorems.
Theorem 1.3.23 (Chidume and Ali, [54]) Let E be a real uniformly convex
Banach space and K be a closed convex nonempty subset of E which is also
a nonexpansive retract with retraction P. Let T1; T2 : : : ; Tm : K ! E be
asymptotically nonexpansive mappings of K into E with sequences fking1 n=1
and fng1 n=1 satisfying kin ! 1; as n ! 1 and
P1
n=1(kin 1) < 1; i =
1; 2; : : : ;m. Let fng be a sequence in [; 1] for some 2 (0; 1). If one of
General Introduction 20
fTigmi
=1
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