ABSTRACT
It is well know that many physically signicant problems in dierent areas of research
can be transformed into an equation of the form
Au = 0; (0.0.1)
where A is a nonlinear monotone operator from a real Banach space E into its dual
E. For instance, in optimization, if f : E ! R [ f+1g is a convex, G^ateaux
dierentiable function and x is a minimizer of f, then f0(x) = 0. This gives a
criterion for obtaining a minimizer of f explicitly. However, most of the operators
that are involved in several signicant optimization problems are not dierentiable.
For instance, the absolute value function x 7! jxj has a minimizer, which, in fact,
is 0. But, the absolute value function is not dierentiable at 0. So, in a case where
the operator under consideration is not dierentiable, it becomes dicult to know
a minimizer even when it exists. Thus, the above characterization only works for
dierentiable operators.
A generalization of dierentiability called subdierentiability allows us to recover
the above result for non dierentiable maps.
For a convex lower semi-continuous function which is not identically +1, the subdi
erential of f at x is given by
@f(x) = fx 2 E : hx; y xi f(y) f(x) 8 y 2 Eg: (0.0.2)
Observe that @f maps E into the power set of its dual space, 2E
. Clearly, 0 2 @f(x)
if and only if x minimizes f. If we set A = @f, then the inclusion problem becomes
0 2 Au
which also reduces to (0.0.1) when A is single-valued. In this case, the operator
maps E into E. Thus, in this example, approximating zeros of A, is equivalent to
the approximation of a minimizer of f.
In chapter three and four of the thesis, we give convergence results for approximating
zeros of equation (0.0.1) in Lp spaces, 1 < p < 1, where the operator A
vi
Abstract vii
is Lipschitz strongly monotone and generalised -strongly monotone and bounded
maps respectively.
As remarked by Charles Byrne [23], most of the maps that arise in image reconstruction
and signal processing are nonexpansive in nature. A more general class of
nonexpansive operators is the class of k-striclty pseudo-contractive maps. In chapter
ve of this thesis, we prove some convergence results for a xed point of nite
family of k-striclty pseudo-contractive maps in CAT(0) spaces. We also prove a
convergence result for a countable family of k-striclty pseudo-contractive maps in
Hilbert spaces in chapter six of the thesis.
Let
Rn be bounded. Let k :
! R and f :
R ! R be measurable
functions. An integral equation of Hammerstein has the form
u(x) +
Z
k(x; y)f(y; u(y))dy = w; (0.0.3)
where the unknown function u and inhomogeneous function w lie in the function
space E. In abstract form, the equation (0.0.3) can be written in the form
u + AFu = w (0.0.4)
where A : E ! E and F : E ! E are monotone operators.
In general, every elliptic boundary value problem whose linear part posses a Green’s
function (e.g., the problem of forced oscillation of nite amplitude pendulum) can
be transformed into an equation of Hammerstein type. Thus, approximating zeros
of the Hammerstein-type equation in (0.0.4) (when w = 0) is equivalent to
the approximation of solutions of some boundary value problems. Hammerstein
equations also play crucial role in variational calculus and xed point theory. In
chapter seven of this thesis, we give convergence results for approximating solutions
of Hammerstein-type equations in LP spaces, 1 < p < 1.
In particular, we prove the following results in this thesis.
Let E = Lp; 1 < p < 2. Let A : E ! E be a strongly monotone and
Lipschitz map. For x0 2 E arbitrary, let the sequence fxng be dened by:
xn+1 = J1(Jxn Axn); n 0;
where 2
0;
. Then, the sequence fxng converges strongly to x 2 A1(0)
and x is unique.
Let E= Lp; 2 p < 1. Let A : E ! E be a Lipschitz map. Assume that
there exists a constant k 2 (0; 1) such that A satises the following condition:
Ax Ay; x y
kkx yk
p
p1 ;
Abstract viii
and that A1(0) 6= ;: For arbitrary x0 2 E, dene the sequence fxng iteratively
by:
xn+1 = J1(Jxn Axn); n 0;
where 2 (0; p). Then, the sequence fxng converges strongly to the unique
solution of the equation Ax = 0:
Let E = Lp; 1 < p < 2. Let A : E ! E be a generalized -strongly
monotone and bounded map with A1(0) 6= ;. For arbitrary x1 2 E, dene a
sequence fxng iteratively by:
xn+1 = J1(Jxn nAxn); n 1;
where fng1 n=1 (0; 1) satises the following conditions:
P1
P n=1 n = 1 and 1
n=1 2n
< 1. Suppose there exists 0 > 0 such that if n 0 for all n 1.
Then, the sequence fxng1 n=1 converges strongly to a solution of the equation
Ax = 0:
Let E = Lp; 2 p < 1. Let A : E ! E be a generalized -strongly
monotone and bounded map with A1(0) 6= ;. For arbitrary x1 2 E, dene a
sequence fxng iteratively by:
xn+1 = J1(Jxn nAxn); n 1;
where fng1 n=1 (0; 1) satises the following conditions:
P1
n=1 n = 1 and
P1
n=1
p
p1
n < 1. Then, there exists 0 > 0 such that if n 0, the sequence
fxng1 n=1 converges strongly to a solution of the equation Ax = 0:
Let K be a nonempty closed convex subset of a complete CAT(0) space X. Let
Ti : K ! CB(K); i = 1; 2; : : : ; m; be a family of demi-contractive mappings
with constants ki 2 (0; 1); i = 1; 2; : : : ;m, such that
Tm
i=1 F(Ti) 6= ;. Suppose
that Ti(p) = fpg for all p 2
Tn
i=1 F(Ti). For arbitrary x1 2 K, dene a
sequence fxng by
xn+1 = 0xn 1y1n
2y2n
mym
n ; n 1;
where yin
2 Tixn; i = 1; 2; : : : ; m; 0 2 (k; 1); i 2 (0; 1); i = 1; 2; : : : ; m; such
that
Pm
i=0 i = 1, and k := maxfki; i = 1; 2; : : : ;mg. Then, lim
n!1
fd(xn; p)g
exists for all p 2
Tn
i=1 F(Ti), and lim
n!1
d(xn; Tixn) = 0 for all i = 1; 2; : : : ;m.
Let K be a nonempty closed and convex subset of a real Hilbert space H, and
Ti : K ! CB(K) be a countable family of multi-valued ki-strictly pseudocontractive
mappings; ki 2 (0; 1); i = 1; 2; ::: such that
T1
i=1 F(Ti) 6= ;; and
supi1 ki 2 (0; 1). Assume that for p 2
T1
i=1 F(Ti), Ti(p) = fpg: Let fxng1 n=1
be a sequence dened iteratively for arbitrary x0 2 K by
xn+1 = 0xn +
1X
i=1
iyin
;
Abstract ix
where yin
2 Tixn; n 1 and 0 2 (k; 1);
P1
i=0 i = 1 and k := supi1 ki.
Then, limn!1 d(xn; Tixn) = 0, i = 1; 2; ::::
Let E = Lp; 1 < p < 2. Let F : E ! E and K : E ! E be strongly
monotone and bounded maps. For (u0; v0) 2 E E, dene the sequences
fung and fvng in E and E respectively by
un+1 = J1(Jun n(Fun vn)); n 0;
vn+1 = J1
(Jvn n(Kvn + un)); n 0;
where fng1 n=1 (0; 1) satises the following conditions:
P1
n=1 n = 1,
P1
n=1 2n
< 1 and
P1
n=1
q
q1
n < 1, where q is such that 1
p + 1
q = 1. Assume
that the equation u+KFu = 0 has a solution. Then, there exists 0 > 0 such
that if n 0 for all n 1, the sequences fung1 n=1 and fvng1 n=1 converge
strongly to u and v, respectively, where u is the solution of u + KFu = 0
with v = Fu.
Let E = Lp; 2 p < 1. Let F : E ! E and K : E ! E be strongly
monotone and bounded maps. For (u0; v0) 2 E E, dene the sequences
fung and fvng in E and E, respectively, by
un+1 = J1(Jun n(Fun vn)); n 0;
vn+1 = J1
(Jvn n(Kvn + un)); n 0;
where fng1 n=1 (0; 1) satises the following conditions:
P1
n=1 n = 1,
P1
n=1 2n
< 1 and
P1
n=1
p
p1
n < 1. Assume that the equation u+KFu = 0
has a solution. Then, there exists 0 > 0 such that if n 0 for all n 1, the
sequences fung1 n=1 and fvng1 n=1 converge strongly to u and v respectively,
where u is the solution of u + KFu = 0 with v = Fu
TABLE OF CONTENTS
Dedication iii
Acknowledgements iv
Abstract vi
1 General introduction 1
General Introduction 1
1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Approximation of zeros of nonlinear mappings of monotonetype
in classical Banach spaces . . . . . . . . . . . . . . . . . 1
1.2 Approximation Methods for the Zeros of Nonlinear Mappings of
Accretive-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Iterative methods for zeros of monotone-type mappings . . . . . . . 7
1.4 Approximation of xed points of a nite family of k-strictly pseudocontractive
mappings in CAT(0) spaces . . . . . . . . . . . . . . . . 8
1.5 Fixed point of multivalued maps . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Game Theory and Market Economy . . . . . . . . . . . . . . 10
1.5.2 Non-smooth Dierential Equations . . . . . . . . . . . . . . . 11
1.6 Iterative methods for xed points of some nonlinear multi-valued
mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Hammerstein Integral Equations . . . . . . . . . . . . . . . . . . . . 14
1.8 Approximating solutions of equations of Hammerstein-type . . . . . 16
2 Preliminaries 19
2.1 Duality Mappings and Geometry of Banach Spaces . . . . . . . . . . 19
2.2 Some Nonlinear Functionals and Operators . . . . . . . . . . . . . . 23
2.3 Some Important Results about Geodesic Spaces . . . . . . . . . . . . 27
xii
Abstract xiii
3 Krasnoselskii-Type Algorithm For Zeros of Strongly Monotone
Lipschitz Maps in Classical Banach Spaces 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Convergence in LP spaces, 1 < p < 2 . . . . . . . . . . . . . . . . . . 32
3.3 Convergence in Lp spaces, 2 p < 1. . . . . . . . . . . . . . . . . . 33
4 An Algorithm for Computing Zeros of Generalized Phi-Strongly
Monotone and bounded Maps in Classical Banach Spaces 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Convergence Theorems in Lp spaces, 1 < p < 2 . . . . . . . . . . . . 36
4.3 Convergence Theorems in Lp spaces, 2 p < 1 . . . . . . . . . . . . 38
5 Strong and -Convergence Theorems for Common Fixed Point
of a Finite Family of Multivalued Demi-Contractive Mappings in
CAT(0) Spaces 41
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Convergence Theorem for a Countable Family of Multi-Valued
Strictly Pseudo-Contractive Mappings in Hilbert Spaces 47
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7 Approximation of Solutions of Hammerstein Equations with Strongly
Monotone and Bounded Operators in Classical Banach Spaces 53
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Convergence Theorems in Lp spaces, 1 < p < 2 . . . . . . . . . . . . 54
7.3 Convergence Theorems in Lp spaces, p 2 . . . . . . . . . . . . . . . 57
CHAPTER ONE
General introduction
1.1 Some Motivation
The contents of this thesis fall within the general area of nonlinear functional analysis,
an area which has attracted the attention of prominent mathematicians due
to its diverse applications in numerous elds of sciences. The contributions of this
thesis concentrate mainly on the following three important topics. Namely;
Approximation of zeros of nonlinear monotone mappings in classical Banach
spaces.
Approximation of xed points of a nite family of k-strictly pseudo-contractive
mappings in CAT(0) spaces, and a countable family of k-strictly pseudocontractive
maps in Hilbert spaces.
Approximating solutions of Integral equations of Hammerstein-type with monotone
operators in Banach spaces.
1.1.1 Approximation of zeros of nonlinear mappings of monotone-
type in classical Banach spaces
It is well known that many physically signicant problems in dierent areas of
research can be transformed into an equation of the form
Au = 0; (1.1.1)
where A is a nonlinear monotone operator dened on a real Banach space E. Let H
be a real inner product space. A mapping A : D(A) H ! H is called monotone
if for each x; y 2 D(A); the following inequality holds:
hAx Ay; x yi 0;
1
General Introduction 2
and is called strongly monotone if there exists k 2 (0; 1) such that for all x; y 2 D(A);
the following inequality holds:
hAx Ay; x yi kkx yk2:
Monotone mappings were studied in Hilbert spaces by Zarantonello [118], Minty
[83], Kacurovskii [69] and a host of other authors. Interest in such mappings stems
mainly from their usefulness in numerous applications. Consider, for example, the
following: Let f : H ! R [ f1g be a proper convex function. The sub-dierential
of f at x 2 H is dened by
@f(x) =
x 2 H : f(y) f(x)
y x; x
8 y 2 H
:
It is easy to check that @f : H ! 2H is a monotone operator on H, and that
0 2 @f(x) if and only if x is a minimizer of f. Setting @f A, it follows that
solving the inclusion 0 2 Au, in this case, is solving for a minimizer of f. In a case
where the operator A is single valued, the inclusion 0 2 Au reduces to equation
(1.1.1).
The extention of the monotonicity denition to operators from real Banach space
into its dual has been the beginning of nonlinear functional analysis as remarked
by Pascali and Sburian [91] as follows:
The extension of the monotonicity denition to operators from a Banach
space into its dual has been the starting point for the development
of nonlinear functional analysis …: The monotone maps constitute
the most manageable class, because of the very simple structure of the
monotonicity condition. The monotone mappings appear in a rather
wide variety of contexts, since they can be found in many functional
equations. Many of them appear also in calculus of variations, as subdi
erential of convex funtions (Pascali and Sburian [91], p.101).
Unlike as in the case of Hilbert spaces, where the operator A maps H to H (in
this case H = H by the virtue of Reiz representation theorem), in arbitrary real
Banach space E, the extension of monotonicity is split into two cases; a case where
A maps E to E in which A shall be called accretive, and the other case where A
maps E to E (the dual of E) in which it retains its name as monotone.
Let E be a real normed space with dual E. An operator A : E ! E is said
to be accretive if and only if 8 x; y 2 E, there exists j(x y) 2 J(x y) such that
hAx Ay; j(x y)i > 0;
where J is the normalized duality mapping on E dened by
J(x) = fj(x) 2 E : hj(x); xi = kj(x)kkxk; kj(x)k = kxkg:
and is called strongly accretive if and only if there exists k 2 (0; 1), and for each
x; y 2 D(A) there exists j(x y) 2 J(x y) such that the following inequality
holds:
hAx Ay; j(x y)i kkx yk2:
General Introduction 3
An operator A : E ! E is said to be monotone if and only if
hAx Ay; x yi > 0 8 x; y 2 E:
and is called strongly monotone if and only if there exists k 2 (0; 1) such that for
each x; y 2 E; the following inequality holds:
hAx Ay; x yi kkx yk2:
In equation (1.1.1), setting T = I A we obtain that zeros of A are precisely the
xed points of the operator T (i.e., Au = 0 if and only if Tu = u). In the case that
A maps E to E the operator T is called pseudo-contractive whenever the operator
A is accretive.
Accretive operators were introduced independently in 1967 by Browder [14] and
Kato [73]. Interest in such mappings stems mainly from their rm connection with
the existence theory for nonlinear equations of evolution in Banach spaces. For
accretive-type operator A, solutions of the equation Au = 0, in many cases, represent
equilibrium state of some dynamical system. The examples below show how
some problems in applications can be transformed into an equation of the form
(1.1.1).
Evolution Equations: Consider the following diussion equation
8<
:
@u
@t (t; x) = 4u(t; x) + g(u(t; x)); t 0; x 2
;
u(t; x) = 0; t 0; x 2 @
;
u(0; x) = u0(x); u0 2 L2(
);
(1.1.2)
where
is an open smooth subset of Rn.
By simple transformation i.e., by setting v(t) = u(t; :); where
v : [0;+1) ! L2(
)
is dened by v(t)(x) = u(t; x) and f(‘)(x) = g(‘(x)); where
f : L2(
) ! L2(
);
we see that equation (1.1.2) is equivalent to
v0(t) = Av(t) + f(v(t)); t 0;
v(0) = u0:
(1.1.3)
Setting f to be identically zero, at an equilibrium state (i.e., when the system
becomes independent of time) we see that equation (1.1.3) reduces to
Av = 0:
General Introduction 4
Thus, approximatig zeros of equation (1.1.1) or equivalently xed points of T, where
T = I A, is equivalent to the approximation of solutions of the diusion equation
(1.1.2) at an equilibrium state.
Optimization: Consider the following optimization problem:
nd x 2 E such that f(x) f(x) 8 x 2 E; (1.1.4)
where f : E ! R [ f+1g is a map and E is a real normed linear space. It is well
known that if the function f is dierentiable and x exists, then f0(x) = 0. This
gives a criterion for obtaining a minimizer explicitly. However, most of the operators
that are involved in several signicant optimization problems are not dierentiable
in the usual sense. For instance, the absolute value function x 7! jxj has a minimizer,
which, in fact, is 0. But, the absolute value function is not dierentiable at 0. So,
in a case where the operator under consideration is not dierentiable, it becomes
dicult to compute a minimizer even when it exists. Thus, the above result only
works for dierentiable operators.
A generalization of dierentiability called subdierentiability allows us to recover in
a sense, the above result for non dierentiable maps.
Let E be a real normed linear space and f : E ! R [ f+1g be a convex and
proper function (i.e., f is not identically 1). Then, the sub-dierential of f at x
denoted by @f(x) is dened by
@f(x) = fx 2 E : hx; y xi f(y) f(x) 8 y 2 Eg: (1.1.5)
It is easy to see that 0 2 @f(x) if and only if x minimizes f. If we set A = @f, then
the optimization problem (1.1.4) reduces to the inclusion problem
0 2 Au
which also reduces to (1.1.1) when A is single-valued. In this case, the operator A
maps E into E. Thus, approximating zeros of A, is equivalent to the approximation
of a minimizer of f.
1.2 Approximation Methods for the Zeros of Nonlinear
Mappings of Accretive-type
We recall that in Hilbert spaces accretive and monotone operators coincide. A
monotone operator from a real Hilbert space H into itself is said to be maximal
monotone if R(I + A) = H 8 > 0. For the approximation of zeros of maximal
monotone operators in Hilbert space, assuming existence, Martinet [82], introduced
the so-called proximal point algorithm which was further studied by Rockafellar [99]
and a host of other authors (see e.g., Reich [24, 94, 98], Ishikawa [80], Takahashi
and Ueda [108] ). Specically, given xk 2 H, an approximation of a solution of
General Introduction 5
(1.1.1), the proximal point algorithm generates the next iterate xk+1 by solving the
following equation:
xk+1 =
1 +
1
k
A
1
(xk) + ek; (1.2.1)
where k > 0 is a regularizing parameter. If the sequence fkg1k
=1 is bounded from
above, then the resulting sequence fxkg1k
=1 of proximal point iterates converges
weakly to a solution of (1.1.1), provided that a solution exists (Rockafellar [99]).
Rockafellar then posed the following question.
Does the proximal point algorithm always converge strongly?
This question was resolved in the negative by Guler [66], who produced a proper
closed convex function g in the innite-dimensional Hilbert space l2 for which the
proximal point algorithm converges weakly but not strongly. This naturally raises
the following question.
Can the proximal point algorithm be modied to guarantee strong convergence?
Solodov and Svaiter [105] were the rst to propose a modication of the proximal
point algorithm which guarantees strong convergence in a real Hilbert space. Their
algorithm is as follows:
Choose arbitrary x0 2 H and 2 [0; 1). At iteration k, having xk choose k > 0
and nd (yk; vk) an inexact solution of 0 2 Tx+k(xxk), with tolerance . Dene
Ck := fz 2 H : hz yk; vki 0g;
Qk := fz 2 H : hz xk; x0 xki 0g:
Take
xk+1 = PCk\Qk (x0):
The authors themselves noted ([105], p.195) that \. . . at each iteration, there are
two subproblems to be solved. . . “: Firstly, to nd an inexact solution of the proximal
point algorithm. Secondly, to nd the projection of x0 onto Ck \ Qk, the
intersection of the two half spaces. They also acknowledged that these two subproblems
constitute a serious drawback in their algorithm. This method of Solodov
and Svaiter is part of the so-called CQ-method which has been studied by various
authors.
Several authors have successfully extended the results of Martinet [82], and Rockafellar
[99], to a more general space than Hilbert space in a case where the operator
A is accretive. (see e.g., Reich [24, 94, 98], Bruck [19], Browder [14, 18], Takahashi
[71], and a host of other authors).
Remark 1.2.1 We remark here that while many convergence results have appeared
on the extention of the results of Martinet [82], and Rockafellar [99] to a more
General Introduction 6
general space than Hilbert space in a case where the operator A, is accretive, most
of the convergence results obtained are weak convergence, in a case where strong
convergence is obtained, virtually all the algorithms use CQ-method introduced by
Solodov and Svaiter [105], which is not suitable for implementation in applications
as Solodov and Svaiter acknowledged themselves.
We recall that a point x 2 E is said to be xed point of a map T : E ! E if
Tx = x. The set of of xed points of T is denoted by F(T). A map T is said to be
Lipschitz if there exists L > 0 such that kTx Tyk Lkx yk for all x; y 2 E: If
L = 1, then T is called nonexpansive. Also, a map T : E ! E is said to be strongly
pseudo-contractive if (I T) is strongly accretive.
In 1986, Chidume [30], proved the following strong convergence theorem for Lipschitz
strongly pseudo-contractive mappings in LP spaces, 2 p < 1.
Theorem 1.2.2 Let E = Lp; 2 p < 1, and K E be nonempty closed convex
and bounded. Let T : K ! K be a strongly pseudo-contractive and Lipschitz map.
For arbitrary x0 2 K, let a sequence fxng be dened iteratively by
xn+1 = (1 n)xn + nTxn; n 0; (1.2.2)
where fng (0; 1) satises the following conditions: (i)
P1
n=1 n = 1;
(ii)
P1
n=1 2n
< 1. Then, fxng converges strongly to the unique xed point of T
The iteration formula (1.2.2) is the so-called Mann iteration formula in the light
of Mann in [81], to approximate xed points of nonexpansive maps. Replacing T
by I A in Theorem 1.2.2, the following theorem for approximating the unique
solution of Au = 0 when A : E ! E is a strongly accretive and Lipschitz map is
easily proved.
Theorem 1.2.3 Let E = Lp; 2 p < 1. Let A : E ! E be a strongly accretive
and Lipschitz map. For arbitrary x0 2 K, let a sequence fxng be dened iteratively
by
xn+1 = xn nAxn; n 0; (1.2.3)
where fng (0; 1) satises the following conditions: (i)
P1
n=1 n = 1;
(ii)
P1
n=1 2n
< 1. Then, fxng converges strongly to the unique solution of Au = 0.
The main tool used in the proof of Theorem 1.2.2 is an inequality of Bynum [22].
This theorem signalled the return to extensive research eorts on inequalities in
Banach spaces and their applications to iterative methods for solutions of nonlinear
equations. Consequently, Theorem 1.2.2 has been generalized and extended in
various directions, leading to ourishing areas of research, for the past thirty years
or so, for numerous authors (see e.g., Censor and Riech [24], Chidume [26, 27],
Chidume and Ali [31], Chidume and Chidume [36, 37], Chidume and Osilike [48],
Deng [56], Mouda [84, 85, 86, 87], Zhou and Jia [120], Liu [80], Qihou [92], Berinde
et al. [7], Reich [94, 95, 96], Reich and Sabach [97, 98], Weng [109], Xiao [111], Xu
[113, 116, 117], Xu and Roach [114], Xu [115], Zhu [121] and a host of other authors).
General Introduction 7
Recent monographs emanating from these researches include those by Berinde [6],
Chidume [29], Goebel and Reich [65], and William and Shahzad [110].
Taking into account the references mentioned above (and the references contained
therein), it is readily clear that much has been done on the approximation of zeros
of mappings of accretive-type. However, little has been done in the case where the
operator A is monotone (i.e., A maps E into E). This is, perhaps, because of the
following two major diculties.
Well deneness of the scheme: If we consider, for instance, the Mann recursion
formula for approximatig zeros of accretive operators which is given by, x0 2 E
and
xn+1 = xn nAxn; n 0;
we see that in the case of monotone operators this formula is not applicable,
simply because of the fact that it is not well dened (i.e., we are adding two
elements from two dierent vector spaces. i.e., xn 2 E and Axn 2 E). So
there is a need to develope a scheme that is well dened and simple to implement
in any possible application.
Inequalities: Most of the inequalities developed for proving convergence results
for iterative schemes for zeros of accretive operators are not applicable
in the case of monotone operators as they involved the generalized duality
mappings, where as the denition of monotone operators does not involve the
generalized duality mappings.
1.3 Iterative methods for zeros of monotone-type map-
pings
In trying to overcome these two major diculties, recently, many authors have
successfully employed the notion of suppressive operators introduced by Alber [2]
and Bregman [8] respectively, to approximate zeros of monotone operators (see e.g.,
Aoyama et al. [5], Kamimura et al. [72], Takahashi [71], Zegeye and Shahzad [119]
and the references contained therein). A typical example of the algorithms used by
most of these authors is contained in the following result of Zegeye and Shahzad
[119]. We rst remark that a map A : E ! E is said to be -inverse strongly
monotone if there exists 2 (0; 1) such that for all x; y 2 E the following inequality
holds
hAx Ay; (x y)i kAx Ayk2:
Theorem 1.3.1 (Zegeye and Shahzad [119]) Let E be uniformly smooth and 2-
uniformly convex real Banach space with dual E. Let A : E ! E be a -inverse
strongly monotone mapping and T : E ! E be relatively weak nonexpansive mapping
with A1(0) \ F(T) 6= ;: Assume that 0 < n b0 := c2
2 ; where c is the
General Introduction 8
constants from the Lipschitz property of J1, then the sequence generated by
8>>>>>>>>>><
>>>>>>>>>>:
x0 2 K; choosen arbitrary;
yn = J1(Jxn nAxn);
zn = Tyn;
H0 =
v 2 K : (v; z0) (v; y0) (v; x0)
;
Hn =
v 2 Hn1 \Wn1 : (v; zn) (v; yn) (v; xn)
;
W0 = E;
Wn =
v 2 Wn1 \ Hn1 :
xn v; jx0 jxn
0
;
xn+1 = Hn\Wn(x0); n 0;
(1.3.1)
converges strongly to F(T)\A1(0)x0, where F(T)\A1(0) is the generalised projection
from E onto F(T) \ A1(0):
In the above theorem J is the duality mapping on E and : E E ! R is the
suppressive operator introduced by Alber in [2], which is given by
(x; y) = kxk2 2hx; j(y)i + kyk2:
Remark 1.3.2 We point out the major weaknesses in scheme (1.3.1).
The duality mapping J (resp. J1) is not known precisely in any space more
general than Lp spaces, 1 < p < 1. Therefore, the value of J (resp. J1)
cannot be computed in spaces more general than LP spaces.
At each step of scheme (1.3.1), one has to compute the inverse of the duality
mapping which like the duality mapping itself, is not known in spaces more
general than LP spaces. One has to compute some sets (e.g., Hn and Wn)
which are quite dicult to obtain as they involve generalized projections.
Even though the approximation method used in Thoerem 1.3.1 yields strong convergence
to a solution of the problem under consideration, it is clear that it is not
easy to be used in application.
In chapter three and four of this thesis we shall give one-step iterative algorithm that
does not involve projections for approximating zeros of Lipschitz strongly monotone
operators and bounded generalised -monotone operators, respectively, in Lp
spaces, 1 < p < 1.
1.4 Approximation of xed points of a nite family of
k-strictly pseudo-contractive mappings in CAT(0)
spaces
An important class of nonlinear operators is the class of nonexpansive mappings.
We recall that an operator T : D(T) E ! E is said to be nonexpansive if
kTx Tyk 6 kx yk for all x; y 2 D(T);
General Introduction 9
where D(T) is the domain of T. Nonexpansive operators surface in many important
real world applications such as image reconstruction, signal processing, e.t.c. The
following quotation further shows the importance of iterative methods for approximating
xed points of nonexpansive mappings.
\Many well known algorithms in signal processing and image reconstruction are
iterative in nature. A wide variety of iterative procedures used in signal processing
and image reconstruction and elsewhere are special cases of the KM iteration procedure,
for particular choice of ne operator: : ::” (Charles Byrne, [23]).
Note that KM in the above quotation stands for Krasnoselskii method and ne stands
for nonexpansive.
For x0 2 E, the recursion formula dened by
xn+1 = (1 )xn + Txn; n > 0; (1.4.1)
is called the Krasnoselskii formula, while the formula dened by
xn+1 = (1 n)xn + nTxn; n > 0; (1.4.2)
is called the Mann iteration formula. The Mann iterative method has been successfully
employed in approximating xed points (when they exist) of nonexpansive
mappings. This success does not carry over to the more general class of Lipschitz
pseudo-contractions (see Chidume and Mutangadura [45]). An important superclass
of the class of nonexpansive mappings and a subclass of the class of Lipschitz
pseudo-contractive mappings is the class of k-strictly pseudo-contractive mappings
introduced by Browder and Petryshyn in Hilbert spaces in [18]. They dened the
map in Hilbert and Banach spaces, respectively, as follows.
Let K be a nonempty subset of a real Hilbert space H. A map T : K ! H is
called k-strictly pseudo-contractive if there exists k 2 (0; 1) such that
kTx Tyk2 kx yk2 + kkx y (Tx Ty)k2 8 x; y 2 K: (1.4.3)
It is easy to see that every nonexpansive map is also pseudo-contractive.
Let K be a nonempty subset of a real normed space E. A map T : K ! E
is called k-strictly pseudo-contractive (see, e.g., [29], p.145; [17] ) if there exists
k 2 (0; 1) such that for all x; y 2 K, there exists j(x y) 2 J(x y) such that
hTx Ty; j(x y)i kx yk2 kkx y (Tx Ty)k2: (1.4.4)
It can be trivially shown that in Hilbert spaces (1.4.3) and (1.4.4) are equivalent.
The class of k-strictly pseudo-contractive operators is important for the following
two reasons; rstly, it is an important generalization of nonexpansive maps, and
secondly, it helps to have better understanding of the class of Lipschitz pseudocontractive
mappings.
General Introduction 10
1.5 Fixed point of multivalued maps
Interest in xed point theory for multi-valued nonlinear mappings stems, perhaps,
mainly from their usefulness in real-world applications, such as in Game Theory
and Market Economy and in other areas of mathematics, such as in Non-Smooth
Dierential Equations. We give below some examples that show the connection
between xed point theory and some of the areas of applications in sciences.
1.5.1 Game Theory and Market Economy
In game theory and market economy, the existence of equilibrium was uniformly
obtained by the application of a xed point theorem. In fact, Nash [88, 89] showed
the existence of equilibria for non-cooperative static games as a direct consequence
of Brouwer [13] or Kakutani [70] xed point theorem. More precisely, under some
regularity conditions, given a game, there always exists a multi-valued map whose
xed points coincide with the equilibrium points of the game. A model example of
such an application is the Nash equilibrium theorem (see, e.g., [88]).
Consider a game G = (un;Kn) with N players denoted by n, n = 1; ;N, where
Kn Rmn is the set of possible strategies of the n’th player and is assumed to be
nonempty, compact and convex and un : K := K1 K2 KN ! R is the payo
(or gain function) of the player n and is assume to be continuous. The player n can
take individual actions, represented by a vector n 2 Kn. All players together can
take a collective action, which is a combined vector = (1; 2; ; N). For each
n, 2 K and zn 2 Kn, we will use the following standard notations:
Kn := K1 Kn1 Kn+1 KN;
n := (1; ; n1; n+1; ; N);
(zn; n) := (1; ; n1; zn; n+1; ; N):
A strategy n 2 Kn permits the n’th player to maximize his gain under the condition
that the remaining players have chosen their strategies n if and only if
un(n; n) = max
zn2Kn
un(zn; n):
Now, let Tn : Kn ! 2Kn be the multi-valued map dened by
Tn(n) := Arg max
zn2Kn
un(zn; n) 8 n 2 Kn:
Denition. A collective action = (1; ; N) 2 K is called a Nash equilibrium
point if, for each n, n is the best response of the n’th player to the action n
made by the remaining players. That is, for each n,
un() = max
zn2Kn
un(zn; n) (1.5.1)
General Introduction 11
or equivalently,
n 2 Tn(n): (1.5.2)
This is equivalent to is a xed point of the multi-valued map T : K ! 2K dened
by
T() := [T1(1); T2(2); ; TN(N)]:
From the point of view of social recognition, game theory is perhaps the most
successful area of application of xed point theory of multi-valued mappings. However,
it has been remarked that the applications of this theory to equilibrium are
mostly static: they enhance understanding conditions under which equilibrium may
be achieved but do not indicate how to construct a process starting from a nonequilibrium
point and convergent to equilibrium solution. This is part of the problem
that is being addressed by iterative methods for xed point of multi-valued
mappings.
1.5.2 Non-smooth Dierential Equations
The mainstream of applications of xed point theory for multi-valued maps has
been initially motivated by the problem of dierential equations (DEs) with discontinuous
right-hand sides which gave birth to the existence theory of dierential
inclusion (DIs). Here is a simple model for this type of application.
Consider the initial value problem
du
dt
= f(t; u); a:e: t 2 I := [a; a]; u(0) = u0: (1.5.3)
If f : IR ! R is discontinuous with bounded jumps, measurable in t, one looks for
solutions in the sense of Filippov [63] which are solutions of the dierential inclusion
du
dt
2 F(t; u); a:e: t 2 I; u(0) = u0; (1.5.4)
where
F(t; x) = [lim inf
y!x
f(t; y); lim sup
y!x
f(t; y)]: (1.5.5)
Now, set H := L2(I) and let NF : H ! 2H be the multi-valued Nemystkii operator
dened by
NF (u) := fv 2 H : v(t) 2 F(t; u(t)) a:e: on Ig:
Finally, let T : H ! 2H be the multi-valued map dened by T := NF L1, where
L1 is the inverse of the derivative operator Lu = u0 given by
L1v(t) := u0 +
Z t
0
v(s)ds:
One can see that problem (1.5.4) reduces to the xed point problem: u 2 Tu.
General Introduction 12
Finally, a variety of xed point theorems for multi-valued maps, with non empty and
convex values is available to conclude the existence of solution. We used a rst order
dierential equation as a model for simplicity of presentation but this approach is
most commonly used with respect to second order boundary value problems for ordinary
dierential equations or partial dierential equations. For more about these
topics, one can consult [25, 55, 61, 64] and references therein as examples.
1.6 Iterative methods for xed points of some nonlinear
multi-valued mappings
Let E be a real normed linear space and K be a nonempty subset of E. The set K
is called proximinal (see e.g., [90, 101, 106]) if for each x 2 E, there exists u 2 K
such that
d(x; u) = inffkx yk : y 2 Kg = d(x;K);
where d(x; y) = kxyk for all x; y 2 E. Let CB(K) and P(K) denote the families of
nonempty closed bounded subsets of K and nonempty proximinal bounded subsets
of K, respectively. The Hausdor metric on CB(K) is dened by:
H(A;B) = max
n
sup
a2A
d(a;B); sup
b2B
d(b;A)
o
for all A;B 2 CB(K). Let T : D(T) E ! CB(E) be a multi-valued mapping on
E. A point x 2 D(T) is called a xed point of T if and only if x 2 Tx. The xed
point set of T is denoted by F(T) := fx 2 D(T) : x 2 Txg.
A multi-valued mapping T : D(T) E ! CB(E) is called L- Lipschitzian if there
exists L > 0 such that
H(Tx; Ty) Lkx yk 8 x; y 2 D(T): (1.6.1)
When L 2 (0; 1) in (1.6.1), we say that T is a contraction, and T is called nonexpansive
if L = 1.
Several results on the approximation of xed points of multi-valued nonexpansive
mappings in real Hilbert spaces have appeared in the literature (see e.g., Abbas et
al. [1], Khan et al. [74], Panyanak [90], Sastry and Babu [101], Song and wong [106]
and the references contained therein). For their generalizations (see e.g., Chidume
et al. [39], Chidume and Ezeora [41] and the references contained therein). In
[101], Sastry and Babu proved the following result for multi-valued nonexpansive
mappings:
Theorem 1.6.1 (Sastri and Babu [101]) Let H be real Hilbert space, K be a nonempty,
compact and convex subset of H, and T : K ! CB(K) be a multi-valued nonexpansive
map with a xed point p. Assume that (i) 0 n; n < 1; (ii) n ! 0 and
General Introduction 13
(iii)
P
nn = 1: where n and n are sequences of real numbers. Let x 2 F(T),
then the sequence dened by
8<
:
yn = (1 n)xn + nzn; zn 2 Txn; kzn xk = (x; Txn);
xn+1 = (1 n)xn + nun; un 2 Tyn; kun xk = d(yn; x);
(1.6.2)
converges strongly to a xed point of T.
In [90], Panyanak extended the result of Sastry and Babu to a uniformly convex
real Banach spaces. He proved the following result.
Theorem 1.6.2 (Panyanak, [90]) Let E be a uniformly convex real Banach space,
K be a nonempty, closed, bounded and convex subset of E, and T : D(T) E !
CB(K) a multi-valued nonexpansive map with a xed point p. Assume that (i) 0
n; n < 1; (ii) n ! 0 and (iii)
P
nn = 1: where n and n are sequences
of real numbers. Then, the sequence dened by (1.6.2) converges strongly to a xed
point of T.
Remark 1.6.3 In the recursion formular (1.6.2) the authors imposed condition
that, zn 2 Txn such that kzn xk = (x; Txn). The existence of such zn in each
step of the iteration process is guaranteed when Txn is proximinal. In general to
pick zn is very dicult and hence this makes the iterative process to be inconvenient
in any possible application.
Chidume et al., [39], introduced multi-valued k-strictly pseudo-contractive mappings.
They gave the following denition.
Denition 1.6.4 A multi-valued map T : D(T) H ! CB(H) is called k-strictly
pseudo-contractive if there exists k 2 (0; 1) such that for all x; y 2 D(T),
H(Tx; Ty)
2 kx yk2 + kkx y (u v)k2 8u 2 Tx; v 2 Ty:
They constructed a Krasnoselskii-type algorithm and showed that it converges
strongly to a xed point of T under some additional mild condition. More precisely,
they proved the following result.
Theorem 1.6.5 (Chidume et al. [39]) Let K be a nonempty, closed and convex
subset of a real Hilbert space H. Suppose that T : K ! CB(K) is a multi-valued
k-strictly pseudo-contractive mapping such that F(T) 6= ;. Assume that Tp = fpg
for all p 2 F(T). Suppose that T is semi-compact and continuous. Let fxng be a
sequence dened iteratively from x0 2 K by
xn+1 = (1 )xn + yn; n 0; (1.6.3)
where yn 2 Txn and 2 (0; 1 k). Then, the sequence fxng converges strongly to
a xed point of T .
General Introduction 14
Remark 1.6.6 This result of Chidume et al. is an important improvement of
several results in the literature. It deals with the class of multi-valued k-strictly
pseudo-contractive mappings which is an important generalization of the class of
multi-valued nonexpansive mappings. Moreover, the condition zn 2 Txn such that
kzn xk = (x; Txn) imposed by Sastry and Babu in the recusion formular (1.6.2)
is dispensed with in the theorem of Chidume et al. [39].
Later on, Chidume et al. [40] extended their result to q-uniformly smooth real
Banach space. The following is their main result.
Theorem 1.6.7 (Chidume et al. [40]) Let q > 1 be a real number and K be a
nonempty, closed and convex subset of a q-uniformly smooth real Banach space E.
Let T : K ! CB(K) be a multi-valued k-strictly pseudo-contractive mapping with
F(T) 6= ; and such that Tp = fpg for all p 2 F(T). Suppose that T is continuous
and semi-compact. Let fxng be a sequence dened iteratively from x1 2 K by
xn+1 = (1 )xn + yn; (1.6.4)
where yn 2 Txn and 2 (0; ). Then, the sequence fxng converges strongly to a
xed point of T.
This leads us to the following important question.
Question: Can an iterative algorithm be obtained to approximate xed points of
multi-valued k-strictly pseudo-contractive mappings in a more general metric space?
That is, can we obtain the analogue of the results of [39] in important space that do
necessarily have a norm?
In chapter ve of this thesis, we answer the above question in the armative by
constructing a Krasnoselskii-type algorithm that converges strongly to a xed point
of T in a complete CAT(k) space, k 0; which has been studied by various worldclass
mathematicians (see e.g., Bridson and Hae iger [12], Bruhat [20], Burago et
al. [21], Kirk [75, 76, 77]).
In chapter six of this thesis, we also prove a convergence result for a countable
family of k-strictly pseudo-contractive mappings in Hilbert spaces.
1.7 Hammerstein Integral Equations
Let
Rn be bounded. Let k :
! R and f :
R ! R be measurable realvalued
functions. An integral equation (generally nonlinear) of Hammerstein-type
has the form
u(x) +
Z
k(x; y)f(y; u(y))dy = w(x); (1.7.1)
where the unknown function u and inhomogeneous function w lie in a Banach space
E of measurable real-valued functions. If we dene F : F(
;R) ! F(
;R) and
General Introduction 15
K : F(
;R) ! F(
;R) by
Fu(y) = f(y; u(y)); y 2
;
and
Kv(x) =
Z
k(x; y)v(y)dy; x 2
;
respectively, where F(
;R) is a space of measurable real-valued functions dened
from
to R, then equation (1.7.1) can be put in an abstract form
u + KFu = w: (1.7.2)
Without loss of generality we can assume that w 0 so that (1.7.2) becomes
u + KFu = 0: (1.7.3)
Indeed, if w 6= 0, then u w + KFu = 0. setting h = u w we obtain that
h + KFh = 0;
where F(h) = F(h + w).
Interest in (1.7.1) stems mainly from the fact that several problems that arise in
dierential equations, for instance, elliptic boundary value problems whose linear
part posses Green’s function can, as a rule, be transformed into the form (1.7.1)
(see e.g., Pascali and Sburian [91], chapter 4, p. 164). Among these, we mention
the problem of the forced ocsillation of nite amplitude of a pendulum.
Example. We consider the problem of the pendulum
8<
:
d2v(t)
dt2 + a2 sin v(t) = z(t); t 2 [0; 1];
v(0) = v(1) = 0:
(1.7.4)
where the driving force z is odd. The constant a (a 6= 0) depends on the length of
the pendulum and gravity. Since the Green’s function of the problem
v00(t) = 0; v(0) = v(1) = 0
is the function dened by
k(t; s) =
(
t(1 s); 0 t s 1;
s(1 t); 0 s t 1;
(1.7.5)
it follows that problem (1.7.4) is equivalent to the nonlinear integral equation
v(t) =
Z 1
0
k(t; s)[z(s) a2 sin v(s)]ds; t 2 [0; 1]: (1.7.6)
General Introduction 16
Setting g(t) =
R 1
0 k(t; s)z(s)ds and u(t) = v(t) g(t), then we have
u(t) +
Z 1
0
k(t; s)a2 sin(u(s) + g(s))ds = 0
which is in Hammerstein equation form
u(t) +
Z 1
0
k(t; s)f(s; u(s))ds = 0;
where f(s; t) = a2 sin(t + g(s)).
Equations of Hammerstein-type play a crucial role in the theory of optimal control
system and in automation and network theory (see e.g., Dolezale [60]). Several existence
results have been proved for equations of Hammerstein-type (see e.g., Brezis
and Browder [9, 10, 11], Browder [15], Browder, De Figueiredo and Gupta [16]).
1.8 Approximating solutions of equations of Hammerstein-
type
In general, equations of Hammerstein-type are nonlinear and there is no known
method to nd a close form solutions for them. Consequently, methods of approximating
solutions of such equations are of interest.
Let H be a real Hilbert space. A nonlinear operator A : H ! H is said to be
angle-bounded with angle > 0 if and only if
hAx Ay; z yi hAx Ay; x yi (1.8.1)
for any triple elements x; y; z 2 H. For y = z inequality (1.8.1) implies the monotonicity
of A.
A monotone linear operator A : H ! H is said to be angle bounded with angle
> 0 if and only if
jhAx; yi hAy; xij 2hAx; xi
1
2 hAy; yi
1
2 (1.8.2)
for all x; y 2 H. In the special case where the operator is angle bounded Brezis
and Browder [9, 11] proved the strong convergence of a suitably dened Galerkin
approximation to a solution of (1.7.2). In fact, they prove the following theorem.
Theorem 1.8.1 (Brezis and Browder [11]) Let H be a separable Hilbert space
and C be a closed subspace of H. Let K : H ! C be a bounded continuous monotone
operator and F : C ! H be angle-bounded and weakly compact mapping. For a given
f 2 C, consider the Hammerstein equation
(I + KF)u = f (1.8.3)
General Introduction 17
and its nth Galerkin approximation given by
(I + KnFn)un = Pf; (1.8.4)
where Kn = P
nKPn : H ! C and Fn = PnFP
n : Cn ! H.
Then, for each n 2 N, the Galerkin approximation (1.8.4) admits a unique
solution un in Cn and fung converges strongly in H to the unique solution u 2 C
of the equation (1.8.3).
In the theorem above all the symbols used have their usual meanings (see e.g., [91]).
It is obvious that if an iterative algorithm can be developed for the approximation
of solutions of equation of Hammerstein-type (1.7.3), this will certainly be
preferred.
Attempts have been made to approximate solutions of equations of Hammersteintype
using Mann-type iteration scheme. However, the results obtained were not
satisfactory (see e.g., [49]). The recurrance formulas used in early attempts involved
K1 which is also required to be strongly monotone, and this, apart from
limiting the class of mappings to which such iterative schemes are applicable, it is
also not convenient in applications. Part of the diculty is the fact that the composition
of two monotone operators need not to be monotone. It suces to take
K : R2 ! R2; F : R2 ! R2; where
K =
1 2
2 1
and F =
0 1
1 2
:
The rst satisfactory results on iterative methods for approximating solutions of
Hammerstein equations, as far as we know, were obtained by Chidume and Zegeye
[51, 52, 53]. Under the setting of a real Hilbert space H, for F;K : H ! H, they
dened an auxillary map on the Cartesian product E := H H; T : E ! E by
T[u; v] = [Fu v;Kv + u]:
We note that
T[u; v] = 0 () u solves (1:7:3) and v = Fu:
With this, they were able to obtain strong convergence of an iterative scheme dened
in the Cartesian product space E to a solution of Hammerstein equation (1.7.3).
Extensions to a real Banach space setting were also obtained.
Let X be a real Banach space and F;K : X ! X be accretive-type mappings.
Let E := X X. The same authors (see [51, 52]) dened T : E ! E by
T[u; v] = [Fu v;Kv + u]
and obtained strong convergence theorems for solutions of Hammerstein equations
under various continuity conditions in the Cartesian product space E.
The method of proof used by Chidume and Zegeye provided the clue to the establishement
of the following couple explicit algorithm for computing a solution of the
General Introduction 18
equation u + KFu = 0 in the original space X. With initial vectors u0; v0 2 X,
sequences fung and fvng in X were dened iteratively as follows:
un+1 = un n(Fun vn); n 0; (1.8.5)
vn+1 = vn n(Kvn + un); n 0; (1.8.6)
where fng is a sequence in (0; 1) satisfying appropriate conditions. The recursion
formulas (1.8.5) and (1.8.6) had been used successfully to approximate solutions
of Hammerstein equations involving nonlinear accretive-type mappings. Following
this, Chidume and Djitte [43, 44] studied this explicit couple iterative algorithm
and proved several strong convergence theorems.
We remark here that even though monotone-type operators have more applications
than accretive-type operators in Banach spaces, virtually all the results on the approximation
of solutions of Hammerstein equations are either proved in Hilbert
spaces or in a Banach space in the case where the operators K and F are accretivetype
mappings (see [42], [46], [48] and [50]). To the best of our knowledge, there is
no single result on the approximation of solutions of Hammerstein-type equations
in Banach spaces (in the case where the operators K and F are monotone-type
operators) that has appeared in the literature. Perhaps, part of the problem is that
since the operator F maps E to E and K maps E to E the recursion formulas
used for accretive-type mappings may no longer make sense.
In chapter seven, we proved convergence results for solutions of equations of
Hammerstein-type in Lp spaces, 1 < p < 1, in the case where the operators K and
F are of monotone-type using Mann-type
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