**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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