## ABSTRACT

Let X be a uniformly convex and uniformly smooth real Banach space with dual

space X. Let F : X ! X and K : X ! X be bounded monotone mappings

such that the Hammerstein equation u + KFu = 0 has a solution in X. An explicit

iteration sequence is constructed and proved to converge strongly to a solution of the

equation. This is achieved by combining geometric properties of uniformly convex

and uniformly smooth real Banach spaces recently introduced by Alber with our

method of proof which is also of independent interest.

** **

## TABLE OF CONTENTS

ertication ii

1 Introduction and literature review 2

1.0.1 Hammerstein equations . . . . . . . . . . . . . . . . . . . . . . 3

1.0.2 Approximation of solutions of Hammerstein integral equations 10

2 PRELIMINARIES 13

2.1 Denition of some terms and concepts. . . . . . . . . . . . . . . . . . 13

2.2 Results of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Some interesting properties of Normalized Duality map . . . . . . . . 19

3 A Strong convergence theorem 21

3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1

## CHAPTER ONE

Introduction and literature review

The contents of this thesis fall within the general area of nonlinear operator theory,

a ourishing area of research for numerous mathematicians. In this thesis, we con-

centrate on an important topic in this area-approximation of solutions of nonlinear

integral equations of Hammaerstein type involving monotone-type mappings.

Let H be a real inner product space. A map A : H ! 2H is called monotone if

for each x; y 2 H,

; x y

0; 8 2 Ax; 2 Ay: (1.1)

If A is single-valued, the map A : H ! H is monotone if

Ax Ay; x y

0 8 x; y 2 H: (1.2)

Monotone mappings were rst studied in Hilbert spaces by Zarantonello [50], Minty

[42], Kacurovskii [37] and a host of other authors. Interest in such mappings stems

mainly from their usefulness in numerous applications. Consider, for example, the

following:

Example 1. Let g : H ! R [ f1g be a proper convex function. The subdierential

of g at x 2 H, @g : H ! 2H , is dened by

@g(x) =

x 2 H : g(y) g(x)

y x; x

8 y 2 H

:

It is easy to check that @g is a monotone operator on H, and that 0 2 @g(u) if and

only if u is a minimizer of g. Setting @g A, it follows that solving the inclusion

0 2 Au, in this case, is solving for a minimizer of g.

Example 2. Again, let A : H ! H be a monotone map. Consider the evolution

equation

du

dt

+ Au = 0: (1.3)

At equilibrium state, du

dt = 0 so that

Au = 0: (1.4)

Consequently, solving the equation Au = 0, in this case, corresponds to solving for

the equilibrium state of the system described by (1.3).

Monotone maps also appear in Hammerstein equations. Since this thesis focuses

on this class of equations, we give a brief review.

2

3 Chapter 1. Introduction and literature review

1.0.1 Hammerstein equations

Let

Rn be bounded. Let k :

! R and f :

R ! R be measurable real-

valued 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); (1.5)

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

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 (1.5) can be put in the abstract form

u + KFu = 0: (1.6)

where, without loss of generality, we have assumed that w 0. The operators K

and F are generally of the monotone-type. A closer look at equation (1.6) reveals

that it is a special case of (1.4), where

A := I + KF:

Interest in (1.6) stems mainly from the fact that several problems that arise in

dierential equations, for instance, elliptic boundary value problems whose linear

parts possess Green’s function can, as a rule, be transformed into the form (1.5) (see

e.g., Pascali and Sburlan [43], chapter IV, p. 164). Among these, we mention the

problem of the forced ocsillation of nite amplitude of a pendulum.

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)

where the driving force z is odd. The constant a (a 6= 0) depends on the length of

the pendulum and gravity. Since Green’s function for the problem

v00(t) = 0; v(1) = v(0) = 0;

is the triangular function

K(t; x) =

8<

:

t(1 x); if 0 t x

x(1 t); if x t 1;

(1.8)

4

it follows that problem (1.7) is equivalent to the nonlinear integral equation

v(t) =

Z 1

0

K(t; x)[z(x) a2 sin v(x)]dx: (1.9)

If g(t) =

R 1

0 K(t; x)z(x)dx and v(t) + g(t) = u(t); then (1.9) can be written as the

Hammerstein equation

u(t) =

Z 1

0

K(t; x)f(x; u(x))dx = 0; (1.10)

where

f(x; u(x)) = a2sin[u(x) g(x)];

(see e.g., [43])

Equations of Hammerstein-type also play a crucial role in the theory of optimal

control systems and in automation and network theory (see e.g., Dolezale [33]).

Several existence results have been proved for equations of Hammerstein-type (see

e.g., Brezis and Browder [4, 5, 6], Browder [7], Browder, De Figueiredo and Gupta

[8]).

The concept of monotone maps has been extended to arbitrary real normed spaces.

There are two well-studied extensions of Hilbert-space monotonicity to arbitrary

normed spaces. We brie y explore the two.

The rst is the class of accretive operators.

A map A : E ! 2E is called accretive if for each x; y 2 E, there exists j(xy) 2

J(x y) such that

; j(x y)

0; 8 2 Ax; 2 Ay; (1.11)

Where J is the normalized duality map on E: If A is single-valued, the map A : E !

E is accretive if for each x; y 2 E, there exists j(x y) 2 J(x y) such that

Ax Ay; j(x y)

0: (1.12)

In a Hilbert space, the normalized duality map is the identity map, so that (1.11)

and (1.12) reduce to (1.1) and (1.2) respectively, where E = H and so, accretivity is

one extension of Hilbert space monotonicity to general normed spaces.

A result of Kato [39] shows that (1.12) holds if and only if for all x; y 2 D(A);

the following inequality holds

jjx yjj jjx y + s(Ax Ay)jj 8s > 0: (1.13)

The map A is called generalized -strongly accretive if there exists a stritly

increasing function : [0;1) ! [0;1) with (0) = 0 such that for each x; y 2 E,

there exists j(x y) 2 J(x y) such that

hAx Ay; j(x y)i (kx yk): (1.14)

5 Chapter 1. Introduction and literature review

It is called -strongly accretive if there exists a stritly increasing function : [0;1) !

[0;1) with (0) = 0 such that for each x; y 2 E, there exists j(x y) 2 J(x y)

such that

hAx Ay; j(x y)i jjx yjj(kx yk): (1.15)

Finally, A is called strongly accretive if there exists k 2 (0; 1) such that for each

x; y 2 E, there exists j(x y) 2 J(x y) such that

hAx Ay; j(x y)i kkx yk2: (1.16)

Clearly, the class of strongly accretive mappings is a sub-class of the class of -strongly

accretive maps (one takes (t) = kt); and the class of -strongly accretive maps is

a sub-class of that of generalized -strongly accretive (one takes (t) = t(t)). It is

well known that the inclusions are proper. For the equation Au = 0, when A is of

accretive-type, existence theorems have been proved by various authors, in various

Banach spaces and under various continuity conditions (see Browder [9], [10], [11],

[12] ). It is well known that the class of generalized -strongly accretive maps is the

largest, among the classes of accretive-type mappings, for which, if a solution exists,

it is necessarily unique.

For approximating a solution of equation (1.4), where A : E ! E is of accretive-type,

Browder [13] dened an operator T := I A, where I is the identity map on E. He

called such an operator pseudo-contractive. It is trivial to observe that zeros of A

correspond to xed points of T.

Consequently, solving the equation Au = 0 when A is an accretive-type operator

is reduced to nding xed points of pseudo-contractive-type mappings.

An important class of pseudo-contractive mappings is the class of nonexpansive

maps, where a map T : E ! E is called nonexpansive if for each x; y 2 E, the

following inequality holds; jjTx Tyjj jjx yjj.

Being an obvious generalization of contraction mappings(mappings T : E ! E satis-

fying jjTxTyjj jjxyjj8x; y 2 E and some k 2 (0; 1)), is not all that makes them

important. They are also important, as has been observed by Bruck [16], mainly for

the following two reasons:

Nonexpansive maps are intimately connected with the monotonicity methods

developed since the early 1960’s and constitute one of the rst classes of non-

linear mappings for which xed point theorems were obtained by using the ne

geometric properties of the underlying Banach spaces instead of compactness

properties.

They appear in applications as the transition operators for initial value prob-

lems of dierential inclusions of the form

0 2

du

dt

+ T(t)u;

where the operators fT(t)g are in general set-valued and are accretive or dissi-

pative and minimally continuous.

6

The following xed point theorem has been proved for nonexpansive maps on

uniformly convex spaces.

Theorem 1.0.1. Let E be a re exive Banach space and let K be a nonempty closed

bounded and convex subset of E with normal structure. Let T : K ! K be a nonex-

pansive mapping of K into itself. Then, T has a xed point.

While contractions guarantee existence and uniquness, nonexpansions do not. Trivial

examples show that the sequence of successive approximations

xn+1 = Txn; x0 2 K; n 0 (1.17)

(where K is a nonempty closed convex and bounded subset of a real Banach space

E), for a nonexpansive mapping T : K ! K even with a unique xed point, may

not converge to the xed point. It is enough, for example, to take for T; a rotation

of the unit ball in the plane around the origin of coordinates. Specically, we have

the following example.

Example 3. Let B := fx 2 R2 : jjxjj 1g and let T denote an anticlockwise rotation

of

4 about the origin of coordinates. Then T is nonexpansive with the origin as the

only xed point. Moreover, the sequence dened by (1.17) where B = K does not

converge to zero.

Krasnoselskii [40], however, showed that in this example, a convergent sequence of

succesive approximations can be obtained if instead of T; the auxilliary nonexpansive

mapping 1

2 (I +T); is used, where I denotes the identity transformation of the plane,

i.e., if the sequence of succesive approximations is denedd by

xn+1 =

1

2

xn + Txn

n = 0; 1; :: (1.18)

instead of by the usual so-called Picard iterates, xn+1 = Txn x0 2 K n 0: It is easy

to see that the mappings T and 1

2 (I + T) have the same set of xed points, so that

the limit of the convergent sequence dened by (1.18) is necessarily a xed point of T.

Generally, if X is a normed linear space and K a convex subset of X; a general-

ization of equation (1.18) which has proved successful in the approximation of xed

points of nonexpansive mappings T : K ! K (when they exist), is the following

scheme: x0 2 K;

xn+1 = (1 )xn + Txn; n = 0; 1; 2; ::: 2 (0; 1); (1.19)

constant (see, e.g., Schaefer [47]). However, the most general Mann-type iterative

scheme now studied is the following: x0 2 K

xn+1 = (1 cn)xn + cnTxn; n = 0; 1; 2; ::: (1.20)

where fCng1 n=1 (0; 1) is a real sequence satisfying appropriate conditions

(see, e.g., Chidume [21], Edelstein and O’Brian [34], Ishikawa [35]). Under the fol-

lowing additional assumptions

7 Chapter 1. Introduction and literature review

(i) limCn = 0; and

(ii)

P1

n=1 Cn = 1;

the sequence fxng generated by (1.20) is generally referred to as the Mann sequence

[41]. The recurssion formula (1.19) is known as the Krasnoselskii-Mann (KM) for-

mula for nding xed points of nonexpansive mappings (when they exist).

Let K be a nonempty convex subset of a normed space E and T : K ! K be a

nonexpansive map. Let the sequence fxng1 n=0 in K be dened iteratively by x0 2 K,

xn+1 = (1 cn)xn + cnTxn; n 1; (1.21)

where fcng is a sequence in (0; 1) satisfying the following conditions:

(i)

X1

n=0

cn = 1;

(ii) lim

n!1

cn = 0: Ishikawa [35] proved that If the sequence fxng1n

=0 is bounded, then

it is an approximate xed point sequence in the sense that

lim

n!1

jjxn Txnjj = 0: (1.22)

Edelstein and O’Brian [34] considered the recursion formula

xn+1 = (1 )xn + Txn; x0 2 K; n 2 N; 2 (0; 1); (1.23)

where T maps K into K and proved that if K is bounded, then the convergence in

(1.22) is uniform.

Chidume [21] considered the recursion formular (1.21), introduced the concept of an

admissible sequence and proved that if K is bounded, then the convergence in (1.22)

is uniform.

Remark 1. We therefore note that the best mode of convergence we can get from

recursion formula (1.21) is weak convergence to a xed point of T(see e.g, Reich

[46]). It is always desirable to establish that the sequence is an approximate xed

point sequence i.e., that the sequence dened by (1.21) satises (1.22). In general,

the iteration problem does not yield strong convergence of the sequence to a xed

point of T. To obtain convergence to a xed point of T, some type of compactness

condition must be imposed either on K or on the map T (e.g, T may be required to

be demicompact, or (I T) may be required to map closed bounded subsets of E into

closed subsets of E, etc, see e.g, Chidume [20]).

For the more general class of Lipschitz pseudo-contractive maps, attempts to use

the Mann formula, which has been successfully employed for nonexpansive mappings,

to approximate a xed point of a Lipschitz pseudo-contractive map even on a compact

convex domain in a real Hilbert space proved abortive. In 1974, Ishikawa [36] proved

the following theorem.

Theorem 1.0.2. Let K be a nonempty compact convex subset of a real Hilbert space

H and T : K ! K be a Lipschitz pseudo-contractive map. Let the sequence fxng1n

=0

be dened by x0 2 K;

xn+1 = (1 n)xn + nTyn; (1.24)

8

yn = (1 n)xn + nTxn; n 1; (1.25)

where fng and fng are real sequences satisfying the following conditions: (i) 0

n n < 1 8n 1; (ii)

P

nn = 1; (iii) lim

n!1

n = 0. Then, fxng1n

=0

converges strongly to a xed point of T.

Even though the recursion formulas (1.18) and (1.19) of the Ishikawa scheme have

been successfully used in approximating xed points of Lipschitz pseudo-contractive

mappings in real Hilbert spaces, when the domain of T is compact and convex the

following question remained open.

OPEN QUESTION: Has the simpler and more ecient Mann sequence failed

to converge strongly to some xed point of a Lipschitz pseudo-contractive map in a

Hilbert space, even when the domain of T is compact and convex?

This question remained open for many years until 2001, when Chidume and Mu-

tangadura constructed a counter example to show a Lipschitz pseudo-contractive

map dened on a compact convex subset of R2 with a unique xed point for which

no Mann sequence converges (see Chidume and Mutangadura, [19]).

Browder and Petryshyn [15], (1967) introduced a class of Lipschitz pseudo-contractive

maps which contains the class nonexpansive maps called the class of strictly pseudo-

contractive maps. Let H be a real Hilbert space. A map T : H ! H is called strictly

pseudo-contractive if for each x; y 2 H, jjTxTyjj2 jjxyjj2+kjjxy(TxTy)jj2,

for some k 2 (0; 1). For a real normed space E, T : E ! E is called strictly pseudo-

contractive if each x; y 2 H, hTx Ty; x yi jjx yjj2 jjx y (Tx Ty)jj2:

In 2004, a striking result was proved by Chidume et.al. That for this class of

strictly pseudo-contractive maps, the sequence given by (1.21) is an approximate

xed point sequence for strictly pseudo-contractive maps. Furthermore, under an

additional condition that T is demicompact, they proved that the sequence dened

by (1.21) converges strongly to some xed point T. Infact, they proved the following

theorem.

Theorem 1.0.3. Let E be a real Banach space. Let K be a nonempty closed and

convex subset of E. Let T : K ! K be a strictly pseudo-contractive map in the sense

of Browder and Petryshyn with F(T) := fx 2 K : Tx = xg 6= ;: For a xed x0 2 K,

dene a sequence fxng by

xn+1 = (1 n)xn + nTxn; n 0;

where fng is a real sequence satisfying the following conditions: (i)

1P

n=1

n = 1 and

(ii)

1P

n=1

2n

< 1. If T is demicompact, then fxng converges strongly to some xed

point of T in K.

9 Chapter 1. Introduction and literature review

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 reconstruc-

tion 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 choices of the opera-

tor. . . .”(Charles Byrne [18]).

So far, we have seen the successes in approximating the solutions of (1.4) when the

operator A is accretive. This is perhaps a result of numerous geometric properties

of Banach spaces which play a crucial role.For example, Hilbert spaces have the

nicest geometric properties. The availability of the inner product, the fact that the

proximity map of a real Hilber space H onto a closed convex subset K of H is

Lipschitzian with constant 1, and the following two identities:

jjx + yjj2 = jjxjj2 + 2hx; yi + jjyjj2; (1.26)

jjx + (1 )yjj2 = jjxjj2 + (1 )jjyjj2 (1 )jjx yjj2; (1.27)

which hold for all x; y 2 H; are some of the geometric properties that characterize

Hilbert spaces and also make certain problems posed in ahilbert spaces more man-

agable than those in general Banach spaces. However, as has been rightly observed

by M. Hazewinkel,

\. . .many, and probably most mathematical objects and models do not

naturally live in Hilbert spaces\.

Consequently, to extend some of the Hilbert space techniques to more general Banach

spaces, analogues of the identities (1.26) and (1.27) have to be developed. For this

development, the duality map which has become a most important tool in nonlineaar

functional analysis plays a central role. In 1976, Bynum [17] obtained the following

analogue of (1.26) for lp spaces, 1 < p < 1:

jjx + yjj2 (p 1)jjxjj2 + jjyjj2 + 2hx; j(y)i; 2 p < 1 (1.28)

(p 1)jjx + yjj2 jjxjj2 + jjyjj2 + 2hx; j(y)i; 1 < p < 2 (1.29)

Analogues of (1.27) were also obtained by Bynum. In 1979, Reich [45] obtained

an analogue of (1.26) in uniformly smooth Banach spaces. Other analogues of (1.26)

and (1.27) obtained in 1991 and later can be found, for example, in Xu [48] and Xu

and Roach [49].

For the past 30 years or so, the study of Krasnoselskii-Mann iterative procedures for

the approximation of xed points of nonexpansive maps and xed points of some

of their generalizations, and approximation of zeros of accretive-type operators have

been ourishing areas of research for many mathematicians. Numerous applications

of analogues (1.26) and (1.27) to nonlinear iterations involving various classes of

nonlinear operators have since then been topics of intensive research. Today, sub-

stantial denitive results have been proved, some of the methods have reached their

10

boundaries while others are still subjects of intensive research activity. However, it

is apparent that the theory has now reached a level of maturity appropriate for an

examination of its central themes.

Unfortunately, attempts to use these properties in approximating the solutions

of (1.4) when A is of the monotone type have proved abortive and this is perhaps

because of lack of the geometric properties in the spaces suitable for approaching

such problems. Another reason is that only maps such as A : E ! E are related to

xed points, also, the scheme is not well dened for A : E ! E: Fortunately, Alber

[3] recently introduced the Lyapunov functional which has helped to develope new

geometric properties suitable for monotone mappings. In this thesis, we use these

properties to approximate solutions of Hammaerstein equations when the operators

are maximal monotone. In the next section, we shall see how to approximate the

solutions of Hammerstein equations (assuming existence).

1.0.2 Approximation of solutions of Hammerstein integral

equations

In general, equations of Hammerstein-type are nonlinear and there is no known

method to nd close form solutions for them. Consequently, methods of approximat-

ing solutions of such equations, where solutions are known to exist, 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

hAx Ay; z yi hAx Ay; x yi (1.30)

for any triple elements x; y; z 2 H. For y = z inequality (1.30) implies the mono-

tonicity of A. A monotone linear operator A : H ! H is said to be angle bounded

with angle > 0 if

jhAx; yi hAy; xij 2hAx; xi

1

2 hAy; yi

1

2 (1.31)

for all x; y 2 H.

In the special case where one of the operators is angle-bounded, and the other is

bounded, Brezis and Browder [4, 6] proved the strong convergence of a suitably dened

Galerkin approximation to a solution of equation (1.6). In fact, they proved the

following theorem.

Theorem 1.0.4 (Brezis and Browder [6]). 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 (1.32)

and its nth Galerkin approximation given by

(I + KnFn)un = Pf; (1.33)

11 Chapter 1. Introduction and literature review

where Kn = P

nKPn : H ! C and Fn = PnFP

n : Cn ! H, the symbols have their

usual meanings (see [6]). Then, for each n 2 N, the Galerkin approximation (1.33)

admits a unique solution un in Cn and fung converges strongly in H to the unique

solution u 2 C of the equation (1.32).

It is obvious that if an iterative algorithm can be developed for the approximation

of solutions of equation of Hammerstein-type (1.6), this will certainly be a welcome

complement to the Galerkin approximation method. Attempts had been made to

approximate solutions of equations of Hammerstein-type using Mann-type iteration

scheme (see e.g., Mann [41]). However, the results obtained were not satisfactory (see

[31]). The recurrence formulas used in these attempts, even in real Hilbert spaces,

involved K1 which is required to be strongly monotone when K is, and this, apart

from limiting the class of mappings to which such iterative schemes are applicable,

is also not convenient in any possible applications.

Part of the diculties in establishing iterative algorithms for approximating solu-

tions of Hammerstein equations is that the composition of two monotone maps need

not be monotone. If A is linear, (1.2) reduces to

hAx; xi 0;

for all x 2 R2: Now let E = R2, take

F =

1 1

1 1

; K =

1 2

2 1

; and x =

2

1

: Clearly, KF =

1 3

3 1

:

Now,

hFx; xi = 5; hKx; xi = 5 but hKFx; xi = 3: This shows that although F and K

are monotone, KF is not.

The rst satisfactory results on iterative methods for approximating solutions of

Hammerstein equations involving accretive-type mappings, as far as we know, were

obtained by Chidume and Zegeye [28, 29, 30].

Let X be a real Banach space and F;K : X ! X be accretive-type mappings.

Let E := X X. Then, Chidume and Zegeye (see [29, 30]) dened A : E ! E by

A[u; v] = [Fu v;Kv + u] for [u; v] 2 E:

We note that A[u; v] = 0 if and only if u solves (1:6) and v = Fu: The authors dened

an iterative sequence and obtained strong convergence theorems in the Cartesian

product space E, for solutions of Hammerstein equations under various continuity

conditions on F and K, for special classes of real Banach spaces, X. It turns out that,

in the case of a real Hilbert space, H, the operator A dened on H H is monotone

whenever F and K are. The method of proof used by Chidume and Zegeye provided

the authors a clue for the establishment of the following coupled explicit iterative

algorithm for computing a solution of the equation u + KFu = 0 in the original

space, X. With initial vectors u0; v0 2 X, sequences fung and fvng in X are dened

iteratively as follows:

un+1 = un n(Fun vn); n 0; (1.34)

12

vn+1 = vn n(Kvn + un); n 0; (1.35)

where fng is a sequence in (0; 1) satisfying appropiate conditions. The recursion for-

mulas (1.34) and (1.35) have been used successfully to approximate solutions of Ham-

merstein equations involving nonlinear accretive-type mappings (see e.g., Chidume

and Djitte [25, 26], Chidume and Ofoedu [24], Chidume and Yekini [23], Chidume

[20], and the references contained in them). In particular, the following theorems

have been proved as generalizations of recent important results.

Theorem 1.0.5. [Chidume, [27]] Let E be a uniformly smooth real Banach space

with modulus of smoothness E, and let A : E ! 2E be a multi-valued bounded

maccretive operator with D(A) = E such that the inclusion 0 2 Au has a solution.

For arbitrary x1 2 E, dene a sequence fxng1n

=1 by,

xn+1 = xn nun nn(xn x1); un 2 Axn; n 1;

where fng1n

=1 and fng1 n=1 are sequences in (0; 1) satisfying the folliowing conditions:

(i) limn!1 n = 0; fng1 n=1 is decreasing;

(ii)

P

nn = 1;

P

E(nM1) < 1, for some constant M1 > 0;

(iii) limn!1

h

n1

n

1

i

nn

= 0. There exists a constant 0 > 0 such that

E(n)

n

0n. Then, the sequence fxng1 n=1 converges strongly to a zero of A.

It is our purpose in this paper to use an analogue of Theorem 1.0.5 and approxi-

mate a solution of (1.6) in the case where F and K are bounded maximal monotone

mappings from E to E and E to E; respectively.