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(x?y) 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 jjTx?Tyjj jjx?yjj8x; 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:
ˆ 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.
ˆ 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, jjTx?Tyjj2 jjx?yjj2+kjjx?y?(Tx?Ty)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 K?1 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
m?accretive 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
n?1
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.

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