C:/Documents and Settings/Profesor/Escritorio/Cubo/Cubo/Cubo/Cubo 2011/Cubo2011-13-01/Vol13 n\2721/Art N\2602 .dvi


CUBO A Mathematical Journal
Vol.13, No¯ 01, (11–24). March 2011

Weak Convergence Theorems for Maximal Monotone
Operators with Nonspreading mappings in a Hilbert space

Hiroko Manaka1 and Wataru Takahashi2

Department of Mathematical and Computing Sciences,

Tokyo Institute of Technology,

Ohokayama, Meguroku, Tokyo 152-8552, Japan.

email: hiroko.Manaka@is.titech.ac.jp

email: wataru@is.titech.ac.jp

ABSTRACT

Let C be a closed convex subset of a real Hilbert space H. Let T be a nonspreading

mapping of C into itself, let A be an α-inverse strongly monotone mapping of C into

H and let B be a maximal monotone operator on H such that the domain of B is in-

cluded in C. We introduce an iterative sequence of finding a point of F (T )∩(A+B)−10,

where F (T ) is the set of fixed points of T and (A + B)−10 is the set of zero points of

A + B. Then, we obtain the main result which is related to the weak convergence of

the sequence. Using this result, we get a weak convergence theorem for finding a com-

mon fixed point of a nonspreading mapping and a nonexpansive mapping in a Hilbert

space. Further, we consider the problem for finding a common element of the set of so-

lutions of an equilibrium problem and the set of fixed points of a nonspreading mapping.

RESUMEN

Sea C un subconjunto convexo cerrado de un espacio real de Hilbert H. Sea T una

asignación de C en śı mismo, sea A una asignación monótona α-inversa de C en H y



12 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

sea B un operador monotono máximal en H tal que el dominio de B está incluido en C.

Se introduce una secuencia iterativa para encontrar un punto de F (T ) ∩ (A + B)−10,

donde F (T ) es el conjunto de puntos fijos de T y (A + B)−10 es el conjunto de los

puntos cero de A + B. Entonces, se obtiene el resultado principal que se relaciona con

la convergencia débil de la secuencia.

Utilizando este resultado, obtenemos un teorema de convergencia para encontrar un

punto común de una asignación fija y una asignación en un espacio de Hilbert. Además,

consideramos el problema para encontrar un elemento común del conjunto de soluciones

de un problema de equilibrio y el conjunto de puntos fijos de una asignación.

Keywords: Nonspreading mapping, maximal monotone operator, inverse strongly-monotone map-

ping, fixed point, iteration procedure.

Mathematics Subject Classification: 46C05.

1 Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm ‖·‖ and let C be a

nonempty closed convex subset of H. For a constant α > 0, the mapping A : C → H is said to be

α-inverse strongly monotone if for any x, y ∈ C,

〈x − y, Ax − Ay〉 ≥ α ‖Ax − Ay‖
2
.

It is well-known that an α-inverse strongly monotone mapping is also Lipschitz continuous with a

Lipschitz constant 1
α
. Let S be a mapping of C into itself. We denote by F (S) the set of fixed

points of S. A mapping S of C into itself is nonexpansive if

‖Su − Sv‖ ≤ ‖u − v‖, ∀u, v ∈ C.

If S : C → C is a nonexpansive mapping, then I − S is 1
2
-inverse strongly monotone, where I is

the identity mapping on H; see, for instance, [18]. A mapping S of C into itself is nonspreading if

2‖Su − Sv‖2 ≤ ‖Su − v‖2 + ‖Sv − u‖2, ∀u, v ∈ C;

see [6, 7]. A multi-valued mapping B ⊂ H × H is said to be monotone if 〈x − y, u − v〉 ≥ 0 for all

x, y ∈ H, u ∈ Bx and v ∈ By. A monotone operator B on H is said to be maximal if its graph

is not properly contained in the graph of any other monotone operator on H. Recently, in the

case when S : C → C is a nonexpansive mapping, A : C → H is an α-inverse strongly monotone

mapping and B ⊂ H × H is a maximal monotone operator, Takahashi, Takahashi and Toyoda [15]

proved a strong convergence theorem for finding a point of F (S) ∩ (A + B)−10, where F (S) is the

set of fixed points of S and (A + B)−10 is the set of zero points of A + B.



CUBO
13, 1 (2011)

Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 13

In this paper, motivated by Takahashi, Takahashi and Toyoda [15], we introduce an iteration

sequence of finding a common point of the set F (S) of fixed points of a nonspreading mapping S

and the set (A+B)−10 of zero points of A+B, where A : C → H is an α-inverse strongly monotone

mapping and B ⊂ H × H is a maximal monotone operator. Then, we prove a weak convergence

theorem. Using this result, we get a weak convergence theorem for finding a common fixed point

of a nonspreading mapping and a nonexpansive mapping in a Hilbert space. Further, we obtain a

weak convergence theorem for finding a common element of the set of solutions of an equilibrium

problem and the set of fixed points of a nonspreading mapping.

2 Preliminaries

Throughout this paper, let N be the set of positive integers and let H be a real Hilbert space with

inner product 〈 · , · 〉 and norm ‖ · ‖. A Hilbert space satisfies Opial’s condition [10], that is,

lim inf
n→∞

‖xn − u‖ < lim inf
n→∞

‖xn − v‖

if xn ⇀ u and u 6= v; see [10]. Let C be a nonempty closed convex subset of a Hilbert space H.

The nearest point projection of H onto C is denoted by PC , that is, ‖x − PC x‖ ≤ ‖x − y‖ for all

x ∈ H and y ∈ C. Such PC is called the metric projection of H onto C. We know that the metric

projection PC is firmly nonexpansive, i.e.,

‖PC x − PC y‖
2
≤ 〈PC x − PC y, x − y〉

for all x, y ∈ H. Further 〈x − PC x, y − PC x〉 ≤ 0 holds for all x ∈ H and y ∈ C; see, for instance,

[16].

Let α > 0 be a given constant. A mapping A : C → H is said to be α-inverse strongly monotone

if 〈x − y, Ax − Ay〉 ≥ α ‖Ax − Ay‖
2

for all x, y ∈ C. We have that ‖Ax − Ay‖ ≤ (1/α) ‖x − y‖ for

all x, y ∈ C if A is α-inverse strongly monotone. Let B be a mapping of H into 2H . The effective

domain of B is denoted by D(B), that is, D(B) = {x ∈ H : Bx 6= ∅}. A multi-valued mapping B is

said to be a monotone operator on H if 〈x − y, u − v〉 ≥ 0 for all x, y ∈ D(B), u ∈ Bx, and v ∈ By.

A monotone operator B on H is said to be maximal if its graph is not properly contained in the

graph of any other monotone operator on H. For a maximal monotone operator B on H and r > 0,

we may define a single-valued operator Jr = (I + rB)
−1 : H → D(B), which is called the resolvent

of B for r > 0. Let B be a maximal monotone operator on H and let B−10 = {x ∈ H : 0 ∈ Bx}.

It is known that the resolvent Jr is firmly nonexpansive and B
−10 = F (Jr) for all r > 0.

We give the crucial lemmas in order to prove the main theorem.

Lemma 2.1 ([12]). Let H be a real Hilbert space, let {αn} be a sequence of real numbers such that

0 < a ≤ αn ≤ b < 1 for all n ∈ N and let {vn} and {wn} be sequences in H such that for some

c, lim supn→∞ ‖vn‖ ≤ c, lim supn→∞ ‖wn‖ ≤ c and lim supn→∞ ‖αnvn + (1 − αn)wn‖ = c. Then

limn→∞ ‖vn − wn‖ = 0.



14 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

Lemma 2.2 ([19]). Let H be a Hilbert space and let S be a nonempty closed convex subset of H.

Let {xn} be a sequence in H. If ‖xn+1 − x‖ ≤ ‖xn − x‖ for all n ∈ N and x ∈ S, then {PS (xn)}

converges strongly to some z ∈ S, where PS stands for the metric projection on H onto S.

Using Opial’s theorem [10], we can also prove the following lemma; see, for instance, [18].

Lemma 2.3. Let H be a Hilbert space and let {xn} be a sequence in H such that there exists a

nonempty subset S ⊂ Hsatisfying (i) and (ii):

(i) For every x∗ ∈ S, limn→∞ ‖xn − x
∗‖ exists:

(ii) if a subsequence {xnj } ⊂ {xn} converges weakly to x
∗, then x∗ ∈ S.

Then there exists x0 ∈ S such that xn ⇀ x0.

Let C be a nonempty closed convex subset of a real Hilbert space H, let f : C × C → R

be a bifunction and let A : C → H be a nonlinear mapping. Then, we consider the following

equilibrium problem [8]: Find z ∈ C such that

f (z, y) + 〈Az, y − z〉 ≥ 0, ∀y ∈ C. (2.1)

The set of such z ∈ C is denoted by EP (f, A), i.e.,

EP (f, A) = {z ∈ C : f (z, y) + 〈Az, y − z〉 ≥ 0, ∀y ∈ C}.

In the case of A ≡ 0, EP (f, A) is denoted by EP (f ). In the case of F ≡ 0, EP (f, A) is also

denoted by V I(C, A). For solving the equilibrium problem, let us assume that the bifunction f

satisfies the following conditions:

(A1) f (x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e., f (x, y) + f (y, x) ≤ 0 for all x, y ∈ C;

(A3) for all x, y, z ∈ C, lim sup
t↓0

f (tz + (1 − t)x, y) ≤ f (x, y);

(A4) f (x, ·) is convex and lower semicontinuous for all x ∈ C.

We know the following lemmas; see, for instance, [1] and [2].

Lemma 2.4 ([1]). Let C be a nonempty closed convex subset of H, let f be a bifunction from

C × C to R satisfying (A1)-(A4) and let r > 0 and x ∈ H. Then, there exists z ∈ C such that

f (z, y) +
1

r
〈y − z, z − x〉 ≥ 0

for all y ∈ C.



CUBO
13, 1 (2011)

Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 15

Lemma 2.5 ([2]). For r > 0 and x ∈ H, define the resolvent Tr : H → C of f for r > 0 as

follows:

Trx =

{

z ∈ C : f (z, y) +
1

r
〈y − z, z − x〉 ≥ 0, ∀y ∈ C

}

for all x ∈ H. Then, the following hold:

(i) Tr is single-valued;

(ii) Tr is firmly nonexpansive, i.e., for all x, y ∈ H,

‖Trx − Try‖
2 ≤ 〈Trx − Try, x − y〉;

(iii) F (Tr) = EP (f );

(iv) EP (f ) is closed and convex.

3 Main result

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Then, a

mapping S of C into itself is nonspreading if

2‖Su − Sv‖2 ≤ ‖Su − v‖2 + ‖Sv − u‖2, ∀u, v ∈ C;

see [6, 7]. We know from [6, 7, 3] that if the bifunction f : C × C → R satisfies the conditions

(A1), (A2), (A3) and (A4), then for any r > 0, Tr is a nonspreading mapping of C into itself.

Further, we can give the following example of nonspreading mappings in a Hilbert space. Let H

be a real Hilbert space; see [4]. Set E = {x ∈ H : ‖x‖ ≤ 1}, D = {x ∈ H : ‖x‖ ≤ 2} and

C = {x ∈ H : ‖x‖ ≤ 3}. Define a mapping S : C → C as follows:

Sx

{

0, x ∈ D,

PE x, x /∈ D.

Then, this mapping S is not nonexpansive but nonspreading because it is not continuous. This

implies that the class of nonexpansive mappings does not contain the class of nonspreading map-

pings. Now, we can prove a weak convergence theorem. Before proving it, we give the following

lemma.

Lemma 3.1. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.

Let α > 0. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal

monotone operator on H such that the domain of B is included in C. Let Jλ = (I + λB)
−1 be the

resolvent of B for any λ > 0. Then, the following hold:

(i) If u, v ∈ (A + B)−10, then Au = Av;



16 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

(ii) for any λ > 0, u ∈ (A + B)−1(0) if and only if u = Jλ(I − λA)u.

Proof. (i) If u, v ∈ (A + B)−1(0), then 0 ∈ Au + Bu and 0 ∈ Av + Bv. Then, we have −Au ∈ Bu

and −Av ∈ Bv. Since B is monotone, we have 〈u − v, −Au − (−Av)〉 ≥ 0. On the other hand,

since A is α-inverse strongly monotone, we have 〈u − v, Au − Av〉 ≥ ‖Au − Av‖2. So, we have

〈u − v, −Au − (−Av)〉 = 0 and hence Au = Av.

(ii) For any λ > 0, we have that

u = Jλ(I − λA)u

⇔ u − λAu ∈ u + λBu

⇔ 0 ∈ λAu + λBu

⇔ 0 ∈ Au + Bu

⇔ u ∈ (A + B)−1(0).

This completes the proof.

Now, we can prove the main theorem.

Theorem 3.1. Let C be a nonempty convex closed subset of a real Hilbert space H, let A :

C → H be α-inverse strongly monotone, let B : D(B) ⊂ C → 2H be maximal monotone, let

Jλ = (I + λB)
−1 be the resolvent of B for any λ > 0, and let T : C → C be a nonspreading

mapping. Assume that F (T ) ∩ (A + B)−1(0) 6= ∅. For any x = x1 ∈ C, define

xn+1 = βnxn + (1 − βn)T (Jλn (I − λnA)xn), ∀n ∈ N,

where {βn} and {λn} satisfy the following conditions (∗):

0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 2α. (∗)

Then, xn ⇀ z0 ∈ F (T ) ∩ (A + B)
−1(0), where z0 = limn→∞ PF (T )∩(A+B)−1(0)(xn).

Proof. Set E = F (T ) ∩ (A + B)−1(0). Let yn = Jλn (I − λnA)xn for all n ∈ N and let z ∈ E. Since

z = Jλn (I − λnA)z from Lemma 3.1 and A is α-inverse strongly monotone, we have that

‖yn − z‖
2

= ‖Jλn (I − λnA)xn − Jλn (I − λnA)z‖
2

(3.1)

≤ ‖xn − λnAxn − z + λnAz‖
2

= ‖xn − z‖
2
− 2λn〈xn − z, Axn − Az〉 + λ

2
n ‖Axn − Az‖

2

≤ ‖xn − z‖
2
− 2λnα ‖Axn − Az‖

2
+ λ2n ‖Axn − Az‖

2

= ‖xn − z‖
2

+ λn(λn − 2α) ‖Axn − Az‖
2
.

From (∗), we have that

‖yn − z‖
2
≤ ‖xn − z‖

2
, ∀n ∈ N



CUBO
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Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 17

and hence

‖xn+1 − z‖ = ‖βnxn + (1 − βn)T yn − z‖

≤ βn ‖xn − z‖ + (1 − βn) ‖T yn − z‖

≤ βn ‖xn − z‖ + (1 − βn) ‖yn − z‖

≤ ‖xn − z‖ .

This means that the condition (i) of Lemma 2.3 holds for S = E. We also obtain that limn→∞ ‖xn − z‖

exists. Thus, {xn}, {Axn}, {yn} and {T yn} are bounded. By the inequality (2),

‖xn+1 − z‖
2
≤ βn ‖xn − z‖

2
+ (1 − βn) ‖yn − z‖

2

≤ βn ‖xn − z‖
2

+ (1 − βn){‖xn − z‖
2

+ λn(λn − 2α) ‖Axn − Ax‖
2
}

≤ ‖xn − z‖
2

+ λn(λn − 2α)(1 − βn) ‖Axn − Az‖
2
.

Thus we have

0 ≤ (1 − d)a(2α − d) ‖Axn − Az‖
2

≤ ‖xn − z‖
2
− ‖xn+1 − z‖

2
→ 0,

as n → ∞. This means that

lim
n→∞

‖Axn − Az‖ = 0. (3.2)

On the other hand, since Jλn is firmly nonexpansive, we have that

‖yn − z‖
2

= ‖Jλn (I − λnA)xn − Jλn (I − λnA)z‖
2

≤〈yn − z, (I − λnA)xn − (I − λnA)z〉

=
1

2
{‖yn − z‖

2
+ ‖(I − λnA)xn − (I − λnA)z‖

2

− ‖yn − z − (I − λnA)xn + (I − λnA)z‖
2
}

=
1

2
{‖yn − z‖

2
+ ‖xn − z‖

2

− ‖yn − z − (I − λnA)xn + (I − λnA)z‖
2
}

=
1

2
{‖yn − z‖

2
+ ‖xn − z‖

2
− ‖yn − xn‖

2

− 2λn〈yn − xn, Axn − Az〉 − λn
2 ‖Axn − Az‖

2
}.

Therefore we have

‖yn − z‖
2
≤ ‖xn − z‖

2
− ‖yn − xn‖

2

− 2λn〈yn − xn, Axn − Az〉 − λn
2 ‖Axn − Az‖

2



18 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

and hence

‖xn+1 − z‖
2
≤βn ‖xn − z‖

2
+ (1 − βn) ‖T yn − z‖

2

≤βn ‖xn − z‖
2

+ (1 − βn) ‖yn − z‖
2

≤βn ‖xn − z‖
2

+ (1 − βn){‖xn − z‖
2
− ‖yn − xn‖

2

− 2λn〈yn − xn, Axn − Az〉 − λn
2 ‖Axn − Az‖

2
}

≤ ‖xn − z‖
2
− (1 − d) ‖yn − xn‖

2
− λn

2(1 − βn) ‖Axn − Az‖
2

− 2λn(1 − βn)〈yn − xn, Axn − Az〉.

This means that

(1 − d) ‖yn − xn‖
2
≤ ‖xn − z‖

2
− ‖xn+1 − z‖

2

+ ‖Axn − Az‖{2b(1 − c) ‖yn − xn‖ + b
2(1 − c) ‖Axn − Az‖}.

Since {yn} and {xn} are bounded, limn→∞ ‖Axn − Az‖ = 0 and limn→∞ ‖xn − z‖ exists, we have

lim
n→∞

‖yn − xn‖ = 0.

Since A is Lipschitz continuous, we also have

lim
n→∞

‖Ayn − Axn‖ = 0.

Let x∗ be a weak cluster point of {xn}. First, we prove that x
∗ ∈ (A + B)−1(0). Since yn =

Jλn (I − λnA)xn, we have that

yn = (I + λnB)
−1(I − λnA)xn

⇔ (I − λnA)xn ∈ (I + λnB)yn = yn + λnByn

⇔ xn − yn − λnAxn ∈ λnByn

⇔
1

λn
(xn − yn − λnAxn) ∈ Byn.

Since B is monotone, we have that for (u, v) ∈ B,

〈

yn − u,
1

λn
(xn − yn − λnAxn) − v

〉

≥ 0

and hence

〈yn − u, xn − yn − λn(Axn + v)〉 ≥ 0.

Suppose that a subsequence {xnj } ⊂ {xn} satisfies xnj ⇀ x
∗. Then, since A is α-inverse strongly

monotone and Axn → Az by (3),

〈

xnj − x
∗, Axnj − Ax

∗
〉

≥ α
∥

∥Axnj − Ax
∗
∥

∥

2



CUBO
13, 1 (2011)

Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 19

implies that Axnj → Ax
∗ as j → ∞. Moreover, since limn→∞ ‖yn − xn‖ = 0 implies ynj ⇀ x

∗, we

have

lim
j→∞

〈

ynj − u, xnj − ynj − λnj (Axnj + v)
〉

≥ 0

and hence 〈x∗ − u, −Ax∗ − v〉 ≥ 0. Since B is maximal monotone, (−Ax∗) ∈ Bx∗. That is,

x∗ ∈ (A + B)−1(0).

Next, we show x∗ ∈ F (T ). Putting c = limn→∞ ‖xn − z‖, we have

lim sup
n→∞

‖T yn − z‖ = lim sup
n→∞

‖T yn − T z‖

≤ lim sup
n→∞

‖yn − z‖

≤ lim sup
n→∞

‖xn − z‖ ≤ c.

On the other hand, we have

lim
n→∞

‖xn+1 − z‖ = lim
n→∞

‖βnxn + (1 − βn)T yn − z‖ = c.

From Lemma 2.1, we have

lim
n→∞

‖(xn − z) − (T yn − z)‖ = lim
n→∞

‖xn − T yn‖ = 0. (3.3)

We have also

‖yn − T yn‖ ≤ ‖yn − xn‖ + ‖xn − T yn‖.

Hence, we have

lim
n→∞

‖yn − T yn‖ = 0.

Since xnj ⇀ x
∗ and xn − yn → 0, we have ynj ⇀ x

∗. Now we shall show that T x∗ = x∗. Since T

is nonspreading, we have

0 ≤(‖T yn − x
∗‖

2
− ‖T yn − T x

∗‖
2
) + (‖T x∗ − yn‖

2
− ‖T yn − T x

∗‖
2
)

=2 〈T yn, T x
∗ − x∗〉 + ‖x∗‖

2
− ‖T x∗‖

2
+ 2 〈T yn − yn, T x

∗〉 + ‖yn‖
2
− ‖T yn‖

2

≤ 2 〈T yn − yn, T x
∗ − x∗〉 + 2 〈yn, T x

∗ − x∗〉 + ‖x∗‖
2
− ‖T x∗‖

2

+ 2 〈T yn − yn, T x
∗〉 + (‖yn‖ + ‖T yn‖)(‖yn − T yn‖).

Thus, we have that for all j ∈ N,

0 ≤2
〈

T ynj − ynj , T x
∗ − x∗

〉

+ 2
〈

ynj , T x
∗ − x∗

〉

+ ‖x∗‖
2
− ‖T x∗‖

2

+ 2
〈

T ynj − ynj , T x
∗
〉

+ (
∥

∥ynj
∥

∥ +
∥

∥T ynj
∥

∥)(
∥

∥ynj − T ynj
∥

∥).



20 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

Since limn→∞
∥

∥T ynj − ynj
∥

∥ = 0 and ynj ⇀ x
∗ as j → ∞, the above inequality implies that

0 ≤2 〈x∗, T x∗ − x∗〉 + ‖x∗‖
2
− ‖T x∗‖

2

=2 〈x∗, T x∗〉 − ‖x∗‖
2
− ‖T x∗‖

2

= − ‖x∗ − T x∗‖
2
.

So, we have T x∗ = x∗, i.e., x∗ ∈ F (T ). Therefore we obtain that

x∗ ∈ E = F (T ) ∩ (A + B)−1(0).

This implies that the condition (ii) of Lemma 2.3 holds for S = E. We also know that limn→∞ ‖xn − z‖

exists for z ∈ S = E. So, we have from Lemma 2.3 that there exists z∗ ∈ E such that xn ⇀ z
∗ as

n → ∞. Moreover, since for any z ∈ S = E,

‖xn+1 − z‖ ≤ ‖xn − z‖ , ∀n ∈ N,

by Lemma 2.2 there exists some z0 ∈ S such that PS (xn) → z0. The property of metric projection

implies that

〈z∗ − PS (xn), xn − PS (xn)〉 ≤ 0.

Therefore, we have

〈z∗ − z0, z
∗ − z0〉 = ‖z

∗ − z0‖
2
≤ 0.

This means that z∗ = z0, i.e., xn ⇀ z
∗ = limn→∞ PE (xn).

4 Applications

Let H be a Hilbert space and let f be a proper lower semicontinuous convex function of H into

(−∞, ∞]. Then the subdifferential ∂f of f is defined as follows:

∂f (x) = {z ∈ H : f (x) + 〈z, y − x〉 ≤ f (y), ∀y ∈ H}

for all x ∈ H. By Rockafellar [11], it is shown that ∂f is maximal monotone. Let C be a nonempty

closed convex subset of H and let iC be the indicator function of C, i.e.,

iC (x)

{

0, if x ∈ C,

∞, if x 6∈ C.

Further, for any u ∈ C, we also define the normal cone NC (u) of C at u as follows;

NC (u) = {z ∈ H : 〈z, y − u〉 ≤ 0, ∀y ∈ C}.

Then iC : H → (−∞, ∞] is a proper lower semicontinuous convex function on H and ∂iC is a

maximal monotone operator. Let Jλx = (I + λ∂iC )
−1x for λ > 0 and x ∈ H. Since

∂iC (x) = {z ∈ H : iC (x) + 〈z, y − x〉 ≤ iC (y), ∀y ∈ H}

= {z ∈ H : 〈z, y − x〉 ≤ 0, ∀y ∈ C}

= NC (x)



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Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 21

for x ∈ C, we have

u = Jλx ⇔ (I + λ∂iC )
−1x = u

⇔ x ∈ u + λ∂iC (u)

⇔ x ∈ u + λNC (u)

⇔ x − u ∈ λNC (u)

⇔ 〈x − u, y − u〉 ≤ 0, ∀y ∈ C

⇔ PC (x) = u.

Similarly, we have that for x ∈ C,

x ∈ (A + ∂iC )
−1(0) ⇔ 〈−Ax, y − x〉 ≤ 0, ∀y ∈ C

⇔ x ∈ V I(A, C).

Thus, putting B = ∂iC , we have Jλn = PC for any n ∈ N. Thus, we have the following theorem

from Theorem 3.1.

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H, let A be an

α-inverse strongly monotone mapping of C into H and let T : C → C be a nonspreading mapping.

Assume F (T ) ∩ (A + ∂iC )
−1(0) = F (T ) ∩ V I(A, C) 6= ∅. Define a sequence {xn} in C as follows:

x = x1 ∈ C and

xn+1 = βnxn + (1 − βn)T (PC (I − λnA)xn)

for all n ∈ N, where the sequences {βn} and {λn} satisfy the condition (∗):

0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 2α. (∗)

Then, xn ⇀ z0 ∈ F (T ) ∩ V I(A, C) and z0 = limn→∞ PF (T )∩V I(A,C)(xn).

Let S : C → C be nonexpansive. Then, I − S is 1
2
-inverse strongly monotone. So, we obtain

the following result.

Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H, let S : C →

C be a nonexpansive mapping and let T : C → C be a nonspreading mapping. Assume that

F (T ) ∩ F (S) 6= ∅. Let x = x1 ∈ C and define

xn+1 = βnxn + (1 − βn)T ((1 − λn)xn + λnSxn)

for all n ∈ N , where {λn} and {βn} satisfy the condition (∗):

0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 1. (∗)

Then, xn ⇀ z0 ∈ F (T ) ∩ F (S) and z0 = limn→∞ PF (T )∩F (S)(xn).



22 Hiroko Manaka and Wataru Takahashi CUBO
13, 1 (2011)

Proof. Put A = I − S. Then we have

PC (xn − λnAxn) = PC (xn − λn(I − S)xn)

= PC ((1 − λn)xn + λnSxn)

= (1 − λn)xn + λnSxn.

For u ∈ C, we have Su ∈ C and

u ∈ (A + ∂iC )
−1(0) ⇔ 0 ∈ Au + NC (u)

⇔ Su − u ∈ NC (u)

⇔ 〈Su − u, v − u〉 ≤ 0, ∀v ∈ C

⇔ PC (Su) = u

⇔ Su = u.

Thus, we obtain (A + ∂iC )
−1(0) = V I(A, C) = F (S). So, by Theorem 4.1 we have the desired

result.

Next, we deal with the equilibrium problem with nonspreading mappings in a Hilbert space.

Takahashi, Takahashi and Toyoda [15] showed the following.

Theorem 4.3 ([15]). Let C be a nonempty closed convex subset of a Hibert space H and let

f : C × C → R be a bifunction satisfying the conditions (A1)-(A4). Define Af as follows:

Af (x)

{

{z ∈ H : f (x, y) ≥ 〈y − x, z〉 , ∀y ∈ C}, if x ∈ C,

∅, if x 6∈ C.

Then, EP (f ) = A−1
f

(0) and Af is maximal monotone with the domain of Af in C. Furthermore,

Tr(x) = (I + rAf )
−1(x), ∀r > 0.

We obtain the following theorem from Theorem 3.1.

Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H, let f : C ×C →

R satisfy the conditions (A1)-(A4) and let Tλ be the resolvent of f for λ > 0. Let S : C → C be a

nonspreading mapping. Assume that F (T ) ∩ EP (f ) 6= ∅. For x = x1 ∈ C, define

xn+1 = βnxn + (1 − βn)STλn xn, ∀n ∈ N,

where {βn} and {λn} satisfy the following conditions:

0 < c ≤ βn ≤ d < 1, 0 < a ≤ λn ≤ b < ∞.

Then, xn ⇀ z0 ∈ F (T ) ∩ EP (f ) and z0 = limn→∞ PF (S)∩EP (f )(xn).



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Weak Convergence Theorems for Maximal Monotone Operators
with Nonspreading mappings in a Hilbert space 23

Proof. Suppose A = 0. Then, we have that

〈x − y, Ax − Ay〉 ≥ α ‖Ax − Ay‖
2

= 0, ∀α ∈ R.

So, we can choose α = ∞ in Theorem 3.1. Since Tλn = (I + λnAf )
−1 is the resolvent of Af and Af

is maximal monotone, Theorem 3.1 implies that xn ⇀ z0 ∈ F (T ) ∩ A
−1
f

(0). Moreover, we know

A−1
f

(0) = EP (f ). So, we have the desired result.

Received: June 2009. Revised: September 2009.

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