báo cáo hóa học: " Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense" pot

RESEA R C H Open Access
Strong convergence theorems for system of
equilibrium problems and asymptotically strict
pseudocontractions in the intermediate sense
Peichao Duan
*
and Jing Zhao
* Correspondence:

College of Science, Civil Aviation
University of China, Tianjin 300300,
PR China
Abstract
Let
{S
i
}
N
i
=
1
be N uniformly continuous asymptotically l
i
-strict pseudocontractions in
the intermediate sense defined on a nonempty closed convex subset C of a real
Hilbert space H. Consider the problem of finding a common element of the fixed
point set of these mappings and the solution set of a system of equilibrium
problems by using hybrid method. In this paper, we propose new iterative schemes
for solving this problem and prove these schemes converge strongly.
MSC: 47H05; 47H09; 47H10.
Keywords: asymptotically strict pseudocontraction in the intermediate sense, system

of equilibrium problem, hybrid method, fixed point
1. Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.
A nonlinear mapping S : C ® C is a self mapping of C.Wedenotethesetoffixed
points of S by F(S) (i.e., F(S)={x Î C : Sx = x}). Recall the following concepts.
(1) S is uniformly Lipschitzian if there exists a constant L > 0 such that
||S
n
x −−S
n
y
|| ≤ L||x −
y
|| for all inte
g
ers n ≥ 1andx,
y
∈ C
.
(2) S is nonexpansive if
||Sx − S
y
|| ≤ ||x −
y
|| for all x,
y
∈ C
.
(3) S is asymptotically nonexpansive if there exists a sequence k
n

of positive num-
bers satisfying the property lim
n®∞
k
n
= 1 and
||S
n
x − S
n
y
|| ≤ k
n
||x −
y
|| for all inte
g
ers n ≥ 1andx,
y
∈ C
.
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>© 2011 Duan and Zhao; licensee Spring er. This is an Open Access article di stributed under the term s of the Creative Commons
Attribution License ( .0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
(4) S is asymptotically nonexpansive in the intermediate sense [1] provided S is
continuous and the following inequality holds:
lim sup
n→∞
sup

x,
y
∈C
(||S
n
x − S
n
y|| − ||x − y||) ≤ 0
.
(5) S is asymptotically l-strict pseudocontractive mapping [2] with sequence {g
n
}if
there exists a constant l Î [0, 1) and a sequence {g
n
}in[0,∞) with lim
n®∞
g
n
=0
such that
||S
n
x − S
n
y||
2
≤
(
1+γ
n

)
||x − y||
2
+ λ||x − S
n
x −
(
y − S
n
y
)
||
2
for all x, y Î C and n Î N.
(6) S is asymptotically l-strict pseudocontractive mapping in the intermediate sense
[3,4] with sequence {g
n
} if there exists a constant l Î [0, 1) and a sequence {g
n
}in
[0, ∞) with lim
n®∞
g
n
= 0 such that
lim sup
n→∞
sup
x,
y

∈C
(||S
n
x − S
n
y||
2
− (1 + γ
n
)||x − y||
2
− λ||x − S
n
x − (y − S
n
y)||
2
) ≤
0
(1:1)
for all x, y Î C and n Î N.
Throughout this paper, we assume that
c
n
=max{0, sup
x,
y
∈C
(||S
n

x − S
n
y||
2
− (1 + γ
n
)||x − y||
2
− λ||x − S
n
x − (y − S
n
y)||
2
)}
.
Then, c
n
≥ 0 for all n Î N, c
n
® 0asn ® ∞ and (1.1) reduces to the relation
|
|S
n
x − S
n
y||
2
≤
(

1+γ
n
)
||x − y||
2
+ λ||x − S
n
x −
(
y − S
n
y
)
||
2
+ c
n
(1:2)
for all x, y Î C and n Î N.
When c
n
=0foralln Î N in (1.2), then S is an asymptotically l-strict pseudocon-
tractive mapping with sequence {g
n
}. We note that S is not necessarily uniformly L-
Lipschitzian (see [4]), more examples can also be seen in [3].
Let {F
k
} be a countable famil y of bifunctions from C × C to ℝ,whereℝ is the set of
rea l numbers. Combettes and Hirstoaga [5] considered the following system of equili-

brium problems:
Finding x ∈ C such that F
k
(
x, y
)
≥ 0, ∀k ∈  and ∀y ∈ C
,
(1:3)
where Γ is an arbitrary index set. If Γ is a singleton, then problem (1.3) becomes the
following equilibrium problem:
Finding x ∈ C such that F
(
x, y
)
≥ 0, ∀y ∈ C
.
(1:4)
The solution set of (1.4) is denoted by EP(F).
The problem (1.3) is very general in the sense that it includes, as special cases, opti-
miz ation problems, variational inequalities, minimax pro blems, Nash equilibri um pro-
blem in noncooperative games and others; see, for instance, [6,7] and the references
therein. Some methods have been proposed to solve the equilibrium problem (1.3),
related work can also be found in [8-11].
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 2 of 13
For solving the equilibrium problem, let us assume that the bifunction F satisfies th e
following conditions:
(A1) F(x, x) = 0 for all x Î C;
(A2) F is monotone, i.e.F(x, y)+F(y, x) ≤ 0 for any x, y Î C;

(A3) for each x, y, z Î C, lim sup
t®0
F(tz +(1-t)x, y) ≤ F(x, y);
(A4) F(x,·) is convex and lower semicontionuous for each x Î C.
Recall Mann’s iteration algorithm was introduced by Mann [12]. Since then, the con-
struction of fixed points for nonexpansive mappings and asympt otically strict pseudo-
contractions via Mann’ iteration a lgorithm has been extensively investigated by many
authors (see, e.g., [2,6]).
Mann’s iteration algorithm generates a sequence {x
n
} by the following manner:
∀x
0
∈ C, x
n+1
= α
n
x
n
+
(
1 − α
n
)
Sx
n
, n ≥ 0
,
where a
n

is a real sequence in (0, 1) which satisfies certain control conditions.
On the other hand, Qin et al. [13] introduced the following algorithm for a finite
family of asymptotically l
i
-strict pseudocontractions. Let x
0
Î C and
{α
n
}
∞
n
=
0
be a
sequence in (0, 1). The sequence {x
n
} by the following way:
x
1
= α
0
x
0
+(1− α
0
)S
1
x
0

,
x
2
= α
1
x
1
+(1− α
1
)S
2
x
1
,
···
x
N
= α
N−1
x
N−1
+(1− α
N−1
)S
N
x
N−1
,
x
N+1

= α
N
x
N
+(1− α
N
)S
2
1
x
N
,
···
x
2N
= α
2N−1
x
2N−1
+(1− α
2N−1
)S
2
N
x
2N−1
,
x
2N+1
= α

2N
x
2N
+(1− α
2N
)S
3
1
x
2N
,
···
.
It is called the explicit iterative sequence of a finite family of asymptotically l
i
-strict
pseudocontractions {S
1
, S
2
, , S
N
}. Since, for each n ≥ 1, it can be written as n =(h -1)
N + i,wherei = i(n) Î {1, 2, , N}, h = h(n) ≥ 1 is a positive integer and h(n) ® ∞,as
n ® ∞. We can rewrite the above table in the following compact form:
x
n
= α
n−1
x

n−1
+(1− α
n−1
)S
h(
n
)
i
(
n
)
x
n−1
, ∀n ≥ 1
.
Recently, S ahu et al. [4] introduced new iterative schemes for asymptotically strict
pseudocontractive mappings in the intermediate sense. To be more precise, they
proved the following theorem.
Theorem 1.1. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH
and T: C ® C a uniformly continuous asymptotically -strict pseudocont ractive map-
ping in the intermediate sense with sequence g
n
such that F(T) is nonempty and
bou nded. Let a
n
be a sequence in [0, 1] such that 0<δ ≤ a
n
≤ 1- for all n Î N. Let
{x
n

} ⊂ C be sequences generated by the following (CQ) algorithm:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
u = x
1
∈ C chosen arbitrary,
y
n
=(1− α
n
)x
n
+ α
n
T
n
x
n
,
C
n

= {z ∈ C : ||y
n
− z||
2
≤||x
n
− z||
2
+ θ
n
}
,
Q
n
= {z ∈ C : x
n
− z, u − x
n
≥0},
x
n+1
= P
C
n
∩Q
n
(u), for all n ∈ N,
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 3 of 13
where θ

n
= c
n
+ g
n
Δ
n
and Δ
n
= sup {||x
n
- z||: z Î F(T)} < ∞. Then,{x
n
} converges
strongly to P
F(T)
(u).
Very recently, Hu and Cai [3] further considered the asymptotically strict pseudocon-
tractive mappings in the intermediate sense concerning equilibrium problem. They
obtained the following result in a real Hilbert space.
Theorem 1.2. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH
and N ≥ 1 be an integer, j : C ® C be a bifunction satisfying (A1)-(A4) and A : C ®
Hbeana-inverse-strongly monotone mapping. Let for each 1 ≤ i ≤ N, T
i
: C ® Cbea
uniformly continuous k
i
-strictly asymptotically pseudo contractive mapping in the inter-
media te sense for some 0 ≤ k
i

<1with sequences {g
n,i
} ⊂ [0, ∞) such that lim
n®∞
g
n,i
=
0 and {c
n,i
} ⊂ [0, ∞) such that lim
n®∞
c
n,i
=0.Let k = max{k
i
:1≤ i ≤ N}, g
n
= max{g
n,
i
:1≤ i ≤ N} and c
n
= max{c
n,i
:1≤ i ≤ N}. Assume that
F = ∩
N
i
=1
F(T

i
) ∩ E
P
is nonempty
and bounded. Let {a
n
} and {b
n
} b e sequences in [0, 1] such that 0<a ≤ a
n
≤ 1, 0 <δ ≤
b
n
≤ 1-kforallnÎ Nand0<b ≤ r
n
≤ c <2a. Let {x
n
} and {u
n
} be sequences gener-
ated by the following algorithm:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
0
∈ C chosen arbitrary,
u
n
∈ C, such that φ(u
n
, y)+Ax
n
, y − u
n
 +
1
s
y − u
n
, u
n

− x
n
≥0, ∀y ∈ C
,
z
n
=(1− β
n
)u
n
+ β
n
T
h(n)
i(n)
u
n
,
y
n
=(1− α
n
)u
n
+ α
n
z
n
,
C

n
= {v ∈ C : ||y
n
− v||
2
≤||x
n
− v||
2
+ θ
n
},
Q
n
= {v ∈ C : x
n
− v, x
0
− x
n
≥0},
x
n+1
= P
C
n
∩Q
n
x
0

, ∀n ∈ N ∪{0},
where
θ
n
= c
h
(
n
)
+ γ
h
(
n
)
ρ
2
n
→
0
,asn® ∞,wherer
n
= sup{||x
n
- v||: v Î F}<∞.
Then,{x
n
} converges strongly to P
F(T)
x
0

.
Motivated by Hu and Cai [3], Sahu et al. [4], and Duan [8], the main purpose of this
paper is to introduce a new iterative process for finding a common element of the
fixed point set of a finite family of asymptotically l
i
-strict pseudocontractions and the
solution set of the problem (1.3). Using the hybrid method, we obtain strong conver-
gence theorems that extend and improve the corresponding results [3,4,13,14].
We will adopt the following notations:
1. ⇀ for the weak convergence and ® for the strong convergence.
2.
ω
w
(x
n
)={x : ∃x
n
j
 x
}
denotes the weak ω-limit set of {x
n
}.
2. Preliminaries
We need some facts and tools in a real Hilbert space H which are listed below.
Lemma 2.1. Let H be a real Hilbert space. Then, the following identities hold.
(i) ||x - y||
2
=||x||
2

-||y||
2
-2〈x - y, y〉, ∀x, y Î H.
(ii) ||tx +(1 - t)y||
2
= t||x||
2
+(1 - t)||y||
2
- t(1 - t)||x - y||
2
, ∀t Î [0, 1], ∀x, y Î H.
Lemma 2.2. ([10]) LetHbearealHilbertspace.Givenanonemptyclosedconvex
subset C ⊂ H and points x, y, z Î H and given also a real number a Î ℝ, the set
{v ∈ C : ||
y
− v||
2
≤||x − v||
2
+ z, v + a
}
is convex (and closed).
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 4 of 13
Lemma 2.3. ([15] ) Let C be a nonempty, closed and convex subset of H. Let {x
n
} be a
sequence in H and u Î H. Let q = P
C

u. Suppose that {x
n
} is such that ω
w
(x
n
) ⊂ Cand
satisfies the following condition
|
|x
n
− u|| ≤ ||u −
q
||
f
or all n
.
Then, x
n
® q.
Lemma 2.4. ([4]) Let C be a nonem pty closed convex subset of a real Hilbert space H
and T : C ® C a continuous asymptotically -strict pseudocontractive mapping in the
intermediate sense. Then I - T is demiclosed at zero in the sense that if {x
n
} is a
sequence in C such that x
n
⇀ x Î Candlim sup
m®∞
lim sup

n®∞
||x
n
- T
m
x
n
|| = 0,
then (I - T)x =0.
Lemma 2.5. ([4]) Let C be a nonempty subset of a Hilbert space H and T : C ® Can
asymptotically  - strict pseudocontractive mapping in the intermediate sense with
sequence {g
n
}. Then
||T
n
x − T
n
y|| ≤
1
1 −
κ
(κ||x − y|| +

(1+(1− κ)γ
n
)||x − y||
2
+(1− κ)c
n

)
for all x, y Î C and n Î N.
Lemma 2.6. ([6]) Let C be a nonempty closed convex subset of H, let F be bifunction
from C × Ctoℝ satisfying (A1)-(A4) and let r >0and x Î H. Then there exists z Î C
such that
F(z , y)+
1
r
y − z, z − x≥0, for all y ∈ C
.
Lemma 2.7. ([5]) For r >0,x Î H, define a mapping T
r
: H ® C as follows:
T
r
(x)={z ∈ C | F(z, y)+
1
r
y − z, z − x≥0, ∀y ∈ C
}
for all x Î H. Then, the following statements hold:
(i) T
r
is single-valued;
(ii) T
r
is firmly nonexpansive, i.e., for any x, y Î H,
||T
r
x − T

r
y
||
2
≤T
r
x − T
r
y
, x −
y

;
(iii) F(T
r
)=EP(F);
(iv) EP(F) is closed and convex.
3. Main result
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and
N ≥ 1 be an integer, let F
k
,kÎ {1,2, M}, be a bifunction from C × Ctoℝ which
satisfies conditions (A1)-(A4). Let, for each 1 ≤ i ≤ N, S
i
: C ® C be a uniformly contin-
uous asymptotically l
i
-strict pseudocontractive mapping in the intermediate sense for
some 0 ≤ l
i

<1with sequences {g
n,i
} ⊂ [0, ∞) such that lim
n®∞
g
n,i
=0and {c
n,i
} ⊂ [0,
∞) such that lim
n®∞
c
n,i
=0.Let l =max{l
i
:1≤ i ≤ N}, g
n
=max{g
n,i
:1≤ i ≤ N}
and c
n
=max{c
n,i
:1≤ i ≤ N}. Assume that

= ∩
N
i=1
F(S

i
) ∩ (∩
M
k
=1
EP(F
k
)
)
is nonempt y
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 5 of 13
and bounded. Let {a
n
} and {b
n
} b e sequences in [0, 1] such that 0<a ≤ a
n
≤ 1, 0 <δ ≤
b
n
≤ 1-l for all n Î N and {r
k,n
} ⊂ (0, ∞) satisfies lim inf
n®∞
r
k,n
>0for a l l k Î {1,
2, M}. Let {x
n

} and {u
n
} be sequences generated by the following algorithm:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
∈ C chosen arbitrary,
u
n
= T

F
M
r
M,n
T
F
M−1
r
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
,
z
n
=(1− β
n
)u
n
+ β
n

S
h(n)
i(n)
u
n
,
y
n
=(1− α
n
)u
n
+ α
n
z
n
,
C
n
= {v ∈ C : ||y
n
− v||
2
≤||x
n
− v||
2
+ θ
n
}

,
Q
n
= {v ∈ C : x
n
− v, x
1
− x
n
≥0},
x
n+1
= P
C
n
∩
Q
n
x
1
, ∀n ∈ N,
(3:1)
where
θ
n
= c
h
(
n
)

+ γ
h
(
n
)
ρ
2
n
→
0
,asn® ∞,wherer
n
=sup{||x
n
- v|| : v Î Ω}<∞.
Then {x
n
} converges strongly to P
Ω
x
1
.
Proof. Denote

k
n
= T
F
k
r

k,n
T
F
2
r
2
,
n
T
F
1
r
1
,n
for every k Î {1, 2, , M} and

0
n
=
I
for all n Î N.
Therefore
u
n
= 
M
n
x
n
. The proof is divided into six steps.

Step 1. The sequence {x
n
} is well defined.
It is obvious that C
n
is closed and Q
n
is closed and convex for every n Î N.From
Lemma 2.2, we also get that C
n
is convex.
Take p Î Ω, since for each k Î {1, 2, , M},
T
F
k
r
k
,n
is nonexpansive,
p = T
F
k
r
k
,
n
p
and
u
n

= 
M
n
x
n
, we have
||u
n
− p|| = ||
M
n
x
n
− 
M
n
p|| ≤ ||x
n
− p|| for all n ∈ N
.
(3:2)
It follows from the definition of S
i
and Lemma 2.1(ii), we get
||z
n
− p||
2
= ||(1 − β
n

)(u
n
− p)+β
n
(S
h(n)
i(n)
u
n
− p)||
2
=(1− β
n
)||u
n
− p||
2
+ β
n
||S
h(n)
i(n)
u
n
− p||
2
− β
n
(1 − β
n

)||S
h(n)
i(n)
u
n
− u
n
||
2
≤ (1 − β
n
)||u
n
− p||
2
+ β
n

||(1 + γ
h(n)
)||u
n
− p||
2
+ λ||S
h(n)
i(n)
u
n
− u

n
||
2
+ c
h(n)

− β
n
(1 − β
n
)||S
h(n)
i(n)
u
n
− u
n
||
2
≤ (1 + γ
h(n)
)||u
n
− p||
2
− β
n
(1 − β
n
− λ)||S

h(n)
i(n)
u
n
− u
n
||
2
+ β
n
c
h(n)
≤ (1 + γ
h
(
n
)
)||u
n
− p||
2
+ β
n
c
h
(
n
)
.
(3:3)

By virtue of the convexity of ||·||
2
, one has
|
|y
n
− p||
2
= ||
(
1 − α
n
)(
u
n
− p
)
+ α
n
(
z
n
− p
)
||
2
≤
(
1 − α
n

)
||u
n
− p||
2
+ α
n
||z
n
− p||
2
.
(3:4)
Substituting (3.2) and (3.3) into (3.4), we obtain
|
|y
n
− p||
2
≤ (1 − α
n
)||u
n
− p||
2
+ α
n

(1 + γ
h(n)

)||u
n
− p||
2
+ β
n
c
h(n)

≤||u
n
− p||
2
+ γ
h(n)
||u
n
− p||
2
+ β
n
c
h(n)
≤||u
n
− p||
2
+ γ
h(n)
||x

n
− p||
2
+ c
h(n)
≤||u
n
− p||
2
+ θ
n
≤||x
n
−
p
||
2
+ θ
n
.
(3:5)
It follows that p Î C
n
for all n Î N. Thus, Ω ⊂ C
n
.
Next, we prove that Ω ⊂ Q
n
for all n Î N by induction. For n = 1, we have Ω ⊂ C =
Q

1
. Assume that Ω ⊂ Q
n
for some n ≥ 1. Since
x
n+1
= P
C
n
∩
Q
n
x
1
, we obtain
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 6 of 13

x
n+1
− z, x
1
− x
n+1

≥ 0, ∀z ∈
C
n
∩
Q

n
.
As Ω ⊂ C
n
⋂ Q
n
by induction assumption, the inequality holds, in particular, for all z
Î Ω. This together with the definition of Q
n+1
implies that Ω ⊂ Q
n +1
.
Hence Ω ⊂ Q
n
holds for all n ≥ 1. Thus Ω ⊂ C
n
⋂ Q
n
and therefore the sequence
{x
n
} is well defined.
Step 2. Set q = P
Ω
x
1
, then
|
|x
n+1

− x
1
|| ≤ ||
q
− x
1
|| for all n ∈ N
.
(3:6)
Since Ω is a nonempt y closed convex subset of H, there exists a unique q Î Ω such
that q = P
Ω
x
1
.
From
x
n+1
=
P
C
n
∩
Q
n
x
1
, we have
|
|x

n+1
− x
1
|| ≤ ||v − x
1
|| for all v ∈ C
n
∩ Q
n
,foralln ∈ N
.
Since q Î Ω ⊂ C
n
⋂ Q
n
, we get (3.6).
Therefore, {x
n
} is bounded. So are {u
n
} and {y
n
}.
Step 3. The following limits hold:
l
im
n
→∞
||u
n

− u
n+i
|| =0,
l
im
n
→∞
||x
n
− x
n+i
|| =0;∀i = 1, 2, , N.
From the definition o f Q
n
,wehave
x
n
=
P
Q
n
x
1
, which together with the fact that x
n+1
Î C
n
⋂ Q
n
⊂ Q

n
implies that
||
x
n
− x
1
||
≤
||
x
n+1
− x
1
||
,

x
n
− x
n+1
, x
1
− x
n

≥ 0
.
(3:7)
This shows that the sequence {||x

n
- x
1
||} is nondecreasing. Since { x
n
}isbounded,
the limit of {||x
n
- x
1
||} exists.
It follows from Lemma 2.1(i) and (3.7) that
|
|x
n+1
− x
n
||
2
= ||x
n+1
− x
1
− (x
n
− x
1
)||
2
= ||x

n+1
− x
1
||
2
−||x
n
− x
1
||
2
− 2x
n
− x
n+1
, x
1
− x
n

≤
||
x
n+1
− x
1
||
2
−
||

x
n
− x
1
||
2
.
Noting that lim
n®∞
||x
n
- x
1
|| exists, this implies
lim
n
→
∞
||x
n
− x
n+1
|| =0
.
(3:8)
It is easy to get
||
x
n+i
− x

n
||
→ 0, ∀
i
= 1, 2, ,
N
,asn →∞
.
(3:9)
Since x
n+1
Î C
n
, we have
||
y
n
− x
n+1
||
2
≤||x
n
− x
n+1
||
2
+ θ
n
.

So, we get lim
n®∞
||y
n
- x
n+1
|| = 0. It follows that
|
|
y
n
− x
n
|| ≤ ||
y
n
− x
n+1
|| + ||x
n
− x
n+1
|| → 0, as n →∞
.
(3:10)
Next we will show that
lim
n
→∞
||

k
n
x
n
− 
k−1
n
x
n
|| =0, k =1,2, , M
.
(3:11)
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 7 of 13
Indeed, for p Î Ω, it follows from the firmly nonexpansivity of
T
F
k
r
k
,n
that for each k Î
{1, 2, , M}, we have
||
k
n
x
n
− p||
2

= ||T
F
k
r
k,n

k−1
n
x
n
− T
F
k
r
k,n
p||
2
≤
k
n
x
n
− p, 
k−1
n
x
n
− p
=
1

2
(||
k
n
x
n
− p||
2
+ ||
k−1
n
x
n
− p||
2
−||
k
n
x
n
− 
k−1
n
x
n
||
2
)
.
Thus we get

||
k
n
x
n
− p||
2
≤||
k−1
n
x
n
− p||
2
−||
k
n
x
n
− 
k−1
n
x
n
||
2
, k =1,2, , M
,
which implies that for each k Î {1, 2, , M},
|

|
k
n
x
n
− p||
2
≤||
0
n
x
n
− p||
2
−||
k
n
x
n
− 
k−1
n
x
n
||
2
−||
k−1
n
x

n
− 
k−2
n
x
n
||
2
−···−||
2
n
x
n
− 
1
n
x
n
||
2
−||
1
n
x
n
− 
0
n
x
n

||
2
≤||x
n
− p||
2
−||
k
n
x
n
− 
k−1
n
x
n
||
2
.
(3:12)
Therefore, by the convexity of ||·||
2
, (3.5) and the nonexpansivity of
T
F
k
r
k
,n
, we get

|
|y
n
− p||
2
≤||u
n
− p||
2
+ θ
n
= ||
M
n
x
n
− 
M
n
p||
2
+ θ
n
≤||
k
n
x
n
− p||
2

+ θ
n
≤||x
n
− p||
2
−||
k
n
x
n
− 
k−1
n
x
n
||
2
+ θ
n
.
It follows that
||
k
n
x
n
−
k−1
n

x
n
||
2
≤||x
n
−p||
2
−||y
n
−p||
2
+θ
n
≤||x
n
−y
n
||(||x
n
−p||+ ||y
n
−p||)+θ
n
.
(3:13)
From (3.10) and (3.13), we obtain (3.11). Then, we have
||u
n
− x

n
|| ≤ ||u
n
− 
M−1
n
x
n
|| + ||
M−1
n
x
n
− 
M−2
n
x
n
|| + ···+ ||
1
n
x
n
− x
n
|| → 0
.
(3:14)
Combining (3.8) and (3.14), we have
||

u
n+1
− u
n
||
≤
||
u
n+1
− x
n+1
||
+
||
x
n+1
− x
n
||
+
||
x
n
− u
n
||
→ 0, as n →∞
.
(3:15)
It follows that

||
u
n+i
− u
n
||
→ 0, ∀
i
= 1, 2, ,
N
,asn →∞
.
(3:16)
Step 4. Show that ||u
n
- S
i
u
n
|| ® 0, ||x
n
- S
i
x
n
|| ® 0, as n ® ∞; ∀i Î {1, 2, , N}.
Since, for any positive integer n ≥ N,itcanbewrittenasn =(h(n)-1)N + i(n),
where i(n) Î {1, 2, , N}. Observe that
|
|u

n
− S
n
u
n
|| ≤ ||u
n
− S
h(n)
i(n)
u
n
|| + ||S
h(n)
i(n)
u
n
− S
n
u
n
||
= ||u
n
− S
h(n)
i
(
n
)

u
n
|| + ||S
h(n)
i
(
n
)
u
n
− S
i(n)
u
n
||
.
(3:17)
From (3.10), (3.14), the conditions 0 <a ≤ a
n
≤ 1 and 0 <δ ≤ b
n
≤ 1-l, we obtain
||S
h(n)
i(n)
u
n
− u
n
|| =

1
β
n
||z
n
− u
n
||
=
1
α
n
β
n
||y
n
− u
n
||
≤
1
a
δ
(||y
n
− x
n
|| + ||u
n
− x

n
||) → 0, as n →∞
.
(3:18)
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 8 of 13
Next, we prove that
lim
n
→∞
||S
h(n)−1
i(n)
u
n
− u
n
|| =0
.
(3:19)
It is obvious that the relations hold: h(n)=h(n - N)+1,i(n)=i(n - N).
Therefore,
||S
h(
n
)
−1
i(n)
u
n

− u
n
|| ≤ ||S
h(
n
)
−1
i(n)
u
n
− S
h(
n
)
−1
i(n−N)
u
n−N+1
|| + ||S
h(
n
)
−1
i(n−N)
u
n−N+1
− S
h(
n−N
)

i(n−N)
u
n−N
|
|
+ ||S
h(n−N)
i(n−N)
u
n−N
− u
n−N
|| + ||u
n−N
− u
n−N+1
|| + ||u
n−N+1
− u
n
||
= ||S
h(n)−1
i(n)
u
n
− S
h(n)−1
i(n)
u

n−N+1
|| + ||S
h(n−N)
i(n−N)
u
n−N+1
− S
h(n−N)
i(n−N)
u
n−N
||
+ ||S
h(n−N)
i
(
n−N
)
u
n−N
− u
n−N
|| + ||u
n−N
− u
n−N+1
|| + ||u
n−N+1
− u
n

||.
(3:20)
Applying Lemma 2.5 and (3.16), we get (3.19). Using the uniformly continuity of S
i
,
we obtain
lim
n
→∞
||S
h(n)
i(n)
u
n
− S
i(n)
u
n
|| =0
,
(3:21)
this together with (3.17) yields
lim
n
→
∞
||u
n
− S
n

u
n
|| =0
.
We also have
||
u
n
− S
n+i
u
n
||
≤
||
u
n
− u
n+i
||
+
||
u
n+i
− S
n+i
u
n+i
||
+

||
S
n+i
u
n+i
− S
n+i
u
n
||
→ 0, as n →∞
,
for any i =1,2, N, which gives that
lim
n
→
∞
||u
n
− S
i
u
n
|| =0;∀i =1,2, N
.
(3:22)
Moreover, for each i Î {1, 2, N}, we obtain that
||
x
n

− S
i
x
n
||
≤
||
x
n
− u
n
||
+
||
u
n
− S
i
u
n
||
+
||
S
i
u
n
− S
i
x

n
||
→ 0, as n →∞
.
(3:23)
Step 5. The following implication holds:
ω
w
(
x
n
)
⊂ 
.
(3:24)
We first show that
ω
w
(x
n
) ⊂∩
N
i
=1
F(S
i
)
.Tothisend,wetakeω Î ω
w
(x

n
) and assume
that
x
n
j
ω
as j ® ∞ for some subsequence
{x
n
j
}
of x
n
.
Note that S
i
is uniformly continuous and (3.23), we see that
||x
n
− S
m
i
x
n
|| →
0
, for all
m Î N. So by Lemma 2.4, it follows that
ω ∈∩

N
i
=1
F(S
i
)
and hence
ω
w
(x
n
) ⊂∩
N
i
=1
F(S
i
)
.
Next we will show that
ω ∈∩
M
k
=1
EP(F
k
)
. Indeed, by Lemma 2.6, we have that for each
k = 1, 2, , M,
F

k
(
k
n
x
n
, y)+
1
r
n
y − 
k
n
x
n
, 
k
n
x
n
− 
k−1
n
x
n
≥0, ∀y ∈ C
.
From (A2), we get
1
r

n
y − 
k
n
x
n
, 
k
n
x
n
− 
k−1
n
x
n
≥F
k
(y, 
k
n
x
n
), ∀y ∈ C
.
Hence,
y − 
k
n
j

x
n
j
,

k
n
j
x
n
j
− 
k−1
n
j
x
n
j
r
n
j
≥F
k
(y, 
k
n
j
x
n
j

), ∀y ∈ C
.
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 9 of 13
From (3.11), we obtain that

k
n
j
x
n
j

ω
as j ® ∞ for each k = 1, 2, , M (especially,
u
n
j
= 
M
n
j
x
n
j
). Together with (3.11) and (A4) we have, for each k = 1, 2, , M, that
0 ≥ F
k
(
y, ω

)
, ∀y ∈ C
.
For any, 0 <t ≤ 1 and y Î C, let y
t
= ty +(1-t)ω. Since y Î C and ω Î C, we obtain
that y
t
Î C and hence F
k
(y
t
, ω) ≤ 0. So, we have
0=F
k
(
y
t
, y
t
)
≤ tF
k
(
y
t
, y
)
+
(

1 − t
)
F
k
(
y
t
, ω
)
≤ tF
k
(
y
t
, y
).
Dividing by t, we get, for each k = 1, 2, , M, that
F
k
(
y
t
, y
)
≥ 0, ∀y ∈ C
.
Letting t ® 0 and from (A3), we get
F
k
(

ω, y
)
≥
0
for all y Î C and ω Î EP(F
k
) for each k = 1, 2, , M, i.e.,
ω ∈∩
M
k
=1
EP(F
k
)
.
Hence (3.24) holds.
Step 6. Show that x
n
® q = P
Ω
x
1
.
From (3.6), (3.24) and Lemma 2.3, we conclude that x
n
® q, where q = P
Ω
x
1
. □

Corollary 3.2. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH
and N ≥ 1 be an integer, let F be a bifunction from C × Ctoℝ which satisfies condi-
tions (A1)-(A4).Let,foreach1 ≤ i ≤ N, S
i
: C ® C be a uniformly continuous l
i
-strict
asymptotically pseudocontractive mapping in the in termediate sense for some 0 ≤ l
i
<1
with sequences {g
n,i
} ⊂ [0, ∞) such that lim
n®∞
g
n,i
=0and {c
n,i
} ⊂ [0, ∞) such that
lim
n®∞
c
n
,
i
=0.Letl =max{l
i
:1≤ i ≤ N}, g
n
=max{g

n, i
:1≤ i ≤ N} and c
n
=max
{c
n,i
:1≤ i ≤ N}. Assume that

= ∩
N
i
=1
F(S
i
) ∩ EP(F
)
is non empty and bounded. Let {a
n
}
and {b
n
} be sequences in [0, 1] such that 0 < a ≤ a
n
≤ 1,0 <δ ≤ b
n
≤ 1-l for all n Î
N and {r
n
} ⊂ (0,∞) satisfies lim inf
n®∞

r
n
> 0 for all k Î {1, 2, M}.
Let {x
n
} and {u
n
} be sequences generated by the following algorithm:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x

1
∈ C chosen arbitrary,
u
n
= T
F
r
n
x
n
,
z
n
=(1− β
n
)u
n
+ β
n
S
h(n)
i(n)
u
n
,
y
n
=(1− α
n
)u

n
+ α
n
z
n
,
C
n
= {v ∈ C : ||y
n
− v||
2
≤||x
n
− v||
2
+ θ
n
}
,
Q
n
= {v ∈ C : x
n
− v, x
1
− x
n
≥0},
x

n+1
= P
C
n
∩Q
n
x
1
, ∀n ∈ N,
(3:25)
where
θ
n
= c
h
(
n
)
+ γ
h
(
n
)
ρ
2
n
→
0
,asn ® ∞, where r
n

= sup{||x
n
- v|| : v Î Ω}<∞ .
Then {x
n
} converges strongly to P
Ω
x
1
.
Proof. Putting M = 1, we can draw the desired conclusion from Theorem 3.1.
□
Remark 3.3. Corollary 3.2 extends the theorem of Tada and Takahashi [14] from a
nonexpansive mapping to a finite family of asymptotically l
i
-strict pseudocontractive
mappings in the intermediate sense.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and
N ≥ 1 be an integer, let, for each 1 ≤ i ≤ N, S
i
:C® C be a uniformly continuous l
i
-strict
asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ l
i
<1
with sequences {g
n,i
} ⊂ [0, ∞) such that lim
n®∞

g
n,i
=0and {c
n,i
} ⊂ [0, ∞) such that
lim
n®∞
c
n,i
=0. Let l= max{l
i
:1≤ i ≤ N}, g
n
= max{g
n,i
:1≤ i ≤ N} and c
n
= max{c
n,i
:1
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 10 of 13
≤ i ≤ N}. Assume that

= ∩
N
i
=1
F(S
i

)
is nonempty and bounded. Let {a
n
} and {b
n
} be
sequences in [0, 1] such that 0 <a≤ a
n
≤ 1, 0 < δ ≤ b
n
≤ 1-l for all n Î N. Let {x
n
} and
{u
n
} be sequences generated by the following algorithm:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
x
1
∈ C chosen arbitrary,

y
n
=(1− β
n
)x
n
+ β
n
S
h(n)
i(n)
x
n
,
C
n
= {v ∈ C : ||y
n
− v||
2
≤||x
n
− v||
2
+ θ
n
}
,
Q
n

= {v ∈ C : x
n
− v, x
1
− x
n
≥0},
x
n+1
= P
C
n
∩
Q
n
x
1
, ∀n ∈ N,
(3:26)
where
θ
n
= c
h
(
n
)
+ γ
h
(

n
)
ρ
2
n
→
0
, as n ® ∞, where r
n
=sup{||x
n
- v|| : v Î Ω } <∞.
Then {x
n
} converges strongly to P
Ω
x
1
.
Proof.IfF
k
(x, y)=0,a
n
= 1 in Theorem 3.1, we can draw the conclusion easily. □
Remark 3.5. Corollary 3.4 extends the Theorem 4.1 of [4] and Theorem 2.2 of [ 13],
respectively.
4. Numerical result
In this section, in order to demonstrate the effectiveness, realization and convergence
of the algorithm in Theorem 3.1, we consider the following simple example ever
appeared in the reference [4]:

Example 4.1. Let x = R and C = [0, 1] For each x Î C, we define
Tx =

kx, if x ∈ [0,
1
2
],
0, if x ∈ (
1
2
,1]
,
where 0<k <1.
Set C
1
: = [0, 1/2] and C
2
: = (1/2, 1]. Hence,
|
T
n
x − T
n
y
| = k
n
|x −
y
|≤|x −
y

| for all x,
y
∈ C
1
and n ∈
N
and
|
T
n
x − T
n
y
| =0≤|x −
y
| for all x,
y
∈ C
2
and n ∈ N
.
For x Î C
1
and y Î C
2
, we have
|T
n
x − T
n

y
| = |k
n
x − 0|≤k
n
|x −
y
| + k
n
|
y
|≤|x −
y
| + k
n
for all n ∈ N
.
Thus
|
T
n
x − T
n
y|
2
≤
(
|x − y| + k
n
)

2
≤|x − y|
2
+ k|x − T
n
x −
(
y − T
n
y
)
|
2
+ k
n
K
.
for all x, y Î C, n Î N and some K >0.Therefore,T is an asymptotically k-strict
pseudocontractive mapping in the intermediate sense.
In the algorithm (3.1), set
F
k
(x, y)=0,N =1,β
n
=1− k, α
n
=
n +1
2
n

.Weapplyitto
find the fixed point of T of Example 4.1.
Under the above assumptions, (3.1) is simplified as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
∈ C chosen arbitrary ,
z

n
= kx
n
+(1− k)T
n
x
n
,
y
n
=
n − 1
2n
x
n
+
n +1
2n
z
n
,
C
n
= {v ∈ C : |y
n
− v|
2
≤|x
n
− v|

2
+ θ
n
}
,
Q
n
= {v ∈ C :(x
n
− v)(x
1
− x
n
) ≥ 0},
x
n+1
= P
C
n
∩
Q
n
x
1
, ∀n ∈ N,
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 11 of 13
In fact, in one dimensional case, the C
n
⋂ Q

n
is an closed interval. If we set [a
n
, b
n
]:
= C
n
⋂ Q
n
, then the projection point x
n+1
of x
1
Î C onto C
n
⋂ Q
n
can be expressed as:
x
n+1
= P
C
n
∩Q
n
x
1
=
⎧

⎪
⎨
⎪
⎩
x
1
,ifx
1
∈ [a
n
, b
n
]
,
b
n
,ifx
1
> b
n
,
a
n
,ifx
1
< a
n
.
Since the conditions of Theorem 3.1 are satisfied in Example 4.1, the conclusion
holds, i.e., x

n
® 0 Î F (T).
Now we turn to reali zing (3.1) for approximati ng a fixed point of T. Take the initial
guess x
1
= 1/2, 1/5 and 5/8, respectively. All the numerical results a re given in Tables
1, 2 and 3. The corresponding graph appears in Figure 1a,b,c.
Table 1 x
1
= 0.5
n (iterative number) x
1
(initial guess) Errors (n)
5 0.2471 2.471 × 10
-1
20 0.0527 5.27 × 10
-2
50 0.0028 2.8 × 10
-3
93 0.0000 0
Table 2 x
1
= 0.2
n (iterative number) x
1
(initial guess) Errors (n)
5 0.0998 9.98 × 10
-2
20 0.0211 2.11 × 10
-2

50 0.0011 1.1 × 10
-3
83 0.0000 0
Table 3
x
1
=
5
8
n (iterative number) x
1
(initial guess) Errors (n)
5 0.2636 2.636 × 10
-1
20 0.0562 5.62 × 10
-2
50 0.0030 3.0 × 10
-3
93 0.0000 0
(a)
0 10 20 30 40 50 60 70 80 90
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4

0
.45
(b)
0 10 20 30 40 50 60 70 80 90
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0
.
18
(c)
0 10 20 30 40 50 60 70 80 9
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
.45

Figure 1 The iteration comparison chart of different initial values. (a) x
1
=0.5;(b) x
1
=0.2;(c)
x
1
=
5
8
.
Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13
/>Page 12 of 13
Acknowledgements
The authors would like to thank the reviewers for their good suggestions. This research is supported by Fundamental
Research Funds for the Central Universities (ZXH2011C002).
Authors’ contributions
PD carried out the proof of convergence of the theorems and realization of numerical examples. JZ carried out the
check of the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 22 January 2011 Accepted: 5 July 2011 Published: 5 July 2011
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doi:10.1186/1687-1812-2011-13
Cite this article as: Duan and Zhao: Strong convergence theorems for system of equilibrium problems and
asymptotically strict pseudocontractions in the intermediate sense. Fixed Point Theory and Applications 2011
2011:13.
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