WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS

The Pure and Applied Mathematics.
2014.
Aug,
21(3):
195-206

- Received : April 10, 2014
- Accepted : June 19, 2014
- Published : August 31, 2014

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In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically
k
-strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al.
[1]
.
x
∈
C
such that
where
ϕ
:
C
×
C
→ ℝ is a bifunction.
On the other hand, the problem of finding a common fixed point of a family of mappings is a classical problem in nonlinear analysis. Finding an optimal point in the set of common fixed points of a family of mappings is a task that occurs frequently in various areas of mathematical sciences and engineering. For example, the convex feasibility problem reduces to finding a point of the set of common fixed points of a family of nonexpansive mappings
[4]
.
In 2009, Qin et al. considered the following weak convergence theorem to a common fixed point of a finite family of asymptotically
k
-strictly pseudo-contractive mappings under a hybrid iterative scheme.
Theorem 1.1
(
[5]
).
Assume the following conditions;
Assume that F
:= (
F
(
T_{i}
)) ≠
.
For any x_{0}
∈
C
,
let
{
x_{n}
}
be a sequence generated by
where
{
a_{n}
}
is a sequence in
(0, 1)
such that k
+
Ɛ
≤
a_{n}
≤ 1 −
Ɛ
for some
Ɛ
∈ (0, 1)
and n
= (
h
− 1)
N
+
i
(
n
≥ 1),
where i
=
i
(
n
) ∈ {1, 2, ⋯,
N
},
h
=
h
(
n
) ≥ 1
is a positive integer and h
(
n
) → ∞
as n
→ ∞.
Then
{
x_{n}
}
converges weakly to an element of F
.
The existence of solutions of equilibrium problems and common fixed points of finite mappings are very important in nonlinear analysis with applications. Moreover, to find the intersection of solution sets of equilibrium problems and common fixed points of finite mappings and to apply the intersection are also important. Recently, there have been a few works for the intersection of the two sets to be the set of weakly convergent points of two given sequences in Hilbert spaces.
In 2000, Kumam et al. considered the following weak convergence theorem to a given common element of the set of common xed points of a finite family of asymptotically
k
-strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction under a hybrid iterative scheme.
Theorem 1.2
(
[1]
). Assume the conditions (1)-(5) in Theorem 1.1. Let
ϕ
:
C
×
C
→ ℝ be a bifunction satisfying the followings;
Assume that
F
:= (
F
(
T_{i}
)) ∩
EP
(
ϕ
) ≠
, where
EP
(
ϕ
) is a set of solutions of equilibrium problem (1.1). For any
x
_{0}
∈
C
, let {
x_{n}
} and {
υ_{n}
} be sequences generated by
where
n
= (
h
− 1)
N
+
i
(
n
≥ 1),
i
=
i
(
n
) ∈ {1, 2,⋯,
N
},
h
=
h
(
n
) ≥ 1 is a positive integer and
h
(
n
) → ∞ as
n
→∞. Let {
a_{n}
} and {
r_{n}
} satisfy the following conditions:
Then {
x_{n}
} and {
υ_{n}
} converges weakly to an element of
F
.
On the other hand, the fixed point iterative scheme with errors was introduced by Liu
[6]
. The idea of considering fixed point iterative scheme problems with errors which comes from practical numerical computation usually concerns the approximation fixed point and is related to the stability of fixed point iterative schemes. The idea of considering iterative scheme procedures with errors leads to finding the approximate solution to equilibrium problems. In 2005, Combettes and Hirstoga
[3]
introduced an iterative scheme for a problem of finding best approximate solutions to equilibrium problem and proved the strong convergence result.
In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically
k
-strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al.
[1]
.
The following results will be needed in the main result.
Lemma 1.1
(
[7
,
8]
).
Let H be a real Hilbert space. There hold the following identities
Lemma 1.2
(
[3]
).
Assume that ϕ
:
C
×
C
→ ℝ
satisfies (A1)-(A4). For r
> 0
and x
∈
H
,
define a mapping S_{r}
:
H
→
C as follows;
for all z
∈
H. Then the following hold;
Definition 2.1.
A mapping
T
:
C
→
C
is said to be
asymptotically k-strictly pseudo-contractive
if there exist a sequence {
k_{n}
} ⊂ [1, ∞) with
k_{n}
= 1 and
k
∈ [0, 1) such that
∥
T^{n}x
−
T^{n}y
∥
^{2}
≤
∥
x
−
y
∥
^{2}
+
k
∥(
I
−
T^{n}
)
x
− (
I
−
T^{n}
)
y
∥
^{2}
, ∀
x
,
y
∈
C
and
n
∈ ℕ.
The following proposition by Osilike and Igbokwe
[8]
was considered by using infinite terms of the given sequences based on Lemma 1 in
[9]
, but our proof is considered by using only finite terms of the given sequences based on the basic concepts of limit superior and limit inferior.
Proposition 2.1.
Let
{
a_{n}
}, {
c_{n}
}
and
{
δ_{n}
}
be nonnegative real sequences satisfying the following condition:
if
and
then
a_{n}
exists
.
Proof
. Consider
Thus
Since
and
for any
ε
> 0, take
N
∈ ℕ such that
and
for n ≥
N
. Thus, lim sup
_{m→∞}
a_{m}
≤
e^{ε}a_{n}
+
εe^{ε}
. Letting
ε
→ 0, we have the wanted result. Hence lim sup
_{m→∞}
a_{m}
≤ lim inf
_{n→∞}
a_{n}
, which shows the existence of
a_{n}
⧠
Putting
δ_{n}
= 0( ∀
n
∈ ℕ), we have the following known lemma as a corollary;
Lemma 2.1
(
[9]
).
Let
{
a_{n}
}
and
{
b_{n}
}
be nonnegative real sequences satisfying the following condition:
If
then
a_{n} exists
.
Now, we prove our main result.
Theorem 2.1.
Assume the conditions (1)-(5) in Theorem 1.1. Let ϕ
:
C
×
C
→ ℝ
be a bifunction satisfying (A1)-(A4). Assume that F
:= (
F
(
T_{i}
))∩
EP
(
ϕ
)≠
.
For any x
_{0}
∈
C
,
let
{
x_{n}
} and {
v_{n}
}
be sequences generated by
where
{
a_{n}
}, {
b_{n}
}
and
{
c_{n}
}
are sequences in
[0; 1)
such that a_{n}
+
b_{n}
+
c_{n}
= 1,
a_{n}
≥
k
+
ε
,
b_{n}
≥
ε for some ε
∈ (0, 1),
{
u_{n}
}
is a bounded sequence in C
, {
r_{n}
}
is a sequence in
(0, ∞)
such that
inf
r_{n}
≥ 0
and n
= (
h
−1)
N
+
i
(
n
≥ 1),
where i
=
i
(
n
) ∈ {1, 2,⋯,
N
},
h
=
h
(
n
) ≥ 1
is a positive integer and h
(
n
) → ∞
as
n
→ ∞.
Then
{
x_{n}
}
and
{
v_{n}
}
converges weakly to an element of F
.
Proof
. Let
p
∈
F
. From (2.1) and Lemma 1.2, we have
v
_{n−1}
=
S
_{rn−1}
x
_{n−1}
and
From (2.1) and Lemma 1.1(ii),
From Proposition 2.1,
∥
x_{n}
−
p
∥ exists. Observe (2.2) again
Since
a_{n}
≥
k
+
ε
,
b_{n}
≥
ε
for all
n
≥ 0 and some
ε
∈ (0, 1),
Taking the limits as
n
→ ∞, we have
Observe that
It follows that
Since
S
_{rn−1}
is firmly nonexpansive, we have
and hence
Using (2.2) and (2.7), we have
hence
Since
∥
x_{n}
−
p
∥ exists and
k
_{h(n)}
= 1,
From (2.6) and (2.8), we have
It follows that
Applying (2.8) and (2.9), we obtain
which implies that
∥
x_{n}
−
x
_{n+j}
∥ = 0, ∀
j
∈ {1, ⋯,
N
}. Since, for any positive integer
n
>
N
, it can be written as
n
= (
k
(
n
)−1)
N
+
i
(
n
), where
i
(
n
) ∈ {1, 2,⋯,
N
}, observe that
Since, for each
n
>
N
,
n
≡
n
−
N
(mod
N
) and
n
= (
k
(
n
) − 1)
N
+
i
(
n
), we have
n
−
N
= (
k
(
n
)−1)
N
+
i
(
n
) = (
k
(
n
−
N
)−1)
N
+
i
(
n
−
N
), that is,
i
(
n
−
N
) =
i
(
n
),
k
(
n
−
N
) =
k
(
n
) − 1. Observe that
and
It follows from (2.11)-(2.13) that
Applying (2.5) and (2.10) to (2.14), we obtain
From (2.9) and (2.15),
Also, we have
for any
j
= 1, ⋯,
N
, which gives that
Moreover, for each
l
∈ {1, 2,⋯,
N
}, we have
Put
W
(
x_{n}
) = {
x
∈
H
:
x_{ni}
⇀
x
for some subsequence {
x_{ni}
} of {
x_{n}
}}.
Firstly,
W
(
x_{n}
) ≠ø. Indeed, since {
x_{n}
} is bounded and
H
is reflexive,
W
(
x_{n}
) ≠ø.
Secondly, we claim that
Let
w
∈
W
(
x_{n}
) be an arbitrary element. Then there exists a subsequence {
x_{ni}
} of {
x_{n}
} converging weakly to
w
. Applying (2.8), we can obtain that
v_{ni}
⇀
w
as
i
→ ∞. It follows from
∥
v_{n}
−
T_{l}v_{n}
∥ = 0 that
T_{l}v_{ni}
→
w
, ∀
l
∈ {1, ⋯,
N
}. Let us show that
w
∈
EP
(
ϕ
). Since
v_{n}
=
T_{rn}
v_{n}
, we have
From (A2), we have
and hence
Since
→ 0 and
as
i
→ ∞, from (A4), we have
For
t
∈ (0; 1] and
y
∈
C
, let
y_{t}
=
t_{y}
+(1−
t
)
w
. Since
y
∈
C
and
w
∈
C
,
y_{t}
∈
C
, and hence
ϕ
(
y_{t}
,
w
) ≤ 0. So, from (A1) and (A4),
and hence 0 ≤
ϕ
(
y_{t}
,
y
). From (A3), 0 ≤
ϕ
(
w
,
y
), ∀
y
∈
C
and hence
w
∈
EP
(
ϕ
)
Next, we prove that
w
∈ (
F
(
T_{i}
)). Suppose that
w
∉ (
F
(
T_{i}
)). Then there exists
l
∈ {1, ⋯,
N
} such that
w
∉
F
(
T_{l}
). From (2.16) and the Opial’s condition,
which derives a contradiction. Hence
w
∈ (
F
(
T_{i}
)).
Finally, we show that {
x_{n}
} and {
v_{n}
} converge weakly to an element of
F
. Indeed, it is sufficient to show that
W
(
x_{n}
) is a single point set. We take
w
_{1}
,
w
_{2}
∈
W
(
x_{n}
) arbitrarily and let {
x_{ni}
} and {
x_{nj}
} be subsequences of {
x_{n}
} such that
x_{ni}
⇀
w
_{1}
and
x_{nj}
⇀
w
_{2}
. Since
∥
x_{n}
−
p
∥ exists for each
p
∈
F
and
w
_{1}
,
w
_{2}
∈
F
, by Lemma 1.1 (iii), we obtain
Hence
w
_{1}
=
w
_{2}
, which shows that
W
(
x_{n}
) is single point set. ⧠
Remark 2.1.
(1) If
c_{n}
= 0(∀
n
∈
N
) in Theorem 2.1, then we obtain Theorem 1.2. (2) If
ϕ
(
x
,
y
) = 0(∀
x,y
∈
C
) and
v_{n}
=
x_{n}
, ∀
n
∈ ℕ in (2.1), then we obtain Theorem 1.1.

equilibrium problems
;
fixed point problems
;
asymptotically k-strictly pseudo-contractive mappings
;
hybrid iterative scheme

1. Introduction and Preliminaries

The equilibrium problems were introduced by Blum and Oettli
[2]
in 1994. Numerous problems in applied sciences, for example optimization problems, saddle point problems, variational inequality problems and Nash equilibria in noncoopera- tive games, are reduced to find a solution of the following equilibrium problem
[2
,
3]
; finding
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- (1)C is a closed convex subset of a Hilbert spaceH,
- (2)Ti:C→Cis an asymptotically ki-strictly pseudo-contractive mapping,
- where1 ≤i≤N for some natural number N and0 ≤ki< 1,
- (3) {kn,i}is a sequence in[1, ∞)such that< ∞
- (4)k= max{ki: 1 ≤i≤N}and
- (5) {kn}is a sequence defined by kn= max{kn,i: 1 ≤i≤N}for n∈ ℕ.

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- xn=an−1xn−1+ (1 −an−1)∀n≥ 0,

- (A1)ϕ(x,x) = 0, ∀x∈C;
- (A2)ϕis monotone, i.e.,ϕ(x,y) +ϕ(y,x) ≤ 0 for anyx,y∈C;
- (A3)ϕis upper-hemicontinuous, i.e., for eachx,y,z∈C,
- ϕ(tz+ (1 −t)x,y) ≤ϕ(x,y);
- (A4)ϕ(x, · ) is convex and lower semicontinuous for eachx∈C.

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- (1) {an} ⊂ [α,β], for someα,β∈ (k; 1) and
- (2) {rn} ⊂ (0, ∞) and

- (i) ∥x−y∥2=∥x∥2−∥y∥2− 2 ⟨x−y,y⟩, ∀x,y∈H
- (ii) ∥ax+by+cz∥2=a∥x∥2+b∥y∥2+c∥z∥2−ab∥x−y∥2−bc∥y−z∥2−ca∥z−x∥2, ∀x,y∈H,where a,b,c∈ [0, 1]with a+b+c= 1,
- (iii)If{xn}is a sequence inHweakly converging to z, then
- sup∥xn−y∥2=sup ∥xn−z∥2+ ∥z−y∥2, ∀y∈H.

- Sr(x) = {z∈C:ϕ(z,y)+⟨y−z,z−x⟩ ≥ 0, ∀y∈C},

- (i)Sris single-valued;
- (ii)Sris firmly nonexpansive, i.e., for any x,y∈H,
- ∥Srx−Sry∥2≤ ⟨Srx−Sry,x−y⟩;(iii)F(Sr) =EP(ϕ);
- (iv)EP(ϕ)is closed and convex.

2. Main Results

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- an+1≤ (1 +δn)an+cn, ∀n∈ ℕ.

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- an+1≤an+bn, ∀n∈ ℕ

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- ∥vn−1−p∥ = ∥Srn−1xn−1−Srn−1p∥ ≤ ∥xn−1−p∥, ∀n≥ 0.

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- ∥xn−p∥2≤∥vn−1−p∥2+cn-1∥un−1−p∥2
- ≤∥xn−1−p∥2− ∥xn−1−vn−1∥2+cn−1∥un−1−p∥2,

- ∥xn−1−vn−1∥2≤∥xn−1−p∥2− ∥xn-1−p∥2+cn−1∥un−1−p∥2.

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- ∥xn−xn−1∥ = ∥xn−vn+vn−vn−1+vn−1−xn−1∥
- ≤ ∥xn−vn∥+∥vn−vn−1∥+∥vn−1−xn−1∥ →0 asn→∞,

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- ∥vn−Tnvn∥ ≤ ∥vn−vn−1∥ + ∥vn−1−Tnvn−1∥ + ∥Tnvn−1−Tnvn∥
- ≤ (1 +L) · ∥vn−vn−1∥ + ∥vn−1−Tnvn−1∥ → 0 asn→ ∞

- ∥vn−Tn+jvn∥ ≤ ∥vn−vn+j∥ + ∥vn+j−Tn+jvn+j∥ + ∥Tn+jvn+j−Tn+jvn∥
- ≤ (1 +L) · ∥vn−vn+j∥ + ∥vn+j−Tn+jvn+j∥ → 0 asn→ ∞

- ∥vn−Tlvn∥ = 0, ∀l∈ {1, ⋯,N}.

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- ϕ(vn,y) +⟨y−vn,vn−xn⟩ ≥ 0, ∀y∈C.

- ⟨y−vn,vn−xn⟩ ≥ϕ(y,vn)

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- ϕ(y,w) ≤ 0, ∀y∈C.

- 0 =ϕ(yt,yt) ≤tϕ(yt,y) + ( 1−t)ϕ(yt,w) ≤tϕ(yt,y)

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Kumam P.
,
Petrot N.
,
Wangkeeree R.
2010
A hybrid iterative scheme for equilibrium problems and fixed point problems of asymptotically k-strict pseudo-contractions
J. Comp. Appl. Math.
233
2013 -
2026
** DOI : 10.1016/j.cam.2009.09.036**

Blum E.
,
Oettli W.
1994
From optimization and variational inequalities to equilibrium problems
Math. Student
63
123 -
145

Combettes P.L.
,
Hirstoaga S.A.
2005
Equilibrium programming in Hilbert spaces
J. Nonlinear Convex Anal.
6
117 -
136

Bauschke H.H.
,
Borwein J.M.
1996
On projection algorithms for solving convex feasibility problems
SIAM Rev.
38
367 -
426
** DOI : 10.1137/S0036144593251710**

Qin X.
,
Cho Y.J.
,
Kang S.M.
,
Shang M.
2009
A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces
Nonlinear Anal.
70
1902 -
1911
** DOI : 10.1016/j.na.2008.02.090**

Liu L.S.
1995
Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces
J. Math. Anal. Appl.
194
114 -
115
** DOI : 10.1006/jmaa.1995.1289**

Marino G.
,
Xu H.K.
2007
Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space
J. Math. Anal. Appl.
329
336 -
346
** DOI : 10.1016/j.jmaa.2006.06.055**

Osilike M.O.
,
Igbokwe D.I.
2000
Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations
Comput. Math. Appl.
40
559 -
567
** DOI : 10.1016/S0898-1221(00)00179-6**

Tan K.K.
,
Xu H.K.
1993
Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process
J. Math. Anal. Appl.
178
(2)
301 -
308
** DOI : 10.1006/jmaa.1993.1309**

Citing 'WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS
'

@article{ SHGHCX_2014_v21n3_195}
,title={WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS}
,volume={3}
, url={http://dx.doi.org/10.7468/jksmeb.2014.21.3.195}, DOI={10.7468/jksmeb.2014.21.3.195}
, number= {3}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={KIM, SEUNG-HYUN
and
LEE, BYUNG-SOO}
, year={2014}
, month={Aug}