- Research Article
- Open access
- Published:
Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems
Fixed Point Theory and Applications volume 2011, Article number: 852789 (2011)
Abstract
We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of -strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature.
1. Introduction
Let be a nonempty closed and convex subset of a real Hilbert space
. A mapping
is called monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ1_HTML.gif)
A mapping is called inverse-strongly monotone (see, e.g., [1, 2]) if there exists a positive real number
such that
, for all
. For such a case,
is called
-inverse-strongly monotone. A
-inverse-strongly monotone is sometime called
-cocoercive. A mapping
is said to be relaxed
-cocoercive if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ2_HTML.gif)
is said to be relaxed
-cocoercive if there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ3_HTML.gif)
A mapping is said to be
-Lipschitzian if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ4_HTML.gif)
Let be a nonlinear mapping. The variational inequality problem is to find an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ5_HTML.gif)
(See, e.g., [3, 4].) We will denote the set of solutions of the variational inequality problem (1.5) by .
A monotone mapping is said to be maximal if the graph
is not properly contained in the graph of any other monotone map, where
for a multivalued mapping
. It is also known that
is maximal if and only if for
for every
implies
. Let
be a monotone mapping defined from
into
and
a normal cone to
at
, that is,
. Define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ6_HTML.gif)
Then, is maximal monotone and
(see, e.g., [5]).
A mapping is said to be
-strictly pseudocontractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ7_HTML.gif)
for all . If
, then the mapping
is nonexpansive. A point
is called a fixed point of
if
. The fixed points set of
is the set
. Iterative approximation of fixed points of
-strictly pseudocontractive mappings have been studied extensively by many authors (see, e.g., [1, 6–9] and the references contained therein).
Let be a real-valued function and
a nonlinear mapping. Suppose
into
is an equilibrium bifunction. That is,
, forall
. The generalized mixed equilibrium problem is to find
(see, e.g., [10–12]) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ8_HTML.gif)
for all . We shall denote the set of solutions of this generalized mixed equilibrium problem by
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ9_HTML.gif)
If , then problem (1.8) reduces to equilibrium problem studied by many authors (see, e.g., [8, 13–17]), which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ10_HTML.gif)
for all . The set of solutions of (1.10) is denoted by
.
If , then problem (1.8) reduces to generalized equilibrium problem studied by many authors (see, e.g., [18–20]), which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ11_HTML.gif)
for all . The set of solutions of (1.11) is denoted by EP.
If , then problem (1.8) reduces to mixed equilibrium problem considered by many authors (see, e.g., [21–23]), which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ12_HTML.gif)
for all . The set of solutions of (1.12) is denoted by MEP.
The generalized mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [24]). Numerous problems in Physics, optimization, and economics reduce to find a solution of problem (1.8). Several methods have been proposed to solve the fixed-point problems, variational inequality problems and equilibrium problems in the literature (see, e.g., [5, 11, 12, 20, 25–30]).
Recently, Ceng and Yao [25] introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space . In their results, they imposed the following condition on a nonempty closed and convex subset
of
:
(E) is
-strongly convex and its derivative
is sequentially continuous from weak topology to the strong topology.
We remark here that this condition (E) has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space (see, e.g., [31, 32]). However, it is observed that the condition (E) does not include the case and
. Furthermore, Peng and Yao [21], R. Wangkeeree and R. Wangkeeree [30], and many other authors replaced condition (E) with the following conditions:
(B1) for each and
, there exists a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ13_HTML.gif)
or
(B2) is a bounded set.
Consequently, conditions (B1) and (B2) have been used by many authors in approximating solution to generalized mixed equilibrium (mixed equilibrium) problems in a real Hilbert space (see, e.g., [21, 30]).
Recently, Takahashi et al. [33] proved the following convergence theorem using hybrid method.
Theorem 1.1 (Takahashi et al. [33]).
Let be a nonempty closed and convex subset of a real Hilbert space
. Let
be a nonexpansive mapping of
into itself such that
. For
, define sequences
and
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ14_HTML.gif)
Assume that satisfies
. Then,
converges strongly to
.
Motivated by the results of Takahashi et al. [33], Kumam [28] studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem.
Theorem 1.2 (Kumam, [28]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
satisfying (A1)–(A4) and let
be a
-inverse-strongly monotone mapping of
into
. Let
be a nonexpansive mapping of
into
such that
. For
, define sequences
and
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ15_HTML.gif)
Assume that and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ16_HTML.gif)
Then, converges strongly to
.
Motivated by the ongoing research and the above-mentioned results, we introduce a new iterative scheme for finding a common element of the set of fixed points of an infinite family of-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the three afore mentioned sets. In our results, we use conditions (B1) and (B2) mentioned above. Our result extends many important recent results. Finally, we give some applications of our results.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
and let
be a nonempty closed and convex subset of
. The strong convergence of
to
is denoted by
as
.
For any point , there exists a unique point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ17_HTML.gif)
is called the metric projection of
onto
. We know that
is a nonexpansive mapping of
onto
. It is also known that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ18_HTML.gif)
for all . Furthermore,
is characterized by the properties
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ19_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ20_HTML.gif)
In the context of the variational inequality problem, (2.3) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ21_HTML.gif)
If is
-inverse-strongly monotone mapping of
into
, then it is obvious that
is
-Lipschitz continuous. We also have that for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ22_HTML.gif)
So, if , then
is a nonexpansive mapping of
into
.
For solving the generalized mixed equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
(A1) for all
,
(A2) is monotone, that is,
for all
,
(A3) for each ,
(A4)for each is convex and lower semicontinuous.
We need the following technical result.
Lemma 2.1 (R. Wangkeeree and R. Wangkeeree [30]).
Assume that satisfies (A1)–(A4) and let
be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ23_HTML.gif)
for all . Then, the following hold:
(1)for each ,
(2) is single-valued,
(3) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ24_HTML.gif)
(4),
(5) is closed and convex.
3. Main Results
Theorem 3.1.
Let be a nonempty closed and convex subset of a real Hilbert space
. For each
, let
be a bifunction from
satisfying (A1)–(A4),
a proper lower semicontinuous and convex function with assumption (
) or (
),
an
-inverse-strongly monotone mapping of
into
,
a
-inverse-strongly monotone mapping of
into
and for each
, let
be a
-strictly pseudocontractive mapping for some
such that
. Let
be a μ-Lipschitzian, relaxed
-cocoercive mapping of
into
. Suppose
. Let
,
,
,
and
be generated by
,
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ25_HTML.gif)
Assume that and
satisfy
(i),
(ii),
(iii),
(iv).
Then, converges strongly to
.
Proof.
For all and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ26_HTML.gif)
This shows that is nonexpansive for each
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ27_HTML.gif)
Since both and
are nonexpansive for each
and
, from (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ28_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ29_HTML.gif)
Let , then
is closed convex for each
. Now assume that
is closed convex for some
. Then, from definition of
, we know that
is closed convex for the same
. Hence,
is closed convex for
and for each
. This implies that
is closed convex for
. Furthermore, we show that
. For
,
. For
, let
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ30_HTML.gif)
which shows that , for all
, for all
. Thus,
, for all
, for all
. Hence, it follows that
, for all
. Therefore,
is well defined. Since
, for all
and
, for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ31_HTML.gif)
Also, as , by (2.1) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ32_HTML.gif)
From (3.7) and (3.8), we have that exists. Hence,
is bounded and so are
,
,
,
,
,
,
and
,
. For
, we have that
. By (2.4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ33_HTML.gif)
Letting and taking the limit in (3.9), we have
,
, which shows that
is Cauchy. In particular,
. Since,
is Cauchy and
is closed, there exists
such that
,
. Since
, therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ34_HTML.gif)
and it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ35_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ36_HTML.gif)
Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ37_HTML.gif)
Since ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ38_HTML.gif)
Hence, . From (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ39_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ40_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ41_HTML.gif)
Putting (3.17) into (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ42_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ43_HTML.gif)
Therefore, . Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ44_HTML.gif)
Since and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ45_HTML.gif)
Hence, . From (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ46_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ47_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ48_HTML.gif)
Putting (3.24) into (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ49_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ50_HTML.gif)
Therefore, . Then, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ51_HTML.gif)
Since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ52_HTML.gif)
But implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ53_HTML.gif)
Putting (3.29) into (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ54_HTML.gif)
Thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ55_HTML.gif)
But
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ56_HTML.gif)
Putting (3.32) into (3.31) and rearranging, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ57_HTML.gif)
Hence, ,
. Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ58_HTML.gif)
Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ59_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ60_HTML.gif)
By conditions (iii) and (iv), we have that . Now, (2.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ61_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ62_HTML.gif)
Using this last inequality, we obtain from (3.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ63_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ64_HTML.gif)
Since , we have
. Also since
and
, we have that
. By
and
,
, we have that
.Since
,
, we have for any
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ65_HTML.gif)
Furthermore, from the last inequality and using (A2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ66_HTML.gif)
Let for all
and
. This implies that
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ67_HTML.gif)
Since ,
, we obtain
,
. Furthermore, by the monotonicity of
, we obtain
. Then, by (A4) we obtain (noting that
,
since
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ68_HTML.gif)
Using (A1), (A4) and (3.44), we also obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ69_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ70_HTML.gif)
Letting , we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ71_HTML.gif)
This implies that . By following the same arguments, we can show that
.
Next, we show . Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ72_HTML.gif)
Since is relaxed
-cocoercive and by condition (iv), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ73_HTML.gif)
which shows that is monotone. Thus,
is maximal monotone. Let
. Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ74_HTML.gif)
On the other hand, from , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ75_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ76_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ77_HTML.gif)
which implies that . We have
and hence
. Therefore,
.
Noting that , we have by (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ78_HTML.gif)
for all . Since
and by the continuity of inner product, we obtain from the above inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ79_HTML.gif)
for all . By (2.3) again, we conclude that
. This completes the proof.
Corollary 3.2.
Let be a nonempty closed and convex subset of a real Hilbert space
. For each
, let
be a bifunction from
satisfying (A1)–(A4),
a proper lower semicontinuous and convex function with assumption (B1) or (B2),
an
-inverse-strongly monotone mapping of
into
,
a
-inverse-strongly monotone mapping of
into
and for each
, let
be a nonexpansive mapping such that
. Let
be a
-Lipschitzian, relaxed
-cocoercive mapping of
into
. Suppose
. Let
,
,
,
and
be generated by
,
,
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ80_HTML.gif)
Assume that , and
satisfy
(i),
(ii),
(iii),
(iv).
Then, converges strongly to
.
Let be a nonempty closed and convex cone in
and
an operator of
into
. We define the polar of
in
to be the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ81_HTML.gif)
Then, the element is called a solution of the complementarity problem if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ82_HTML.gif)
The set of solutions of the complementarity problem is denoted by . We shall assume that
satisfies the following conditions:
(E1) is
-inverse strongly monotone,
(E2).
Also, we replace conditions (B1) and (B2) with
(D1) for each and
there exist a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ83_HTML.gif)
(D2) is a bounded set.
Theorem 3.3.
Let be a nonempty closed and convex cone of a real Hilbert space
. For each
, let
be a bifunction from
satisfying (A1)–(A4),
a proper lower semicontinuous and convex function with assumption (
) or (
),
an
-inverse-strongly monotone mapping of
into
,
a
-inverse-strongly monotone mapping of
into
and for each
, let
be a
-strictly pseudocontractive mapping for some
such that
. Let
be a
-Lipschitzian, relaxed
-cocoercive mapping of
into
. Suppose
. Let
, and
be generated by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F852789/MediaObjects/13663_2010_Article_1434_Equ84_HTML.gif)
Assume that , and
satisfy
(i),
(ii),
(iii),
(iv).
Then, converges strongly to
.
Proof.
Using Lemma  7.1.1 of [34], we have that . Hence, by Theorem 3.1, we obtain the desired conclusion.
Remark 3.4.
Our Corollary 3.2 extends Theorems 1.1 and 1.2.
Remark 3.5.
Our iterative scheme (3.1) is simpler than the iterative schemes (5.1) and (5.11) of Acedo and Xu [6]. Furthermore, in our results, we use iterative scheme (3.1) to approximate a common fixed point of an infinite family of-strictly pseudocontractive mappings while the iterative schemes (5.1) and (5.11) of Acedo and Xu [6] are used to approximate a common fixed point of a finite family of
-strictly pseudocontractive mappings.
Remark 3.6.
Our results also hold for infinite family of uniformly continuous quasistrict pseudocontractions. Hence, we can adapt our results for an infinite family of uniformly continuous quasi-nonexpansive mappings in a real Hilbert space.
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Acknowledgments
The author is extremely grateful to Professor S. Al-Homidan and the anonymous referees for their valuable comments and useful suggestions which improve the presentation of this paper. This research work is dedicated to Professor C. E. Chidume with admiration and respect.
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Shehu, Y. Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems. Fixed Point Theory Appl 2011, 852789 (2011). https://doi.org/10.1155/2011/852789
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DOI: https://doi.org/10.1155/2011/852789