- Research Article
- Open access
- Published:
An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces
Journal of Inequalities and Applications volume 2011, Article number: 173430 (2011)
Abstract
We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.
1. Introduction
Let be a real Hilbert space and
a closed convex subset of
, and let
be a bifunction of
into
, where
is the set of real numbers. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ1_HTML.gif)
denoted the set of solution by . Given a mapping
, let
for all
, then
if and only if
, that is,
is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem, see, for instance, [1, 2].
A mapping of
into itself is nonexpansive if
, for all
. The set of fixed points of
is denoted by
. In 2007, Plubtieng and Punpaeng [3], S. Takahashi and W. Takahashi [4], and Tada and W. Takahashi [5] considered iterative methods for finding an element of
.
Recall that an operator is strongly positive if there exists a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ2_HTML.gif)
In 2006, Marino and Xu [6] introduced the general iterative method and proved that for a given , the sequence
is generated by the algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ3_HTML.gif)
where is a self-nonexpansive mapping on
,
is a contraction of
into itself with
and
satisfies certain conditions, and
is a strongly positive bounded linear operator on
and converges strongly to a fixed-point
of
which is the unique solution to the following variational inequality:
, for
, and is also the optimality condition for some minimization problem. 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%2F173430/MediaObjects/13660_2010_Article_2325_Equ4_HTML.gif)
Note that the class of -strict pseudo-contraction strictly includes the class of nonexpansive mapping, that is,
is nonexpansive if and only if
is 0-srictly pseudocontractive; it is also said to be pseudocontractive if
. Clearly, the class of
-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions.
The set of fixed points of is denoted by
. Very recently, by using the general approximation method, Qin et al. [7] obtained a strong convergence theorem for finding an element of
. On the other hand, Ceng et al. [8] proposed an iterative scheme for finding an element of
and then obtained some weak and strong convergence theorems. Based on the above work, Y. Liu [9] introduced two iteration schemes by the general iterative method for finding an element of
.
In 2001, Yamada [10] introduced the following hybrid iterative method for solving the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ5_HTML.gif)
where is
-Lipschitzian and
-strongly monotone operator with
,
,
, then he proved that if
satisfyies appropriate conditions, the
generated by (1.5) converges strongly to the unique solution of variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ6_HTML.gif)
Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of , where
is a
-strictly pseudocontractive non-self mapping, and then obtained two strong convergence theorems.
2. Preliminaries
Throughout this paper, we always assume that is a nonempty closed convex subset of a Hilbert space
. We write
to indicate that the sequence
converges weakly to
.
implies that
converges strongly to
. For any
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ7_HTML.gif)
Such a is called the metric projection of
onto
. It is known that
is nonexpansive. Furthermore, for
and
,
, for all
.
It is widely known that satisfies Opial's condition [11], that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ8_HTML.gif)
holds for every with
. In order to solve the 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 all .
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ9_HTML.gif)
(A4) For each fixed , the function
is convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 (see [12]).
Let be a bifunction from
into
satisfying (A1), (A2),(A3) and (A4) then, for any
and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ10_HTML.gif)
Further, if , then the following hold:
(1) is single-valued,
(2) is firmly nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ11_HTML.gif)
(3),
(4) is nonempty, closed and convex.
Lemma 2.2 (see [13]).
If is a
-strict pseudo-contraction, then the fixed-point set
is closed convex, so that the projection
is well difened.
Lemma 2.3 (see [14]).
Let be a
-strict pseudo-contraction. Define
by
for each
, then, as
, T is nonexpansive mapping such that
.
Lemma 2.4 (see [15]).
In a Hilbert space , there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ12_HTML.gif)
Lemma 2.5 (see [16]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ13_HTML.gif)
where is a sequence in (0,1) and
is a sequence in
, such that
(i),
(ii) or
Then
.
3. Main Results
Throughout the rest of this paper, we always assume that is a
-lipschitzian continuous and
-strongly monotone operator with
and assume that
.
. Let
be mappings defined as Lemma 2.1. Define a mapping
by
, for all
, where
, then, by Lemma 2.3,
is nonexpansive. We consider the mapping
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ14_HTML.gif)
where . By Lemmas 2.1 and 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ15_HTML.gif)
It is easy to see that is a contraction. Therefore, by the Banach contraction principle,
has a unique fixed-point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ16_HTML.gif)
For simplicity, we will write for
provided no confusion occurs. Next, we prove that the sequence
converges strongly to a
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ17_HTML.gif)
Equivalently, .
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
into
satisfying (A1), (A2), (A3), and (A4). Let
be a
-strictly pseudocontractive nonself mapping such that
. Let
be an
-Lipschitzian continuous and
-
monotone operator on
with
and
,
. Let
be asequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ18_HTML.gif)
where ,
, and
satisfy
if
and
satisfy the following conditions:
(i),
,
(ii) and
,
then converges strongly to a point
which solves the variational inequality (3.4).
Proof.
First, take . Since
and
, from Lemma 2.1, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ19_HTML.gif)
Then, since , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ20_HTML.gif)
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ21_HTML.gif)
It follows that .
Hence, is bounded, and we also obtain that
and
are bounded. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ22_HTML.gif)
By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ23_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ24_HTML.gif)
Thus, from Lemma 2.4, (3.7), and (3.11), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ25_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ26_HTML.gif)
Since , therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ27_HTML.gif)
From (3.9), we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ28_HTML.gif)
Define by
, then
is nonexpansive with
by Lemma 2.3. We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ29_HTML.gif)
So by (3.15) and , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ30_HTML.gif)
Since is bounded, so there exists a subsequence
which converges weakly to
. Next, we show that
. Since
is closed and convex,
is weakly closed. So we have
. Let us show that
. Assume that
, Since
and
, it follows from the Opial's condition that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ31_HTML.gif)
This is a contradiction. So, we get and
.
Next, we show that . Since
, for any
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ32_HTML.gif)
From (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ33_HTML.gif)
Replacing by
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ34_HTML.gif)
Since and
, it follows from (A4) that
, for all
. Let
for all
and
, then we have
and hence
. Thus, from (A1) and (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ35_HTML.gif)
and hence . From (A3), we have
for all
and hence
. Therefore,
. On the other hand, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ36_HTML.gif)
Hence, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ37_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ38_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ39_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ40_HTML.gif)
Since , it follows from (3.27) that
as
. Next, we show that
solves the variational inequality (3.4).
As a matter of fact, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ41_HTML.gif)
and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ42_HTML.gif)
Hence, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ43_HTML.gif)
Since is monotone (i.e.,
, for all
. This is due to the nonexpansivity of
).
Now replacing in (3.30) with
and letting
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ44_HTML.gif)
That is, is a solution of (3.4). To show that the sequence
converges strongly to
, we assume that
. Similiary to the proof above, we derive
. Moreover, it follows from the inequality (3.31) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ45_HTML.gif)
Interchange and
to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ46_HTML.gif)
Adding up (3.32) and (3.33) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ47_HTML.gif)
Hence, , and therefore
as
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ48_HTML.gif)
This is equivalent to the fixed-point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ49_HTML.gif)
Theorem 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
into
satisfying (A1), (A2), (A3) and (A4). Let
be a
-strictly pseudocontractive nonself mapping such that
. Let
be an
-Lipschitzian continuous and
-strongly monotone operator on
with
. Suppose that
,
. Let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ50_HTML.gif)
where ,
if
,
, and
satisfy the following conditions:
(i),
,
,
,
(ii) and
,
,
(iii),
and
,
then and
converge strongly to a point
which solves the variational inequality(3.4).
Proof.
We first show that is bounded. Indeed, pick any
to derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ51_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ52_HTML.gif)
and hence is bounded. From (3.6) and (3.7), we also derive that
and
are bounded. Next, we show that
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ53_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ54_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ55_HTML.gif)
From and
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ56_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ57_HTML.gif)
Putting in (3.43) and
in (3.44), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ58_HTML.gif)
So, from (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ59_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ60_HTML.gif)
Since , without loss of generality, let us assume that there exists a real number a such that
for all
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ61_HTML.gif)
where . Next, we estimate
. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ62_HTML.gif)
From (3.48), (3.49), and (3.42), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ63_HTML.gif)
where is an appropriate constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ64_HTML.gif)
From (3.41) and (3.50), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ65_HTML.gif)
where . Hence, few by Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ66_HTML.gif)
From (3.48) and (3.50), and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ67_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ68_HTML.gif)
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ69_HTML.gif)
From and (3.53), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ70_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ71_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ72_HTML.gif)
Then, from (3.7) and (3.59), we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ73_HTML.gif)
Since ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ74_HTML.gif)
From (3.57) and (3.61), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ75_HTML.gif)
Define by
, then
is nonexpansive with
by Lemma 2.3. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ76_HTML.gif)
By (3.62) and , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ77_HTML.gif)
Next, we show that , where
is a unique solution of the variational inequality (3.4). Indeed, take a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ78_HTML.gif)
Since is bounded, there exists a subsequence
of
which converges weakly to
.
Without loss of generality, we can assume that . From (3.61) and (3.64), we obtain
and
. By the same argument as in the proof of Theorem 3.1, we have
. Since
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ79_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ80_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173430/MediaObjects/13660_2010_Article_2325_Equ81_HTML.gif)
where ,
, and
.
It is easy to see that ,
, and
by (3.66). Hence by Lemma 2.5, the sequence
converges strongly to
.
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Acknowledgments
M. Tian was supported in part by the Science Research Foundation of Civil Aviation University of China (no. 2010kys02). He was also Supported in part by The Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021).
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Tian, M. An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces. J Inequal Appl 2011, 173430 (2011). https://doi.org/10.1155/2011/173430
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DOI: https://doi.org/10.1155/2011/173430