An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

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

(11)

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

(12)

In 2006, Marino and Xu [6] introduced the general iterative method and proved that for a given , the sequence is generated by the algorithm

(13)

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

(14)

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:

(15)

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

(16)

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.

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

(21)

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

(22)

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 .

(23)

(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

(24)

Further, if , then the following hold:

(1) is single-valued,

(2) is firmly nonexpansive, that is,

(25)

(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

(26)

Lemma 2.5 (see [16]).

Assume that is a sequence of nonnegative real numbers such that

(27)

where is a sequence in (0,1) and is a sequence in , such that

(i) ,

(ii) or Then .

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

(31)

where . By Lemmas 2.1 and 2.3, we have

(32)

It is easy to see that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed-point such that

(33)

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

(34)

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

(35)

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

(36)

Then, since , we obtain that

(37)

Further, we have

(38)

It follows that .

Hence, is bounded, and we also obtain that and are bounded. Notice that

(39)

By Lemma 2.1, we have

(310)

It follows that

(311)

Thus, from Lemma 2.4, (3.7), and (3.11), we obtain that

(312)

It follows that

(313)

Since , therefore

(314)

From (3.9), we derive that

(315)

Define by , then is nonexpansive with by Lemma 2.3. We note that

(316)

So by (3.15) and , we obtain that

(317)

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

(318)

This is a contradiction. So, we get and .

Next, we show that . Since , for any , we obtain

(319)

From (A2), we have

(320)

Replacing by , we have

(321)

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

(322)

and hence . From (A3), we have for all and hence . Therefore, . On the other hand, we note that

(323)

Hence, we obtain

(324)

It follows that

(325)

This implies that

(326)

In particular,

(327)

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

(328)

and we have

(329)

Hence, for ,

(330)

Since is monotone (i.e., , for all . This is due to the nonexpansivity of ).

Now replacing in (3.30) with and letting , we obtain

(331)

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

(332)

Interchange and to obtain

(333)

Adding up (3.32) and (3.33) yields

(334)

Hence, , and therefore as ,

(335)

This is equivalent to the fixed-point equation

(336)

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

(337)

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

(338)

By induction, we have

(339)

and hence is bounded. From (3.6) and (3.7), we also derive that and are bounded. Next, we show that . We have

(340)

where

(341)

On the other hand, we have

(342)

From and , we note that

(343)

(344)

Putting in (3.43) and in (3.44), we have

(345)

So, from (A2), we have

(346)

and hence

(347)

Since , without loss of generality, let us assume that there exists a real number a such that for all . Thus, we have

(348)

where . Next, we estimate . Notice that

(349)

From (3.48), (3.49), and (3.42), we obtain that

(350)

where is an appropriate constant such that

(351)

From (3.41) and (3.50), we obtain

(352)

where . Hence, few by Lemma 2.5, we have

(353)

From (3.48) and (3.50), and , we have

(354)

Since

(355)

it follows that

(356)

From and (3.53), we have

(357)

For , we have

(358)

This implies that

(359)

Then, from (3.7) and (3.59), we derive that

(360)

Since , , we have

(361)

From (3.57) and (3.61), we obtain that

(362)

Define by , then is nonexpansive with by Lemma 2.3. Notice that

(363)

By (3.62) and , we obtain that

(364)

Next, we show that , where is a unique solution of the variational inequality (3.4). Indeed, take a subsequence of such that

(365)

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

(366)

From , we have

(367)

This implies that

(368)

where , , and .

It is easy to see that , , and by (3.66). Hence by Lemma 2.5, the sequence converges strongly to .