We begin this section by proving a strong convergence theorem of an implicit iterative sequence obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

Throughout the rest of this paper, we always assume that is a contraction of into itself with coefficient , and is a strongly positive bounded linear operator with coefficient and . Let be a nonexpansive mapping of into . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping and let be defined as in (2.10). Let be a sequence of mappings defined as Lemma 2.1. Consider a sequence of mappings on defined by

where By Lemma 2.2, we note that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point such that

Theorem 3.1.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

where , and satisfy and . Then, , and converges strongly to a point in which solves the variational inequality:

Equivalently, we have

Proof.

First, we assume that . By Lemma 2.2, we obtain . Let Since we have

We note from that . As is nonexpansive, we have

for all Thus, we have

It follows that Hence is bounded and we also obtain that ,, , and are bounded. Next, we show that . Since , we note that

Moreover, it follows from Lemma 2.1 that

and hence Therefore, we have

and hence

Since is bounded and it follows that as

Put . From (3.10), it follows by the nonexpansive of and the inverse strongly monotonicity of that

which implies that

Since , we have as . Since is –inverse-strongly monotone and is nonexpansive, we have

Thus, we have

From (3.5), (3.10), and (3.15), we have

Thus, we get

Since , as , we have as . It follows from the inequality that as . Moreover, we have as .

Put . Since both and are nonexpansive, we have is a nonexpansive mapping on and then we have for all . It follows by Theorem 3.1 of Plubtieng and Punpaeng [6] that converges strongly to , where and , for all . We will show that . Since converges strongly to , we also have . Let us show Assume Since and , we have Since it follows by the Opial's condition that

This is a contradiction. Hence We now show that . In fact, since is –inverse-strongly monotone, is an -Lipschitz continuous monotone mapping and . It follows from Lemma 2.4 that is maximal monotone. Let , that is, . Again since , we have that is,

By the maximal monotonicity of , we have

and so

It follows from , and that

Since is maximal monotone, this implies that that is, . Hence, . Since , we have . It implies that is the unique solution of the variational inequality (3.4).

Now we prove the following theorem which is the main result of this paper.

Theorem 3.2.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a sequence of nonexpansive mappings on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

for all , where , and satisfy

Suppose that for any bounded subset of . Let be a mapping of into itself defined by and suppose that . Then, , and converges strongly to , where is a unique solution of the variational inequalities (3.4).

Proof.

Since , we may assume that for all . First we will prove that is bonded. Let Then, we have

It follows from (3.25) and induction that

Hence is bounded and therefore , and are also bounded. Next, we show that . Since is nonexpansive, it follows that

Then, we have

where . On the other hand, we note that

Putting in (3.29) and in (3.30), and By (A2), we have

and hence

Since we assume that there exists a real number such that for all Thus, we have

and hence

where . From (3.28), we have

Since is bounded, it follows that . Hence, by Lemma 2.3, we have as . From (3.34) and , we have . Moreover, we have from (3.27) that .

We note from (3.23) that . Then, we have

Since , and , we get . From the proof of Theorem 3.1, we have

for all . Therefore, we have

and hence

Since is bounded, and , it follows that as

Put . It follows from (3.38), the nonexpansive of and the inverse strongly monotonicity of that

This implies that

Since and , we have as .From (3.5), (3.15) and (3.38), we have

Thus, we obtain

Since , and , we have as . It follows from the inequality that as . Moreover, we note that as . Since

for all , it follows that . Next, we show that

where is a unique solution of the variational inequality (3.4). To show this inequality, we choose a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From we obtain . Let us show . It follows by (3.23) and (A2) that

and hence

Since and it follows by (A4) that for all For with and let Since and we have and hence So, from (A1) and (A4) we have

and hence . From (A3), we have for all and hence By the same argument as in proof of Theorem 3.1, we have and hence . This implies that

Finally we prove that . From (3.23), we have

This implies that

where and . It easily verified that , and . Hence, by Lemma 2.1, the sequence converges strongly to .

As in [10, Theorem 4.1], we can generate a sequence of nonexpansive mappings satisfying condition . for any bounded of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point.

Corollary 3.3.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping, be a maximal monotone mapping. Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let be a family of nonnegative numbers with indices with such that

(i)

(ii)

(iii)

Let be a sequence of nonexpansive mappings on with and let , and be sequences generated by and

for all , where , and satisfy

Then, , and converges strongly to in which solves the variational inequality:

If and in Theorem 3.2, we obtain the following corollary.

Corollary 3.4 (see Peng et al. [9]).

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant . Let , and be sequences generated by and

for all , where , and satisfy

Then, , and converges strongly to , where .

If and in Theorem 3.2, we obtain the following corollary.

Corollary 3.5 (see S. Plubtieng and R. Punpaeng [6]).

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

for all , where and satisfy

Then, and converges strongly to a point in which solves the variational inequality: