Theorem 3.1.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let be a maximal monotone mapping, be a sequence of nonexpansive mappings with , be the nonexpansive mapping defined by (2.5), and be a bifunction satisfying conditions . Let be the sequence defined by

where the mapping is defined by (2.18), and are two constants with , and

If

where and GEP is the set of solutions of variational inclusion (1.2) and generalized equilibrium problem (1.6), respectively, then the sequence defined by (3.1) converges strongly to , which is the unique solution of the following quadratic minimization problem:

Proof.

We divide the proof of Theorem 3.1 into four steps.

Step 1 (The sequence is bounded).

Set

Taking , then it follows from Lemma 2.11 that

Since both and are nonexpansive, and are -inverse strongly monotone and -inverse strongly monotone, respectively, from Proposition 2.2, we have

This implies that

It follows from (3.1) and (3.9) that

where . This shows that is bounded. Hence, it follows from (3.9) that the sequence and are also bounded.

It follows from (3.5), (3.6), and (3.9) that

This shows that is bounded.

Step 2.

Now, we prove that

Since is nonexpansive, from (3.5) and (3.9), we have that

Let in (3.14), in view of condition , we have

By virtue of Lemma 2.7, we have

This implies that

We derive from (3.17) that

From (3.1) and (3.8), we have

where

that is,

Let , noting the assumptions that , , from (3.2) and (3.18), we have

By virtue of Lemma 2.10(i) and (3.1), we have

Simplifying it, we have

Similarly, in view of Proposition 2.5(ii) and (3.1), we have

Simplifying it, from (3.24), we have

From (3.19) and (3.26), we have

Let nd in view of (3.18) and (3.22), we have

This shows that

Then, we have

Step 3 (sequence converges strongly to ).

Because is bounded, without loss of generality, we can assume that . In view of (3.12), it yields that and . From Lemma 2.9 and (3.30), we know that .

Next, we prove that .

Since , we have

It follows from condition that

Therefore,

For any and , then . From (3.33), we have

Since is -inverse strongly monotone, from Proposition 2.2(i) and (3.12), we have

Let in (3.34), in view of condition and , we have

It follows from conditions , and (3.36) that

that is,

Let to 0 in (3.38), we have

This shows that .

Step 4 (now, we prove that ).

Since is -inverse strongly monotone, from Proposition 2.2 (i), we know that is an -Lipschitz continuous and monotone mapping and , where is the domain of . It follows from Lemma 2.8 that is maximal monotone. Let , that is, . Since , we have , that is, . By virtue of the maximal monotonicity of , we have

Therefore we have

Since is monotone, this implies that

Since

from (3.42), we have

Since is maximal monotone, , that is, .

Summing up the above arguments, we have proved that

On the other hand, for any , we have

and so we have

Put in (3.47), we have

where and . Since , it is easy to see that and . By Lemma 2.12, we conclude that as , where is the unique solution of the following quadratic minimization problem:

This completes the proof of Theorem 3.1.

In Theorem 3.1, if , then the following corollary can be obtained immediately.

Corollary 3.2.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let be a maximal monotone mapping, be a nonexpansive mappings with . Let be a bifunction satisfying conditions . Let be the sequence defined by

where the mapping is defined by (2.18), and are two constants with , and

If

where and GEP are the sets of solutions of variational inclusion (1.2) and generalized equilibrium problem (1.6), then the sequence defined by (3.50) converges strongly to , which is the unique solution of the following quadratic minimization problem:

In Theorem 3.1, if , where is the indicator function of , then the variational inclusion problem (1.2) is equivalent to variational inequality (1.5), that is, to find such that , for all . Since . Consequently, we have the following corollary.

Corollary 3.3.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let and be a nonexpansive mappings with . Let be a bifunction satisfying conditions . Let be the sequence defined by

where the mapping is defined by (2.18), and are two constants with , and

If

where and GEP are the sets of solutions of variational inclusion (1.5) and generalized equilibrium problem (1.6), then the sequence defined by (3.54) converges strongly to , which is the unique solution of the following quadratic minimization problem: