Let be a real Hilbert space, be a lower semicontinuous and convex real-valued function, be an equilibrium bifunction. Let be a mapping and be a mapping. In this section, we first introduce the following new iterative algorithm.

Algorithm 3.1.

Let be a positive parameter. Let be a sequence in and be a sequence in . Define the sequences , and by the following manner:

Now we give a strong convergence result concerning Algorithm 3.1 as follows.

Theorem 3.2.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a demiclosed and -demicontractive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that

(a),

(b) is affine in the first variable,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iii)For each ; there exist a bounded subset and such that, for any ,

(iv) for some , and .

Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.

Proof.

First, we prove that , , and are all bounded. Without loss of generality, we may assume that . Given and , we have

that is,

Take . From (3.1), we have

Therefore,

where .

Note that and are firmly nonexpansive. Hence, we have

which implies that

From (2.2) and (3.1), we have

From (3.6)–(3.9), we have

This implies that is bounded, so are and .

From (3.1), we can write . Thus, from (3.9), we have

Since , . Therefore, from (3.8) and (3.11), we obtain

We note that and are bounded. So there exists a constant such that

Consequently, we get

Now we divide two cases to prove that converges strongly to .

Case 1.

Assume that the sequence is a monotone sequence. Then is convergent. Setting .

(i)If , then the desired conclusion is obtained.

(ii)Assume that . Clearly, we have

this together with and (3.14) implies that

that is to say

Let be a weak limit point of . Then there exists a subsequence of , still denoted by which weakly converges to . Noting that , we also have

Combining (3.1) and (3.17), we have

Since is demiclosed, then we obtain .

Next we show that . Since , we derive

From the monotonicity of , we have

and hence

Since and weakly, from the weak lower semicontinuity of and in the second variable , we have

for all . For and , let . Since and , we have and hence . From the convexity of equilibrium bifunction in the second variable , we have

and hence . Then, we have

for all and hence .

Therefore, we have

Thus, if is a solution of problem (2.8), we have

Suppose that there exists another subsequence which weakly converges to . It is easily checked that and

Therefor, we have

Since is -strongly monotone, we have

By (3.17)–(3.30), we get

From (3.12), for , we deduce that there exists a positive integer number large enough, when ,

This implies that

Since and is bounded, hence the last inequality is a contraction. Therefore, , that is to say, .

Case 2.

Assume that is not a monotone sequence. Set and let be a mapping for all by

Clearly, is a nondecreasing sequence such that as and for . From (3.14), we have

thus

Therefore,

Since , for all , from (3.12), we get

which implies that

Since is bounded, there exists a subsequence of , still denoted by which converges weakly to . It is easily checked that . Furthermore, we observe that

Hence, for all ,

Therefore

which implies that

Thus,

It is immediate that

Furthermore, for , it is easily observed that if (i.e., ), because for . As a consequence, we obtain for all ,

Hence , that is, converges strongly to . Consequently, it easy to prove that and converge strongly to . This completes the proof.

Remark 3.3.

The advantages of these results in this paper are that less restrictions on the parameters are imposed.

As direct consequence of Theorem 3.2, we obtain the following.

Corollary 3.4.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a nonexpansive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that;

(a),

(b) is affine in the first variable,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iii)For each ; there exist a bounded subset and such that, for any ,

(iv) for some , and .

Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.

Corollary 3.5.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a strictly pseudo-contractive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that

(a),

(b) is affine in the first variable,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iii)For each ; there exist a bounded subset and such that, for any ,

(iv) for some , and .

Then the sequences , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.