In this section, we present a new general composite iterative scheme for inversestrongly monotone mappings and a nonexpansive mapping.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space such that . Let be an inversestrongly monotone mapping of into and a nonexpansive mapping of into itself such that . Let and let be a strongly positive bounded linear operator on with constant and a contraction of into itself with constant . Assume that and . Let be a sequence generated by
where , , and . Let , , and satisfy the following conditions:
(i); ;
(ii) for all and for some ;
(iii) for some , with ;
(iv), , .
Then converges strongly to , which is a solution of the optimization problem
where is a potential function for .
Proof.
We note that from the control condition (i), we may assume, without loss of generality, that . Recall that if is bounded linear selfadjoint operator on , then
Observe that
which is to say that is positive. It follows that
Now we divide the proof into several steps.
Step 1.
We show that is bounded. To this end, let and for every . Let . Since is nonexpansive and from (2.4), we have
Similarly, we have
Now, set . Let . Then, from (IS) and (3.4), we obtain
From (3.5) and (3.6), it follows that
By induction, it follows from (3.7) that
Therefore, is bounded. So , , , , , , and are bounded. Moreover, since and , and are also bounded. And by the condition (i), we have
Step 2.
We show that and . Indeed, since and are nonexpansive and , we have
Similarly, we get
Simple calculations show that
So, we obtain
Also observe that
By (3.11), (3.13), and (3.14), we have
where , , and . From the conditions (i) and (iv), it is easy to see that
Applying Lemma 2.3 to (3.15), we obtain
Moreover, by (3.10) and (3.13), we also have
Step 3.
We show that and . Indeed,
which implies that
Obviously, by (3.9) and Step 2, we have as . This implies that
By (3.9) and (3.21), we also have
Step 4.
We show that and . To this end, let . Since and , we have
So we obtain
Since from the condition (i) and from Step 3, we have . Moreover, from (2.4) we obtain
and so
Thus
Then, we have
Since , and , we get . Also by (3.21)
Step 5.
We show that . In fact, since
from (3.9) and (3.29), we have .
Step 6.
We show that
where is a solution of the optimization problem (OP1). First we prove that
Since is bounded, we can choose a subsequence of such that
Without loss of generality, we may assume that converges weakly to .
Now we will show that . First we show that . Assume that . Since and , by the Opial condition and Step 5, we obtain
which is a contradiction. Thus we have .
Next, let us show that . Let
Then is maximal monotone. Let . Since and , we have
On the other hand, from , we have and hence
Therefore, we have
Since in Step 4 and is inversestrongly monotone, we have as . Since is maximal monotone, we have and hence .
Therefore, . Now from Lemma 2.5 and Step 5, we obtain
By (3.9) and (3.39), we conclude that
Step 7.
We show that and , where is a solution of the optimization problem (OP1). Indeed from (IS) and Lemma 2.2, we have
that is,
where , , and
From (i), in Steps 3, and 6, it is easily seen that , , and . Hence, by Lemma 2.3, we conclude as . This completes the proof.
As a direct consequence of Theorem 3.1, we have the following results.
Corollary 3.2.
Let , and be as in Theorem 3.1. Let be a sequence generated by
where and . Let and satisfy the conditions (i), (ii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem
where is a potential function for .
Corollary 3.3.
Let , , , , , , , , , and be as in Theorem 3.1. Let be a sequence generated by
where , , and . Let , and satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem
where is a potential function for .
Remark 3.4.

(1)
Theorem 3.1 and Corollary 3.3 improve and develop the corresponding results in Chen et al. [6], Iiduka and Takahashi [8], and Jung [10].

(2)
Even though for , the iterative scheme (3.44) in Corollary 3.2 is a new one for fixed point problem of a nonexpansive mapping.