Lemma 2.1.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each and a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

Then

Proof.

First, we show that the sequence generated in (1.18) is well defined. For each , define a mapping as follows:

Notice that

From the restriction (a), we see that is a contraction for each . From Banach contraction mapping principle, we can prove that the sequence generated in (1.18) is well defined.

Fixing , we see that

Notice that . We see from the restrictions (a) and (b) that there exists a positive integer such that

where . It follows from (2.4) that

where is an appropriate constant such that . In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that exists. It follows that the sequence is bounded. In view of Lemma 1.3, we see that

where and are appropriate constants such that and . This implies that

In view of the restrictions (a) and (b), we obtain that

Since is a continuous, strictly increasing, and convex function with , we obtain that

Next, we show that

From Lemma 1.3, we also see that

This implies that

In view of the restrictions (a) and (b), we obtain that

Since is a continuous, strictly increasing, and convex function with , we obtain that (2.11) holds. Notice that

In view of (2.10) and (2.11), we see from the restriction (b) that

which implies that

Since for any positive integer , it can be written as , where , observe that

Since for each , (mod ), on the other hand, we obtain from that . That is,

Notice that

Substituting (2.20) into (2.18), we arrive at

In view of (2.11) and (2.17), we obtain that

Notice that

It follows from (2.16) and (2.22) that

Notice that

From (2.17) and (2.24), we arrive at

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

Letting , we have

In view of (2.10) and (2.16), we see that

Observe that

In view of

we arrive at

In view of (2.10), (2.17), and (2.29), we obtain that

Notice that

From (2.16) and (2.33), we see that

On the other hand, we have

It follows from (2.17) and (2.35) that

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

This completes the proof.

Recall that a mapping is said to be *semicompact* if for any bounded sequence in such that as , then there exists a subsequence such that .

Next, we give strong convergence theorems with the help of the semicompactness.

Theorem 2.2.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

If one of or one of is semicompact, then the sequence converges strongly to some point in .

Proof.

Without loss of generality, we may assume that is semicompact. From (2.38), we see that there exits a subsequence of converging strongly to . For each , we get that

Since is Lipshcitz continuous, we obtain from (2.38) that . Notice that

Since is Lipshcitz continuous, we obtain from (2.27) that . This means that . In view of Lemma 2.1, we obtain that exists. Therefore, we can obtain the desired conclusion immediately.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.3.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

If one of is semicompact, then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.4.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , for all ;

(b).

If one of is semicompact, then the sequence converges strongly to some point in .

In 2005, Chidume and Shahzad [11] introduced the following conception. Recall that a family with is said to satisfy *Condition* on if there is a nondecreasing function with and for all such that for all

Based on Condition , we introduced the following conception for two finite families of mappings. Recall that two families and with are said to satisfy Condition on if there is a nondecreasing function with and for all such that for all

Next, we give strong convergence theorems with the help of Condition .

Theorem 2.5.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

If and satisfy Condition , then the sequence converges strongly to some point in .

Proof.

In view of Condition , we obtain from (2.27) and (2.38) that , which implies . Next, we show that the sequence is Cauchy. In view of (2.6), for any positive integers , where , we see that

where . It follows that

It follows that is a Cauchy sequence in and so converges strongly to some . Since and are Lipschitz for each , we see that is closed. This in turn implies that . This completes the proof.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.6.

Let be a real uniformly convex uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

If satisfies Condition , then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.7.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , and , for all ;

(b).

If satisfies Condition , then the sequence converges strongly to some point in .

Finally, we give a strong convergence theorem criterion.

Theorem 2.8.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

Then converges strongly to some point in if and only if .

Proof.

The necessity is obvious. We only show the sufficiency. Assume that

For each , we see that

Notice that . We see from the restrictions (a) and (b) that there exists a positive integer such that

where . Notice that the sequence is bounded. It follows from (2.46) that

where is an appropriate constant such that . In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that exists. This implies that

In view of Theorem 2.5, we can conclude the desired conclusion easily.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.9.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

Then converges strongly to some point in if and only if .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.10.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , for all ;

(b).

Then converges strongly to some point in if and only if .