Theorem 3.1 Let E be a real Banach space and K be a nonempty, closed and convex subset of E. Let be N asymptotically nonexpansive mappings with . Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
or ;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges strongly to a common fixed point of if and only if .
Proof The necessity of Theorem 3.1 is obvious. Now we prove the sufficiency of Theorem 3.1.
For arbitrary , it follows from (17) in Lemma 2.6 that
where and . Hence, we have
(18)
It follows from (18) and Lemma 2.5 that limit exists. By the assumption, we have . Consequently, for any given , there exists a positive integer () such that
and there exists such that , . By (18) and the inequality (), for any and all , we have
Hence, is a Cauchy sequence in E. By the completeness of E, we can assume that . Next we prove that is a close subset of K. Let is a sequence in which converges strongly to some p, then we have for any
Thus, , and is closed. Since , then . Consequently, defined by (7) converges strongly to a common fixed point of in K. This completes the proof of Theorem 3.1. □
Theorem 3.2 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with . Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges weakly to a common fixed point of in K.
Proof First, we prove that for all .
For any , it follows from Lemma 2.6 that exists. Suppose that , we have from (7)
(19)
Since , then be a bounded sequence. By virtue of the condition (iii) and the boundedness of sequences and , we have
It follows from that . We have
Therefore, by (19), (20), (21), (ii) and Lemma 2.4, we obtain that
Hence,
(22)
which implies that for all . On the other hand, we also have
(23)
It follows from (22), (23), conditions (ii) and (iv) that
(24)
Since for each , , , hence , i.e. and . Therefore, we have
(25)
and
(26)
In view of (25) and (26), we have
(27)
From (24) and (27), it is obviously that , which implies that
Consequently, we obtain that for all
(28)
By virtue of (28), we have for all .
Since E is uniformly convex, every bounded subset of E is weakly compact. Again since is a bounded subset in K, there exists a subsequence of such that converges weakly to q in K, and for all . By Lemma 2.2, we have that . Hence, for all . Therefore, .
Next, we prove that converges weakly to q. Suppose that contrary, then there exists a subsequence of such that converges weakly to and . Using the same method, we can prove that and limit exists. Without loss generality, we assume that , , where , are two nonnegative constants. By virtue of the Opial’s condition of E, we have
This is contradictory. Hence, , which implies that converges weakly to q. The proof of Theorem 3.2 is completed. □
Theorem 3.3 Let E be a real uniformly convex Banach space and K be a nonempty, closed and convex subset of E. Let be N asymptotically nonexpansive mappings with and at least there exists (), it is semicompact. Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
, ;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges strongly to a common fixed point of in K.
Proof Without loss of generality, we assume that is semicompact. By Theorem 3.2, we have . Hence, there exists a subsequence of such that as . Therefore, we have for all
(29)
It follows from (29) that for all . This implies that . Therefore, be a common fixed point of . By virtue of Lemma 2.6, exists. It follows from that . Hence, the perturbed composites implicit iterative sequence generated by (7) strongly converges to a common fixed point of . This completes the proof of Theorem 3.3. □
Corollary 3.4 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with and let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
Then the perturbed implicit iterative sequence defined by (9) converges strongly to a common fixed point of if and only if .
Proof It is enough to take , for all in Theorem 3.1. □
Corollary 3.5 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with and let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, .
Then the Mann type iterative sequence defined by (8) converges weakly to a common fixed point of in K.
Proof It is sufficient to take for all in Theorem 3.2. □
Corollary 3.6 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with and at least there exists (), it is semicompact. Let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, .
Then the Mann type iterative sequence defined by (8) converges strongly to a common fixed point of in K.
Proof It is enough to take for all in Theorem 3.3. □