- Open Access
General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings
© Thakur et al.; licensee Springer. 2014
- Received: 3 January 2014
- Accepted: 25 March 2014
- Published: 7 April 2014
In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space.
- implicit iteration process
- asymptotically pseudo-contractive mapping
- common fixed points
where denotes the dual space of E and denotes the generalized duality pairing.
It is well known that if is strictly convex, then J is single valued.
In the sequel, we shall denote the single valued normalized duality mapping by j.
Let K be a nonempty subset of E. A mapping is said to be L-Lipschitzian if there exists a constant such that for all , we have . It is said to be nonexpansive if , for all . T is called asymptotically nonexpansive  if there exists a sequence with such that , for all integers and all .
We use to denote the set of fixed points of T; that is, .
Hence every asymptotically nonexpansive mapping is asymptotically pseudo-contractive.
It can be observed from the definition that an asymptotically nonexpansive mapping is uniformly L-Lipschitzian, where .
Hence T is asymptotically pseudo-contractive mapping with constant sequence .
The iterative approximation problems for a nonexpansive mapping, an asymptotically nonexpansive mapping, and an asymptotically pseudo-contractive mapping were studied extensively by Browder , Kirk , Goebel and Kirk , Schu , Xu [9, 10], Liu  in the setting of Hilbert space or uniformly convex Banach space.
where be a sequence in and . They proved in  that the sequence converges weakly to a common fixed point of , .
Later on Osilike and Akuchu , and Chen et al.  extended the iteration process (1.3) to a finite family of asymptotically pseudo-contractive mapping and a finite family of continuous pseudo-contractive self-mapping, respectively. Zhou and Chang  studied the convergence of a modified implicit iteration process to the common fixed point of a finite family of asymptotically nonexpansive mappings. Then Su and Li , and Su and Qin  introduced the composite implicit iteration process and the general iteration algorithm, respectively, which properly include the implicit iteration process. Recently, Beg and Thakur  introduced a modified general composite implicit iteration process for a finite family of random asymptotically nonexpansive mapping and proved strong convergence theorems.
The purpose of this paper is to consider a finite family of asymptotically pseudo-contractive mappings and to establish convergence results in Banach spaces based on the modified general composite implicit iteration:
for each , which can be written as , where and is a positive integer, with as . The sequences , , , and are in such that for all .
In what follows we shall use the following results.
Lemma 2.1 
Let E be a Banach space, K be a nonempty closed convex subset of E, and be a continuous and strong pseudo-contraction. Then T has a unique fixed point.
Lemma 2.2 
where is some nonnegative integer, and .
if, in addition, there exists a subsequence such that , then as .
Lemma 2.3 
imply that , where is some constant.
Lemma 2.4 
Let E be a reflexive smooth Banach space with a weakly sequential continuous duality mapping J. Let K be a nonempty bounded and closed convex subset of E and be a uniformly L-Lipschitzian and asymptotical pseudo-contraction. Then is demiclosed at zero, where I is the identical mapping.
We shall denote weak convergence by ⇀ and strong convergence by →.
We know that a Banach space with a sequentially continuous duality mapping satisfies Opial’s condition (for details, see ).
Throughout this section, E is a uniformly convex Banach space, K a nonempty closed convex subset of E. ℕ denotes the set of natural numbers and , the set of the first N natural numbers. () are N uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K. Let .
Since () are uniformly Lipschitzian, there exist constants such that , for all , and . Also, since () are asymptotically pseudo-contractive; therefore there exist sequences such that for all and .
Take and .
Before presenting the main results, we first show that the proposed iteration (1.4) is well defined.
where , with .
Thus the implicit iteration (1.4) is defined in K for a finite family of uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K, provided , where , for all , .
exists, for all ,
exists, where ,
where for all , by condition , we have .
Since , there exists a M such that .
Thus, by Lemma 2.2, and exist.
Since for any positive integer , we can write , .
This completes the proof. □
3.1 Strong convergence theorems
First, we prove necessary and sufficient conditions for the strong convergence of the modified general composite implicit iteration process to a common fixed point of a finite family of asymptotically pseudo-contractive mappings.
Theorem 3.1 Let E, K, and () be as defined above and be a sequence of real numbers as in Lemma 3.1. Then the sequence generated by (1.4) converges strongly to a member of ℱ if and only if .
Proof The necessity of the condition is obvious. Thus, we will only prove the sufficiency.
Let . Then from (ii) in Lemma 3.1, we have .
Next, we show that is a Cauchy sequence in K. For any given , since , there exists a natural number such that when .
Since exists for all , we have , for all and some positive number .
This implies that is a Cauchy sequence. Since E is complete, therefore is convergent.
Since K is closed, we get , then converges strongly to q.
It remains to show that .
since and , we obtain .
This completes the proof. □
Corollary 3.1 Suppose that the conditions are the same as in Theorem 3.1. Then the sequence generated by (1.4) converges strongly to if and only if has a subsequence which converges strongly to .
for at least one , or, in other words, at least one of the ’s satisfies condition (A).
Condition (B) reduces to condition (A) when all but one of the ’s are identities. Also condition (B) and condition () are equivalent (see ).
Senter and Dotson  established a relation between condition (A) and demicompactness that the condition (A) is weaker than demicompactness for a nonexpansive mapping T defined on a bounded set. Every compact operator is demicompact. Since every completely continuous mapping is continuous and demicompact, it satisfies condition (A).
Therefore in the next result, instead of complete continuity of mappings , we use condition ().
Theorem 3.2 Let E and K be as defined above, () be N asymptotically pseudo-contractive mappings as defined above and satisfying condition () and be a sequence of real numbers as in Lemma 3.1. Then the sequence generated by (1.4) converges strongly to a member of ℱ.
Proof By Lemma 3.1, we see that and exist.
Let one of the ’s, say , , satisfy condition (A).
By Lemma 3.1, we have . Therefore we have . By the nature of f and the fact that exists, we have . By Theorem 3.1, we find that converges strongly to a common fixed point in ℱ.
This completes the proof. □
A mapping is said to be semicompact, if the sequence in K such that , as , has a convergent subsequence.
Theorem 3.3 Let E and K be as defined above, and let () be N asymptotically pseudo-contractive mappings as defined above such that one of the mappings in is semicompact, and let be a sequence of real numbers as in Lemma 3.1. Then the sequence generated by (1.4) converges strongly to a member of ℱ.
for all . By continuity of , we have for all .
Thus for all . This implies that . Also, . By Theorem 3.1, we find that converges strongly to a common fixed point in ℱ. □
3.2 Weak convergence theorem
Theorem 3.4 Let E be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, and let K and () be as defined above and be a sequence of real numbers as in Lemma 3.1. Then the sequence generated by (1.4) converges weakly to a member of ℱ.
By Lemma 2.4, we find that is demiclosed at zero, i.e. , so that . By the arbitrariness of , we know that .
Next we prove that converges weakly to q.
If has another subsequence which converges weakly to , then we must have , and since exists and since the Banach space E has a weakly sequentially duality mapping, it satisfies Opial’s condition, and it follows from a standard argument that . Thus converges weakly to . □
Remark 3.1 Our results improve and generalize the corresponding results of Browder , Kirk , Goebel and Kirk , Schu , Xu [9, 10], Liu , Zhou and Chang , Osilike , Osilike and Akuchu , Su and Li , Su and Qin , and many others.
k-set-contractive if , for each bounded subset D of K and some constant ;
k-set-condensing if , for each bounded subset D of K with ;
k-ball-contractive if , for each bounded subset D of K and some constant ;
k-ball-condensing if , for each bounded subset D of K with .
compact if is compact whenever is bounded;
completely continuous if it maps weakly convergence sequences into strongly convergent sequences;
a generalized contraction if for each there exists such that for all ;
a mapping is called uniformly strictly contractive (relative to E) if for each there exists such that for all . Every k-set-contractive mapping with is set-condensing and also every compact mapping is set-condensing.
T is strictly semicontractive if there exists a continuous mapping with for such that for each , is a k-contraction with and is compact;
T is of strictly semicontractive type if there exists a continuous mapping with , for such that for each , is a k-contraction with some independent of x and is compact from K into the space of continuous mapping of K into E with the uniform metric;
T is of strongly semicontractive type relative to X if there exists a mapping with , for such that , is uniformly strictly contractive on K relative to E and is a completely continuous from K to E, uniformly for .
T is either set-condensing or ball-condensing (or compact);
T is a generalized contraction;
T is uniformly strictly contractive;
T is strictly semicontractive;
T is of strictly semicontractive type;
T is of strongly semicontractive type.
Remark 3.2 In view of the above, it is possible to replace the semicompactness assumption in Theorem 3.3 with any of the contractive assumptions (i)-(vi).
We now give an example of asymptotically pseudo-contractive mapping with nonempty fixed point set.
Example 3.1 
for all . That is, T is an asymptotically pseudo-contractive mapping with sequence .
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