Abstract
In recent works, many authors generated strongly relatively nonexpansive sequences of mappings by the sequences of firmly nonexpansive-like mappings. In this paper, we introduce a new method for construction of strongly relatively nonexpansive sequences from firmly nonexpansive-like mappings.
1. Introduction and Preliminaries
The class of firmly nonexpansive-like mappings has been introduced in []. Fixed point theory for such mappings can be applied to several nonlinear problems such as zero point problems for monotone operators, convex feasibility problems, convex minimization problems, equilibrium problems (see, [,,,,] for more details).
Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space X, J be a normalized duality mapping from X into dual , and are firmly nonexpansive-like mappings. The set of all fixed points of T is denoted by . It is known that if C is a bounded subset, then is nonempty ([], Theorem 7.4). We investigate asymptotic behavior of the following sequence in a uniformly smooth and 2-uniformly convex Banach space X.
for all , where , denotes the uniform convexity constant of and denotes the generalized projection of X onto C. If X is a Hilbert space, then (1) is reduced to
Throughout the present paper, we denote by the set of all positive integers, the set of all real numbers, X a real Banach space with dual , the norms of X and , the value of at , strong convergence of a sequence of X to , weak convergence of a sequence of X to , the unit sphere of X, and the closed unit ball of X.
Now, we present some definitions which are needed in the sequel. The normalized duality mapping of X into is defined by
for all . The space X is said to be smooth if
exists for all . The space X is said to be uniformly smooth, if (4) converges uniformly in . It is said to be strictly convex, if whenever and . It is said to be uniformly convex, if for all , where is the modulus of convexity of X defined by
for all .
The space X is said to be 2-uniformly convex, if there exists such that for all .
It is obvious that every 2-uniformly convex Banach space is uniformly convex. It is known that all Hilbert spaces are uniformly smooth and 2-uniformly convex. It is also known that all the Lebesgue spaces are uniformly smooth and 2-uniformly convex whenever .
For a smooth Banach space, J is said to be weakly sequentially continuous if converges weak to , whenever is a sequence of X such that .
Define by
for all . It is known that
for all .
Definition 1
([]). The metric projection from X onto C and the generalized projection from X onto C are defined by
for all , respectively.
Obviously, for and ,
Also, for and ,
Definition 2
([]). A mapping is said to be a firmly nonexpansive-like mapping, if
for all .
Definition 3
([]). Let be a mapping. A point is said to be an asymptotic fixed point of if there exists a sequence of C such that and . The set of all asymptotic fixed points of T is denoted by .
Definition 4
([]). The mapping T is said to be of type if is nonempty and for all and .
It is known that if T is a mapping of type , then is closed and convex.
Definition 5
([]). The mapping T is said to be of type if T is of type and , whenever is a bounded sequence of C such that for some .
Definition 6
([]). The sequence is said to satisfy the condition if every weak subsequential limit of belongs to , whenever is a bounded sequence of C such that .
Now, we give some results which will be used in our main results.
Theorem 1
([]). The space X is 2-uniformly convex if and only if there exists such that
Lemma 1
([], Lemma 2.2). Suppose that X is 2-uniformly convex. Then
Lemma 2
([]). If is a firmly nonexpansive-like mapping, then is a closed convex subset of X and .
Lemma 3
([]). Suppose that X is uniformly convex. If and are mappings of type such that is nonempty and S or T is of type , then is of type and . Further, if both S and T are of type , then so is .
Lemma 4
([]). Suppose that X is uniformly convex. Let be a sequence of mappings of X into itself and a sequence of mappings of C into X such that is nonempty, both and are of type and or is of type for all . Then the following holds:
- (i)
- is of type ;
- (ii)
- if X is uniformly smooth and both and satisfy the condition , then so does .
Theorem 2
([]). Let X be a smooth and uniformly convex Banach space, C a nonempty closed convex subset of X, and a sequence of mappings of C into X such that is of type and satisfies the condition . If for all and J is weakly sequentially continuous, then the sequence defined by and for all converges weakly to the strong limit of .
Now, we construct a new strongly relatively nonexpansive sequence from a given sequence of firmly nonexpansive-like mappings with a common fixed point in Banach spaces.
2. Main Results
The following results will be used in the sequel of the paper.
Lemma 5.
Let C be a nonempty closed convex subset of a smooth, strictly convex, 2-uniformly convex and reflexive Banach space X. Suppose that is a pair of firmly nonexpansive-like mappings of C into X and let . Let U be a mapping of C into X defined by , where and I denotes the identity mapping on C. Then
for all and .
Proof.
Now, we present the construction of strongly relatively nonexpansive sequences in the following.
Theorem 3.
Let C be a nonempty closed convex subset of a smooth and 2-uniformly convex Banach space X;
- (i)
- , are sequences of firmly nonexpansive-like mappings from C into X such that is nonempty;
- (ii)
- is a sequence of mappings from C into X defined by
Then and is of type . Also, if X is uniformly smooth and satisfies the condition , then satisfies the condition .
Proof.
We can easily see that . At first, we show that is of type .
Note that is nonempty. By Lemma 5, we also know that each is a mapping of type from C into X.
Suppose that is a bounded sequence of C such that
for some . Then, it follows from Lemma 5 that
Thus, it follows from that . Consequently, we have and hence is of type . Now, we present the proof of part . Suppose that X is uniformly smooth and satisfies the condition . Let p be a weak subsequential limit of a bounded sequence of C such that . By the definition of , we have
for all . Since J is uniformly norm-to-norm continuous on each nonempty bounded subset of X and , it follows from (20) that
From our assumptions, we know that . Therefore, satisfies the condition . □
Remark 1.
It is notable that every nonexpansive mapping T is a mapping of type , but the converse is not necessarily satisfied in a Hilbert space. For instance, let be defined by , then T is of type and is neither nonexpansive nor of type . Also, let be defined by . Then T is a mapping of type .
Remark 2.
For a mapping T from C into X, the following assertions hold:
- (a)
- T is of type if and only if is of type ;
- (b)
- if and only if satisfies the condition .
Corollary 1.
Let be a pair of firmly nonexpansive-like mappings from C into X such that are nonempty and U be a mapping from C into X which is defined by
where . Then the following assertions hold:
- (i)
- and U is of type ;
- (ii)
- if X is uniformly smooth, then .
Theorem 4.
Let be a sequence of mappings from C into itself which are defined by
for all . Then the following consequences hold:
- (i)
- and is of type ;
- (ii)
- if X is uniformly smooth and satisfies the condition , then so does .
Proof.
We know that for all and hence . We first show that is of type . From of Corollary 1, we know that each is of type . Since is of type from X into itself and
Lemma 3 implies that each is also of type .
Since is of type by Remark 2, is of type by Theorem 3, and
the part of Lemma 4 implies that is of type .
We finally show the part . Suppose that X is uniformly smooth and satisfies the condition . Since C is weakly closed, we can easily see that . This implies that satisfies the condition . From Theorem 3, we know that satisfies the condition . Thus, the part of Lemma 4 implies the conclusion. □
As a direct consequence of Theorems 2 and 4, we obtain the following result.
Theorem 5.
Let X be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of X, be two sequences of firmly nonexpansive-like mappings from C into X such that is nonempty and satisfies the condition , be a sequence of real numbers such that
and be a sequence defined by and
for all . If J is weakly sequentially continuous, then converges weakly to the strong limit of .
Author Contributions
H.I. analyzed and prepared/edited the manuscript, M.R.H. analyzed and prepared the manuscript, V.P. analyzed and prepared/edited the manuscript, C.P. analyzed and prepared/edited the manuscript, and S.K. analyzed and prepared the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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