Abstract
In this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify Halpern’s iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to an open problem posed by Akashi and Takahashi in a recent work of (Appl. Math. Comput. 219(4):2035-2040, 2012) for nonexpansive and nonspreading mappings. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problems, and multiple sets split feasibility problems. Our results contain many original results on multiple sets split feasibility problem in the literature. Our results also improve and generalize many well-known results in the current literature.
MSC:47H10, 37C25.
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1 Introduction
Throughout this paper, we denote the set of real numbers and the set of positive integers by ℝ and ℕ, respectively. Let H be a Hilbert space with the norm and C a nonempty subset of H. Let be a mapping. We denote by the set of fixed points of T and by the set of attractive points (see [1]) of T, i.e.,
A mapping is said to be nonexpansive if for all . A mapping is said to be quasi-nonexpansive if and for all and . A mapping is said to be strongly monotone if there exists such that for all .
Recall that the one-step Halpern iteration (see [2]) is given by the following formula:
Here, is a real sequence in satisfying some appropriate conditions. A more general iteration scheme of one-step Halpern iteration is two-step Halpern iteration given by
where the sequences and satisfy some appropriate conditions. In particular, when all , the Halpern iteration (1.2) becomes the standard Halpern iteration (1.2).
Definition 1.1 Let C be a nonempty subset of a Hilbert space H. Then C is called star-shaped if there exists such that for any and any ,
Such is called a center of the star-shaped set C.
Recently, Takahashi and Takeuchi [1] introduced the concept of attractive points. Akashi and Takahashi [3] proved the following strongly convergence attractive point theorem for nonexpansive mappings on a star-shaped set C of a Hilbert space.
Theorem 1.1 Let H be a Hilbert space and C be a star-shaped subset of H with center . Let be a nonexpansive mapping with . Suppose that is a sequence generated by and
where , , and . Then converges strongly to , where is metric projection of H onto .
Akashi and Takahashi [3] posed the following open problem in their final remark.
Question 1.1 Is there any strong convergence theorem of Halpern’s type for a wide class of nonlinear mappings which contains nonexpansive mappings and nonspreading mappings in a real Hilbert space H?
Definition 1.2 ([4])
Let C be a nonempty subset of a Banach space X. For , we say that a mapping satisfies condition () on C if there exists such that for all ,
We say that T satisfies condition (E) on C whenever T satisfies () on C for some .
The split feasibility problem (SFP) is to find a point
where C is a nonempty closed convex subset of a Hilbert space , Q is a nonempty closed convex subset of a Hilbert space , and is an operator. The split feasibility problem in finite dimensional Hilbert spaces was first introduced by Censor et al. [5] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. The split feasibility problem has applications in signal processing, image reconstruction, approximation theory, control theory, biomedical engineering, communications, and geophysics. One may refer to [6–9].
Let be nonempty closed convex subsets of a Hilbert space , let be nonempty, closed convex subsets of Hilbert space and let be linear operators. The well known multiple sets split feasibility problem (MSSFP) is to find such that and for all .
Multiple sets split feasibility problem (MSSFP) contains convex feasibility problem (CFP) and split feasibility problems (SFP) as special cases [5, 10, 11].
In this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify the Halpern iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to open Question 1.1. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iterative scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problem, multiple sets split feasibility problem. To the best of our knowledge, there is no result on multiple sets split monotone inclusion for inverse strongly monotone mappings and multiple sets split optimization problem in the literature. Our results also improve and generalize many well-known results in the current literature; see, for example, [3].
2 Preliminaries
Following Kohsaka and Takahashi [12], a mapping is said to be nonspreading if
for all .
Let C be a nonempty, closed convex subset of a Hilbert space H and . Then there exists a unique nearest point such that . We denote such a correspondence by . The mapping is called a metric projection of H onto C.
Definition 2.1 ([13])
Let C be a nonempty subset of a Banach space X. We say that a mapping satisfies condition (C) on C if for all ,
implies
Let .
-
(i)
It is obvious that if T is nonexpansive, then T satisfies () on C for some , but the converse is not true.
-
(ii)
If T satisfies condition (C), then T satisfies () on C for some , but the converse is not true.
-
(iii)
If T satisfies condition (E), it is easy to see that
In this section, we collect some lemmas which will be used in the proofs for the main results in next sections. We start with the following well-known lemma.
Lemma 2.1 ([14])
Let H be a real Hilbert space and C a nonempty convex subset of H. For given :
-
(i)
if and only if
-
(ii)
if and only if
-
(iii)
, . Consequently, is a nonexpansive mapping.
Lemma 2.2 ([15])
In a Hilbert space H, we have
-
(i)
for all and
-
(ii)
for all
-
(iii)
for all
-
(iv)
for all ,
Takahashi and Takeuchi [1] proved the following useful lemmas related to attractive points of a nonempty set C in a Hilbert space H.
Lemma 2.3 Let C be a nonempty subset of a real Hilbert space H and be a mapping. Then is a closed convex subset of H.
Lemma 2.4 Let C be a nonempty subset of a real Hilbert space H and be a mapping. If be a sequence in H such that
for all . If a subsequence of , converges weakly to , then .
Lemma 2.5 Let C be a nonempty subset of a real Hilbert space H and be a nonexpansive mapping. Then the following assertions are equivalent.
-
(1)
The attractive point set .
-
(2)
There exists such that the sequence is bounded.
Proposition 2.1 Let C be a nonempty subset of a Banach space X and be a mapping which satisfies condition () for some . Then the following statements hold.
-
(i)
For all ,
-
(ii)
For all ,
-
(iii)
For all ,
Proof
-
(i)
Since T satisfies condition (),
-
(ii)
By (i),
-
(iii)
This follows immediately from (ii). This completes the proof.
□
Example 2.1 Let be defined by
Then T is a nonspreading mapping with . Indeed, for any and , we have and . Observe now that
The other cases can be verified similarly. It is worth mentioning that T is neither nonexpansive nor continuous.
Proposition 2.2 Let C be a nonempty subset of a Banach space X and be a mapping. If T is a nonspreading mapping, then it satisfies condition () on C.
Proof Since T is a nonspreading mapping, then we have
This implies that, for any ,
If (1) holds, then we have
If (2) holds, then we obtain
This completes the proof. □
Let us give an example of a generalized nonexpansive mapping which is not a nonspreading mapping.
Example 2.2 Let be defined by
It could easily be verified that T satisfies condition (E) on ; for more details, see [4]. However, the mapping T is not a nonspreading mapping. Indeed, for and , we have and . Thus we obtain
and
If T is a nonspreading mapping, then, in view of (2.2), we have
This is a contradiction. Therefore, T is not a nonspreading mapping.
Lemma 2.6 (see [[16], Lemma 2.1])
Let be a sequence of nonnegative real numbers satisfying the inequality:
where and satisfy the conditions:
-
(i)
and , or equivalently, ;
-
(ii)
, or
(ii)′ .
Then .
To prove our main result, we need the following lemma.
Lemma 2.7 ([17])
Let be a sequence of nonnegative real numbers, let be a sequence of with , let be a sequence of nonnegative real numbers with and let be a sequence of real numbers with . Suppose that
Then .
The following lemma has been proved in [18].
Lemma 2.8 Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a subsequence such that and the following properties are satisfied by all (sufficiently large) numbers :
In fact, .
Let X be a real Banach space. The modulus δ of convexity of X is denoted by
for every ϵ with . A Banach space X is said to be uniformly convex if for every . It is well that any Hilbert space is a uniformly convex Banach space; see, for more details [14].
We know the following result from [19].
Lemma 2.9 Let X be a uniformly convex Banach space and , . Then there exists a continuous strictly increasing convex function with such that
for all and all with .
The following result has been proved in [20].
Lemma 2.10 Let X be a uniformly convex Banach space, be a constant. Then there exists a continuous, strictly increasing and convex function such that
for all , , and with .
3 Strong convergence theorems
The following result presents an existence theorem of attractive points of a generalized nonexpansive mapping T on a nonempty subset C of a Hilbert space H.
Theorem 3.1 Let C be a nonempty subset of a real Hilbert space H. Let be a mapping satisfying condition (E) on C which is uniformly asymptotically regular, i.e., for all . Then the following assertions are equivalent.
-
(1)
The attractive point set .
-
(2)
There exists such that the sequence is bounded.
Proof The implication (1) ⟹ (2) is obvious. For the converse implication, suppose that there exists such that the sequence is bounded. Setting for all , the uniformly asymptotically regularity of T assures that
Since is bounded and C is a nonempty subset of the Hilbert space H, there exists a subsequence of such that as . Next, we denote by for all . This, together with (3.1), implies that
Thus we have
The Opial property implies that , which completes the proof. □
The following strong convergence result provides an affirmative answer to open Question 1.1 in the case where the mapping T is a generalized nonexpansive mapping.
Theorem 3.2 Let H be a Hilbert space and C be a star-shaped subset of H with center . Let be a mapping satisfying condition (E) on C such that . Suppose that is a sequence generated by and
where , , and . Then converges strongly to , where is metric projection of H onto .
Proof Let and . Following the same argument as in Theorem 3.1 [3], we can show that the sequences and are bounded.
Let , with the same argument as in Theorem 3.1 [3], we see that
Since , , and , we have from Lemma 2.7 that
This last result together with (3.2) amounts to
In view of Lemma 2.2(iv), we get, for any ,
where . Thus we obtain
Since is bounded, there exists a subsequence of such that , and
By Lemma 2.4, . This, together with Lemma 2.1(ii), implies that
From Lemma 2.2(iii) and (3.2), we have
Then Theorem 3.2 follows from Lemma 2.7. □
Applying Theorems 3.1 and 3.2 and following the same arguments as Theorem 3.2 [3], we have the following fixed point theorem, which generalizes Theorem 3.2 [3].
Theorem 3.3 Let H be a Hilbert space and C be a closed star-shaped subset of H. Let be a mapping satisfying condition (E) on C such that T is uniformly asymptotically regular and is bounded for some . Then .
For the special case of Theorem 3.2, we have the following fixed point theorem.
Corollary 3.1 Let H be a Hilbert space and C be a closed star-shaped subset of H. Let be a mapping satisfying condition (C) on C such that T is uniformly asymptotically regular and is bounded for some . Then .
Applying Theorem 3.2 and following the same arguments as Theorem 3.4 [3], we have the following fixed point convergence theorem, which generalizes Theorem 3.4 [3].
Theorem 3.4 Let H be a Hilbert space and C be a closed star-shaped subset of H with center . Let be a mapping satisfying condition (E) on C such that . Suppose that is a sequence generated by and
where , , and . Then converges strongly to some , where
Corollary 3.2 Let H be a Hilbert space and C be a closed star-shaped subset of H with center . Let be a mapping satisfying condition (C) on C such that . Suppose that is a sequence generated by and
where , , and . Then converges strongly to some , where
Remark 3.1 The two-step Halpern iteration process is a generalization of the one-step Halpern iteration process. It provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective.
In the following, we prove strong convergence theorems of common attractive points for generalized nonexpansive mappings in a Hilbert space.
Theorem 3.5 Let H be a Hilbert space and C be a convex subset of H and . Let be a mapping satisfying condition () on C and be a mapping satisfying condition () on C such that . Let , , , and be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, .
Let be a sequence generated by
Then the sequence defined in (3.3) converges strongly to , where is the metric projection from H onto A.
Proof We divide the proof into several steps. Set
Step 1. We prove that the sequences , , , and are bounded.
We first show that is bounded.
Let be fixed. In view of Lemma 2.9, there exists a continuous strictly increasing convex function with such that
This together with (3.3) entails
Consequently, by induction, we deduce that
for all . This implies that the sequence is bounded and hence the sequence is bounded. Then, by (3.4), , , and are bounded.
Step 2. We prove that for any
Let us show (3.5). For each and , in view of (3.4), we obtain
This implies that
Let . It follows from (3.6) that
In view of Lemma 2.2(ii) and (3.4), we obtain
Step 3. We prove that as .
We discuss the following two possible cases.
Case 1. Suppose that there exists such that is nonincreasing. Then the sequence is convergent. Thus we have as . This, together with conditions (c), (d), and (3.7), imply that
On the other hand, we have
This implies that
By the triangle inequality, we conclude that
It follows from (3.9) that
Using Proposition 2.1, Lemma 2.2(iv), and (3.8), we obtain for any
where . Thus we obtain
Similarly, we have
where .
Thus we obtain
Since is bounded, there exists a subsequence of such that , and
By Lemma 2.4, (3.10), and (3.11), . By Lemma 2.1(ii), we show that
Thus we have the desired result by (3.5) and Lemma 2.6.
Case 2. Suppose that there exists a subsequence of such that
for all . Then, by Lemma 2.8, there exists a nondecreasing sequence such that ,
for all . This, together with (3.7), imply that
for all and . By conditions (a), (c), and (d), we have
By the same argument, as in Case 1, we arrive at
It follows from (3.7) that
Since , we have
In particular, since , we obtain
In view of (3.15), we deduce that
This, together with (3.14), implies that
On the other hand, we have for all which implies that as . Thus, we have as . We thus complete the proof. □
Corollary 3.3 Let H be a Hilbert space and C be a convex subset of H and . Let a mapping satisfying condition (E) on C such that . Let and be two sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
.
Let be a sequence generated by
Then the sequence defined in (3.16) converges strongly to , where is the metric projection from H onto .
Applying Theorem 3.5, we study the approximation of common fixed points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings.
Theorem 3.6 Let H be a Hilbert space and C be a closed convex subset of H and . Let be a mapping satisfying condition () on C and a mapping satisfying condition () on C such that . Let , , , be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, .
Let be a sequence generated by
Then the sequence defined in (3.17) converges strongly to some , where
Proof Since and are mappings satisfying condition (E), for any and , we have
and
This implies that . Thus we obtain . It follows from Theorem 3.6, that converges strongly to . Since C is closed, we have . We follow the same argument as in the proof of Theorem 3.3 [3], we can prove Theorem 3.7. □
Using Lemma 2.10 and Theorem 3.5, we can prove the following result.
Theorem 3.7 Let H be a Hilbert space and C be a convex subset of H and . For any , let be a mapping satisfying condition () on C such that . Let , be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, .
Let be a sequence generated by
Then the sequence defined in (3.18) converges strongly to , where is the metric projection from H onto A.
Remark 3.2 Theorem 3.7 improves Theorem 1.1 and many fixed point results in the literature.
4 Applications to multiple sets split feasibility problems
Let , and be Hilbert spaces, , and be nonempty, closed convex subsets of , and , respectively. Let be a multivalued mapping. The effective domain of G is denoted by , that is, . Then is called
-
(i)
a monotone operator on if for all , , and ;
-
(ii)
a maximal monotone operator on if G is a monotone operator on and its graph is not properly contained in the graph of any other monotone operator on .
A mapping is called α-inverse strongly monotone on (in short α-ism), if
Let I and denote the identity functions on , and , respectively. For each , let
-
(i)
κ, , and , be a -inverse strongly monotone mapping of into , be a -inverse strongly monotone mapping of into , L be a κ-inverse strongly monotone mapping of into ;
-
(ii)
M and be maximal monotone mappings on such that the domains of M and are included in , be a maximal monotone mapping on such that the domain of includes ;
-
(iii)
, , ;
-
(iv)
and be bounded linear operators, A and be the adjoints of A and , respectively;
-
(v)
R and be the spectral radii of and , respectively.
Throughout this section, we use these notations and assumptions unless specified otherwise.
A mapping is said to be averaged if , where and is nonexpansive. In this case, we also say that T is α-averaged. A firmly nonexpansive mapping is -averaged.
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a mapping. Then the following are satisfied:
-
(i)
T is nonexpansive if and only if the complement is a -ism.
-
(ii)
If S is υ-ism, then for , γS is a -ism.
-
(iii)
S is averaged if and only if the complement is a υ-ism for some .
-
(iv)
If S and T are both averaged, then the composite ST of S and T is averaged.
-
(v)
If the mappings are averaged and have a common fixed point, then .
In order to study the convergence theorems for the solutions set of multiple sets split problems, we must give an essential result in this paper. We study the following essential problem (SFP-1):
Recently, Yu, Lin and Chuang [23] proved the following useful result.
Lemma 4.2 ([23])
Given any , we have the following.
-
(i)
If is a solution of (SFP-1), then , where , , and .
-
(ii)
Suppose that , , , then , and are averaged. We assume further that the solution set of (SFP-1) is nonempty, and . Then is a solution of (SFP-1).
In the following theorem, we study the following multiple sets variational inclusion problems (MSSVIP-1):
and
where and .
Let denote the solution set of the problem (MSSVIP-1).
Theorem 4.1 Let . Let be a mapping defined by
and be a mapping defined by
where , , and .
Suppose that .
Let , , , be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, ;
-
(e)
for each , , , and .
Let be a sequence generated by
Then the sequence defined in (4.1) converges strongly to some , where .
Proof By Lemma 4.2, for each , and are averaged. Since , there exists , such that
and
and .
By Lemma 4.2,
We also see that
By Lemma 4.1, we see that
and
Therefore and .
By Lemma 4.1 again, we see that and are averaged. Therefore and are nonexpansive mapping. Then by Theorem 3.7, the sequence defined in (4.1) converges strongly to some , where . Since , it follows from Lemma 4.1 that . This completes the proof. □
Remark 4.1 Moudafi [24] studied a weak convergence of split monotone variational inclusion problem, while Theorem 4.1 studied a strong convergence theorem for the multiple sets split monotone variational inclusion problem.
In the following theorem, we study the following multiple sets split inclusion problems for inverse strongly monotone mappings (MSSVIP-2):
and
where and .
Let denote the solution set of the problem (MSSVIP-2).
Theorem 4.2 Let . Let be a mapping defined by
and be a mapping defined by
where , , and .
Suppose that .
Let , , , be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, ;
-
(e)
for each , , , and .
Let be a sequence generated by
Then the sequence defined in (4.1) converges strongly to some , where .
Proof By Lemma 4.2, for each , are averaged and is inverse strongly monotone for some . Following the same argument as in Theorem 3.1 [24], we can show that for each , , is -inverse strong monotone. Following the same argument as in Theorem 4.1, we show that the sequence defined in (4.1) converges strongly to some , where . Since , it follows from Lemma 4.1, it is easy to show that . This completes the proof of Theorem 4.2. □
Remark 4.2 To the best of our knowledge, there are many results on inclusion problems for maximum monotone mappings, but there are no results on inclusion problem for inverse strongly monotone mappings or split inclusion problems for inverse strongly monotone mappings.
As an application of the split inclusion problem for inverse strongly monotone mappings, we study the following split optimization problem.
Let and be nonempty open convex sets in and , respectively, , . For each , let and be convex Gâteaux differential functions. In the following theorem, we study the following multiple sets split optimization problem (MSSVIP-3):
and
where and .
Let denote the solution set of the problem (MSSVIP-3).
Theorem 4.3 In Theorem 4.2, we assume further that and are nonempty open convex sets in and , respectively, , . For each , let and be convex Gâteaux differential functions. For each , suppose that and are strongly monotone and Lipschitz continuous on and , respectively, and and be the Gâteaux derivatives of and , respectively. Then the sequence defined in (4.1) converges strongly to some .
Proof Since for each , and are Lipschitz and strongly monotone, it is easy to see that and are inverse strongly monotone. By Theorem 4.2, the sequence defined in (4.1) converges strongly to some . Therefore for each , , , . Since for each , and are convex Gâteaux differential functions with Gâteaux derivatives and , respectively, we obtain
for all .
Then, for each , for all and .
Similarly, for each , for all and .
This shows that . □
Let f be a proper lower semicontinuous convex function of into . The subdifferential ∂f of f is defined as follows:
for all . From Rockafellar [25], we know that ∂f is a maximal monotone operator. Let C be a nonempty closed convex subset of a real Hilbert space , and be the indicator function of C, i.e.
Then is a proper lower semicontinuous convex function on H, and the subdifferential of is a maximal monotone operator. We define the resolvent for all . We have .
For each , and , let be nonempty closed convex subsets of and be nonempty closed convex subsets of .
In the following theorem, we study the following multiple sets split feasibility problems (MSSVIP-4):
and , , …, , where , .
Let denote the solution set of the problem (MSSVIP-4).
Theorem 4.4 For each , and , let be the nonempty closed convex subsets of and be nonempty closed convex subsets of . Let be a mapping defined by
and a mapping defined by
where , and .
Suppose that .
Let , , , be sequences in satisfying the following control conditions:
-
(a)
;
-
(b)
;
-
(c)
, ;
-
(d)
, ;
-
(e)
for each , .
Let be a sequence generated by
Then the sequence defined in (4.3) converges strongly to some , where
Proof Let , , , in Theorem 4.1. Then Theorem 4.4 follows from Theorem 4.1. □
Example 4.1 Let be defined by
Then, T is a nonspreading mapping. Indeed, for any and , we have and . Observe now that
Therefore, T satisfies condition with . Let , , , then , , …, . We see that the sequence converges strongly to .
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Naraghirad, E., Lin, LJ. Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications. Fixed Point Theory Appl 2014, 72 (2014). https://doi.org/10.1186/1687-1812-2014-72
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DOI: https://doi.org/10.1186/1687-1812-2014-72