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Fixed point theorems of -monotone mappings in Hilbert spaces
Fixed Point Theory and Applications volume 2012, Article number: 131 (2012)
Abstract
We propose a new class of nonlinear mappings, called -monotone mappings, and show that this class of nonlinear mappings contains nonspreading mappings, hybrid mappings, firmly nonexpansive mappings, and -generalized hybrid mappings with . We also give an example to show that a -monotone mapping is not necessary to be a quasi-nonexpansive mapping. We establish an existence theorem of fixed points and the demiclosed principle for the class of -monotone mappings. As a special case of our result, we give an existence theorem of fixed points for -generalized hybrid mappings with . We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for -monotone mappings. We give an example of -monotone mappings which assures the Mann’s type weak convergence.
1 Introduction
Let H be a real Hilbert space with a nonempty closed convex subset C. Let be a self-mapping defined on C. We denote by the set of fixed points of T. The mapping T is called quasi-nonexpansive if and
Takahashi et al. [1–7] gave the following definitions of nonlinear mappings and studied the existence and convergence theorems of fixed points for these mappings.
Definition 1.1 A mapping is called
-
(i)
nonspreading [1] if for every ,
-
(ii)
TY [3] if for every ,
-
(iii)
hybrid [4] if for every ,
-
(iv)
λ-hybrid () [5] if for every ,
-
(v)
-generalized hybrid () [6] if for every ,
-
(vi)
α-nonexpansive () [7] if for every ,
It is obvious that the mappings mentioned in Definition 1.1 are quasi-nonexpansive. Recently, Lin et al. [8] gave the following definition of a new class of nonlinear mappings.
Definition 1.2 [8]
Let , , and . A mapping is called a -generalized hybrid mapping if for every ,
This class of mappings are not necessary to be quasi-nonexpansive and contains nonexpansive mappings, nonspreading mappings, hybrid mappings, and TY mappings. Lin et al. [8] studied weak and strong convergence theorems of -generalized hybrid mappings, but existence theorems of fixed points for -generalized hybrid mapping are not discussed in [8]. On the other hand, Aoyama and Kohsaka [7] characterized the existence of fixed points of α-nonexpansive mappings in uniformly convex Banach spaces.
Motivated by the literatures above, we study existence theorems of fixed points for the mappings mentioned in Definitions 1.1 and 1.2 in an unified method. Precisely, we propose a new class of nonlinear mappings in Hilbert spaces.
Definition 1.3 Let and . A mapping is called an -monotone mapping if for every ,
or equivalently,
Remark 1.1 Let C be a nonempty, closed, and convex subset of a Hilbert space, and let . Recall that a mapping is called α-inverse strongly monotone if
A firmly nonexpansive mapping is an α-inverse strongly monotone mapping with . Note that a firmly nonexpansive mapping (1-inverse strongly monotone mapping with ) is a (1,0)-monotone mapping.
Next, we give an example to show that a -monotone mapping is not necessary to be a quasi-nonexpansive mapping.
Example 1.1 Let . Let and be defined by
for all and for all . Then the following statements hold:
-
(i)
T is a -monotone mapping;
-
(ii)
T is not a quasi-nonexpansive mapping.
Proof It’s obvious that . We first prove part (i). For each ,
Then for each , we have
-
(1)
,
-
(2)
,(3)
Then
By parallelogram law, we have
Take and . Then
Then T is a -monotone mapping. Next we want to prove part (ii). Since , T is not a quasi-nonexpansive mapping. The proof of part (ii) is complete. □
Remark 1.2 Since T in Example 1.1 is not a quasi-nonexpansive mapping, T is not nonspreading, TY, hybrid, λ-hybrid, -generalized hybrid, and α-nonexpansive. This example shows that an -monotone mapping is not necessary to be a quasi-nonexpansive mapping, TY mapping, hybrid mapping, λ-hybrid mapping, -generalized hybrid mapping, and α-nonexpansive mapping.
In this paper, we first show that the class of -monotone mappings contains nonspreading mappings, hybrid mappings, TY mappings, firmly nonexpansive mappings, and -generalized hybrid mappings with . We also give an example to show that this class of mappings are not necessary to be quasi-nonexpansive mappings. We establish an existence theorem of fixed points and the demiclosed principle for the class of -monotone mappings. As a special case of our result, we give an existence theorem of fixed points for -generalized hybrid mappings with . We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for -monotone mappings. An example of -monotone mappings is given to show the Mann’s type weak convergence.
2 Preliminaries
In this paper, we use the following notations:
-
(i)
⇀ for weak convergence and → for strong convergence.
-
(ii)
denotes the weak ω-limit set of .
Let us recall some known results, which will be used later.
Proposition 2.1 [8]
Let C be a nonempty, closed, and convex subset of a Hilbert space H. A mapping be a mapping.
-
(i)
If T is a nonexpansive mapping, then T is a -generalized hybrid mapping;
-
(ii)
If T is a nonspreading mapping, then T is a -generalized hybrid mapping;
-
(iii)
If T is a hybrid mapping, then T is a -generalized hybrid mapping;
-
(iv)
If T is a TY mapping, then T is a -generalized hybrid mapping;
-
(v)
If T is an -generalized hybrid mapping with and , then T is a -generalized hybrid mapping.
Lemma 2.1 [3]
Let C be a nonempty, closed and convex subset of a Hilbert space H. Let be a mapping. Suppose that there exist and a Banach limit μ such that is bounded and
Then T has a fixed point.
Lemma 2.2 [9]
Let H be a real Hilbert space. Let C be a closed convex subset of H, let and let a be a real number. The set
is closed and convex.
Lemma 2.3 Let K be a closed convex subset of a real Hilbert space H and let be the metric projection from H onto K. Let and . Then if and only if
Lemma 2.4 [9]
Let K be a closed convex subset of a real Hilbert space H. Let be a sequence in H and . Let . Suppose that and
Then .
Lemma 2.5 Let H be a real Hilbert space. Then
Theorem 2.1 [10]
Let H be a Hilbert space and let be a bounded sequence in H. Then is weakly convergent if and only if each weakly convergent subsequence of has the same weak limit, that is, for ,
3 Fixed point theorem of -monotone mappings
Proposition 3.1 Let C be a nonempty, closed, and convex subset of a Hilbert space H. If is a -generalized hybrid mapping with , then T is a -monotone mapping, where .
Proof If T is an -generalized hybrid mapping with , then for every ,
and
where .
Note that
We have
Without loss of generality, we may assume that .
Since , we have that
that is, . Take and , we see that T is an -monotone mapping. □
The following proposition follows immediately from Propositions 2.1 and 3.1.
Proposition 3.2 Let C be a nonempty, closed, and convex subset of a Hilbert space H. A mapping be a mapping.
-
(i)
If T is a nonspreading mapping, then T is a -monotone mapping;
-
(ii)
If T is a hybrid mapping, then T is a -monotone mapping;
-
(iii)
If T is a TY mapping, then T is a -monotone mapping;
-
(vi)
If T is an -generalized hybrid mapping with , and , then T is an -monotone mapping.
Proposition 3.3 Let C be a closed convex subset of a Hilbert space. Let T be a -monotone mapping defined on C. Then
Proof Since T is a -monotone mapping, we have that for each and ,
that is,
Then
that is,
□
Now we give a demiclosed principle of -monotone mappings:
Theorem 3.1 Let C be a closed convex subset of a Hilbert space. Let T be a -monotone mapping defined on C. If a sequence with and . Then .
Proof Since T is a -monotone mapping, we have that
that is,
Since and , and are bounded. Taking limit on the inequality above, we have
Since , we have that , that is, . □
Let C be a closed convex subset of a Hilbert space. Let T be a self-mapping defined on C and satisfies one of the following:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
If a sequence with and , then .
Theorem 3.2 Let C be a closed convex subset of a Hilbert space. Let T be a -monotone mapping defined on C. If is nonempty, then is closed and convex.
Proof First, we show that is closed. For each , there exists a sequence with . Since and for all , we have that and for all . By Theorem 3.1, . Next, we want to show that is a convex subset of C. Take any and . Let . By Proposition 3.3, we have
Since , we have that . □
Theorem 3.3 Let C be a nonempty subset of a Hilbert space H. Let be a -monotone mapping with . Suppose that is bounded for some . Then for all Banach limits μ and for all .
Proof Let μ be a Banach limit and let be given. Since T is a -monotone mapping with , we have that
that is,
Then
Hence . Since , we have that
□
As a direct consequence of Theorem 3.3 and Lemma 2.1, we have the following existence theorem of fixed points for -monotone mappings.
Theorem 3.4 Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let be a -monotone mapping with . Then if and only if there exists such that is bounded.
Let C be a closed convex subset of a Hilbert space. Let T be a self-mapping defined on C and satisfies one of the following:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
Then if and only if there exists such that is bounded.
4 Convergence theorems
In this section, we first prove a weak convergence theorem of Mann’s type for -monotone mappings in a Hilbert space.
Theorem 4.1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let be a -monotone mapping satisfies . If a sequence with and , then for each , the sequence with for all weakly converges to some fixed point of T.
Proof We first show that there exists a sequence satisfies our assumptions. Since , , we have , there exists a constant such that . If we take for all , then such that and . Since T is a -monotone mapping, by Proposition 3.3, we have that for each and ,
Since , we have that
Then exists and sequence is bounded. Further, from the inequality above, we have that
Since , we have .
Therefore, . Since is bounded, there exist a subsequence of and a point such that . Since T is a -monotone mapping, by Theorem 3.1, we have .
For each subsequence of with for some , we follow the same argument as above, we see that . We have to show that . Otherwise, if , then by Optial condition,
This leads to a contradiction. Therefore . By Theorem 2.1, we have that . □
Example 4.1 Let H, ϕ, T be the same as in Example 1.1. For any fixed , take a sequence as in Theorem 4.1 with for all , that is,
Then
and hence
Therefore, , and hence .
Corollary 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a mapping with and satisfies one of the following:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
If a sequence satisfies , then for each , the sequence with for all weakly converges to some fixed point of T.
Proof Since T is an -monotone mapping with , we have . Then Corollary 4.1 follows from Theorem 4.1. □
Corollary 4.2 Let C be a nonempty, closed, and convex subset of real Hilbert space H. Let be a mapping with and satisfies one of the following:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
Then for each , the sequence with for all weakly converges to some fixed point of T.
Proof Take . Then Corollary 4.2 follows from Corollary 4.1. □
Next we prove a strong convergence theorem by hybrid method for -monotone mappings in a Hilbert space.
Theorem 4.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a -monotone mapping with . Suppose that is a sequence generated by the following scheme:
If the sequence satisfies , then .
Proof By Lemma 2.2, we see that is closed and convex for all . For any , by Proposition 3.3, we have
Hence, . Then we have that for all .
Next, we show that for all . We prove this by induction. For , we have . Assume that . Since is the projection of onto , by Lemma 2.3, we have for all . As by the induction assumption, the last inequality holds, in particular, for all . This together with the definition of implies that . Hence for all . Then the sequence is well defined.
The definition of and Lemma 2.3 imply that , which in turn implies that for all , in particular, is bounded and with .
That asserts that
It follows from Lemma 2.5 and the inequality above that
The last inequality implies that is increasing. Since is bounded, we have that exists and . Since ,
Note that
Then
Then
Without loss of generality, we may assume that for all . Otherwise, for some and we complete the proof. Therefore,
Hence,
By the choice of , . Therefore, . Consequently, by Theorem 3.1. Hence, applying Lemma 2.4 (to and ), one can conclude that . □
Corollary 4.3 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let be a mapping with and satisfies one of the following conditions:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
Suppose that is a sequence generated by the following scheme:
If the sequence satisfies . Then .
Proof Since T is a -monotone mapping with , we have , then Corollary 4.3 follows from Theorem 4.2. □
Corollary 4.4 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let be a mapping with and satisfies one of the following conditions:
-
(i)
T is a nonspreading mapping;
-
(ii)
T is a hybrid mapping;
-
(iii)
T is a TY mapping.
Suppose that is a sequence generated by the following scheme:
Then .
Proof Take for all , then Corollary 4.4 follows from Corollary 4.3. □
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The authors declare no competing interests, except Prof. LJL was supported by the National Science Council of Republic of China while he worked on the publish.
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LJL responsible for the problem resign, coordinator, discussion, revised the manuscript and submission, SYW carried out this problem, complete the draft the manuscript. All the authors read and approved the manuscript.
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Lin, LJ., Wang, SY. Fixed point theorems of -monotone mappings in Hilbert spaces. Fixed Point Theory Appl 2012, 131 (2012). https://doi.org/10.1186/1687-1812-2012-131
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DOI: https://doi.org/10.1186/1687-1812-2012-131