Fixed point theorems of ( a , b ) -monotone mappings in Hilbert spaces

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.

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.

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]

LetCbe a nonempty, closed, and convex subset of a Hilbert spaceH. A mappingbe a mapping.

(i) IfTis a nonexpansive mapping, thenTis a-generalized hybrid mapping;

(ii) IfTis a nonspreading mapping, thenTis a-generalized hybrid mapping;

(iii) IfTis a hybrid mapping, thenTis a-generalized hybrid mapping;

(iv) IfTis aTYmapping, thenTis a-generalized hybrid mapping;

(v) IfTis an-generalized hybrid mapping withand, thenTis a-generalized hybrid mapping.

Lemma 2.1[3]

LetCbe a nonempty, closed and convex subset of a Hilbert spaceH. Letbe a mapping. Suppose that there existand a Banach limitμsuch thatis bounded and

ThenThas a fixed point.

Lemma 2.2[9]

LetHbe a real Hilbert space. LetCbe a closed convex subset ofH, letand letabe a real number. The set

is closed and convex.

Lemma 2.3LetKbe a closed convex subset of a real Hilbert spaceHand letbe the metric projection fromHontoK. Letand. Thenif and only if

Lemma 2.4[9]

LetKbe a closed convex subset of a real Hilbert spaceH. Letbe a sequence inHand. Let. Suppose thatand

Then.

Lemma 2.5LetHbe a real Hilbert space. Then

Theorem 2.1[10]

LetHbe a Hilbert space and letbe a bounded sequence inH. Thenis weakly convergent if and only if each weakly convergent subsequence ofhas the same weak limit, that is, for,

Proposition 3.1LetCbe a nonempty, closed, and convex subset of a Hilbert spaceH. Ifis a-generalized hybrid mapping with, thenTis 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 3.2LetCbe a nonempty, closed, and convex subset of a Hilbert spaceH. A mappingbe a mapping.

(i) IfTis a nonspreading mapping, thenTis a-monotone mapping;

(ii) IfTis a hybrid mapping, thenTis a-monotone mapping;

(iii) IfTis aTYmapping, thenTis a-monotone mapping;

(vi) IfTis an-generalized hybrid mapping with, and, thenTis an-monotone mapping.

Proposition 3.3LetCbe a closed convex subset of a Hilbert space. LetTbe a-monotone mapping defined onC. 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 3.1LetCbe a closed convex subset of a Hilbert space. LetTbe a-monotone mapping defined onC. If a sequencewithand. 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, . □

Corollary 3.1[2-4]

LetCbe a closed convex subset of a Hilbert space. LetTbe a self-mapping defined onCand satisfies one of the following:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

If a sequencewithand, then.

Theorem 3.2LetCbe a closed convex subset of a Hilbert space. LetTbe a-monotone mapping defined onC. Ifis nonempty, thenis 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 3.3LetCbe a nonempty subset of a Hilbert spaceH. Letbe a-monotone mapping with. Suppose thatis bounded for some. Thenfor all Banach limitsμand 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 3.4LetCbe a nonempty, closed, and convex subset of a Hilbert spaceH. Letbe a-monotone mapping with. Thenif and only if there existssuch thatis bounded.

Corollary 3.2[1,3,4]

LetCbe a closed convex subset of a Hilbert space. LetTbe a self-mapping defined onCand satisfies one of the following:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

Thenif and only if there existssuch thatis bounded.

In this section, we first prove a weak convergence theorem of Mann’s type for -monotone mappings in a Hilbert space.

Theorem 4.1LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Letbe a-monotone mapping satisfies. If a sequencewithand, then for each, the sequencewithfor allweakly converges to some fixed point ofT.

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 4.1LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbe a mapping withand satisfies one of the following:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

If a sequencesatisfies, then for each, the sequencewithfor allweakly converges to some fixed point ofT.

Proof Since T is an -monotone mapping with , we have . Then Corollary 4.1 follows from Theorem 4.1. □

Corollary 4.2LetCbe a nonempty, closed, and convex subset of real Hilbert spaceH. Letbe a mapping withand satisfies one of the following:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

Then for each, the sequencewithfor allweakly converges to some fixed point ofT.

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 4.2LetCbe a nonempty, closed and convex subset of a real Hilbert spaceH. Letbe a-monotone mapping with. Suppose thatis a sequence generated by the following scheme:

If the sequencesatisfies, 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 4.3LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Letbe a mapping withand satisfies one of the following conditions:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

Suppose thatis a sequence generated by the following scheme:

If the sequencesatisfies. Then.

Proof Since T is a -monotone mapping with , we have , then Corollary 4.3 follows from Theorem 4.2. □

Corollary 4.4LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Letbe a mapping withand satisfies one of the following conditions:

(i) Tis a nonspreading mapping;

(ii) Tis a hybrid mapping;

(iii) Tis aTYmapping.

Suppose thatis a sequence generated by the following scheme:

Then.

Proof Take for all , then Corollary 4.4 follows from Corollary 4.3. □

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.

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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