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Journal of Mathematics
Volume 2019, Article ID 5874305, 12 pages
https://doi.org/10.1155/2019/5874305
Research Article

Fixed Point and Acute Point Theorems for New Mappings in a Banach Space

1College of Engineering, Nihon University, Fukushima 963–8642, Japan
2Faculty of Engineering, Tamagawa University, Tokyo 194–8610, Japan

Correspondence should be addressed to Toshiharu Kawasaki; pj.en.ytfin@ikasawak.urahihsot

Received 5 September 2018; Accepted 24 April 2019; Published 9 June 2019

Academic Editor: Tepper L. Gill

Copyright © 2019 Toshiharu Kawasaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In a Hilbert space, concepts of attractive point and acute point are studied by many researchers. Moreover, these concepts extended to Banach space. In this paper, we introduce a new class of mappings on Banach space corresponding to the class of all widely more generalized hybrid mappings on Hilbert space. Moreover, we introduce some extensions of acute point and prove some acute point theorems.

1. Introduction

In [1] Takahashi and Takeuchi introduced a concept of attractive point in a Hilbert space. Let be a real Hilbert space, let be a nonempty subset of , and let be a mapping from into . is called an attractive point of if for any . LetMoreover, they proved that the Baillon type ergodic theorem [2] for generalized hybrid mappings [3] without convexity of . A mapping from into is said to be generalized hybrid if there exist such that for any . Such a mapping is said to be -generalized hybrid. The class of all generalized hybrid mappings is a new class of nonlinear mappings including nonexpansive mappings, nonspreading mappings [4], and hybrid mappings [5]. A mapping from into is said to be nonexpansive if for any ; nonspreading if for any ; hybrid if for any . Any nonexpansive mapping is -generalized hybrid; any nonspreading mapping is -generalized hybrid; any hybrid mapping is -generalized hybrid.

Motivated these mappings, in [6] Kawasaki and Takahashi introduced a new very wider class of mappings, called widely more generalized hybrid mappings, than the class of all generalized hybrid mappings. A mapping from into is widely more generalized hybrid if there exist such that for any . Such a mapping is said to be -widely more generalized hybrid. This class includes the class of all generalized hybrid mappings and also the class of all -pseudocontractions [7] for . A mapping from into is called a -pseudocontraction if for any . Any -generalized hybrid mapping is -widely more generalized hybrid; any -pseudocontraction is -widely more generalized hybrid. Moreover, they proved some fixed point theorems and some ergodic theorems; for instance, see [6, 810].

There are some studies on Banach space related to these results. In [11] Takahashi, Wong and Yao introduced the generalized nonspreading mapping and the skew-generalized nonspreading mapping in a Banach space. Let be a smooth Banach space and let be a nonempty subset of . A mapping from into is said to be generalized nonspreading if there exist such that for any , where is the duality mapping on and Such a mapping is said to be -generalized nonspreading. A mapping from into is said to be skew-generalized nonspreading if there exist such that for any . Such a mapping is said to be -skew-generalized nonspreading. These classes include the class of generalized hybrid mappings in a Hilbert space; however, it does not include the class of widely more generalized hybrid mappings. Moreover, they introduced some extensions of attractive point and proved some attractive point theorems. is an attractive point of if for any ; is a skew-attractive point of if for any . Let They also proved the following.

Theorem 1. Let be a reflexive and smooth Banach space, let be a nonempty subset of , and let be an -generalized nonspreading mapping from into itself satisfying and . Then has an attractive point if and only if there exists such that is bounded.
Additionally, if is strictly convex and is closed and convex, then has a fixed point if and only if there exists such that is bounded.

Theorem 2. Let be a strictly convex, reflexive, and smooth Banach space, let be a nonempty subset of , and let be an -skew-generalized nonspreading mapping from into itself satisfying and . Then has a skew-attractive point if and only if there exists such that is bounded.
Additionally, if is strictly convex and is closed and convex, then has a fixed point if and only if there exists such that is bounded.

On the other hand, in [12] Atsushiba, Iemoto, Kubota, and Takeuchi introduced a concept of acute point as an extension of attractive point in a Hilbert space. Let be a real Hilbert space, let be a nonempty subset of , and let be a mapping from into and . is called a -acute point of if for any . Let Moreover, using a concept of acute point, they proved convergence theorems without convexity of .

Motivated by these results, in this paper, we introduce a new class of mappings on Banach space corresponding to the class of all widely more generalized hybrid mappings on Hilbert space. Moreover, we introduce some extensions of acute point and prove some acute point theorems. Moreover, in the next paper [13], we show some mean convergence theorems for the new class of mappings.

2. Preliminaries

We know that the following hold; for instance, see [14].(T1)Let be a normed space, let be a nonempty, closed, and convex subset in , and let be a nonempty, compact, and convex subset in . Suppose that . Then (T2)Let be a normed space and let and be nonempty and convex subsets with . Then there exists , where is the topological dual space of , such that and (T3)Let be a Banach space, let be the topological dual space of , and let be the duality mapping on defined by for any . Then is strictly convex if and only if is injective; that is, implies .(T4)Let be a Banach space, let be the topological dual space of , and let be the duality mapping on . Then is reflexive if and only if is surjective; that is, .(T5)Let be a Banach space and let be the duality mapping on . Then is smooth if and only if is single-valued.(T6)Let be a Banach space and let be the duality mapping on . If is single-valued, then is norm-to-weak continuous.(T7)Let be a Banach space and let be the duality mapping on . Then is strictly convex if and only if for any with and and for any .(T8)Let be a Banach space and let be the topological dual space of . Then is reflexive if and only if is reflexive.(T9)Let be a Banach space and let be the topological dual space of . If is strictly convex, then is smooth. Conversely, is reflexive and smooth, then is strictly convex.(T10)Let be a Banach space and let be the topological dual space of . If is smooth, then is strictly convex. Conversely, is reflexive and strictly convex, then is smooth.

Let be a smooth Banach space, let be the duality mapping on , and let be the mapping from into defined by for any . Since by (T5) is single-valued, is well-defined. It is obvious that implies . Conversely, by (T7)(T11)If is also strictly convex, then implies .

Let be a strictly convex and smooth Banach space. By (T3) an (T5) is a bijective mapping from onto . In particular, if is also reflective, then by (T4) is a bijective mapping from onto . Suppose that is strictly convex, reflective, and smooth. Let be the mapping from into defined by for any . Thenholds. Therefore if and only if .

Let be the Banach space consists of all bounded sequences and . Sometimes we denote by the value . If satisfies , where , then is called a mean. If a mean satisfies , then is called a Banach limit. We know that there exists some Banach limits. If and is a mean, then the following holds:(T12).

The following lemma is introduced in [11]. For completeness we write its proof.

Lemma 3. Let be a Banach space, let be the dual space of , let be a bounded sequence in , and let be a mean on . Then there exists a unique such that for any .

Proof. Define a mapping from into by for any . Then is linear and bounded clearly and hence . Since is reflexive, we can put .
Lastly, we show that . Assume that . Since is nonempty, close, and convex and is nonempty, compact, and convex, by (T1) and (T2) there exists such that and On the other hand, by (T12) we obtain It is a contradiction. Therefore .

3. Fixed Point Theorems

Let be a smooth Banach space and let be a nonempty subset of . A mapping from into is called a generalized pseudocontraction if there exist such thatfor any . Such a mapping is called an -generalized pseudocontraction.

If all parameters are negative, of course, all mappings satisfy the above inequality. However, in the following argument, such cases are properly excluded.

Theorem 4. Let be a strictly convex, reflexive and smooth Banach space, let be a nonempty, closed, and convex subset of and let be an -generalized pseudocontraction from into itself. Then the following hold. (1)Suppose that satisfies , , , , , and one of the following(1-1) and ;(1-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(2)Suppose that satisfies , , , , , and one of the following:(2-1) and ;(2-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(3)Suppose that satisfies , , , and one of the following:(3-1) and ;(3-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if , then the fixed point is unique.

Proof. If has a fixed point , then is bounded.
Conversely, suppose that there exists such that is bounded. Moreover, we suppose that (1) holds. Sincefor any , we obtain Since , , , , and , we obtain Therefore Replacing by , we obtain Applying a Banach limit to both sides of this inequality, we obtain By Lemma 3 there exists a unique such thatTherefore we obtain for any . Putting , we obtain In the case of (1-1), since , and is strictly convex, by (T11) we obtain . In the case of (1-2), since , , and is strictly convex, by (T11), we obtain .
In the case of (2), changing the variables and in (26), we obtainSuppose that (2) holds. Using the result in the case of (1), we can show that has a fixed point.
In the case of (3), adding (26) and (36), we obtainSuppose that (3) holds. Using the result in the case of (1), we can show that has a fixed point.
Lastly, suppose that or in the case of (1). Let and be fixed points of . Then By (T11) we obtain holds. The cases of (2) and (3) are similar.

Let be the dual space of a strictly convex, reflexive, and smooth Banach space and let be a nonempty subset of . A mapping from into is called a -generalized pseudocontraction if there exist such that for any . Such a mapping is called an --generalized pseudocontraction.

Theorem 5. Let be the dual space of a strictly convex, reflexive, and smooth Banach space , let be a nonempty, closed, and convex subset of , and let be an --generalized pseudocontraction from into itself. Then the following hold.(1)Suppose that satisfies , , , , , and one of the following:(1-1) and ;(1-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(2)Suppose that satisfies , , , , , and one of the following:(2-1) and ;(2-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(3)Suppose that satisfies , , , and one of the following:(3-1) and ;(3-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if , then the fixed point is unique.

Proof. If has a fixed point , then is bounded.
Conversely, suppose that there exists such that is bounded. Moreover, we suppose that (1) holds. From (23) and (27) we obtain for any . Similarly to the proof of Theorem 4 we obtainReplacing by , we obtain Applying a Banach limit to both sides of this inequality, we obtain By Lemma 3 there exists a unique such that Therefore we obtain for any . Putting , we obtain In the case of (1-1), since , , and is strictly convex, by we obtain . In the case of (1-2), since , , and is strictly convex, by we obtain .
In the cases of (2) and (3) we can show similarly.

By Theorem 5 we obtain the following.

Theorem 6. Let be a strictly convex, reflexive, and smooth Banach space, let be a nonempty subset of satisfying is closed and convex, and let be an -generalized pseudocontraction from into itself. Then the following hold. (1)Suppose that satisfies , , , , , and one of the following:(1-1) and ;(1-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(2)Suppose that satisfies , , , , , and one of the following:(2-1) and ;(2-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if or , then the fixed point is unique.(3)Suppose that satisfies , , , and one of the following:(3-1) and ;(3-2) and .Then has a fixed point if and only if there exists such that is bounded.Moreover, if , then the fixed point is unique.

Proof. If has a fixed point , then is bounded.
Conversely, suppose that there exists such that is bounded. Since is strictly convex, reflexive, and smooth, by (T8), (T9), and (T10) is also so. By assumption is nonempty, close, and convex. Let . Then is a mapping from into itself. Putting , since , , and hence is bounded. Putting and , (26) is equivalent to