Journal of Mathematics
Volume 2019 (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

Toshiharu Kawasaki1,2

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; toshiharu.kawasaki@nifty.ne.jp

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 . Let

Moreover, 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, 8–10].

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 . Then

holds. 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 that

for 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. Since

for 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 that

Therefore 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 obtain

Suppose 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 obtain

Suppose 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 obtain

Replacing 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

from (23). Therefore is an --generalized pseudocontraction from into itself. By Theorem 5 we have that has a fixed point . Then is a fixed point of .

4. Acute Point and Skew-Acute Point

Let be a smooth Banach space, let be a nonempty subset of , let be a mapping from into , and let . is called a -acute point of if

for any . is called a -skew-acute point of if

for any . Let

It is obvious that

for any with and .

The following lemmas are important property characterizing them.

Lemma 7. Let be a smooth Banach space, let be a nonempty subset of , let be a mapping from into , and let . Then is closed and convex.

Proof. Since (48) is equivalent to

from (27), is closed and convex.

Lemma 8. Let be a smooth Banach space, let be a nonempty subset of , let be a mapping from into , and let . Then is closed.

Proof. Inequality (49) is equivalent to

from (27). Moreover, by (T6) is norm-to-weak continuous. Therefore is closed.

Let be the dual space of a strictly convex, reflexive, and smooth Banach space , let be a nonempty subset of , let be a mapping from into , and let . is called a --acute point of if

for any . is called a --skew-acute point of if

for any . Let

Lemma 9. Let be the dual space of a strictly convex, reflective, and smooth Banach space , let be a nonempty subset of , let be a mapping from into , and let . Then is closed and convex.

Proof. Since

is equivalent to

from (27) and (23), is closed and convex.

Lemma 10. Let be the dual space of a strictly convex, reflexive, and smooth Banach space , let be a nonempty subset of , let be a mapping from into , and let . Then is closed.

Proof. Since

is equivalent to

from (27) and (23), moreover, by (T6) is norm-to-weak continuous. Therefore is closed.

Lemma 11. Let be a strictly convex, reflective, and smooth Banach space, let be a nonempty subset of , let be a mapping from into , let , and let . Then

In particular, is closed and convex and is closed.

Proof. Let . Then

for any . From (23)

for any . Since , putting , we obtain

Therefore and hence .
can be shown similarly.
Moreover, by Lemma 9 is closed and convex and by Lemma 10 is closed.

Lemma 12. Let be a strictly convex and smooth Banach space, let be a nonempty subset of , let be a mapping from into , and let . Then the following hold.

(1) If , then is included in the set of all fixed points of .(2) If , then is included in the set of all fixed points of .

Proof. Let . Then (48) holds for any . Putting , by (T11) we obtain . If , then by (T11) we obtain .
Let . Then (49) holds for any . Putting , by (T11) we obtain . If , then by (T11) we obtain .

5. Acute Point and Skew-Acute Point Theorems

Theorem 13. Let be a reflexive and smooth Banach space, let be a nonempty subset of , and let be an -generalized pseudocontraction from into itself. Suppose that there exists such that is bounded. Then the following hold.

(1)If , , , , , and , then there exists a -acute point.(2)If , , , , , and , then there exists a -acute point.(3)If , , , , and , then there exists a -acute point.

Proof. Suppose that there exists such that is bounded. In the case of (1), similarly to the proof of Theorem 4, we have that there exists a unique such that

for any . Since

for any , we obtain

Since , we obtain

Therefore is a -acute point.
In the cases of (2), changing the variables and in (26), we have that there exists a unique such that

Since , we obtain

Therefore is a -acute point.
In the case of (3), adding (26) and (36), we have that there exists a unique such that

Since , we obtain

Therefore is a -acute point.

Theorem 14. Let be the dual space of a strictly convex, reflexive, and smooth Banach space , let be a nonempty subset of , and let be an --generalized pseudocontraction from into itself. Suppose that there exists such that is bounded. Then the following hold.

(1)If , , , , , and , then there exists a --acute point.(2)If , , , , , and , then there exists a --acute point.(3)If , , , , and , then there exists a --acute point.

Proof. Similar to the proof of Theorem 5 we obtain

for any . Since

for any , we obtain

Since , we obtain

Therefore is a --acute point.
In the cases of (2) and (3) we can show similarly.

By Theorem 14 we obtain the following.

Theorem 15. Let be a strictly convex, reflexive, and smooth Banach space, let be a nonempty subset of , and let be an -generalized pseudocontraction from into itself. Suppose that there exists such that is bounded. Then the following hold.

(1)If , , , , , and , then there exists a -skew-acute point.(2)If , , , , , and , then there exists a -skew-acute point.(3)If , , , , and , then there exists a -skew-acute point.

Proof. By Theorem 14, has a -acute point . By Lemma 11 is a skew-acute point of .

6. Applications

By Theorem 13 we can show easily the first part of Theorem 1. Note that .

Proof of the First Part of of 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 .
Suppose that has an attractive point . Then it is clear that is bounded.
Conversely, suppose that there exists such that is bounded. is an -generalized pseudocontraction. These parameters satisfy the conditions of Theorem 13 (1). Therefore there exists a -acute point, that is, an attractive point.

By Theorem 15 we can show easily the first part of Theorem 2. Note that .

Proof of the First Part of 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 .
Suppose that has a skew-attractive point . Then it is clear that is bounded.
Conversely, suppose that there exists such that is bounded. is an -generalized pseudocontraction. These parameters satisfy the conditions of Theorem 15 (2). Therefore there exists a -skew-acute point, that is, a skew-attractive point.

By Theorem 13 and Lemma 12 (1) we obtain the following.

Theorem 16. 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. If satisfies , , , , , and , then by Theorem 13 there exits such that it is a -acute point. Moreover,

 If (1-1) holds, then and ; If (1-2) holds, then and .

By Lemma 12 (1) is a fixed point of .
In the cases of (2) and (3) we can show similarly.

Remark 17. Theorem 16 is the same as Theorem 4. The only difference is that Theorem 16 needs more assumptions than Theorem 4.

By Theorem 16 we can show easily the second part of Theorem 1.

Proof of the Second Part of Theorem 1. Let be a strictly convex, reflexive, and smooth Banach space, let be a nonempty, closed, and convex subset of , and let be an -generalized nonspreading mapping from into itself satisfying and .
Suppose that has a fixed point . Then it is clear that is bounded.
Conversely, suppose that there exists such that is bounded. is an -generalized pseudocontraction. These parameters satisfy the conditions of Theorem 16 (1). Therefore there exists a fixed point.

By Theorem 15 and Lemma 12 (2) we obtain the following.

Theorem 18. 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. If satisfies , , , , , and , then by Theorem 15 there exits such that it is a -skew-acute point. Moreover,

 If (1-1) holds, then and ; If (1-2) holds, then and .

By Lemma 12 (2) is a fixed point of .
In the cases of (2) and (3) we can show similarly.

Remark 19. Theorem 18 is the same as Theorem 6. The only difference is that Theorem 18 needs more assumptions than Theorem 6.
By Theorem 18 we can show easily the second part of Theorem 2.

Proof of the Second Part of Theorem 2. 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 -skew-generalized nonspreading mapping from into itself satisfying and .
Suppose that has a fixed point . Then it is clear that is bounded.
Conversely, suppose that there exists such that is bounded. is an -generalized pseudocontraction. These parameters satisfy the conditions of Theorem 18 (2). Therefore there exists a fixed point.

Data Availability

No data were used to support this study.

Disclosure

An earlier version of this study was presented as an oral presentation in The 6th Asian Conference on Nonlinear Analysis and Optimization 5–9 November 2018.

Conflicts of Interest

The author declares that he has no conflicts of interest.

References

W. Takahashi and Y. Takeuchi, “Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 12, no. 2, pp. 399–406, 2011.

J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéairs dans un espace de Hilbert,” in Comptes Rendus de L’Académie des Sciences, vol. 280 of Séries A et B, pp. 1511–1514, 1975.

P. Kocourek, W. Takahashi, and J.-C. Yao, “Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 14, no. 6, pp. 2497–2511, 2010.

F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008.

W. Takahashi, “Fixed point theorems for new nonlinear mappings in a Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 11, pp. 79–88, 2010.

T. Kawasaki and W. Takahashi, “Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 14, no. 1, pp. 71–87, 2013.

F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp. 197–228, 1967.

T. Kawasaki, “Fixed point theorems for widely more generalized hybrid mappings in metric spaces, Banach spaces and Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 19, pp. 1675–1683, 2018.

T. Kawasaki and T. Kobayashi, “Existence and mean approximation of fixed points of generalized hybrid non-self mappings in Hilbert spaces,” Scientiae Mathematicae Japonicae 7, pp. 13–26, 2014.

T. Kawasaki and W. Tkahashi, “Fixed point theorems for generalized hybrid demicontinuous mappings in Hilbert spaces,” Linear and Nonlinear Analysis, vol. 1, pp. 125–138, 2015.

W. Takahashi, N.-C. Wong, and J.-C. Yao, “Attractive point and mean convergence theorems for new generalized nonspreading mappings in banach spaces,” in Infinite Products of Operators and their Applications, S. Reich and A. J. Zaslavski, Eds., vol. 636 of Contemporary Mathematics, pp. 225–248, American Mathematical Society, Providence, RI, USA, 2015.

S. Atsushiba, S. Iemoto, R. Kubota, and Y. Takeuchi, “Convergence theorems for some classes of nonlinear mappings in Hilbert spaces,” Linear and Nonlinear Analysis, vol. 2, no. 1, pp. 125–153, 2016.

T. Kawasaki, “Mean convergence theorems for new mappings in a Banach space,” Journal of Nonlinear and Variational Analysis, vol. 3, pp. 61–78, 2019.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.