Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces

Atsumasa Kondo

doi:10.1515/dema-2022-0167

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Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces

Atsumasa Kondo

Published/Copyright: November 2, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

In this article, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. From the main theorem, a common fixed point theorem for commutative generalized hybrid mappings is derived as a special case. Our novel approach significantly expands the applicable range of mappings for well-known fixed point theorems to be effective. Examples are presented to explicitly illustrate this contribution.

Keywords: common fixed point theorem; generalized hybrid mappings; nonexpansive mappings; nonspreading mappings; Hilbert space

MSC 2010: 47H10

1 Introduction

Let H be a real Hilbert space endowed with an inner product ⟨ ⋅ , ⋅ ⟩ and the induced norm ∥ ⋅ ∥ . A mapping S : C → H is called nonexpansive if

(1.1) ∥ S x − S y ∥ ≤ ∥ x − y ∥ for all x , y ∈ C ,

where C is a nonempty subset of H . A simple version of the common fixed point theorem for nonexpansive mappings by Browder [1] can be described in the framework of a real Hilbert space as follows:

Theorem 1.1

Let C be a nonempty, closed, convex, and bounded subset of H . Let S , T : C → C be nonexpansive mappings such that S T = T S . Then, S and T have a common fixed point.

See also the studies of Kirk [2] and Göhde [3]. Fixed point theorems have been intensively studied by many researchers. In particular, the applicable classes of mappings for which the existence of fixed points is guaranteed have been expanded. Following the demands of optimizing theory, a nonspreading mapping [4] is defined as

(1.2) 2 ∥ S x − S y ∥ 2 ≤ ∥ x − S y ∥ 2 + ∥ S x − y ∥ 2 for all x , y ∈ C .

Although a nonexpansive mapping is continuous, a nonspreading mapping is not necessarily continuous; see, for example, the studies of Igarashi et al. [5], Kohsaka [6], Hojo et al. [7], and Kondo [8,9]. From conditions (1.1) and (1.2), a hybrid mapping [10] is deduced as

(1.3) 3 ∥ S x − S y ∥ 2 ≤ ∥ x − y ∥ 2 + ∥ x − S y ∥ 2 + ∥ S x − y ∥ 2 for all x , y ∈ C .

Fixed point theorems for the classes of mappings characterized by (1.2) and (1.3) were proved in the studies of Kohsaka and Takahashi [4], Takahashi [10], respectively; see also the study of Takahashi and Yao [11].

In 2010, Kocourek et al. [12] unified the types of mappings indicated by (1.1)–(1.3). If there exist α , β ∈ R such that

(1.4) α ∥ S x − S y ∥ 2 + ( 1 − α ) ∥ x − S y ∥ 2 ≤ β ∥ S x − y ∥ 2 + ( 1 − β ) ∥ x − y ∥ 2

for all x , y ∈ C , then S : C → H is called a generalized hybrid mapping or an ( α , β ) -generalized hybrid. Clearly, a ( 1 , 0 ) -generalized hybrid mapping is nonexpansive. Furthermore, ( 2 , 1 ) - and 3 2 , 1 2 -generalized hybrid mappings are nonspreading (1.2) and hybrid (1.3), respectively. Hence, nonexpansive mappings, nonspreading mappings, and hybrid mappings are special cases of generalized hybrid mappings. It is noteworthy that λ -hybrid mappings [13], which also unify the (1.1)–(1.3) classes of mappings, are also generalized hybrid mappings. Kocourek et al. [12] established a fixed point theorem for a generalized hybrid mapping and investigated how to define sequences that approximate fixed points. The results in [12] can be applied to nonexpansive mappings, nonspreading mappings, hybrid mappings, and λ -hybrid mappings as they are all special cases of generalized hybrid mappings.

The fixed point theorem of Kocourek et al. [12] has further been generalized in various directions. For results on more general classes of mappings, see the studies of Kawasaki and Takahashi [14], Kawasaki and Kobayashi [15], Kondo [16], Rouhani [17], and Takahashi et al. [18]. For multi-valued versions of nonspreading and hybrid mappings, see the studies of Cholamjiak et al. [19] and Cholamjiak and Cholamjiak [20], respectively. Hojo et al. [21] and Hojo [22] established common fixed point theorems for two additional general classes of mappings by assuming that the mappings are commutative; see also the studies of Kohsaka and Takahashi [4] and Kohsaka [6]. The following is a simple version of a common fixed point theorem for generalized hybrid mappings:

Theorem 1.2

Let C be a nonempty, closed, and convex subset of H . Let S and T be generalized hybrid mappings from C into itself with S T = T S . Suppose that there exists z ∈ C such that { S k T l z : k , l ∈ N ∪ { 0 } } is bounded. Then, S and T have a common fixed point.

In this article, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. From the main theorem of this article, Theorem 1.2 is derived as a corollary. In this sense, our main theorem is a generalized common fixed point theorem. Furthermore, our approach significantly expands the applicable range of mappings for a well-known fixed point theorem to be effective. To explicitly illustrate this contribution, we provide some specific examples. In this article, the main theorem is proven in Section 2. In Section 3, derivative results deduced from the main theorem are presented along with examples of the mappings addressed in this article. Section 4 briefly concludes the article.

2 Main result

In this section, we establish the main theorem, which generalizes a common fixed point theorem (Theorem 1.2) for generalized hybrid mappings. In the theorem, we use a convex combination of conditions (1.4) of generalized hybrid mappings. We denote F ( S ) to be the set that collects all the fixed points of a mapping S : C → C , that is, F ( S ) = { x ∈ C : S x = x } .

Theorem 2.1

Let C be a nonempty, closed, and convex subset of H. Let S and T be mappings from C into itself with S T = T S . Suppose that there exists λ ∈ [ 0 , 1 ] and α , β ∈ R such that

(2.1) λ ( α ∥ S x − S y ∥ 2 + ( 1 − α ) ∥ x − S y ∥ 2 ) + ( 1 − λ ) ( α ′ ∥ T x − T y ∥ 2 + ( 1 − α ′ ) ∥ x − T y ∥ 2 ) ≤ λ ( β ∥ S x − y ∥ 2 + ( 1 − β ) ∥ x − y ∥ 2 ) + ( 1 − λ ) ( β ′ ∥ T x − y ∥ 2 + ( 1 − β ′ ) ∥ x − y ∥ 2 )

for all x , y ∈ C . Suppose that there exists z ∈ C such that { S k T l z : k , l ∈ N ∪ { 0 } } is bounded. Then, F ( S ) (resp. F ( T ) ) is nonempty if λ ∈ ( 0 , 1 ] (resp. λ ∈ [ 0 , 1 ) ). In particular, if λ ∈ ( 0 , 1 ) , then F ( S ) ∩ F ( T ) is not empty.

Proof

Define

A n = 1 n 2 ∑ k = 0 n − 1 ∑ l = 0 n − 1 S k T l z

for all n ∈ N . As C is convex, { A n } is a sequence in C . From the hypothesis that { S k T l z : k , l ∈ N ∪ { 0 } } is bounded, { A n } is also bounded. Thus, there exists a subsequence { A n i } of { A n } such that A n i ⇀ v for some v ∈ H . As C is weakly closed, { A n i } ⊂ C , and A n i ⇀ v , we have v ∈ C . Hence, S v and T v ( ∈ C ) exist.

As S T = T S , letting x = S k T l z and y = v in (2.1), we have that

λ ( α ∥ S k + 1 T l z − S v ∥ 2 + ( 1 − α ) ∥ S k T l z − S v ∥ 2 ) + ( 1 − λ ) ( α ′ ∥ S k T l + 1 z − T v ∥ 2 + ( 1 − α ′ ) ∥ S k T l z − T v ∥ 2 ) ≤ λ ( β ∥ S k + 1 T l z − v ∥ 2 + ( 1 − β ) ∥ S k T l z − v ∥ 2 ) + ( 1 − λ ) ( β ′ ∥ S k T l + 1 z − v ∥ 2 + ( 1 − β ′ ) ∥ S k T l z − v ∥ 2 )

for all k , l ∈ N ∪ { 0 } . From this,

λ { α ( ∥ S k + 1 T l z − S v ∥ 2 − ∥ S k T l z − S v ∥ 2 ) + ∥ S k T l z − S v ∥ 2 } + ( 1 − λ ) { α ′ ( ∥ S k T l + 1 z − T v ∥ 2 − ∥ S k T l z − T v ∥ 2 ) + ∥ S k T l z − T v ∥ 2 } ≤ λ { β ( ∥ S k + 1 T l z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + ∥ S k T l z − v ∥ 2 } + ( 1 − λ ) { β ′ ( ∥ S k T l + 1 z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + ∥ S k T l z − v ∥ 2 } .

We have

λ { α ( ∥ S k + 1 T l z − S v ∥ 2 − ∥ S k T l z − S v ∥ 2 ) + ∥ S k T l z − S v ∥ 2 } + ( 1 − λ ) { α ′ ( ∥ S k T l + 1 z − T v ∥ 2 − ∥ S k T l z − T v ∥ 2 ) + ∥ S k T l z − T v ∥ 2 } ≤ λ { β ( ∥ S k + 1 T l z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + ∥ S k T l z − S v ∥ 2 + 2 ⟨ S k T l z − S v , S v − v ⟩ + ∥ S v − v ∥ 2 } + ( 1 − λ ) { β ′ ( ∥ S k T l + 1 z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + ∥ S k T l z − T v ∥ 2 + 2 ⟨ S k T l z − T v , T v − v ⟩ + ∥ T v − v ∥ 2 } ,

which implies that

λ α ( ∥ S k + 1 T l z − S v ∥ 2 − ∥ S k T l z − S v ∥ 2 ) + ( 1 − λ ) α ′ ( ∥ S k T l + 1 z − T v ∥ 2 − ∥ S k T l z − T v ∥ 2 ) ≤ λ { β ( ∥ S k + 1 T l z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + 2 ⟨ S k T l z − S v , S v − v ⟩ + ∥ S v − v ∥ 2 } + ( 1 − λ ) { β ′ ( ∥ S k T l + 1 z − v ∥ 2 − ∥ S k T l z − v ∥ 2 ) + 2 ⟨ S k T l z − T v , T v − v ⟩ + ∥ T v − v ∥ 2 } .

Summing these inequalities with respect to k = 0 , 1 , ⋯ , n − 1 , we obtain

λ α ( ∥ S n T l z − S v ∥ 2 − ∥ T l z − S v ∥ 2 ) + ( 1 − λ ) α ′ ∑ k = 0 n − 1 ∥ S k T l + 1 z − T v ∥ 2 − ∑ k = 0 n − 1 ∥ S k T l z − T v ∥ 2 ≤ λ β ( ∥ S n T l z − v ∥ 2 − ∥ T l z − v ∥ 2 ) + ( 1 − λ ) β ′ ∑ k = 0 n − 1 ∥ S k T l + 1 z − v ∥ 2 − ∑ k = 0 n − 1 ∥ S k T l z − v ∥ 2 + 2 λ ∑ k = 0 n − 1 S k T l z − n S v , S v − v + 2 ( 1 − λ ) ∑ k = 0 n − 1 S k T l z − n T v , T v − v + n λ ∥ S v − v ∥ 2 + n ( 1 − λ ) ∥ T v − v ∥ 2 ,

for all l ∈ N ∪ { 0 } . Summing these inequalities with respect to l = 0 , 1 , ⋯ , n − 1 and dividing by n 2 yield

λ α n 2 ∑ l = 0 n − 1 ∥ S n T l z − S v ∥ 2 − ∑ l = 0 n − 1 ∥ T l z − S v ∥ 2 + ( 1 − λ ) α ′ n 2 ∑ k = 0 n − 1 ∥ S k T n z − T v ∥ 2 − ∑ k = 0 n − 1 ∥ S k z − T v ∥ 2 ≤ λ β n 2 ∑ l = 0 n − 1 ∥ S n T l z − v ∥ 2 − ∑ l = 0 n − 1 ∥ T l z − v ∥ 2 + ( 1 − λ ) β ′ n 2 ∑ k = 0 n − 1 ∥ S k T n z − v ∥ 2 − ∑ k = 0 n − 1 ∥ S k z − v ∥ 2 + 2 λ ⟨ A n − S v , S v − v ⟩ + 2 ( 1 − λ ) ⟨ A n − T v , T v − v ⟩ + λ ∥ S v − v ∥ 2 + ( 1 − λ ) ∥ T v − v ∥ 2 .

Recall that A n i ⇀ v . Replacing n with n i and taking i → ∞ , we have

0 ≤ 2 λ ⟨ v − S v , S v − v ⟩ + 2 ( 1 − λ ) ⟨ v − T v , T v − v ⟩ + λ ∥ S v − v ∥ 2 + ( 1 − λ ) ∥ T v − v ∥ 2 ,

which implies that

λ ∥ S v − v ∥ 2 + ( 1 − λ ) ∥ T v − v ∥ 2 ≤ 0 .

Assume that λ ∈ ( 0 , 1 ] . Subtracting ( 1 − λ ) ∥ T v − v ∥ 2 ( ≥ 0 ) from the left-hand side (LHS), we obtain λ ∥ S v − v ∥ 2 ≤ 0 . Dividing by λ ( > 0 ) yields ∥ S v − v ∥ 2 ≤ 0 , which means that S v = v . Therefore, F ( S ) is not empty in this case. Similarly, if λ ∈ [ 0 , 1 ) , then F ( T ) is not empty. This completes the proof.□

Although Theorem 2.1 implies Theorem 1.2, it has greater potential applicability. We investigate this point in the next section with some examples.

3 Derivative results and examples

In this section, we simplify Theorem 2.1 and derive some of its corollaries, revealing the applicability of the theorem. First, letting α = α ′ = 1 and β = β ′ = 0 in condition (2.1) of mappings S and T , we obtain the following:

Theorem 3.1

Let C be a nonempty, closed, and convex subset of H. Let S and T be mappings from C into itself with S T = T S . Suppose that there exists λ ∈ [ 0 , 1 ] such that

(3.1) λ ∥ S x − S y ∥ 2 + ( 1 − λ ) ∥ T x − T y ∥ 2 ≤ ∥ x − y ∥ 2

From this result, Theorem 1.1, which relates to nonexpansive mappings, is derived. In the following example, we present commutative nonexpansive mappings:

Example 3.1

Let H = C = R , D = [ 0 , 2 ] , and U = [ − 1 , 1 ] . Let P D and P U be metric projections from R onto D and U , respectively. Then, P D and P U are nonexpansive and commutative with a common fixed point v ∈ D ∩ U = [ 0 , 1 ] .

As S and S 2 are commutative, the next corollary also follows from Theorem 3.1:

Corollary 3.1

[16] Let C be a nonempty, closed, and convex subset of H and let S be a mapping from C into itself. Suppose that there exists λ ∈ ( 0 , 1 ] such that

(3.2) λ ∥ S x − S y ∥ 2 + ( 1 − λ ) ∥ S 2 x − S 2 y ∥ 2 ≤ ∥ x − y ∥ 2

for all x , y ∈ C . Suppose that there exists z ∈ C such that { S n z } is bounded. Then, F ( S ) is nonempty.

Corollary 3.1 was also derived using a different approach in a very recent article of Kondo [16]. For a type of mapping with a close condition to that in (3.2), see the study of Goebel and Japon-Pineda [23]. Clearly, a nonexpansive mapping satisfies (3.2). We provide an example of a mapping that is not nonexpansive but satisfies (3.2). This example is a slightly generalized version of that in the study of Kondo [16].

Example 3.2

Let H = C = R . Given λ ∈ ( 0 , 1 ] , define S : R → R as follows:

S x = − 1 λ x if x < 0 , 0 if x ≥ 0 .

Although the mapping S with λ < 1 is not nonexpansive, it satisfies condition (3.2). This can be verified as follows: S 2 x = 0 for all x ∈ R ; thus, it suffices to demonstrate that

(3.3) λ ∥ S x − S y ∥ 2 ≤ ∥ x − y ∥ 2

for all x , y ∈ C . (i) If x , y < 0 , then S x = − x / λ and S y = − y / λ . We have

LHS − RHS of (3.3) = λ x λ − y λ 2 − ∥ x − y ∥ 2 = 0 ,

which shows that condition (3.3) holds. (ii) If x , y ≥ 0 , then S x = S y = 0 . In this case, (3.3) is fulfilled. (iii) Without loss of generality, assume that x < 0 ≤ y . Then, S x = − x / λ and S y = 0 . Consequently, we have

LHS − RHS of (3.3) = λ x λ 2 − ( x 2 − 2 x y + y 2 ) = y ( 2 x − y ) ≤ 0 .

This implies that (3.3) is met.

Despite the fact that the mapping S with λ < 1 in Example 3.2 is not nonexpansive, it is in the applicable range of Corollary 3.1, which asserts the existence of a fixed point. This indicates effectiveness of our approach, which is based on Theorem 2.1 or 3.1.

We present another example of mappings S and T that jointly satisfy condition (3.1) in Theorem 3.1 although they are not included in the types of mappings addressed in Theorem 1.1 or Corollary 3.1.

Example 3.3

Let H = R and C = [ 0 , 1 ] . Define S , T : C → C as S x = x 2 and T x = 1 for all x ∈ C , respectively. Although S and T are commutative, S is not nonexpansive and T ≠ S 2 . Let λ = 1 / 4 . Then, condition (3.1) is satisfied. Indeed, since T x = T y = 1 and x , y ∈ C = [ 0 , 1 ] ,

LHS − RHS of (3.1) = 1 4 ( x 2 − y 2 ) 2 − ( x − y ) 2 = 1 4 ( x − y ) 2 ( ( x + y ) 2 − 4 ) ≤ 0 .

Therefore, the two mappings S and T jointly satisfy condition (3.1) as claimed and have a common fixed point 1 ∈ C .

Example 3.3 implies that the applicability of Theorem 3.1 (or 2.1) is not limited to the ranges of Theorem 1.1 and Corollary 3.1.

As stated in Introduction, a nonspreading mapping is a ( 2 , 1 ) -generalized hybrid mapping characterized by condition (1.2). Substituting α = α ′ = 2 and β = β ′ = 1 into (2.1) in Theorem 2.1, we obtain the following:

Theorem 3.2

Let C be a nonempty, closed, and convex subset of H. Let S and T be mappings from C into itself with S T = T S . Suppose that there exists λ ∈ [ 0 , 1 ] such that

(3.4) 2 λ ∥ S x − S y ∥ 2 + 2 ( 1 − λ ) ∥ T x − T y ∥ 2 ≤ λ ( ∥ x − S y ∥ 2 + ∥ S x − y ∥ 2 ) + ( 1 − λ ) ( ∥ x − T y ∥ 2 + ∥ T x − y ∥ 2 )

Theorem 3.2 yields the next corollary, which was proven in the study of Kohsaka and Takahashi [4] in a setting of a Banach space:

Corollary 3.2

[4] Let C be a nonempty, closed, and convex subset of H. Let S and T be nonspreading mappings from C into itself with S T = T S . Suppose that there exists z ∈ C such that { S k T l z : k , l ∈ N ∪ { 0 } } is bounded. Then, F ( S ) ∩ F ( T ) is nonempty.

To analyze a few examples, we present the following proposition:

Proposition 3.1

[9] Define a mapping S : R → R as

S x = 1 if x > A , 0 if x ≤ A ,

where A ≥ 1 . Then, S is nonspreading (1.2) if and only if A ≥ 2 .

As a proof of Proposition 3.1 is available in the study of Kondo [9], we omit it here. Focusing on the cases of A = 2 and 3 , we obtain the following:

Example 3.4

Define S , T : R → R as

S x = 1 if x > 2 , 0 if x ≤ 2 ,

and

T x = 1 if x > 3 , 0 if x ≤ 3 ,

respectively. From Proposition 3.1, the mappings S and T are nonspreading. Furthermore, they are commutative. Therefore, the mappings are within the class addressed in Corollary 3.2 and have a common fixed point 0 ∈ R .

For other examples of commutative nonspreading mappings, see [5,7, 8,9]. Since mappings S and S 2 are commutative, substituting T = S 2 into (3.4), we have the following:

Corollary 3.3

Let C be a nonempty, closed, and convex subset of H. Let S be a mapping from C into itself. Suppose that there exists λ ∈ ( 0 , 1 ] such that

(3.5) 2 λ ∥ S x − S y ∥ 2 + 2 ( 1 − λ ) ∥ S 2 x − S 2 y ∥ 2 ≤ λ ( ∥ x − S y ∥ 2 + ∥ S x − y ∥ 2 ) + ( 1 − λ ) ( ∥ x − S 2 y ∥ 2 + ∥ S 2 x − y ∥ 2 )

for all x , y ∈ C . Suppose that there exists z ∈ C such that { S n z } is bounded. Then, F ( S ) is nonempty.

Combining this corollary with Proposition 3.1, we can show that the applicable range of mappings for a fixed point theorem to be effective is significantly expanded from the class of nonspreading mappings.

Example 3.5

Define S : R → R as

S x = 1 if x > 1 , 0 if x ≤ 1 .

From Proposition 3.1, the mapping S is not nonspreading. Nonetheless, S satisfies condition (3.5) with λ = 1 / 2 . Indeed, as S 2 x = 0 for all x ∈ R , we aim to demonstrate that

(3.6) 2 λ ∥ S x − S y ∥ 2 ≤ λ ( ∥ x − S y ∥ 2 + ∥ S x − y ∥ 2 ) + ( 1 − λ ) ( ∥ x ∥ 2 + ∥ y ∥ 2 ) .

If x , y ≤ 1 or x , y > 1 , the desired result is true. Without loss of generality, assume that x ≤ 1 < y . Then, S x = 0 and S y = 1 . Therefore, on the one hand, the LHS of (3.6) = 2 λ = 1 . On the other hand,

RHS = 1 2 ( ( x − 1 ) 2 + y 2 ) + 1 2 ( x 2 + y 2 ) ≥ y 2 > 1 = LHS .

Therefore, the mapping S is within the class targeted in Corollary 3.3 and has a fixed point 0 ∈ R .

This example, together with Examples 3.2 and 3.3, demonstrates the effectiveness of the proposed method in comparison with earlier studies.

4 Concluding remarks

In this article, we proposed an approach to prove a common fixed point theorem for nonlinear mappings, where a convex combination of conditions for nonlinear mappings is exploited. From the main theorem, new results (Theorems 3.1, 3.2, and Corollary 3.3) and well-known results (Theorems 1.1, 1.2, and Corollary 3.1, 3.2) are derived. Using specific examples explicitly demonstrates the effectiveness of our approach. As a final remark, the approach presented in this article can be extended to finitely many mappings.

Acknowledgments

The author would like to thank the Institute for Economics and Business Research of Shiga University for financial support. The author would also appreciate the anonymous reviewers for their helpful comments and advice.

Conflict of interest: The author states no conflict of interest.

References

[1] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965), no. 4, 1041. 10.1073/pnas.54.4.1041Search in Google Scholar PubMed PubMed Central

[2] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. 10.2307/2313345Search in Google Scholar

[3] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258. 10.1002/mana.19650300312Search in Google Scholar

[4] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. 91 (2008), no. 2, 166–177. 10.1007/s00013-008-2545-8Search in Google Scholar

[5] T. Igarashi, W. Takahashi and K. Tanaka, Weak convergence theorems for nonspreading mappings and equilibrium problems, in: Nonlinear Analysis and Optimization. S. Akashi, W. Takahashi and T. Tanaka Eds., Yokohama Publishers, Yokohama, 2008, pp. 75–85. Search in Google Scholar

[6] F. Kohsaka, Existence and approximation of common fixed points of two hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 16 (2015), no. 11, 2193–2205. Search in Google Scholar

[7] M. Hojo, W. Takahashi, and I. Termwuttipong, Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces, Nonlinear Anal. 75 (2012), no. 4, 2166–2176. 10.1016/j.na.2011.10.017Search in Google Scholar

[8] A. Kondo, Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces, Adv. Oper. Theory 7 (2022), no. 3, 2610.1007/s43036-022-00190-5Search in Google Scholar

[9] A. Kondo, Strong convergence theorems by Martinez-Yanes-Xu projection method for mean-demiclosed mappings in Hilbert spaces, to appear in Rendiconti di Mat. e delle Sue Appl., Sapienza Università di Roma, Italy, 2023.Search in Google Scholar

[10] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), no. 1, 79–88. Search in Google Scholar

[11] W. Takahashi and J.-C. Yao, Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math. 15 (2011), no. 2, 457–472. 10.11650/twjm/1500406216Search in Google Scholar

[12] P. Kocourek, W. Takahashi, and J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010), no. 6, 2497–2511. 10.11650/twjm/1500406086Search in Google Scholar

[13] K. Aoyama, S. Iemoto, F. Kohsaka, and W. Takahashi, Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 11 (2010), no. 2, 335–343. Search in Google Scholar

[14] T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 14 (2013), no. 1, 71–87. 10.1155/2013/904164Search in Google Scholar

[15] T. Kawasaki and T. Kobayashi, Existence and mean approximation of fixed points of generalized hybrid non-self mappings in Hilbert spaces, Scientiae Math. Japonicae 77 (2014), no. 1, 13–26. Search in Google Scholar

[16] A. Kondo, Fixed point theorem for generic 2-generalizd hybrid mappings in Hilbert spaces, Topol. Meth. Nonlinear Anal. 59 (2022), no. 2B, 833–849. Search in Google Scholar

[17] B. D. Rouhani, Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space, Demonstr. Math. 51 (2018), no. 1, 27–36. 10.1515/dema-2018-0005Search in Google Scholar

[18] W. Takahashi, N.-C. Wong, and J.-C. Yao, Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 13 (2012), no. 4, 745–757. Search in Google Scholar

[19] W. Cholamjiak, S. Suantai, and Y. J. Cho, Fixed points for nonspreading-type multi-valued mappings: existence and convergence results, Ann. Acad. Rom. Sci. Ser. Math. Apll 10 (2018), no. 2, 838–844. Search in Google Scholar

[20] P. Cholamjiak and W. Cholamjiak, Fixed point theorems for hybrid multivalued mappings in Hilbert spaces, J. Fixed Point Theory Appl. 18 (2016), no. 3, 673–688. 10.1007/s11784-016-0302-3Search in Google Scholar

[21] M. Hojo, S. Takahashi, and W. Takahashi, Attractive point and ergodic theorems for two nonlinear mappings in Hilbert spaces, Linear Nonlinear Anal. 3 (2017), no. 2, 275–286. Search in Google Scholar

[22] M. Hojo, Attractive point and mean convergence theorems for normally generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 18 (2017), no. 12, 2209–2120. Search in Google Scholar

[23] K. Goebel and M. Japon-Pineda, A new type of nonexpansiveness, In: Proceedings of 8th International Conference on Fixed Point Theory and its Applications, Yokohama Publishers, Chiang Mai, 2007. Search in Google Scholar

Received: 2022-07-01

Revised: 2022-09-23

Accepted: 2022-09-27

Published Online: 2022-11-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0167

Keywords for this article

common fixed point theorem; generalized hybrid mappings; nonexpansive mappings; nonspreading mappings; Hilbert space

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