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Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications

Fixed Point Theory and Applications20132013:116

https://doi.org/10.1186/1687-1812-2013-116

  • Received: 30 January 2013
  • Accepted: 15 April 2013
  • Published:

Abstract

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem.

MSC:Primary 47H10; secondary 47H05.

Keywords

  • Hilbert space
  • nonexpansive mapping
  • nonspreading mapping
  • hybrid mapping
  • fixed point
  • non-self mapping

1 Introduction

Let be the real line and let [ 0 , π 2 ] be a bounded, closed and convex subset of . Consider a mapping T : [ 0 , π 2 ] R defined by
T x = ( 1 + 1 2 x ) cos x 1 2 x 2

for all x [ 0 , π 2 ] . Such a mapping T has a unique fixed point z [ 0 , π 2 ] such that cos z = z . What kind of fixed point theorems can we use to find such a unique fixed point z of T?

Let H be a real Hilbert space and let C be a non-empty subset of H. Kocourek, Takahashi and Yao [1] introduced a class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings [2] and hybrid mappings [3]. A mapping T : C H is said to be generalized hybrid if there exist α , β R such that
α T x T y 2 + ( 1 α ) x T y 2 β T x y 2 + ( 1 β ) x y 2
(1.1)
for all x , y C . We call such a mapping an ( α , β ) -generalized hybrid mapping. An ( α , β ) -generalized hybrid mapping is nonexpansive for α = 1 and β = 0 , i.e.,
T x T y x y
for all x , y C . It is nonspreading for α = 2 and β = 1 , i.e.,
2 T x T y 2 x T y 2 + y T x 2
for all x , y C . Furthermore, it is hybrid for α = 3 2 and β = 1 2 , i.e.,
3 T x T y 2 x T y 2 + y T x 2 + y x 2
for all x , y C . They proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for generalized hybrid mappings; see also Kohsaka and Takahashi [5] and Iemoto and Takahashi [6]. Very recently, Kawasaki and Takahashi [7] introduced a broader class of nonlinear mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping T from C into H is called widely more generalized hybrid if there exist α , β , γ , δ , ε , ζ , η R such that
α T x T y 2 + β x T y 2 + γ T x y 2 + δ x y 2 + ε x T x 2 + ζ y T y 2 + η ( x T x ) ( y T y ) 2 0
(1.2)

for all x , y C . Such a mapping T is called an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping. In particular, an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek, Takahashi and Yao [1] if α + β = γ δ = 1 and ε = ζ = η = 0 . An ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping is strict pseudo-contractive in the sense of Browder and Petryshyn [8] if α = 1 , β = γ = 0 , δ = 1 , ε = ζ = 0 , η = k , where 0 k < 1 . A generalized hybrid mapping with a fixed point is quasi-nonexpansive. However, a widely more generalized hybrid mapping is not quasi-nonexpansive in general even if it has a fixed point. In [7], Kawasaki and Takahashi proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for such widely more generalized hybrid mappings in a Hilbert space. In particular, they proved directly the Browder and Petryshyn fixed point theorem [8] for strict pseudo-contractive mappings and the Kocourek, Takahashi and Yao fixed point theorem [1] for super hybrid mappings by using their fixed point theorems. However, we cannot use Kawasaki and Takahashi fixed point theorems to solve the above problem. For a nice synthesis on metric fixed point theory, see Kirk [9].

In this paper, motivated by such a problem, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo, Takahashi and Yao [10] and Kawasaki and Takahashi fixed point theorems [7]. Using these fixed point theorems for non-self mappings, we prove the Browder and Petryshyn fixed point theorem [8] for strict pseudo-contractive non-self mappings and the Kocourek, Takahashi and Yao fixed point theorem [1] for super hybrid non-self mappings. In particular, we solve the above problem by using one of our fixed point theorems.

2 Preliminaries

Throughout this paper, we denote by the set of positive integers. Let H be a (real) Hilbert space with the inner product , and the norm , respectively. From [11], we know the following basic equality: For x , y H and λ R , we have
λ x + ( 1 λ ) y 2 = λ x 2 + ( 1 λ ) y 2 λ ( 1 λ ) x y 2 .
(2.1)
Furthermore, we know that for x , y , u , v H ,
2 x y , u v = x v 2 + y u 2 x u 2 y v 2 .
(2.2)
Let C be a non-empty, closed and convex subset of H and let T be a mapping from C into H. Then we denote by F ( T ) the set of fixed points of T. A mapping S : C H is called super hybrid [1, 12] if there exist α , β , γ R such that
α S x S y 2 + ( 1 α + γ ) x S y 2 ( β + ( β α ) γ ) S x y 2 + ( 1 β ( β α 1 ) γ ) x y 2 + ( α β ) γ x S x 2 + γ y S y 2
(2.3)

for all x , y C . We call such a mapping an ( α , β , γ ) -super hybrid mapping. An ( α , β , 0 ) -super hybrid mapping is ( α , β ) -generalized hybrid. Thus the class of super hybrid mappings contains generalized hybrid mappings. The following theorem was proved in [12]; see also [1].

Theorem 2.1 ([12])

Let C be a non-empty subset of a Hilbert space H and let α, β and γ be real numbers with γ 1 . Let S and T be mappings of C into H such that T = 1 1 + γ S + γ 1 + γ I . Then S is ( α , β , γ ) -super hybrid if and only if T is ( α , β ) -generalized hybrid. In this case, F ( S ) = F ( T ) . In particular, let C be a nonempty, closed and convex subset of H and let α, β and γ be real numbers with γ 0 . If a mapping S : C C is ( α , β , γ ) -super hybrid, then the mapping T = 1 1 + γ S + γ 1 + γ I is an ( α , β ) -generalized hybrid mapping of C into itself.

In [1], Kocourek, Takahashi and Yao also proved the following fixed point theorem for super hybrid mappings in a Hilbert space.

Theorem 2.2 ([1])

Let C be a non-empty, bounded, closed and convex subset of a Hilbert space H and let α, β and γ be real numbers with γ 0 . Let S : C C be an ( α , β , γ ) -super hybrid mapping. Then S has a fixed point in C. In particular, if S : C C is an ( α , β ) -generalized hybrid mapping, then S has a fixed point in C.

A super hybrid mapping is not quasi-nonexpansive in general even if it has a fixed point. There exists a class of nonlinear mappings in a Hilbert space defined by Kawasaki and Takahashi [13] which covers contractive mappings and generalized hybrid mappings. A mapping T from C into H is said to be widely generalized hybrid if there exist α , β , γ , δ , ε , ζ R such that
α T x T y 2 + β x T y 2 + γ T x y 2 + δ x y 2 + max { ε x T x 2 , ζ y T y 2 } 0

for any x , y C . Such a mapping T is called ( α , β , γ , δ , ε , ζ ) -widely generalized hybrid. Kawasaki and Takahashi [13] proved the following fixed point theorem.

Theorem 2.3 ([13])

Let H be a Hilbert space, let C be a non-empty, closed and convex subset of H and let T be an ( α , β , γ , δ , ε , ζ ) -widely generalized hybrid mapping from C into itself which satisfies the following conditions (1) and (2):
  1. (1)

    α + β + γ + δ 0 ;

     
  2. (2)

    ε + α + γ > 0 , or ζ + α + β > 0 .

     

Then T has a fixed point if and only if there exists z C such that { T n z n = 0 , 1 , } is bounded. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the condition (1).

Very recently, Kawasaki and Takahashi [7] also proved the following fixed point theorem which will be used in the proofs of our main theorems in this paper.

Theorem 2.4 ([7])

Let H be a Hilbert space, let C be a non-empty, closed and convex subset of H and let T be an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping from C into itself, i.e., there exist α , β , γ , δ , ε , ζ , η R such that
α T x T y 2 + β x T y 2 + γ T x y 2 + δ x y 2 + ε x T x 2 + ζ y T y 2 + η ( x T x ) ( y T y ) 2 0
for all x , y C . Suppose that it satisfies the following condition (1) or (2):
  1. (1)

    α + β + γ + δ 0 , α + γ + ε + η > 0 and ζ + η 0 ;

     
  2. (2)

    α + β + γ + δ 0 , α + β + ζ + η > 0 and ε + η 0 .

     

Then T has a fixed point if and only if there exists z C such that { T n z n = 0 , 1 , } is bounded. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the conditions (1) and (2).

In particular, we have the following theorem from Theorem 2.4.

Theorem 2.5 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping from C into itself which satisfies the following condition (1) or (2):
  1. (1)

    α + β + γ + δ 0 , α + γ + ε + η > 0 and ζ + η 0 ;

     
  2. (2)

    α + β + γ + δ 0 , α + β + ζ + η > 0 and ε + η 0 .

     

Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the conditions (1) and (2).

3 Fixed point theorems for non-self mappings

In this section, using the fixed point theorem (Theorem 2.5), we first prove the following fixed point theorem for widely more generalized hybrid non-self mappings in a Hilbert space.

Theorem 3.1 Let C be a non-empty, bounded, closed and convex subset of a Hilbert space H and let α , β , γ , δ , ε , ζ , η R . Let T : C H be an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping. Suppose that it satisfies the following condition (1) or (2):
  1. (1)

    α + β + γ + δ 0 , α + γ + ε + η > 0 , α + β + ζ + η 0 and ζ + η 0 ;

     
  2. (2)

    α + β + γ + δ 0 , α + β + ζ + η > 0 , α + γ + ε + η 0 and ε + η 0 .

     
Assume that there exists a positive number m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m . Then T has a fixed point in C. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the conditions (1) and (2).

Proof We give the proof for the case of (1). By the assumption, we have that for any x C , there exist y C and t with 0 < t m such that T x = x + t ( y x ) . From this, we have T x = t y + ( 1 t ) x and hence
y = 1 t T x + t 1 t x .
Define U x C as follows:
U x = ( 1 t m ) x + t m y = ( 1 t m ) x + t m ( 1 t T x + t 1 t x ) = 1 m T x + m 1 m x .
Taking λ > 0 with m = 1 + λ , we have that
U x = 1 1 + λ T x + λ 1 + λ x
and hence
T = ( 1 + λ ) U λ I .
(3.1)
Since T : C H is an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping, we have from (3.1) and (2.1) that for any x , y C ,
α ( 1 + λ ) U x λ x ( ( 1 + λ ) U y λ y ) 2 + β x ( ( 1 + λ ) U y λ y ) 2 + γ ( 1 + λ ) U x λ x y 2 + δ x y 2 + ε x ( ( 1 + λ ) U x λ x ) 2 + ζ ( 1 + λ ) U y λ y y 2 + η x ( ( 1 + λ ) U x λ x ) ( y ( ( 1 + λ ) U y λ y ) ) 2 = α ( 1 + λ ) ( U x U y ) λ ( x y ) 2 + β ( 1 + λ ) ( x U y ) λ ( x y ) 2 + γ ( 1 + λ ) ( U x y ) λ ( x y ) 2 + δ x y 2 + ε ( 1 + λ ) ( x U x ) 2 + ζ ( 1 + λ ) ( y U y ) 2 + η ( 1 + λ ) ( x U x ) ( 1 + λ ) ( y U y ) 2 = α ( 1 + λ ) U x U y 2 α λ x y 2 + α λ ( 1 + λ ) x y ( U x U y ) 2 + β ( 1 + λ ) x U y 2 β λ x y 2 + β λ ( 1 + λ ) y U y 2 + γ ( 1 + λ ) U x y 2 γ λ x y 2 + γ λ ( 1 + λ ) x U x 2 + δ x y 2 + ε ( 1 + λ ) 2 x U x 2 + ζ ( 1 + λ ) 2 y U y 2 + η ( 1 + λ ) 2 x U x ( y U y ) 2 = α ( 1 + λ ) U x U y 2 + β ( 1 + λ ) x U y 2 + γ ( 1 + λ ) U x y 2 + ( α λ β λ γ λ + δ ) x y 2 + ( γ λ + ε λ + ε ) ( 1 + λ ) x U x 2 + ( β λ + ζ λ + ζ ) ( 1 + λ ) y U y 2 + ( α λ + η λ + η ) ( 1 + λ ) x y ( U x U y ) 2 0 .
This implies that U is widely more generalized hybrid. Since α + β + γ + δ 0 , α + γ + ε + η > 0 , α + β + ζ + η 0 and ζ + η 0 , we obtain that
α ( 1 + λ ) + β ( 1 + λ ) + γ ( 1 + λ ) α λ β λ γ λ + δ = α + β + γ + δ 0 , α ( 1 + λ ) + γ ( 1 + λ ) + ( γ λ + ε λ + ε ) ( 1 + λ ) + ( α λ + η λ + η ) ( 1 + λ ) = ( 1 + λ ) ( α + γ + ε + η + λ ( γ + ε + α + η ) ) = ( 1 + λ ) 2 ( α + γ + ε + η ) > 0 , ( β λ + ζ λ + ζ ) ( 1 + λ ) + ( α λ + η λ + η ) ( 1 + λ ) = ( ( α + β + ζ + η ) λ + ζ + η ) ( 1 + λ ) 0 .
By Theorem 2.5, we obtain that F ( U ) . Therefore, we have from F ( U ) = F ( T ) that F ( T ) . Suppose that α + β + γ + δ > 0 . Let p 1 and p 2 be fixed points of T. We have that
α T p 1 T p 2 2 + β p 1 T p 2 2 + γ T p 1 p 2 2 + δ p 1 p 2 2 + ε p 1 T p 1 2 + ζ p 2 T p 2 2 + η ( p 1 T p 1 ) ( p 2 T p 2 ) 2 = ( α + β + γ + δ ) p 1 p 2 2 0

and hence p 1 = p 2 . Therefore, a fixed point of T is unique.

Similarly, we can obtain the desired result for the case when α + β + γ + δ 0 , α + β + ζ + η > 0 , α + γ + ε + η 0 and ε + η 0 . This completes the proof. □

The following theorem is a useful extension of Theorem 3.1.

Theorem 3.2 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping from C into H which satisfies the following condition (1) or (2):
  1. (1)

    α + β + γ + δ 0 , α + γ + ε + η > 0 , α + β + ζ + η 0 and [ 0 , 1 ) { λ ( α + β ) λ + ζ + η 0 } ;

     
  2. (2)

    α + β + γ + δ 0 , α + β + ζ + η > 0 , α + γ + ε + η 0 and [ 0 , 1 ) { λ ( α + γ ) λ + ε + η 0 } .

     
Assume that there exists m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m . Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the conditions (1) and (2).

Proof Let λ [ 0 , 1 ) { λ ( α + β ) λ + ζ + η 0 } and define S = ( 1 λ ) T + λ I . Then S is a mapping from C into H. Since λ 1 , we obtain that F ( S ) = F ( T ) . Moreover, from T = 1 1 λ S λ 1 λ I and (2.1), we have that
α ( 1 1 λ S x λ 1 λ x ) ( 1 1 λ S y λ 1 λ y ) 2 + β x ( 1 1 λ S y λ 1 λ y ) 2 + γ ( 1 1 λ S x λ 1 λ x ) y 2 + δ x y 2 + ε x ( 1 1 λ S x λ 1 λ x ) 2 + ζ y ( 1 1 λ S y λ 1 λ y ) 2 + η ( x ( 1 1 λ S x λ 1 λ x ) ) ( y ( 1 1 λ S y λ 1 λ y ) ) 2 = α 1 1 λ ( S x S y ) λ 1 λ ( x y ) 2 + β 1 1 λ ( x S y ) λ 1 λ ( x y ) 2 + γ 1 1 λ ( S x y ) λ 1 λ ( x y ) 2 + δ x y 2 + ε 1 1 λ ( x S x ) 2 + ζ 1 1 λ ( y S y ) 2 + η 1 1 λ ( x S x ) 1 1 λ ( y S y ) 2 = α 1 λ S x S y 2 + β 1 λ x S y 2 + γ 1 λ S x y 2 + ( λ 1 λ ( α + β + γ ) + δ ) x y 2 + ε + γ λ ( 1 λ ) 2 x S x 2 + ζ + β λ ( 1 λ ) 2 y S y 2 + η + α λ ( 1 λ ) 2 ( x S x ) ( y S y ) 2 0 .
Therefore S is an ( α 1 λ , β 1 λ , γ 1 λ , λ 1 λ ( α + β + γ ) + δ , ε + γ λ ( 1 λ ) 2 , ζ + β λ ( 1 λ ) 2 , η + α λ ( 1 λ ) 2 ) -widely more generalized hybrid mapping. Furthermore, we obtain that
α 1 λ + β 1 λ + γ 1 λ λ 1 λ ( α + β + γ ) + δ = α + β + γ + δ 0 , α 1 λ + γ 1 λ + ε + γ λ ( 1 λ ) 2 + η + α λ ( 1 λ ) 2 = α + γ + ε + η ( 1 λ ) 2 > 0 , α 1 λ + β 1 λ + ζ + β λ ( 1 λ ) 2 + η + α λ ( 1 λ ) 2 = α + β + ζ + η ( 1 λ ) 2 0 , ζ + β λ ( 1 λ ) 2 + η + α λ ( 1 λ ) 2 = ( α + β ) λ + ζ + η ( 1 λ ) 2 0 .
Furthermore, from the assumption, there exists m > 1 such that for any x C ,
S x = ( 1 λ ) T x + λ x = ( 1 λ ) ( x + t ( y x ) ) + λ x = t ( 1 λ ) ( y x ) + x ,
where y C and 0 < t m . From 0 λ < 1 , we have 0 < t ( 1 λ ) m . Putting s = t ( 1 λ ) , we have that there exists m > 1 such that for any x C ,
S x = x + s ( y x )

for some y C and s with 0 < s m . Therefore, we obtain from Theorem 3.1 that F ( S ) . Since F ( S ) = F ( T ) , we obtain that F ( T ) .

Next, suppose that α + β + γ + δ > 0 . Let p 1 and p 2 be fixed points of T. As in the proof of Theorem 3.1, we have p 1 = p 2 . Therefore a fixed point of T is unique.

In the case of α + β + γ + δ 0 , α + β + ζ + η > 0 , α + γ + ε + η 0 and [ 0 , 1 ) { λ ( α + γ ) λ + ε + η 0 } , we can obtain the desired result by replacing the variables x and y. □

Remark 1 We can also prove Theorems 3.1 and 3.2 by using the condition
β δ + ε + η > 0 , or γ δ + ε + η > 0
instead of the condition
α + γ + ε + η > 0 , or α + β + ζ + η > 0 ,
respectively. In fact, in the case of the condition β δ + ε + η > 0 , we obtain from α + β + γ + δ 0 that
0 < β δ + ε + η α + γ + ε + η .

Thus we obtain the desired results by Theorems 3.1 and 3.2. Similarly, in the case of γ δ + ε + η > 0 , we can obtain the results by using the case of α + β + ζ + η > 0 .

4 Fixed point theorems for well-known mappings

Using Theorem 3.1, we first show the following fixed point theorem for generalized hybrid non-self mappings in a Hilbert space; see also Kocourek, Takahashi and Yao [1].

Theorem 4.1 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized hybrid mapping from C into H, i.e., there exist α , β R such that
α T x T y 2 + ( 1 α ) x T y 2 β T x y 2 + ( 1 β ) x y 2
for any x , y C . Suppose α β 0 and assume that there exists m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m . Then T has a fixed point.

Proof An ( α , β ) -generalized hybrid mapping T from C into H is an ( α , 1 α , β , ( 1 β ) , 0 , 0 , 0 ) -widely more generalized hybrid mapping. Furthermore, α + ( 1 α ) β ( 1 β ) = 0 , α + ( 1 α ) + 0 + 0 = 1 > 0 , α β + 0 + 0 = α β 0 and 0 + 0 = 0 , that is, it satisfies the condition (2) in Theorem 3.1. Furthermore, since there exists m 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m , we obtain the desired result from Theorem 3.1. □

Using Theorem 3.1, we can also show the following fixed point theorem for widely generalized hybrid non-self mappings in a Hilbert space; see Kawasaki and Takahashi [13].

Theorem 4.2 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an ( α , β , γ , δ , ε , ζ ) -widely generalized hybrid mapping from C into H which satisfies the following condition (1) or (2):
  1. (1)

    α + β + γ + δ 0 , α + γ + ε > 0 and α + β 0 ;

     
  2. (2)

    α + β + γ + δ 0 , α + β + ζ > 0 and α + γ 0 .

     
Assume that there exists m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t R with 0 < t m . Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + β + γ + δ > 0 under the conditions (1) and (2).

Proof Since T is ( α , β , γ , δ , ε , ζ ) -widely generalized hybrid, we obtain that
α T x T y 2 + β x T y 2 + γ T x y 2 + δ x y 2 + max { ε x T x 2 , ζ y T y 2 } 0
for any x , y C . In the case of α + γ + ε > 0 , from
ε x T x 2 max { ε x T x 2 , ζ y T y 2 } ,
we obtain that
α T x T y 2 + β x T y 2 + γ T x y 2 + δ x y 2 + ε x T x 2 0 ,
that is, it is an ( α , β , γ , δ , ε , 0 , 0 ) -widely more generalized hybrid mapping. Furthermore, we have that α + β + γ + δ 0 , α + γ + ε + 0 = α + γ + ε > 0 , α + β + 0 + 0 = α + β 0 and 0 + 0 = 0 , that is, it satisfies the condition (1) in Theorem 3.1. Furthermore, since there exists m 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m , we obtain the desired result from Theorem 3.1. In the case of α + β + γ + δ 0 , α + β + ζ > 0 and α + γ 0 , we can obtain the desired result by replacing the variables x and y. □

We know that an ( α , β , γ , δ , ε , ζ , η ) -widely more generalized hybrid mapping with α = 1 , β = γ = ε = ζ = 0 , δ = 1 and η = k ( 1 , 0 ] is a strict pseudo-contractive mapping in the sense of Browder and Petryshyn [8]. We also define the following mapping: T : C H is called a generalized strict pseudo-contractive mapping if there exist r , k R with 0 r 1 and 0 k < 1 such that
T x T y 2 r x y 2 + k ( x T x ) ( y T y ) 2

for any x , y C . Using Theorem 3.2, we can show the following fixed point theorem for generalized strict pseudo-contractive non-self mappings in a Hilbert space.

Theorem 4.3 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized strict pseudo-contractive mapping from C into H, that is, there exist r , k R with 0 r 1 and 0 k < 1 such that
T x T y 2 r x y 2 + k ( x T x ) ( y T y ) 2
for all x , y C . Assume that there exists m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t R with 0 < t m . Then T has a fixed point. In particular, if 0 r < 1 , then T has a unique fixed point.

Proof A generalized strict pseudo-contractive mapping T from C into H is a ( 1 , 0 , 0 , r , 0 , 0 , k ) -widely more generalized hybrid mapping. Furthermore, 1 + 0 + 0 + ( r ) 0 , 1 + 0 + 0 + ( k ) = 1 k > 0 , 1 + 0 + 0 + ( k ) = 1 k > 0 and [ 0 , 1 ) { λ ( 1 + 0 ) λ + 0 k 0 } = [ k , 1 ) , that is, it satisfies the condition (1) in Theorem 3.2. Furthermore, since there exists m 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m , we obtain the desired result from Theorem 3.2. In particular, if 0 r < 1 , then 1 + 0 + 0 + ( r ) > 0 . We have from Theorem 3.2 that T has a unique fixed point. □

Let us consider the problem in the Introduction. A mapping T : [ 0 , π 2 ] R was defined as follows:
T x = ( 1 + 1 2 x ) cos x 1 2 x 2
(4.1)
for all x [ 0 , π 2 ] . We have that
T x = ( 1 + 1 2 x ) cos x 1 2 x 2 1 1 + 1 2 x T x + 1 2 x 1 + 1 2 x x = cos x .
Thus we have that for any x [ 0 , π 2 ] ,
1 + 1 2 x 1 + π ( 1 1 + 1 2 x T x + 1 2 x 1 + 1 2 x ) + ( 1 1 + 1 2 x 1 + π ) x = 1 + 1 2 x 1 + π cos x + ( 1 1 + 1 2 x 1 + π ) x ,
and hence
1 1 + π T x + π 1 + π x = 1 + 1 2 x 1 + π cos x + π 1 2 x 1 + π x .
Using this, we also have from (2.1) that for any x , y [ 0 , π 2 ] ,
| 1 1 + π T x + π 1 + π x ( 1 1 + π T y + π 1 + π y ) | 2 = | 1 + 1 2 x 1 + π cos x + π 1 2 x 1 + π x ( 1 + 1 2 y 1 + π cos y + π 1 2 y 1 + π y ) | 2
and hence
1 1 + π | T x T y | 2 + π 1 + π | x y | 2 π ( 1 + π ) 2 | x y ( T x T y ) | 2 = | 1 + 1 2 x 1 + π cos x + π 1 2 x 1 + π x ( 1 + 1 2 y 1 + π cos y + π 1 2 y 1 + π y ) | 2 .
(4.2)
Define a function f : [ 0 , π 2 ] R as follows:
f ( x ) = 1 + 1 2 x 1 + π cos x + π 1 2 x 1 + π x
for all x [ 0 , π 2 ] . Then we have
f ( x ) = 1 2 1 + π cos x 1 + 1 2 x 1 + π sin x + π 1 + π x 1 + π
and
f ( x ) = 1 1 + π sin x 1 + 1 2 x 1 + π cos x 1 1 + π .
Since
f ( 0 ) = 1 2 + π 1 + π , f ( π 2 ) = 1 + 1 4 π 1 + π
and f ( x ) < 0 for all x [ 0 , π 2 ] , we have from the mean value theorem that there exists a positive number r with 0 < r < 1 such that
| 1 + 1 2 x 1 + π cos x + π 1 2 x 1 + π x ( 1 + 1 2 y 1 + π cos y + π 1 2 y 1 + π y ) | 2 r | x y | 2
for all x , y [ 0 , π 2 ] . Therefore, we have from (4.2) that
1 1 + π | T x T y | 2 + π 1 + π | x y | 2 r | x y | 2 + π ( 1 + π ) 2 | x y ( T x T y ) | 2
for all x , y [ 0 , π 2 ] . Furthermore, we have from (4.1) that
T x = ( 1 + 1 2 x ) ( cos x x ) + x
for all x [ 0 , π 2 ] . Take m = 1 + π and let t = 1 + 1 2 x and y = cos x for all x [ 0 , π 2 ] . Then we have that
T x = t ( y x ) + x , y = cos x [ 0 , π 2 ] and 0 < t = 1 + 1 2 x 1 + π .
Using Theorem 3.2, we have that T has a unique fixed point z [ 0 , π 2 ] . We also know that z = T z is equivalent to cos z = z . In fact,
z = T z z = ( 1 + 1 2 z ) ( cos z z ) + z 0 = ( 1 + 1 2 z ) ( cos z z ) 0 = cos z z .

Using Theorem 3.2, we can also show the following fixed point theorem for super hybrid non-self mappings in a Hilbert space; see [1].

Theorem 4.4 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a super hybrid mapping from C into H, that is, there exist α , β , γ R such that
α T x T y 2 + ( 1 α + γ ) x T y 2 ( β + ( β α ) γ ) T x y 2 + ( 1 β ( β α 1 ) γ ) x y 2 + ( α β ) γ x T x 2 + γ y T y 2
for all x , y C . Assume that there exists m > 1 such that for any x C ,
T x = x + t ( y x )

for some y C and t with 0 < t m . Suppose that α β 0 or γ 0 . Then T has a fixed point.

Proof An ( α , β , γ ) -super hybrid mapping T from C into H is an ( α , 1 α + γ , β ( β α ) γ , 1 + β + ( β α 1 ) γ , ( α β ) γ , γ , 0 ) -widely more generalized hybrid mapping. Furthermore, α + ( 1 α + γ ) + ( β ( β α ) γ ) + ( 1 + β + ( β α 1 ) γ ) = 0 , α + ( 1 α + γ ) + ( γ ) + 0 = 1 > 0 and α β ( β α ) γ ( α β ) γ + 0 = α β 0 , that is, it satisfies the conditions α + β + γ + δ 0 , α + β + ζ + η > 0 and α + γ + ε + η 0 in (2) of Theorem 3.2. Moreover, we have that
[ 0 , 1 ) { λ ( α + ( β ( β α ) γ ) ) λ + ( ( α β ) γ ) + 0 0 } = [ 0 , 1 ) { λ ( α β ) ( ( 1 + γ ) λ γ ) 0 } .
If α β > 0 , then
[ 0 , 1 ) { λ ( α β ) ( ( 1 + γ ) λ γ ) 0 } = [ 0 , 1 ) { λ ( 1 + γ ) λ γ 0 } = { [ 0 , 1 ) if  γ < 0 , [ γ 1 + γ , 1 ) if  γ 0 ,
that is, it satisfies the condition [ 0 , 1 ) { λ ( α + γ ) λ + ε + η 0 } in (2) of Theorem 3.2. If α β = 0 , then
[ 0 , 1 ) { λ ( α β ) ( ( 1 + γ ) λ γ ) 0 } = [ 0 , 1 ) ,
that is, it satisfies the condition [ 0 , 1 ) { λ ( α + γ ) λ + ε + η 0 } in (2) of Theorem 3.2. If α β < 0 and γ 0 , then
[ 0 , 1 ) { λ ( α β ) ( ( 1 + γ ) λ γ ) 0 } = [ 0 , 1 ) { λ ( 1 + γ ) λ γ 0 } = [ 0 , γ 1 + γ ] ,

that is, it again satisfies the condition [ 0 , 1 ) { λ ( α + γ ) λ + ε + η 0 } in (2) of Theorem 3.2. Then we obtain the desired result from Theorem 3.2. Similarly, we obtain the desired result from Theorem 3.2 in the case of (1). □

We remark that some recent results related to this paper have been obtained in [1417].

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The first author was partially supported by Grant-in-Aid for Scientific Research No. 23540188 from Japan Society for the Promotion of Science. The second and the third authors were partially supported by the grant NSC 99-2115-M-110-007-MY3 and the grant NSC 99-2115-M-037-002-MY3, respectively.

Authors’ Affiliations

(1)
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
(2)
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan
(3)
Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 80702, Taiwan
(4)
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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