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

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.

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

Tx= ( 1 + 1 2 x ) cosx 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 cosz=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:CH is said to be generalized hybrid if there exist α,βR such that

α T x T y 2 +(1α) x T y 2 β T x y 2 +(1β) x y 2
(1.1)

for all x,yC. 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.,

TxTyxy

for all x,yC. 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,yC. 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,yC. 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,ηR 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,yC. 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=γδ=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 0k<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,yH and λR, we have

λ x + ( 1 λ ) y 2 =λ x 2 +(1λ) y 2 λ(1λ) x y 2 .
(2.1)

Furthermore, we know that for x,y,u,vH,

2xy,uv= 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:CH is called super hybrid [1, 12] if there exist α,β,γR 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,yC. 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 γ1. 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 γ0. If a mapping S:CC 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 γ0. Let S:CC be an (α,β,γ)-super hybrid mapping. Then S has a fixed point in C. In particular, if S:CC 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,ζR 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,yC. 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+δ0;

  2. (2)

    ε+α+γ>0, or ζ+α+β>0.

Then T has a fixed point if and only if there exists zC such that { T n zn=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,ηR 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,yC. Suppose that it satisfies the following condition (1) or (2):

  1. (1)

    α+β+γ+δ0, α+γ+ε+η>0 and ζ+η0;

  2. (2)

    α+β+γ+δ0, α+β+ζ+η>0 and ε+η0.

Then T has a fixed point if and only if there exists zC such that { T n zn=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+δ0, α+γ+ε+η>0 and ζ+η0;

  2. (2)

    α+β+γ+δ0, α+β+ζ+η>0 and ε+η0.

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,ηR. Let T:CH 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+δ0, α+γ+ε+η>0, α+β+ζ+η0 and ζ+η0;

  2. (2)

    α+β+γ+δ0, α+β+ζ+η>0, α+γ+ε+η0 and ε+η0.

Assume that there exists a positive number m>1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm. 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 xC, there exist yC and t with 0<tm such that Tx=x+t(yx). From this, we have Tx=ty+(1t)x and hence

y= 1 t Tx+ t 1 t x.

Define UxC as follows:

Ux= ( 1 t m ) x+ t m y= ( 1 t m ) x+ t m ( 1 t T x + t 1 t x ) = 1 m Tx+ m 1 m x.

Taking λ>0 with m=1+λ, we have that

Ux= 1 1 + λ Tx+ λ 1 + λ x

and hence

T=(1+λ)UλI.
(3.1)

Since T:CH 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,yC,

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+δ0, α+γ+ε+η>0, α+β+ζ+η0 and ζ+η0, 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+δ0, α+β+ζ+η>0, α+γ+ε+η0 and ε+η0. 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+δ0, α+γ+ε+η>0, α+β+ζ+η0 and [0,1){λ(α+β)λ+ζ+η0};

  2. (2)

    α+β+γ+δ0, α+β+ζ+η>0, α+γ+ε+η0 and [0,1){λ(α+γ)λ+ε+η0}.

Assume that there exists m>1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm. 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+η0} and define S=(1λ)T+λI. Then S is a mapping from C into H. Since λ1, 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 xC,

Sx=(1λ)Tx+λx=(1λ) ( x + t ( y x ) ) +λx=t(1λ)(yx)+x,

where yC and 0<tm. From 0λ<1, we have 0<t(1λ)m. Putting s=t(1λ), we have that there exists m>1 such that for any xC,

Sx=x+s(yx)

for some yC and s with 0<sm. 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+δ0, α+β+ζ+η>0, α+γ+ε+η0 and [0,1){λ(α+γ)λ+ε+η0}, 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

βδ+ε+η>0,orγδ+ε+η>0

instead of the condition

α+γ+ε+η>0,orα+β+ζ+η>0,

respectively. In fact, in the case of the condition βδ+ε+η>0, we obtain from α+β+γ+δ0 that

0<βδ+ε+ηα+γ+ε+η.

Thus we obtain the desired results by Theorems 3.1 and 3.2. Similarly, in the case of γδ+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,βR such that

α T x T y 2 +(1α) x T y 2 β T x y 2 +(1β) x y 2

for any x,yC. Suppose αβ0 and assume that there exists m>1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm. Then T has a fixed point.

Proof An (α,β)-generalized hybrid mapping T from C into H is an (α,1α,β,(1β),0,0,0)-widely more generalized hybrid mapping. Furthermore, α+(1α)β(1β)=0, α+(1α)+0+0=1>0, αβ+0+0=αβ0 and 0+0=0, that is, it satisfies the condition (2) in Theorem 3.1. Furthermore, since there exists m1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm, 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+δ0, α+γ+ε>0 and α+β0;

  2. (2)

    α+β+γ+δ0, α+β+ζ>0 and α+γ0.

Assume that there exists m>1 such that for any xC,

Tx=x+t(yx)

for some yC and tR with 0<tm. 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,yC. 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+δ0, α+γ+ε+0=α+γ+ε>0, α+β+0+0=α+β0 and 0+0=0, that is, it satisfies the condition (1) in Theorem 3.1. Furthermore, since there exists m1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm, we obtain the desired result from Theorem 3.1. In the case of α+β+γ+δ0, α+β+ζ>0 and α+γ0, 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:CH is called a generalized strict pseudo-contractive mapping if there exist r,kR with 0r1 and 0k<1 such that

T x T y 2 r x y 2 +k ( x T x ) ( y T y ) 2

for any x,yC. 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,kR with 0r1 and 0k<1 such that

T x T y 2 r x y 2 +k ( x T x ) ( y T y ) 2

for all x,yC. Assume that there exists m>1 such that for any xC,

Tx=x+t(yx)

for some yC and tR with 0<tm. Then T has a fixed point. In particular, if 0r<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)=1k>0, 1+0+0+(k)=1k>0 and [0,1){λ(1+0)λ+0k0}=[k,1), that is, it satisfies the condition (1) in Theorem 3.2. Furthermore, since there exists m1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm, we obtain the desired result from Theorem 3.2. In particular, if 0r<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:

Tx= ( 1 + 1 2 x ) cosx 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 Tx+ π 1 + π x= 1 + 1 2 x 1 + π cosx+ 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 cosx+ π 1 2 x 1 + π x

for all x[0, π 2 ]. Then we have

f (x)= 1 2 1 + π cosx 1 + 1 2 x 1 + π sinx+ π 1 + π x 1 + π

and

f (x)= 1 1 + π sinx 1 + 1 2 x 1 + π cosx 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 cosx+ 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

Tx= ( 1 + 1 2 x ) (cosxx)+x

for all x[0, π 2 ]. Take m=1+π and let t=1+ 1 2 x and y=cosx for all x[0, π 2 ]. Then we have that

Tx=t(yx)+x,y=cosx [ 0 , π 2 ] and0<t=1+ 1 2 x1+π.

Using Theorem 3.2, we have that T has a unique fixed point z[0, π 2 ]. We also know that z=Tz is equivalent to cosz=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,γR 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,yC. Assume that there exists m>1 such that for any xC,

Tx=x+t(yx)

for some yC and t with 0<tm. Suppose that αβ0 or γ0. Then T has a fixed point.

Proof An (α,β,γ)-super hybrid mapping T from C into H is an (α,1α+γ,β(βα)γ,1+β+(βα1)γ,(αβ)γ,γ,0)-widely more generalized hybrid mapping. Furthermore, α+(1α+γ)+(β(βα)γ)+(1+β+(βα1)γ)=0, α+(1α+γ)+(γ)+0=1>0 and αβ(βα)γ(αβ)γ+0=αβ0, that is, it satisfies the conditions α+β+γ+δ0, α+β+ζ+η>0 and α+γ+ε+η0 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 αβ>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+η0} in (2) of Theorem 3.2. If αβ=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+η0} in (2) of Theorem 3.2. If αβ<0 and γ0, 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+η0} 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].

References

  1. Kocourek P, Takahashi W, Yao J-C: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 2010, 14: 2497–2511.

    MathSciNet  Google Scholar 

  2. Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. 2008, 91: 166–177. 10.1007/s00013-008-2545-8

    Article  MathSciNet  Google Scholar 

  3. Takahashi W: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 2010, 11: 79–88.

    MathSciNet  Google Scholar 

  4. Baillon J-B: Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C. R. Acad. Sci. Paris Ser. A-B 1975, 280: 1511–1514.

    MathSciNet  Google Scholar 

  5. Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 2008, 19: 824–835. 10.1137/070688717

    Article  MathSciNet  Google Scholar 

  6. Iemoto S, Takahashi W: Approximating fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 2009, 71: 2082–2089. 10.1016/j.na.2009.03.064

    Article  MathSciNet  Google Scholar 

  7. Kawasaki T, Takahashi W: Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert space. J. Nonlinear Convex Anal. 2013, 14: 71–87.

    MathSciNet  Google Scholar 

  8. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6

    Article  MathSciNet  Google Scholar 

  9. Kirk WA: Metric fixed point theory: old problems and new directions. Fixed Point Theory 2010, 11: 45–58.

    MathSciNet  Google Scholar 

  10. Hojo M, Takahashi W, Yao J-C: Weak and strong convergence theorems for supper hybrid mappings in Hilbert spaces. Fixed Point Theory 2011, 12: 113–126.

    MathSciNet  Google Scholar 

  11. Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohoma Publishers, Yokohoma; 2009.

    Google Scholar 

  12. Takahashi W, Yao J-C, Kocourek P: Weak and strong convergence theorems for generalized hybrid nonself-mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2010, 11: 567–586.

    MathSciNet  Google Scholar 

  13. Kawasaki T, Takahashi W: Fixed point and nonlinear ergodic theorems for new nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2012, 13: 529–540.

    MathSciNet  Google Scholar 

  14. Itoh S, Takahashi W: The common fixed point theory of single-valued mappings and multi-valued mappings. Pac. J. Math. 1978, 79: 493–508. 10.2140/pjm.1978.79.493

    Article  MathSciNet  Google Scholar 

  15. Kurokawa Y, Takahashi W: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 1562–1568. 10.1016/j.na.2010.04.060

    Article  MathSciNet  Google Scholar 

  16. Takahashi W, Wong N-C, Yao J-C: Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2012, 13: 745–757.

    MathSciNet  Google Scholar 

  17. Takahashi W, Yao J-C: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces. Taiwan. J. Math. 2011, 15: 457–472.

    MathSciNet  Google Scholar 

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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.

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Takahashi, W., Wong, NC. & Yao, JC. Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications. Fixed Point Theory Appl 2013, 116 (2013). https://doi.org/10.1186/1687-1812-2013-116

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