36

Fixed Point and Convergence Theorems
for Two Nonlinear Mappings in Hilbert Spaces
芝浦工業大学

北條真弓 (Mayumi Hojo)
Shibaura Institute of Technology, Oomiya, Saitama 337‐8570, Japan

Abstract

In this paper, first, we introduce attractive point and fixed point theorems and mean
convergence theorem for two commutative 2‐generalized mappings in Hilbert spaces.
Next we obtain a weak convergence theorem of Mann s type iteration, a strong conver‐gence theorem of Halpern’s type iteration, and a strong convergence theorem by thehybrid method for two commutative 2‐generalized hybrid mappings in a Hilbert space.

2010 Mathematics Subject Classification: 47H10
Keywords and phrases: Attractive point, fixed point, generalized hybrid mapping,
Hilbert space, nonexpansive mapping, 2‐generalized hybrid mapping, hybrid method.

1

introduction

Throughout this paper, we denote by \mathb {N} the set of positive integers and by \mathbb{R} the set of real
numbers. Let H be a real Hilbert space and ıet C be a nonempty subset of H . Let T be amapping of C into H . Then we denote by F(T) the set of fixed points of T and by A(T) theset of attractive points [23] of T , i.e.,(i) F(T)= { z\in C :Tz =z };(ii) A(T)=\{z\in H: \Vert Tx-z\Vert\leq\Vert x-z\Vert, \forall x\in C\}.
We know that A(T) is closed and convex. This property is important for proving our maintheorems. A mapping T : Carrow H is said to be nonexpansive if \Vert Tx-Ty\Vert\leq\Vert x-y\Vert forallx,y\in C . It is well‐known that if C is a bounded, closed and convex subset of H andT:Carrow C is nonexpansive, thenF(T) is nonempty. Furthermore, from Baillon we know thefirst nonlinear ergodic theorem in a Hilbert space: Let C be a bounded, closed and convex
subset of H and let T : Carrow C be nonexpansive. Then for any x\in C,

S_{n}x= \frac{1}{n}\sum_{k=0}^{n-1}T^{k_{X}converges weakly to an element z\in F(T) . In 2010, Kocourek, Takahashi and Yao [10] defineda broad class of nonlinear mappings in a Hilbert space: Let H be a Hilbert space and let C
be a nonempty subset of H . A mapping T : Carrow H is called generalized hybrid if there exist\alpha, \beta\in \mathbb{R} such that\alpha\Vert Tx-Ty\Vert^{2}+(1-\alpha)\Vert x-Ty\Vert^{2}\leq\beta\Vert Tx- y\Vert^{2}+(1-\beta)\Vert x-y\Vert^{2}

(1.1)

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for all x, y\in C . Such a mapping T is called (\alpha, \beta) ‐generalized hybrid. We also know thefollowing mapping: For \lambda\in \mathbb{R} , a mapping U : Carrow H is called \lambda ‐hybrid if\Vert Ux-Uy\Vert^{2}\leq\Vert x-y\Vert^{2}+2(1-\lambda)\{x-Ux, y- Uy\}

(1.2)

for all x, y\in C . Notice that the class of generalized hybrid mappings covers several well‐knownmappings. For example, a(1,0) ‐generalized hybrid mapping is nonexpansive, i.e.,\Vert Tx-Ty\Vert\leq\Vert x-y\Vert, \forall x, y\in C.
It is nonspreading for \alpha=2 and \beta=1 , i.e.,2\Vert Tx-Ty\Vert^{2}\leq\Vert Tx-y\Vert^{2}+\Vert Ty-x\Vert^{2}, \forall x, y\in C.
It is also hybrid for \alpha=\frac{3}{2} and \beta=\frac{1}{2} , i.e.,3\Vert Tx-Ty\Vert^{2}\leq\Vert x-y\Vert^{2}+\Vert Tx-y\Vert^{2}+\Vert Ty- x\Vert^{2}, \forall x, y\in C.
In general, nonspreading and hybrid mappings are not continuous. We also know that \lambda ‐hybrid
mappings are in the class of generalized hybrid mappings. The nonlinear ergodic theorem byBaillon for nonexpansive mappings has been extended to generalized hybrid mappings ina Hilbert space by Kocourek, Takahashi and Yao. Recently, Kohsaka [11] also proved thefollowing theorem:

Theorem 1.1. Let H be a Hilbert space and let C be a nonempty, closed and convex subsetof H. Let S and T be commutative \lambda and \mu ‐hybrid mappings of C into itself such that the set
F(S)\cap F(T) of common fixed points of S and T\iota s nonempty. Then, for any x\in C,

S_{n}x= \frac{ \imath} {(n+1)^{2} \sum_{k=0}^{n}\sum_{l=0}^{n}S^{k}T^{l}x
converges weakly to a point of F(S)\cap F(T) .

On the other hand, Takahashi and Takeuchi [23] proved the following attractive point andmean convergence theorem without convexity in a Hilbert space.
Theorem 1.2. Let H be a Hilbert space and let C be a nonempty subset of H. Let T be a

generalized hybrid mapping from C into itself. Assume that {Tnz} for some z\in C is boundedand define

S_{n}x= \frac{1}{n}\sum_{k=0}^{n}T^{k}x
for all x\in C and n\in \mathbb{N} . Then \{S_{n}x\} converges weakly to u_{0}\in A(T) , where\lim_{narrow\infty}P_{A(T)}T^{n}x and P_{A(T)} is the metric projection of H onto A(T) .

u_{0}=Maruyama, Takahashi and Yao also defined a more broad class of nonlinear mappings called
2‐generalized hybrid which contains generalized hybrid mappings in a Hilbert space. Let C
be a nonempty subset of H and let T be a mapping of C into C . A mapping T : Carrow H is
2‐generalized hybrid [15] if there exist \alpha_{1}, \alpha_{2}, \beta_{1} , \beta_{2}\in \mathbb{R} such that\alphaı | T2 x-Ty\Vert^{2}+\alpha_{2}\Vert Tx-Ty\Vert^{2}+(l-\alpha_{{\imath}}-\alpha_{2}) \Vert x —Tyll2\leq\beta_{l}\Vert T^{2}x-y\Vert^{2}+\beta_{2}\Vert Tx-y\Vert^{2}+(l- \beta_{{\imath}}-\beta_{2})\Vert x-y\Vert^{2}

(1.3)

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for all x, y\in C.

In this paper, using means, we first obtain attractive point and fixed point theorems forcommutative 2‐generalized hybrid mappings in Hilbert spaces. Using these results, we prove
weak and strong convergence theorems for commutative 2‐generalized hybrid mappings inHilbert spaces.

2Preliminaries

In a Hilbert space, it is known that\Vert x+y-z\Vert^{2}-\Vert x-z\Vert^{2}\geq 2\langle y, x-z\rangle(2.1)\Vert\alpha x+(1-\alpha)y\Vert^{2}=\alpha\Vert x\Vert^{2}+(1-\alpha)\Vert y\Vert^{2}-\alpha(1-\alpha)\Vert x-y\Vert^{2}(2.2)

for all x, y, z\in H andfor all x, y\in H and \alpha\in \mathbb{R} ; see [21]. Furthermore, in a Hilbert space, we have that2 \{x-y, z-w\}=\Vert x-w\Vert^{2}+\Vert y-z\Vert^{2}-\Vert x-z\Vert^{2}-\Vert y -w\Vert^{2}(2.3)

for all x, y, z, w\in H . Let H be a Hilbert space and let C be a nonempty subset of H.
mapping T:Carrow H with F(T)\neq\emptyset is called quasi‐nonexpansive if\Vert Tx-u\Vert\leq\Vert x-u\Vert, \forall x\in C, u\in F(T)A.

If C is closed and convex and T:Carrow H with F(T)\neq\emptyset is quasi‐nonexpansive, then F(T) isclosed and convex; see Itoh and Takahashi [9]. For a nonempty, closed and convex subset D of
H , the nearest point projection of H onto D is denoted by P_{D} , that is, \Vert x-P_{D}x\Vert\leq\Vert x-y\Vertfor allx\in H andy\in D . Such a mapping P_{D} is called the metric projection of H ontoD.

We know that the metric projection P_{D} is firmly nonexpansive; \Vert P_{D}x-P_{D}y\Vert^{2}\leq {PDx‐
P_{D}y, x-y\} for all x, y\in H . Furthermore, \{x-P_{D}x , y—PDx) \leq 0 holds for all x\in H andy\in D ; see [20, 21]. Using this inequality and (2.3), we have that\Vert P_{D}x-y\Vert^{2}+\Vert P_{D}x-x\Vert^{2}\leq\Vert x-y\Vert^{2}, \forall x\in H, y\in D .

(2.4)

The following result was proved by Takahashi and Toyoda [24].
Lemma 2.1 ([24]). Let H be a Hilbert space and let C be a nonempty, closed and convex
subset of H. Let \{x_{n}\} be a sequence in H. If \Vert x_{n+{\imath}}-u\Vert\leq\Vert x_{n}-u\Vert for all n\in \mathbb{N} and u\in C,
then \{P_{C}x_{n}\} converges strongly to some z\in C , where P_{C} is the metric projection of H ontoC.

LetH be a Hilbert space and let C be a nonempty subset of H .

A mapping T :Carrow C

was called 2‐generalized hybrid [15] if there exist \alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}\in \mathbb{R} satisfying (1.3). We also
call such a mapping (\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}) ‐generalized hybrid. We know that the class of the map‐
pings above covers well‐known mappings. For example, the class of (0, \alpha_{2},0, \beta_{2}) ‐generalizedhybrid mappings is the class of (\alpha_{2}, \beta_{2}) ‐generalized hybrid mappings in the sense of Kocourek,Takahashi and Yao [ı0]. If x=Tx in (1.3), then for any y\in C,\alpha_{1}\Vert x-Ty\Vert^{2}+\alpha_{2}\Vert x-Ty\Vert^{2}+(1-\alpha_{1}- \alpha_{2})\Vert x-Ty\Vert^{2}\leq\beta_{l}\Vert x-y\Vert^{2}+\beta_{2}\Vert x-y\Vert^{2}+(l-\beta_{{\imath}} -\beta_{2})\Vert x-y\Vert^{2}.

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Hence we have that\Vert x-Ty\Vert\leq\Vert x-y\Vert, \forall x\in F(T), y\in C .

(2.5)

Thus, a 2‐generalized hybrid mapping with a fixed point is quasi‐nonexpansive. Hojo, Taka‐hashi and Takahashi [6] obtained the following attractive point and fixed point theorems fortwo commutative 2‐generalized hybrid mappings in a Hilbert space.

Theorem 2.2 ([6]). Let H be a Hilbert space, let C be a nonempty subset of H and let S andT be commutative 2‐generalized hybrid mappings of C into itself. Suppose that there exists anelement z\in C such that \{S^{k}T^{\iota}z:k, l\in \mathbb{N}\cup\{0\}\} is bounded. Then A(S)\cap A(T) is nonempty.
Additionally, if C is closed and convex, then F(S)\cap F(T) is nonempty.

Also, we prove a mean convergence theorem for commutative 2‐generalized hybrid mappings
without convexity in a Hilbert space.
Let D=\{(k, l) : k, l\in N\cup\{0\}\} . Then D is a directed set by the binary relation:(k, l)\leq(i,j)

ifk\leq i andl\leq j.

Theorem 2.3 ([6]). Let H be a Hilbert space and let C be a nonempty subset of H. Let S andT be commutative 2‐generalized hybrid mappings of C into itself such that A(S)\cap A(T)\neq\emptyset.
Let P be the metric projection of H onto A(S)\cap A(T) . Then, for any x\in C,

S_{n}x= \frac{1}{(n+1)^{2} \sum_{k=0}^{n}\sum_{l=0}^{n}S^{k}T^{\iota_{X}converges weakly to an element q of A(S)\cap A(T) , where q= \lim_{(k,l)\in D}PS^{k}T^{\iota}x . In particular,
if C is closed and convex, \{S_{n}x\} converges weakly to an element q of F(S)\cap F(T) .

3Weak convergence theorem of Mann’s type iteration

In this section, we obtain a weak convergence theorem of Mann’s type iteration for two com‐mutative 2‐generalized hybrid mappings in a Hilbert space. Before proving the theorem, weneed the following lemma.

Lemma 3.1 ([5]). Let C be a nonempty, closed and convex subset of a Hilbert space H andlet S and T be commutative 2‐generalized hybrid mappings of C into itself. Let \{x_{n}\} be abounded sequence of C. Define

S_{n}x_{n}= \frac{1}{(1+n)^{2} \sum_{k=0}^{n}\sum_{l=0}^{n}S^{k}T^{l}x_{n}for all n\in \mathbb{N}\cup\{0\} . Suppose that \Vert S_{n}x_{n}-x_{n}\Vertarrow 0 . Then every weak cluster point of \{x_{n}\} isa point of F(S)\cap F(T) .

Theorem 3.2 ([5]). Let H be a Hilbert space and let C be a nonempty, closed and convex
subset of H. Let S and T be commutative 2‐generalized hybrid mappings of C into itself suchthat F(S)\cap F(T)\neq\emptyset . Let P be the metric projection of H onto F(S)\cap F(T) . Let \{\alpha_{n}\} bea sequence of real numbers such that 0\leq\alpha_{n}<1 and \lim\inf_{narrow\infty}\alpha_{n}(1-\alpha_{n})>0 . Then, asequence \{x_{n}\} generated by x_{1}=x\in C and

x_{n+{\imath} = \alpha_{n}x_{n}+(1-\alpha_{n})\frac{ \imath} {(n+1)^{2} \sum_{k =0}^{n}\sum_{l=0}^{n}S^{k}T^{\iota}x_{n}, \foral n\in \mathb {N}40converges weakly to z\in F(S)\cap F(T) , where z= \lim_{narrow\infty}Px_{n}.

4Strong convergence theorem of Halpern’s type iteration

Using the idea of mean convergence by Shimizu and Takahashi [18, 19], and Kurokawa andTakahashi [14], we prove the folıowing strong convergence theorem of Haıpern’s type iteration
for two commutative 2‐generalized hybrid mappings in a Hilbert space.

Theorem 4.1 ([5]). Let H be a Hilbert space and let C be a nonempty, closed and convex
subset of H. Let S and T be commutative 2‐generalized hybrid mappings of C into itself suchthat F(S)\cap F(T)\neq\emptyset . Let u\in C and define a sequence \{x_{n}\} in C as follows: x_{1}=x\in C
andx_{n+1}= \alpha_{n}u+(1-\alpha_{n})\frac{1}{(n+1)^{2} \sum_{k=0}^{n}\sum_{l=0}^ {n}S^{k}T^{l}x_{n}, \foral n\in \mathb {N},where 0\leq\alpha_{n}\leq 1, \alpha_{n}arrow 0 and \sum_{n=1}^{\infty}\alpha_{n}=\infty . Then \{x_{n}\} converges strongly to Pu, where
P is the metric projection of H onto F(S)\cap F(T) .

5Strong convergence theorems by hybrid methods

In this section, using the hybrid method by Nakajo and Takahashi [17], we first prove a strong
convergence theorem for two commutative 2‐generalized hybrid mappings in a Hilbert space.

Theorem 5.1 ([5]). Let H be a real Hilbert space, let C be a nonempty, convex and closed
Let S, T : Carrow C be commutative 2‐generalized hybrid mappings such that
F(S)\cap F(T)\neq\emptyset . Let \{x_{n}\}\subset C be a sequence generated by x_{1}=x\in C andsubset of H.

\{beginary}{l _n}=\alph_{n}x +(1-\alph_{n})\frac{1}(n+\imath})^{2\sum_{k=0}^n \sum_{l=0}^nS{k}T^lx_{n},C =\{zinC:\Verty_{n}-z\Vertlq\Vertx_{n}-z\Vert},Q_{n=\zinC:\lagex_{n}-z,x_{n}\ragle q0\},x_{n+1}=P_{Cn}\capQ_{n}^X,\foraln\i mathb{N},\end{ary}where P_{C_{n}\cap Q_{n}} is the metric projection of H onto C_{n}\cap Q_{n} and \{\alpha_{n}\}\subset[0,1] satisfies 0\leq\alpha_{n}\leq
a<1 for some a\in \mathbb{R} . Then, \{x_{n}\} converges strongly to z_{0}=P_{F(S)\cap F(T)}x , where P_{F(S)\cap F(T)}is the metric projection of H onto F(S)\cap F(T) .

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