172

数理解析研究所講究録
第2011巻 2016年 172-177

Fixed point theorems for contractively widely moregeneralized hybrid mappings in metric spaces
日本大学 川暗敏治 (toshiharu.kawasaki@nifty.ne.jP)(Toshiharu Kawasaki, Nihon University)
Abstract
In this paper we consider a broad class of mappings containing Kannan mappingsand contratively generalized hybrid mappings.
orems

for such a mapping.

Then we deal with fixed point the‐Using these results, we show directly well‐known fixed

point theorems in complete metric spaces.

1Introduction
Let (X, d) be a metric space. A mapping T from X into itself is said to be contractive

if there exists k wirh k\in[0 , 1) such thatd(Tx Ty )\leq kd(x, y),for any x, y\in X

Such a mapping is called a k ‐contractive mapping. A mapping T from.

X into itself is said to be Kannan [5] if there exists k withk\displaystyle \in[0, \frac{1}{2}) such thatd(Tx Ty )\leq k(d(x, Tx)+d(y, Ty))
,for any x, y\in X A mapping T from X into itself is said to be contractively nonspreading.

[1,4, 9] if there exists k withk\displaystyle \in[0, \frac{1}{2} ) such thatd(Tx Ty )\leq k(d(x, Ty)+d(y,Tx)),for any x, y\in X.

A mapping T from X into itself is said to be contractively hybrid [3] ifthere exists k withk\displaystyle \in[0, \frac{1}{3} ) such thatd(Tx,Ty)\leq k(d(Tx, y)+d(Ty, x)+d(x,y))
for any x, y\in X.

Recently, Hasegawa, Komiya and Takahashi [3] introduced the conceptof contratively generalized hybrid mappings on metric spaces and studied the fixed point
theorems for such mappings on complete metric spaces. A mapping T from X into itself issaid to be contratively generalized hybrid if there exist $\alpha$, $\beta$, r\in \mathbb{R} with r\in[0 , 1) such that$\alpha$ d(Tx, Ty)+(1- $\alpha$)d(x, Ty)\leq r( $\beta$ d(Tx,y)+(1- $\beta$)d(x,y))173( $\alpha,\ \beta$, r) ‐contratively generalized hybridmapping;
For instance, if $\alpha$=1 and $\beta$=0 then an ( $\alpha,\ \beta$,r) ‐contratively generalized hybrid mapping
is contractive; if $\alpha$=1+r and $\beta$=1 then an ( $\alpha,\ \beta$, r) ‐contratively generalized hybrid
mapping is contractively nonspreading; if $\alpha$=1+\displaystyle \frac{r}{2} and $\beta$=\displaystyle \frac{1}{2} then an ( $\alpha,\ \beta$,r) ‐contratively
generalized hybrid mapping is contractively hybrid; see Hasegawa, Komiya and Takahashi
for any x, y\in X
see.

Such a mapping is called analso Kocourek, Takahashi and Yao [7] for such a mapping in Hilbert spaces.
,,,

[3].
In this paper, motivated by Hasegawa, Komiya and Takahashi [3], we consider a broadclass of mappings containing Kannan mappings and contratively generalized hybrid map‐pings. Then we deal with fixed point theorems for such a mapping. Using these results,
directly well‐known fixed point theorems in complete metric spaces.

we showPreliminaries

2

We know the following Caristis fixed point theorem which was generalized by Takahashi[8].
Theorem 2.1. Let (X, d) be a complete metric space, let $\psi$ be a proper, bounded below,

and lower semicontinuous mapping from X into (-\infty, \infty], and let T be a mapping from X

into itself.

Suppose thatd(x,Tx)+ $\psi$(Tx)\leq $\psi$(x)for any x\in X.

Then T has a fxed point.

Let \ell\infty be the Banach space of bounded sequences with supremum norm. Let $\mu$ be anThen we denote by $\mu$(x) the value of $\mu$ atelement of (\ell^{\infty})^{*} , which is the dual space of p\infty

x=(x_{1}, x_{2}, \ldots)\in\ell\infty.

.

Sometimes we denote by $\mu$_{n}(x_{n}) the value $\mu$(x)$\mu$ on \ell\infty is called a mean if $\mu$(e)=\Vert $\mu$\Vert=1 , where e=(1,1,Banach limit on \ell\infty if $\mu$_{n}(x_{n+1})=$\mu$_{n}(x_{n})p\infty

.

.

.

.

.

A linear functional

A mean $\mu$ is called a

We know that there exists a Banach limit on

If $\mu$ is a Banach hmit on p\infty , then\displaystyle \lim_{n\rightar ow}\inf_{\infty}x_{n}\leq$\mu$_{n}(x_{n})\leq\lim_{n\rightar ow}\sup_{\infty}x_{n}holds for any x= (x_{1}, x2, ..)\in\ell\infty

then we obtain $\mu$_{n}(x_{n})=a.

other elementary properties.

.

In particular, if x=(x_{1}, x_{2}, \ldots)\in p\infty and x_{n}\rightarrow a\in \mathbb{R},See [8] for the proof of existence of a Banach limit and its

Moreover we use the following lemma and theorem showed by Hasegawa, Komiya andTakahashi [3].
Lemma 2.1. Let (X, d) be a metric space, let \{x_{n}\} be a bounded sequence in X , let $\mu$ bea mean on \ell\inftyand let g be a mapping from X into \mathbb{R} defined by

g(x)=$\mu$_{n}d(x_{n}, x)for any x\in X.

Then g is continuous.

174

Theorem 2.2. Let (X, d) be a complete metric space, let $\mu$ be a mean on \ell\infty and let T beamapping from X into itself. Suppose that there exist a real number r with 0\leq r<1 andz\in X such that \{T^{n}z|n\in \mathbb{N}\cup\{0\}\} is bounded and$\mu$_{n}d(T^{n}z Tx )\leq r$\mu$_{n}d(T^{n}z,x),for any x\in X.

Then the following hold:(i)

T has a unique fixed point u\in X ;(ii)

u=\displaystyle \lim_{n\rightarrow\infty}T^{r $\iota$}x for any x\in X.

3

Fixed point theorems
In this section we consider an ( $\alpha,\ \beta$, $\gamma$, $\delta,\ \epsilon$, $\zeta$) ‐contractively widely more generalized hy‐brid mapping from a metric space X into itself; see also Kawasaki and Takahashi [6] forsuch a mapping in Hilbert spaces.
Definition 3.1. Let (X, d) be a metric space and let T be a mapping from X into itself.

We say that T is contractively widely more generalized hybrid if T satisfies the following
condition: there exist real numbers $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ and $\zeta$ such thatad (Tx, Ty)+ $\beta$ d(x, Ty)+ $\gamma$ d(Tx, y)+ $\delta$ d(x, y)+ $\epsilon$ d(x, Tx)+ $\zeta$ d(y, Ty)\leq 0for any x, y\in X Such a mapping T is called an ( $\alpha,\ \beta,\ \gamma$, $\delta$, \mathrm{e}, $\zeta$) ‐contractively widely moregeneralized hybrid mapping.
.

Firstly we consider criteria for an ( $\alpha$, $\beta,\ \gamma$, $\delta$, $\epsilon$, $\zeta$) ‐contractively widely more generalized
hybrid mapping T from a metric space X into itself such that \{T^{n}x|n\in \mathrm{N}\cup\{0\}\} is a
Cauchy sequence for any x\in X.
Lemma 3.1. Let (X, d) bemetric space and let T be an ( $\alpha$, $\beta,\ \gamma$, $\delta$, $\epsilon$, $\zeta$) ‐contractively
generalized hybrid mapping from X into itself satisfying (B1), (B2) or (B3):awidely

more

(B1)

$\alpha$+ $\beta$+ $\zeta$\geq 0 and $\alpha$+2\displaystyle \min\{ $\beta$, 0\}+ $\delta$+ $\epsilon$+ $\zeta$>0 ;(B2)$\alpha$+ $\gamma$+ $\epsilon$\geq 0 and $\alpha$+2\displaystyle \min\{ $\gamma$, 0\}+ $\delta$+ $\epsilon$+ $\zeta$>0 ;(B3)

2 $\alpha$+ $\beta$+ $\gamma$+ $\epsilon$+ $\zeta$\geq 0 and $\alpha$+\displaystyle \min\{ $\beta$+ $\gamma$, 0\}+ $\delta$+ $\epsilon$+ $\zeta$>0.

Then \{T^{ $\tau \iota$}x|n\in \mathbb{N}\cup\{0\}\} is a Cauchy sequence for any x\in X.

Using Lemma 3.1, we obtain directly the following theorem.

(X, d) be a complete metric space and let T be an ( $\alpha,\ \beta,\ \gamma$, $\delta,\ \epsilon$, $\zeta$)contractively widely more generalized hybrid mapping from X into itself satisfying (B1),(B2) or (B3). Then for any x\in X there exists \displaystyle \lim_{n\rightarrow\infty}T^{n}x.

Theorem 3.1. Let

175

Remark 3.1. Let (X, d) be a metric space and let \{x_{n}|n\in \mathrm{N}\cup\{0\}\} be a Cauchy sequence

in X

.

Then \{x_{n}|n\in \mathbb{N}\cup\{0\}\} is bounded.

Indeed, since \{x_{n}|n\in \mathbb{N}\cup\{0\}\} is a Cauchy

sequence, for any positive number $\rho$ there exists N\in \mathbb{N} such thatm, n\geq N.

Put M=\displaystyle \max\{d(x_{0},x_{N}), \cdots, d(x_{N-1}, x_{N}), $\rho$\}n\in \mathrm{N}\cup\{0\}.

.

d(x_{m}, x_{n})< $\rho$ for any

Then d(x_{n}, x_{N})\leq \mathrm{M} for any

Using Theorem 2.1, we show the following fixed point theorem.
Theorem 3.2. Let (X, d) bea complete metric space and let T be an( $\alpha$, $\beta,\ \gamma$, $\delta,\ \epsilon$, $\zeta$)contractively widely more generalized hybrid mapping from X into itself satisfying (C1),(C2) or (C3):(C1)$\zeta$>0, a+ $\beta$\geq 0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$+2\displaystyle \min\{ $\epsilon$, 0\}\geq 0 ;(C2)$\epsilon$>0, $\alpha$+ $\gamma$\geq 0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$+2\displaystyle \min\{ $\zeta$, 0\}\geq 0 ;(C3)$\epsilon$+ $\zeta$>0, 2 $\alpha$+ $\beta$+ $\gamma$\geq 0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$\geq 0.

Then T has a fixed point if and only if there exists z\in X such that \{T^{n}z|n\in \mathrm{N}\cup\{0\}\} isbounded. In particular,

if $\alpha$+ $\beta$+ $\gamma$+ $\delta$>0,

then T has a unique fixed point.

Using Lemma 3.1, Remark 3.1 and Theorem 3.2, we obtain the following fixed point
theorem.

(X, d) be a complete metric space and let T be an ( $\alpha$, $\beta,\ \gamma$, $\delta$, $\epsilon$, $\zeta$)contractively widely more generalized hybrid mapping from X into itself satisfying the fol‐lowing:

Theorem 3.3. Let(B)

oneof (B1), (B2) and (B3) holds;(C)

oneof (C1), (C2) and (C3) holds.

Then T has a fixed point. In particular,

if $\alpha$+ $\beta$+ $\gamma$+ $\delta$>0,

then T has a unique fixedpoint.

Using Theorem 2.2, we show the following fixed point theorem.

(X, d) be a complete metric space and let T be an ( $\alpha$, $\beta,\ \gamma$, $\delta,\ \epsilon$, $\zeta$)contractively widely
generalized hybrid mapping from X into itself satisfying (\mathrm{H}1)_{f}
or
(H2)
(H3):

Theorem 3.4. Let

more

(H1)

$\alpha$+ $\beta$+ $\zeta$>0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$+2\displaystyle \min\{ $\epsilon$, 0\}+2\min\{ $\zeta$, 0\}>0 ;(H2)$\alpha$+ $\gamma$+ $\epsilon$>0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$+2\displaystyle \min\{ $\epsilon$, 0\}+2\min\{ $\zeta$, 0\}>0 ;(H3)

2 $\alpha$+ $\beta$+ $\gamma$+ $\epsilon$+ $\zeta$>0 and $\alpha$+ $\beta$+ $\gamma$+ $\delta$+2\displaystyle \min\{ $\epsilon$+ $\zeta$, 0\}>0.

Then T has a fixed point if and only if there exists z\in X such that \{T^{ $\gamma \iota$}z|n\in \mathbb{N}\cup\{0\}\} isbounded. Moreover the following hold:

176

(i)

T has a unique fixed point u\in X ;(ii)

u=\displaystyle \lim_{n\rightarrow\infty}\mathcal{I}^{m}x for any x\in X.

Using Lemma 3.1, Remark 3.1 and Theorem 3.4, we obtain the following fixed point
theorem.

(X, d) be a complete metric space and let T be an ( $\alpha$, $\beta,\ \gamma$, $\delta$, $\epsilon$, $\zeta$)contractively widely more generalized hybrid mapping from X into itself satisfying the fol‐lowing:

Theorem 3.5. Let(B)

oneof (B1), (B2) and (B3) holds;(H)

oneof (H1), (H2) and (H3) holds.

Then the following hold:(i)

T has a unique fixed point u\in X_{f}.

(ii)u=\displaystyle \lim_{n\rightarrow\infty}T^{n}x for any x\in X.

Moreover, if (B) is satisfied, we also show the following fixed point theorem.

(X, d) be a complete metric space and let T be an ( $\alpha$, $\beta,\ \gamma$, $\delta,\ \epsilon$, $\zeta$)contractively widely
generalized hybrid mapping from X into itself satisfying (\mathrm{B})_{f} andone of (M1), (M2) and (M3):

Theorem 3.6. Let

more

(M1)

$\alpha$+ $\beta$+ $\zeta$>0_{J}.

(M2)$\alpha$+ $\gamma$+ $\epsilon$>0 ;(M3) 2 $\alpha$+ $\beta$+ $\gamma$+ $\epsilon$+ $\zeta$>0.
Then T has a fixed point. In particular,

if $\alpha$+ $\beta$+ $\gamma$+ $\delta$>0(i)

T has a unique fixed point u\in X_{f}.

(ii)u=\displaystyle \lim_{n\rightarrow\infty}T^{n}x for any x\in X.

4Applications,

then the following hold:

Theorem 4.1. Let (X, d) be a complete metric space and let T be a contractively generalized

hybrid mapping form X into itself, that is, there exist $\alpha$, $\beta$, r\in \mathbb{R} with 0\leq r<1 such that$\alpha$ d(Tx,Ty)+(1- $\alpha$)d(x, Ty)\leq r( $\beta$ d(Tx, y)+(1- $\beta$)d(x, y))for any x, y\in X Suppose that $\alpha$>r(1+| $\beta$|).

(i)

T has a unique fixed point u\in X ;.

Then the following hold:

177

(ii)

u=\displaystyle \lim_{n\rightarrow\infty}T^{n}x for any x\in X.

Theorem 4.2. Let (X, d) be a complete metric space and let T be a mapping form X into

itself satisfying there exist $\epsilon$, $\zeta$\in \mathbb{R} such that $\epsilon$+ $\zeta$<1 andd(Tx Ty )\leq $\epsilon$ d(x,Tx)+ $\zeta$ d ( y Ty),for any x, y\in X.

,Then the following hold:(i)

T has a unique fixed point u\in X ;(ii)

u=\displaystyle \lim_{n\rightarrow\infty}T^{n}x for any x\in X.

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[3]

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[6] T. Kawasaki and W. Takahashi, Existence and mean approximation offixed points of generalized hybrid
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