数理解析研究所講究録
1071 巻 1998 年 99-104

99

STRONG CONVERGENCE TO FIXED POINTS OF

NON-LIPSCHITZIAN MAPPINGS IN BANACH SPACESGANG-EUN$\mathrm{K}\mathrm{l}\mathrm{M}$ABSTRACT. In this paper, we study the strong convergence of the modified Ishikawa and DasDebata iteration process of non-Lipschitzian mappings which satisfies the property (K) typein a Banach spaces.

1. INTRODUCTIONLet
be a nonempty bounded closed convex subset of a Banach space $E$ and let bea mapping of $C$ into itself. Then is said to be asymptotically nonexpansive [5] if there
exists a sequenceof real numbers with $\lim_{narrow\infty}k_{n}=1$ such that
$C$

$T$

$T$

$\{k_{n}\}$$||T^{n}x-Tny||\leq k_{n}||x-y||$

for $x,$ $y\in C$ and $n=1,2,$. In particular, if $k_{n}=$.
for all $n\geq 1,$
nonexpansive. The weaker definition (cf., Kirk [10]) requires that$1$$\cdots$$T$

is said to be$\lim\sup\sup(||\tau nx-Tn|y|-||x-y||)\leq 0$$narrow\infty y\in C$

for each $x\in C$ , and that
where between these two:is continuous and$T^{N}$

be continuous for some $N\geq 1$ . Consider a definition somesaid to be weakly asymptotically nonexpansive provided$T\dot{\mathrm{i}}\mathrm{s}$$T$$\lim\sup\sup(||T^{n}x-Tn|y|-||x-y||)\leq 0$ .
$narrow\infty x,y\in C$

Compare with the definition of asymptotically nonexpansive mappings in the intermediate
sense initiated by Bruck et al. [1]. For two mappings $S,$ of into itself, we consider thefollowing modified Das-Debata iteration scheme (cf. Das-Debata [3]): $x_{1}\in C$ ,
$C$

$T$

$x_{n+1}=\alpha_{n}S^{n}[\beta nT^{n}x_{n}+(1-\beta_{n})x_{n}]+(1-\alpha_{n})x_{n}$$(*)$

for all $n\geq 1$ , whereandin $[0,1]$ . In this case of $S=T$ , such an iteration scheme
was considered by Tan-Xu [17]; see also Ishikawa [7], Mann [11], Schu [14]. Reich [12],using Mann iteration procedure in a uniformly convex Banach space whose norm is Fr\’echet
differentiable, proved that the iterates
defined by$\{\alpha_{n}\}$$\{\beta_{n}\}$$\{x_{n}\}$$x_{n+1}=(1-\alpha_{n})xn+\alpha_{nn}\tau_{X}$,1991 Mathematics Subject Classification. Primary; Secondary.
Key words and phrases. relaxed Lipschitz, strictly hemi-contractive, (K) type, strong convergence,
approximating fixed points, weakly asymptotically nonexpansive mappings.
$46\mathrm{B}20$$47\mathrm{H}10$

Typeset by$AM^{- \mathrm{I}}\mathrm{E}^{\mathrm{X}}$100$\mathrm{G}.\mathrm{E}$. KIM

for all $n\geq 1$ , converge weakly to a fixed point of nonexpansive mappings: $Carrow C$
under. Tan-Xu [16] improved a result of Reich [12] to the case of
the Ishikawa type iteration. On the other hand, Takahashi-Tamura [15] studied the weak
convergence of iterates
defined by$T$$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$$\{x_{n}\}$$x_{n+1}=\alpha_{n}s[\beta n\tau_{x_{n}}+(1-\beta_{n})x_{n}]+(1-\alpha_{n})x_{n}$for all $n\geq 1$ , in a uniformly convex Banach space which satisfies Opial’s condition or whose
norm is Fr\’echet differentiable. Recently Verma [18] proved the following interesting result
using modified iterative algorithm: Let $H$ be a real Hilbert space andbe a nonempty
closed convex subset of $H$ . Let: $Carrow C$ be a relaxed Lipschitz (see Definition below)
and Lipschitz continuous operator on . Let $r\geq 0$ and $s\geq 1$ be constants for relaxed
Lipschitzity and Lipschitz continuity of , respectively. Let $F=\{x\in C : Tx=x\}$ benonempty, and let
be a sequence in $[0,1]$ such that
. Then for any
in the sequence
defined by
$C$

$T$

$C$

$T$

$\sum_{n=0}^{\infty}\alpha_{n}=\infty$$\{\alpha_{n}\}$$C$$x_{0}$$\{x_{n}\}$$x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}[(1-t)X_{n}+tTx_{n}]$for $n\geq 0,0<k=((1-t)2-2t(1-t)r+t^{2}s^{2})^{\frac{1}{2}}<1$ for all such that $0<t< \frac{2(1+r)}{(1+2r+S^{2})}$and $r\leq s$ , converges to a fixed point of .
In this paper, we first show how to construct (in a uniformly convex Banach space which
neither satisfies the Opial property nor has a Fr\’echet differentiable norm) a unique fixed
point of a non-Lipschitzian mapping $T:Carrow C$ which satisfies the property (K) type (seeDefinition 2.2 below) as the strong limt of a sequence
defined by a modified Ishikawa
iteration of the form
$t$

$T$

$\{x_{n}\}$$x_{n+1}=\alpha_{n}T^{n}[\beta_{nn}\tau nx+(1-\beta_{n})x_{n}]+(1-\alpha_{n})x_{n}$

,

where
and
in $[0,1]$ are chosen so thatand $0\leq\beta_{n}<b$
for some with $0<\text{\’{o}}<1$ . Next, we consider the sequence
defined by $(*)$ converges
strongly to a common fixed point of and under another conditions, that is, in cases whenandare chosen so that
and
or
and
for somewith $0<a\leq b<1$ . Finally, we consider the sequence
defined by $(*)$
converges strongly to a common fixed point of and under another parameter conditions,that is, in cases whenis a sequence in $[0,1]$ such thatandfor all $n\geq 1$ .
$\{\alpha_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$$\{\beta_{n}\}$$b$$\{x_{n}\}$$T$$\{\alpha_{n}\}$$\alpha_{n}\in[a, b]$$\{\beta_{n}\}$

$a,$

$S$

$\beta_{n}\in[0, b]$$\alpha_{n}\in[a, 1]$$b$$\beta_{n}\in[a, b]$$\{x_{n}\}$

$T$

$S$

$\alpha_{n}arrow 0,$$\{\alpha_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$0\leq\beta_{n}\leq 1$

2. PRELIMINARIES AND SOME EXAMPLES
Let $H$ be a real Hilbert space. We denote by
andthe inner product and
$H$
$y\in
H$
$HarrowH$, respectively. An operator
the norm on
for:is said to be relaxed
Lipschitz [18] if, for all $y\in H$ , there exists a constant $r>0$ such that$||x||$$\langle x, y\rangle$

$T$

$x,$

$x,$

$\langle$Tx–Ty, $x-y\rangle$ $\leq-r||x-y||^{2}$ .

Throughout this paper, let $E$ be a Banach space. Recall that $E$ is said to be uniformly$0<\epsilon\leq 2$ , of $E$ defined byconvex if the modulus of convexity$\delta_{E}=\delta_{E}(\epsilon),$$\delta_{E}(\epsilon)=\inf\{1-\frac{||_{X+}y||}{2} :x, y\in E, ||x||\leq 1, ||y||\leq 1, ||x-y||\geq\epsilon\}$

101
STRONG CONVERGENCE TO FIXED POINTS

satisfies the inequality$\delta_{E}(\epsilon)>0$

for every$J(x)=\{x^{*}\in E^{*} :$\epsilon\in(0,2]$. With each $x\in E$ , we associate the set,\langle x, x^{*}\rangle=||x||^{2}=||x^{*}||^{2}\}$at . Then is said to be the duality mapping of $E$ .
denotes the value of
where
Let be a nonempty closed convex subset of $E$ and let be a mapping from into itself.
Then we denote by $F(T)$ the set of all fixed points of , i.e., $F(T)=\{x\in C : Tx=x\}$ .
is a sequence in $E$ , then $x_{n}arrow x(x_{n}arrow x)$ will denote strong (weak) convergence
When
to . We denote by the set of all real numbers.
of the sequence, the mapping $T:Carrow E$
Let be a nonempty closed convex subset of $E$ . If
is said to be strictly hemicontractive [2] if there exists $t>1$ such that for all $x\in C$ and$y\in F(T)$ there exists $j\in J(x-y)-$ such that$x^{*}$$\langle x, x^{*}\rangle$

$J$

$x$

$C$

$T$

$C$

$T$

$\{x_{n}\}$$\{x_{n}\}$$\mathbb{R}$$x$$F(T)\neq\emptyset$$C$${\rm Re} \langle T_{X}-y, j\rangle\leq\frac{1}{t}||x-y||^{2}$.

be a nonempty subset of $E$ . Let be a mappings of into
Definition 2.1 [8]. Let. Then is said to be of (H) type if there exists $t>1$ such that foritself with
$x\inC$
and $y\in F(T)$ , there exists $j\in J(x-y)$ such that
each
$C$

$T$

$C$

$T$

$F(T)\neq\emptyset$$\lim\sup{\rm Re}\langle T^{n}x-y, j\rangle\leq\frac{1}{t}||x-y||^{2}$.

$narrow\infty$

Here we need the following stronger concept than (H) type for constructing an approximating fixed point of a non-Lipschitzian self-mapping in a Banach space.
Definition 2.2. Let $C$ be a nonempty subset of $E$ . Let be a mappings of into itself
with $F(T)\neq\emptyset$ . Then is said to be of (K) type if, for each $x\in C$ and $y\in F(T)$ , there
exists $j\in J(x-y)$ such that
$C$

$T$

$T$

$\lim\sup{\rm Re}\langle T^{n}x-y, j\rangle\leq 0$.

$narrow\infty$

It is obvious that if $T:Carrow C$ is mapping with $F(T)=\{y\}$ and $T^{n}xarrow y$ as $narrow\infty$
for each $x\in C$ , then is of (K) type. Every relaxed Lipschitz mappings are obviously of(K) type.
$T$

Example 2.1 [2]. Take $E=C=\mathbb{R}$ with the usual normby$|\cdot|$. Let $T:Carrow C$ be defined$Tx= \frac{2}{3}x\cos x$

for all $x\in C$ . Clearly $F(T)=\{0\}$ and, since $T^{n}xarrow 0$ for each $x\in C,$
Example 2.2. Take $E=C=\mathbb{R}$ with the usual norm
be defined by$|\cdot|$$T$

is of (K) type.

and let $0<k<1$ . Let $T:Carrow C$$Tx=k_{X}$for all $x\in C$ . Clearly $F(T)=\{0\}$ . Since $T^{n}xarrow 0$ for each $x\in C,$
Example 2.3. Take
defined by$E=\mathbb{R}$with the usual

nor.m$|\cdot|$

and let$T$is also of (K) type.

$C=(\mathrm{O}, 2]$. Let $T:Carrow C$ be$Tx=\sqrt{x}$. Clearly $F(T)=\{1\}$ and, since $T^{n}xarrow 1$ asfor each $x\in C,$
asymptotically nonexpansive which is of (K) type but not Lipschitz mapping.

$\forall x\in C$$narrow\infty$$T$

is weakly

102$\mathrm{G}.\mathrm{E}$. KIM

3. STRONG CONVERGENCE THEOREMS
We first begin with the following:
Lemma 3.1 [1]. Suppose
is a $bo$ unded seq uence of real numbers andis a$dou$ bly-in dexed seq uence of real numbers
which sa tisfy $\lim\sup_{narrow\infty}\lim\sup_{m}arrow\infty^{a_{n}},m\leq 0$,$v_{n+m}\leq v_{n}+a_{n,m}$ for each $n,$ $m\geq 1$ . Thenconverges to an
can be aken
to be independent of $n,$ $a_{n,m}=a_{m}$ , thenfor each .
$\{v_{n}\}$$\{a_{n,m}\}$$\{v_{n}\}$$v\in \mathbb{R};a_{n,m}$$v\leq v_{n}$Lemma 3.2 [6]. For any$x,$ $y\in

E$

$t$

$n$

and $j\in J(x+y)$ , we obtain$||x+y||2\leq||x||^{2}+2R\mathrm{e}\langle y, j\rangle$.

From the proof of Lemma 3 of [16], we note

Lemma 3.3. Let$\lim\inf_{narrow\infty}b_{n}=0$$a_{n},$$b_{n}>0$.

for $n\geq 1$ . If$\sum_{n=1}^{\infty}a_{n}=\infty$

and$\sum_{n=1}^{\infty}a_{n}b_{n}<\infty$, thenUsing Lemma 3.1-3.3, we obtain the following Theorem 3.1.
Theorem 3.1 [9]. Let $E$ be a uniformly convex anspace and let $C$ be a onempty
bounded clos $ed$ convex bset of E. Suppose that $T:Carrow C$ is both weakly asymptoticallynonexpan sive and of $(K)$ type. Put$B$$\mathrm{a}ch$$\mathrm{n}$$s\mathrm{u}$$c_{n}= \sup_{cx,y\in}(||\tau^{n}x-Tny||-||x-y||)\vee \mathrm{o}$so that$\sum_{n=1}^{\infty}c_{n}<\infty$. Then for any$x_{1}$

in$C$$x_{n+1}=\alpha_{n}T^{n}y_{n}+(1-\alpha_{n})x_{n}$which$\{\alpha_{n}\}$, the sequence,,

deti $ned$ by$\{x_{n}\}$$y_{n}=\beta_{n}T^{n_{X_{n}}}+(1-\beta n)x_{n}$andare chosen so that $\alpha_{n}\in[0,1]$ and$n\geq
1$ , converge strongly to the uniquefor all
fixed point of .
$\{\beta_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$$0\leq\beta_{n}<b<1$

Remark. If
for some

and$T$$\{\alpha_{n}\}$$\alpha_{n}\leq b$

,

$a,$

is a sequence in $[0,1]$ which is bounded away fromwith $0<a\leq b<1$ , then
.

$b$

$0$

and 1, i.e., $a\leq$$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$

As a direct consequence of Theorem 3.1 with$\beta_{n}=0$, we have the following result.

Corollary 3.1. Let $E$ be a uniformly conv’ex Banach space and $C$ be a nonempty boundedclosed convex $su$ bset of E. Let : $Carrow C$ be both weakly asymptotically onexpansive
and of $(K)$ type. Put$T$$\mathrm{n}$$c_{n}= \sup_{Cx,y\in}(||T^{n}x-Tny||-||x-y||)\vee \mathrm{o}$so that$\sum_{n=1}^{\infty}c_{n}<\infty$. Then for any$x_{1}$

in$C$, the seq ence$u$$x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}T^{n}Xn$

which
is chosen so th attrongly to the unique tixed point of$\{\alpha_{n}\}$$\mathrm{s}$$\alpha_{n}\in[0,1]$
$T$

.

and

$\{x_{n}\}$,deti$n\mathrm{e}d$by’$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$

for all $n\geq 1$ , converge

103
STRONG CONVERGENCE TO FIXED POINTS

Lemma 3.4 [13]. Let $E$ be a uniformly convex Banach spa ce, $0<b\leq t_{n}\leq c<1$
andare sequences of $E$ such th atfor all $n\geq 1,$ $a\geq 0$ . Suppose th at$\lim\sup_{narrow\infty}||x_{n}||\leq a,$ $\lim\sup_{narrow\infty}||y_{n}||\leq a,$ $md \lim_{narrow\infty}||t_{n}x_{n}+(1-t_{n})y_{n}||=a$ .
Then$\{x_{n}\}_{n=1}^{\infty}$$\lim_{narrow\infty}||X_{n}-yn||=0$$\{y_{n}\}_{n=1}^{\infty}$.

By using Lemma 3.4, we obtain the following Theorem 3.2.

space and $C$ be a nonempty bounded
Theorem 3.2. Let $E$ be a uniformly convex an$T,$$S:CarrowC$
be both weakly asymptotically nonexpansive
clos convex subset of E. Let
and of $(K)$ type with $F(T)\cap F(S)\neq\emptyset$ . Put$B$$\mathrm{a}ch$$\mathrm{e}d$$c_{n}= \max(0,\sup_{x,y\in C}(||TnX-Tn|y|-||x-y||),\mathrm{s}\mathrm{u}\mathrm{p}x,y\in C(||sn_{X}-Sny||-||X-y||))$,. Then for any in , the sequence
defined by $(*)$ , which
so thatandandfor someorare chosen so th atand$0<a\leqb<1$, converge strongly to a common fiixed point of and .
with$\sum_{n=1}^{\infty}c_{n}<\infty$$x_{1}$$\alpha_{n}\in[a, b]$$\beta_{n}$

$a,$

$C$

$\{x_{n}\}$$\{\alpha_{n}\}$$\alpha_{n}\in[a, 1]$$\beta_{n}\in[0, b]$$\beta_{n}\in[a, b]$

$T$

$b$

$S$

The following lemma is very useful to prove the convergence of a sequence to . Comparewith Lemma 1 due to Dunn [4].
$0$Lemma 3.5 [19]. Let$\beta_{n}$be a onnegative sequence sat isfying$\mathrm{n}$$\beta_{n+1}\leq(1-\delta_{n})\beta_{n}+\sigma n$with$\delta_{n}\in[0,1],$$\sum_{i=1}^{\infty}\delta_{i}=\infty$, and$\sigma_{n}=o(\delta_{n})$. Then$\lim_{narrow\infty}\beta_{n}=0$

.

space
Theorem 3.3. Let $C$ be a nonempty bounded closed convex subset of a Ban$(K)$$CarrowC$$T,$nonexpweaklytype
ansiveandofwithasymptotically
be
both
:
E. Let$F(T)\cap F(S)\neq\emptyset$ . Put$\mathrm{a}ch$$S$$c_{n}= \max(0,\sup_{Cx,,y\in}(||TnX-Tn|y|-||x-y||),$ $x,y \sup_{\in C}(||snx-^{s}ny||-||x-y||))$ ,. Then for any in , the sequence
so thatanda seq uence in $[0,1]$ such thatconverge strongly to a common fixed point of and .
$\sum_{n=1}^{\infty}c_{n}<\infty$$x_{1}$$is$$C$$\alpha_{n}arrow 0$

defined by $(*)$ , which
and $0\leq\beta_{n}\leq 1$ for all $n\geq 1$ ,$\{x_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$T$

As a direct consequence of Theorem 3.3 with$\{\alpha_{n}\}$$S$$\beta_{n}=0$, we have the following result.

Corollary 3.2. Let $C$ be a nonempty boun $ded$ closed convex bset of a Banach space
E. Let $T,$ : $Carrow C$ be both weakly asymptotically nonexp ansive and of $(K)$ type with$s\mathrm{u}$$S$$F(T)\cap F(S)\neq\emptyset$. Put$c_{n}= \max(0,\sup_{x,y\in C}(||\tau^{n}X-^{\tau^{n}||-}y||x-y||),\sup_{x,y\in c}(||SnX-^{s^{n}||-}y||x-y||))$

so that$\sum_{n=1}^{\infty}c_{n}<\infty$. Then for any$x_{1}$

in$C$, the sequence$x_{n+1}=(1-\alpha_{n})x_{n}+\alpha n\tau^{n}Xn$

is a seq uence in $[0,1]$ such thatwhichconverge strongly to a common fixed point of$\{\alpha_{n}\}$and$\alpha_{n}arrow 0$

$T$

and

$S$

.

$\{x_{n}\}$deti$\mathrm{n}ed$

,

by

’

$\sum_{n=1}^{\infty}\alpha_{n}=\infty$

for all $n\geq 1$ ,104$\mathrm{G}.\mathrm{E}$. KIM

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$o_{P^{ial}}$

PUKYONG NATIONAL UNIVERSITY, DEPARTMENT OF$\mathrm{A}_{\mathrm{P}\mathrm{P}\mathrm{L}}1\mathrm{E}\mathrm{D}$RIATHEMATICS, PUSAN 608-737, KOREA