D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 7 4 IS S N 1 3 0 3 –5 9 9 1
STRONG AND WEAK CONVERGENCE OF AN ITERATIVE PROCESS FOR A FINITE FAMILY OF MULTIVALUED
MAPPINGS SATISFYING THE CONDITION (C)
ISA YILDIRIM
Abstract. The aim of this paper is to introduce an iterative process with errors for a …nite family of multivalued mappings satisfying the condition (C) which is weaker than nonexpansiveness. We also prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.
1. Introduction
Fixed point theory is one of the most important tool of modern mathematics. This deals with the conditions which guarantee that a singlevalued mapping T of a set X into itself admits one or more …xed points, that is, points x of X which solve an operator equation x = T x, called a …xed point equation. Fixed point theory serves as an essential tool for solving problems arising in various branches of mathematical analysis. These problems can be modeled by the equation T x = x; where T is a nonlinear operator de…ned on a set equipped with some topological or order structure.
The study of …xed points for multivalued contractions and nonexpansive map-pings using the Hausdor¤ metric was initiated by Markin [9] and Nadler [10]. The-ory of multivalued mappings is harder than the corresponding theThe-ory of singleval-ued mappings. Theory of multivalsingleval-ued mappings has applications in control theory, convex optimization, di¤erential equations and economics.
Throughout this paper, the letter N will denote the set of natural numbers. We recall some de…nitions as follows:
Let X be a real Banach space. A subset E is called proximinal if for each x 2 X; there exists an element y 2 E such that
d (x; y) = inffkx zk : z 2 Eg = d (x; E) : Received by the editors: March 03, 2016, Accepted: July 28, 2016. 2010 Mathematics Subject Classi…cation. 47H10, 47H09.
Key words and phrases. Strong and Weak Convergence, Multivalued Mapping, Condition (C).
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
It is known that a weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal. We shall de-note by CB(E); K (E) and P (E) the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of E, respectively. Let H be a Hausdor¤ metric induced by the metric d of X, that is
H(A; B) = maxfsup
x2A
d(x; B); sup
y2Bd(y; A)g
for every A; B 2 CB(E). It is obvious that P (E) CB(E).
Let T : E ! CB(E) be a multivalued mapping. An element x 2 E is said to be a …xed point of T , if x 2 T x. The set of …xed points of T will be denote by F (T ). Moreover, we will write F = \r
i=1F (Ti) for the set of all common …xed points of
the mappings T1; T2; :::; Tr. The mapping T : E ! CB(E) is said to be
(i) nonexpansive if H(T x; T y) kx yk, for all x; y 2 E;
(ii) quasi-nonexpansive if H(T x; T p) kx pk, for all x 2 E and p 2 F (T ). In 2008, Suzuki [17] introduced a condition on mappings, called (C) which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness. A mul-tivalued mapping T : E ! CB(E) is said to satisfy condition (C) provided that
1
2d (x; T x) kx yk ) H(T x; T y) kx yk for all x; y 2 E.
From the above de…nitions, it follows that a nonexpansive mapping must be quasi-nonexpansive mapping. However, the converse of this statement is not true, in general. If T : E ! CB(E) is a multivalued nonexpansive mapping, then T satis…es the condition (C) ([1]). Moreover, if T : E ! CB(E) is a multivalued mapping which satis…es the condition (C) and has a …xed point, then T is a quasi-nonexpansive mapping ([5]).
Di¤erent iterative processes have been used to approximate …xed points of multi-valued nonexpansive mappings. Among these iterative processes, Sastry and Babu [13] considered the following.
Let E be a nonempty convex subset of a Banach space X; T : E ! P (E) a multi-valued mapping with p 2 T p.
(i) The sequences of Mann iterates is de…ned by x12 K;
xn+1= (1 an)xn+ anyn; (1.1)
where for all n 2 N and yn2 T xn;
(ii) The sequence of Ishikawa iterates is de…ned by x12 E,
(
xn+1= (1 n)xn+ nun;
yn= (1 n)xn+ nzn;
(1.2) where for all n 2 N; un2 T yn and zn2 T xn.
They proved that the Mann and Ishikawa iteration processes for multivalued mapping T with a …xed point p converge to a …xed point q of T under certain con-ditions. They also claimed that the …xed point q may be di¤erent from p. Panyanak [12] extended result of Sastry and Babu [13] to uniformly convex Banach spaces. After, Song and Wang [15] noted that there was a gap in the proof of the main result in [12]. They further revised the gap and also gave the a¢ rmative answer to Panyanak’s open question. Shazad and Zegeye [16] extended and improved results already appeared in the papers [12, 13, 15].
Khan and Yildirim [8] further generalized the results of Song and Cho [14] and Shahzad and Zegeye [16] partly by incorporating and unifying their techniques. For results on a three step iteration process, see for example, Khan et al. [7].
Recently, Yildirim and Ozdemir ([19], [20]) proved some strong and weak con-vergence result for nonexpansive and quasi-nonexpansive mappings by using the following multistep iteration process: For an arbitrary …xed order r 2;
8 > > > > > < > > > > > : xn+1= (1 1n) yn+r 2+ 1nT1yn+r 2; yn+r 2= (1 2n) yn+r 3+ 2nT2yn+r 3; .. . yn+1= 1 (r 1)n yn+ (r 1)nTr 1yn; yn= (1 rn) xn+ rnTrxn; (1.3) or, in short, 8 < : xn+1= (1 1n) yn+r 2+ 1nT1yn+r 2; yn+r i= (1 in) yn+r (i+1)+ inTiyn+r (i+1); yn= (1 rn) xn+ rnTrxn; (1.4) where for all n 2 N; f 1ng and f ing ; i = 2; :::r; are real sequences in [0; 1) :
If T1 = T2 = ::: = Tr = T and in = 0 for i = 2; :::r and all n 2 N, then (1.3)
reduces to (1.1).
In 2011, Eslamian and Homaeipour [6] introduced a new three-step iterative process for multivalued mappings in Banach spaces. They also proved some con-vergence theorems for multivalued mappings satisfying condition (C) in uniformly convex Banach spaces. Their iteration process with errors as follows:
Let E be a nonempty convex subset of a Banach space X and T1; T2; T3: E !
CB(E) be three multivalued mappings. Then for x12 E;
8 < : xn+1= (1 n n)xn+ nun+ ns 00 n; yn= (1 cn dn)xn+ cnvn+ dns 0 n; wn= (1 an bn)xn+ anzn+ bnsn; (1.5) where for all n 2 N; un2 T1yn; vn2 T2wn and zn2 T3xn, fang ; fbng ; fcng ; fdng ;
f ng ; f ng [0; 1] and fsng ;
n
s0noandns00noare bounded sequences in X. Inspired by the above works, we introduce the following iterative process for a …nite familiy of multivalued mappings.
Let E be a nonempty convex subset of a Banach space X and Ti: E ! CB(E)
(i = 1; 2; :::; r) be a …nite family of multivalued mappings. For an arbitrary …xed order r 2; 8 > > > > > > < > > > > > > : xn+1= (1 1n 1n) yn+r 2+ 1nzn;1+ 1nu1n; yn+r 2= (1 2n 2n) yn+r 3+ 2nzn;2+ 2nu2n; .. . yn+1= 1 (r 1)n (r 1)n yn+ (r 1)nzn;r 1+ (r 1)nu(r 1)n; yn= (1 rn rn) xn+ rnzn;r+ rnurn; (1.6)
where for all n 2 N; zn;r 2 Tr(xn) and zn;i 2 Ti yn+r (i+1) for i = 1; 2; 3; :::; r
and f ing ; f ing [0; 1] and fuing are bounded sequences in X:
Finding common …xed points of a …nite family of mappings is an important problem. Altough many algorithms have been introduced for various classes of mappings, the existence of common …xed points of a family of mappings are not known in many situations. So, it is natural to consider approximation results for such mappings.
The purpose of this paper is to study convergence of the sequence in (1.6) to a common …xed point of a …nite family of multivalued mappings in uniformly convex Banach spaces. Our work is a signi…cant generalization of the corresponding results in the literature.
2. Preliminaries
Let X be a real normed linear space. The modulus of convexity of X is the function E: (0; 2] ! [0; 1] de…ned by
E(") = inf 1
x + y
2 : kxk = kyk = 1; kx yk = " ; X is called uniformly convex if and only if X(") > 0 for all " 2 (0; 2].
A mapping T : E ! CB(E) is said to be semicompact if, for any sequence fxng
in E such that d (xn; T xn) ! 0 as n ! 1, there exists a subsequence fxnkg of
fxng such that xnk! p 2 E. We note that if E is compact, then every multivalued
mappings T : E ! CB(E) is semicompact.
A mapping T : E ! CB(E) is said to satisfy condition (I) if there is a nonde-creasing function g : [0; 1) ! [0; 1) with g(0) = 0; g(t) > 0 for all t 2 (0; 1) such that
d (x; T x) g (d (x; F (T ))) :
Let Ti: E ! CB(E) (i = 1; 2; :::; r) be a …nite family of mappings. The mappings
Ti for all i (i = 1; 2; :::; r) are said to satisfy condition (II) if there exist a
that r X i=1 d (x; Tix) g (d (x; F)) ; where F = \ri=1F (Ti) :
Throughout this paper, we will denote the weak convergence and the strong convergence by * and !, respectively.
A Banach space E is said to satisfy Opial’s condition [11] if for any sequence fxng in E; xn* x implies that
lim sup
n!1 kxn xk < lim supn!1 kxn yk
for all y 2 E with y 6= x:
Examples of Banach spaces satisfying this condition are Hilbert spaces and all lp spaces (1 < p < 1). On the other hand, Lp[0; 2 ] with 1 < p 6= 2 fail to satisfy
Opial’s condition.
The mapping T : E ! CB(E) is called demi-closed if for every sequence fxng E and any yn 2 T xn such that xn * x and yn ! y, we have x 2 E and
y 2 T x. If the space E satis…es Opial’s condition, then I T is demi-closed at 0, where T : E ! K(E) is a nonexpansive multivalued mapping ([4]).
We use the following lemmas to prove our main results.
Lemma 1. [2] Let E be a nonempty subset of a uniformly convex Banach space X and T : E ! CB(E) be a multivalued mapping with convex-valued and satisfying the condition (C) then
H(T x; T y) 2d (x; T x) + kx yk ; 8x; y 2 E:
Lemma 2. [2] (Demi-closed principle) Let X be a uniformly convex Banach space satisfying the Opial condition, E be a nonempty closed and convex subset of X. Let T : E ! CB(E) be a multi-valued mapping with convex-values and satisfying the condition (C) : Let fxng be a sequence in E such that xn * p 2 E, and let
limn!1d (xn; T xn) = 0, then p 2 T p, i.e., I T is demi-closed at zero.
Lemma 3. [1] Let T : E ! CB(E) be a multivalued nonexpansive mapping, then T satis…es the condition (C) :
Lemma 4. [18] Let fang ; fbng and f ng be sequence of nonnegative real numbers
satisfying the inequality
an+1 (1 + n) an+ bn:
If P1n=1 n < 1 andP1n=1bn< 1, then limn!1an exists. In particular, if fang
has a subsequence converging to 0, then limn!1an= 0.
Lemma 5. [3] Let X be a uniformly convex Banach space and Br= fx 2 X : kxk rg,
r > 0. Then there exists a continuous, strictly increasing and convex function ' : [0; 1) ! [0; 1) with '(0) = 0 such that
for all x; y; z 2 Br and all ; ; 2 [0; 1] with + + = 1.
3. Main Results We start with the following lemma.
Lemma 6. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be a …nite family of
multivalued mappings satisfying the condition (C). Assume that F = \r
i=1F (Ti) 6=
;, Ti(p) = fpg ; (i = 1; 2; :::; r) for each p 2 F and
P1
n=1 in < 1 for each i.
Let fxng be the sequence as de…ned in (1.6). Then limn!1kxn pk exist for any
p 2 F.
Proof. Suppose that p 2 F. Since the sequences fuing are bounded for i = 1; 2; :::; r,
there exists > 0 such that
max fsup ku1n pk ; sup ku2n pk ; :::; sup kurn pkg :
Using (1.6) and the condition (C), we have
kxn+1 pk (1 1n 1n) kyn+r 2 pk + 1nkzn;1 pk + 1nku1n pk (1 1n 1n) kyn+r 2 pk + 1nd (zn;1; T1(p)) + 1nku1n pk (1 1n 1n) kyn+r 2 pk + 1nH (T1(yn+r 2) ; T1(p)) + 1nku1n pk (1 1n 1n) kyn+r 2 pk + 1nkyn+r 2 pk + 1nku1n pk = (1 1n) kyn+r 2 pk + 1nku1n pk kyn+r 2 pk + 1n and kyn+r 2 pk (1 2n 2n) kyn+r 3 pk + 2nkzn;2 pk + 2nku2n pk (1 2n 2n) kyn+r 3 pk + 2nd (zn;2; T2(p)) + 2nku2n pk (1 2n 2n) kyn+r 3 pk + 2nH (T2(yn+r 3) ; T2(p)) + 2nku2n pk (1 2n 2n) kyn+r 3 pk + 2nkyn+r 3 pk + 2nku2n pk = (1 2n) kyn+r 3 pk + 2nku2n pk kyn+r 3 pk + 2n :
Similarly, we have kyn+1 pk 1 (r 1)n (r 1)n kyn pk + (r 1)nkzn;r 1 pk + (r 1)n u(r 1)n p 1 (r 1)n (r 1)n kyn pk + (r 1)nd (zn;r 1; Tr 1(p)) + (r 1)n u(r 1)n p 1 (r 1)n (r 1)n kyn pk + (r 1)nH (Tr 1(yn) ; Tr 1(p)) + (r 1)n u(r 1)n p 1 (r 1)n (r 1)n kyn pk + (r 1)nkyn pk + (r 1)n u(r 1)n p = 1 (r 1)n kyn pk + (r 1)n u(r 1)n p kyn pk + (r 1)n ; and kyn pk (1 rn rn) kxn pk + rnkzn;r pk + rnkurn pk (1 rn rn) kxn pk + rnd (zn;r; Tr(p)) + rnkurn pk (1 rn rn) kxn pk + rnH (Tr(xn) ; Tr(p)) + rnkurn pk (1 rn rn) kxn pk + rnkxn pk + rnkurn pk kxn pk + rn : Therefore kxn+1 pk kxn pk + (1 rn) 1 (r 1)n ::: (1 2n) 1n + (1 rn) 1 (r 1)n ::: (1 3n) 2n + ::: + (1 rn) (r 1)n + rn kxn pk + 1n + 2n + ::: + (r 1)n + rn = kxn pk + ( 1n+ 2n+ ::: + (r 1)n+ rn) kxn pk + 1n+ 2n+ ::: + (r 1)n+ rn = kxn pk + n (3.1)
where n= 1n+ 2n+ ::: + (r 1)n+ rn . Using the fact thatP1n=1 n< 1 and Lemma 4, we conclude that limn!1kxn pk exist for any p 2 F.
Theorem 1. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be a …nite family of
multivalued mappings with nonempty convex-values and satisfying the condition (C). Assume that F = \r
i=1F (Ti) 6= ;, Ti(p) = fpg ; (i = 1; 2; :::; r) for each p 2 F
and Ti (i = 1; 2; :::; r) satisfying the condition (II). Let in+ in 2 [a; b] (0; 1)
for i = 1; 2; :::; r andP1n=1 in< 1 for each i. Then the sequence fxng de…ned in
(1.6) converges strongly to a common …xed point of Ti for i = 1; 2; :::; r.
Proof. We will do our proof in two steps.
Step 1. Assume that p 2 F. By Lemma 6, limn!1kxn pk exists. Since fxng
is bounded, there exists r > 0 such that xn p; yn+r m p 2 Br(0) all for some
positive integer m; 2 m r and n 2 N. As Step 1, there exists > 0 such that maxnsup ku1n pk2; sup ku2n pk2; :::; sup kurn pk2
o : It follows from Lemma 5 that
kxn+1 pk2 1 (r 1)n (r 1)n kyn+r 2 pk2+ 1nkzn;1 pk2 (3.2) + 1nku1n pk2 1n(1 1n 1n) ' (kyn+r 2 zn;1k) (1 1n 1n) kyn+r 2 pk2+ 1nd (zn;1; T1(p))2 + 1nku1n pk2 1n(1 1n 1n) ' (kyn+r 2 zn;1k) (1 1n 1n) kyn+r 2 pk2+ 1nH (T1(yn+r 2) ; T1(p))2 + 1nku1n pk2 1n(1 1n 1n) ' (kyn+r 2 zn;1k) (1 1n) kyn+r 2 pk2+ 1nku1n pk2 1n(1 1n 1n) ' (kyn+r 2 zn;1k) kyn+r 2 pk2+ 1n 1n(1 1n 1n) ' (kyn+r 2 zn;1k) and kyn+r 2 pk2 (1 2n 2n) kyn+r 3 pk2+ 2nkzn;2 pk2 (3.3) + 2nku2n pk2 2n(1 2n 2n) ' (kyn+r 3 zn;2k) (1 2n 2n) kyn+r 3 pk2+ 2nd (zn;2; T2(p))2 + 2nku2n pk2 2n(1 2n 2n) ' (kyn+r 3 zn;2k) (1 2n 2n) kyn+r 3 pk2+ 2nH (T2(yn+r 3) ; T2(p))2 + 2nku2n pk2 2n(1 2n 2n) ' (kyn+r 3 zn;2k) (1 2n 2n) kyn+r 3 pk2+ 2nkyn+r 3 pk2 + 2nku2n pk2 2n(1 2n 2n) ' (kyn+r 3 zn;2k) kyn+r 3 pk2+ 2n 2n(1 2n 2n) ' (kyn+r 3 zn;2k) :
Again, we apply Lemma 5 to conclude that kyn+1 pk2 1 (r 1)n (r 1)n kyn pk2+ (r 1)nkzn;r 1 pk2 (3.4) + (r 1)n u(r 1)n p 2 (r 1)n 1 (r 1)n (r 1)n ' (kyn zn;2k) 1 (r 1)n (r 1)n kyn pk2+ (r 1)nd (zn;r 1; T2(p))2 + (r 1)n u(r 1)n p 2 (r 1)n 1 (r 1)n (r 1)n ' (kyn zn;2k) 1 (r 1)n (r 1)n kyn pk2+ (r 1)nH (Tr 1(yn) ; Tr 1(p))2 + (r 1)n u(r 1)n p 2 (r 1)n 1 (r 1)n (r 1)n ' (kyn zn;2k) 1 (r 1)n (r 1)n kyn pk2+ (r 1)nkyn pk2 + (r 1)n u(r 1)n p 2 (r 1)n 1 (r 1)n (r 1)n ' (kyn zn;2k) 1 (r 1)n kyn pk2+ (r 1)n (r 1)n 1 (r 1)n (r 1)n ' (kyn zn;2k) and kyn pk2 (1 rn rn) kxn pk2+ rnkzn;r pk2+ rnkurn pk2(3.5) rn(1 rn rn) ' (kxn zn;rk) (1 rn rn) kxn pk2+ rnd (zn;r; Tr(p))2+ rnkurn pk2 rn(1 rn rn) ' (kxn zn;rk) (1 rn rn) kxn pk2+ rnH (Tr(xn) ; Tr(p))2 + rnkurn pk2 rn(1 rn rn) ' (kxn zn;rk) (1 rn rn) kxn pk2+ rnkxn pk2+ rn rn(1 rn rn) ' (kxn zn;rk) kxn pk2+ rn rn(1 rn rn) ' (kxn zn;rk) :
By using (3.2), (3.3), (3.4) and (3.5), we obtain
kxn+1 pk2 kxn pk2+ 1n + 2n + ::: + (r 1)n+ rn r Y i=1 in " r X i=1 (1 in in) ' yn+r (i+1) zn;i # :
From the condition in+ in2 [a; b] (0; 1) for i = 1; 2; :::; r, we obtain ar r X i=1 (1 b) ' yn+r (i+1) zn;i r Y i=1 in " r X i=1 (1 in in) ' yn+r (i+1) zn;i # kxn pk2 kxn+1 pk2+ 1n+ 2n+ ::: + (r 1)n+ rn :
This implies that
1 X n=1 " ar r X i=1 (1 b) ' yn+r (i+1) zn;i # kx1 pk2+ 1 X n=1 1n+ 2n+ ::: + (r 1)n+ rn < 1
from which it follows that limn!1' yn+r (i+1) zn;i = 0. Since ' is
contin-uous at 0 and is strictly increasing, we have lim
n!1 yn+r (i+1) zn;i = 0: (3.6)
Hence for i = 1; 2; :::; r, we have lim
n!1kyn+r 2 zn;1k = limn!1kyn+r 3 zn;2k = ::: = limn!1kxn zn;rk = 0: (3.7)
Also, using (1.6), (3:7) andP1n=1 in< 1 for each i, we have
lim n!1kyn xnk = limn!1( rnkzn;r xnk + rnkurn xnk) = 0; (3.8) lim n!1kyn+1 xnk = nlim!1 1 (r 1)n (r 1)n kyn xnk + (r 1)nkzn;r 1 xnk + (r 1)n u(r 1)n xn = 0 .. . lim n!1kxn+1 xnk = nlim!1( rnkzn;r xnk + rnkurn xnk) = 0: From (1.6), we obtain kxn+1 zn;1k = (1 1n 1n) kyn+r 2 zn;1k + 1nku1n zn;1k :
It follows from (3:7) andP1n=1 in< 1 for each i that lim
n!1kxn+1 zn;1k = 0:
From the triangle inequality, we have
Taking the limit of both sides of this inequality and using (3:8), we have lim
n!1kxn zn;1k = 0:
Again, by the triangle inequality, we obtain for each i = 1; 2; :::; r
kxn zn;ik xn yn+r (i+1) + yn+r (i+1) zn;i :
Similarly, for i = 1; 2; :::; r
lim
n!1kxn zn;ik = 0: (3.10)
Hence, it follows from Lemma 1, (3.7), (3.8) and (3.10) that d (xn; T1(xn)) d (xn; T1(yn+r 2)) + H (T1(yn+r 2) ; T1(xn)) d (xn; T1(yn+r 2)) + 2d (yn+r 2; T1(yn+r 2)) + kyn+r 2 xnk kxn zn;1k + 2 kyn+r 2 zn;1k + kyn+r 2 xnk ! 0 as n ! 1; and d (xn; T2(xn)) d (xn; T2(yn+r 3)) + H (T2(yn+r 3) ; T2(xn)) d (xn; T2(yn+r 3)) + 2d (yn+r 3; T2(yn+r 3)) + kyn+r 3 xnk kxn zn;2k + 2 kyn+r 3 zn;2k + kyn+r 3 xnk as n ! 1:
In a similar way, for each i = 1; 2; :::; r we obtain that lim
n!1d (xn; Ti(xn)) = 0:
Step 2. We now show that fxng converges strongly to q 2 F.
From Step 1, we know that limn!1d (xn; Ti(xn)) = 0. Since the condition (II),
limn!1d (xn; F) = 0. Therefore, we can choose a sequence fxnkg of fxng and a
sequence pk in F such that for all k 2 N
kxnk pkk <
1 2k:
From (3:1), we have the following inequality for all p 2 F, xnk+1 p xnk+1 1 p + nk+1 1 xnk+1 2 p + nk+1 2+ nk+1 1 .. . kxnk pk + nk+1Xnk 1 l=1 nk+l
which implies that xnk+1 p kxnk pkk + nk+1Xnk 1 l=1 nk+l < 1 2k + nk+1Xnk 1 l=1 nk+l:
Now, we will show that fpkg is a Cauchy sequence in E. Note that
kpk+1 pkk pk+1 xnk+1 + xnk+1 pk < 1 2k+1 + 1 2k + nk+1Xnk 1 l=1 nk+l < 1 2k 1 + nk+1Xnk 1 l=1 nk+l:
Thus fpkg is a Cauchy sequence in E. Since E is complete, this sequence
is convergent. Let limn!1pk = q. We need to show that q 2 F. Since for
i = 1; 2; :::; r
d (pk; Ti(q)) H (Ti(pk) ; Ti(q)) kpk qk
and pk! q as k ! 1, it follows that d (q; Ti(q)) = 0 for i = 1; 2; :::; r. Hence q 2 F
and fxnkg converges strongly to q. Since limn!1kxn qk exists, we conclude that
fxng converges strongly to q.
Since the condition (II) is weaker than the compactness of K and the semi-compactness of the multivalued mappings fTi: i = 1; 2; :::; rg, therefore we already
have the following theorem.
Theorem 2. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be a …nite family of
multivalued mappings with nonempty convex-values and satisfying the condition (C). Assume that F = \r
i=1F (Ti) 6= ; and Ti(p) = fpg ; (i = 1; 2; :::; r) for each
p 2 F. Let fxng be de…ned in (1.6), and in+ in2 [a; b] (0; 1) for i = 1; 2; :::; r
and P1n=1 in < 1 for each i. Assume that either E is compact or one of the multivalued mappings fTi: i = 1; 2; :::; rg is semicompact. Then fxng converges
strongly to a common …xed point of Ti for i = 1; 2; :::; r.
Proof. As in the proof of Theorem 1, we have limn!1d (xn; Ti(xn)) = 0 for
each i. We assume that either E is compact or one of the multivalued mappings fTi: i = 1; 2; :::; rg is semicompact. Then there exists a subsequence fxnkg of fxng
such that limk!1xnk = z for some z 2 E. By Lemma 1, we have for i = 1; 2; :::; r
d (z; Ti(z)) kz xnkk + d (xnk; Ti(z))
kz xnkk + d (xnk; Ti(xnk)) + H (Ti(xnk) ; Ti(z))
3d (xnk; Ti(xnk)) + 2 kz xnkk ! 0 as k ! 1;
this implies that z 2 F. Since fxnkg converges strongly to z and the limit
limn!1kxn zk exists (as in the proof Theorem 1), it follows that fxng converges
strongly to z.
Theorem 3. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X with the Opial property. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be
a …nite family of multivalued mappings with nonempty convex-values and satisfying the condition (C). Assume that F = \ki=1F (Ti) 6= ; and Ti(p) = fpg ; (i = 1; 2; :::; r)
for each p 2 F. Let fxng be de…ned in (1.6), and in+ in 2 [a; b] (0; 1) for
i = 1; 2; :::; r and P1n=1 in < 1 for each i. Then fxng converges weakly to a
common …xed point of Ti for i = 1; 2; :::; r.
Proof. It follow from Lemma 6 and Theorem 1 that fxng is bounded and limn!1
d (xn; Ti(xn)) = 0 for each i. Since a uniformly convex Banach space is re‡exive,
there exists a subsequence fxnkg fxng such that xnk * q as nk ! 1 for some
q 2 X. We will show that q 2 F. By Lemma 2, I Ti is demi-closed at zero for
each i. Hence from limn!1d (xn; Ti(xn)) = 0, q 2 F (Ti). By the arbitrariness of
i 1, we have q 2 F.
If there exists another subsequence fxnlg fxng such that xnl * q 2 E and
q 6= q . As in the proof above, we can also prove that q 2 F. So by Lemma 6, limn!1kxn wk and limn!1kxn zk exist. Then by using Opial’s property,
lim n!1kxn wk = nklim!1kx nk wk < lim nk!1kx nk zk = lim n!1kxn zk = lim nl!1kxnl zk < limnl!1kxnl wk = lim n!1kxn wk
which is a contradiction. Therefore fxng converges weakly to a common …xed point
of Ti for i = 1; 2; :::; r.
From Lemma 3, we know that if T is a multivalued nonexpansive mapping, then T satis…es the condition (C). So we have the following results:
Corollary 1. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be a …nite family of
multivalued nonexpansive mappings. Assume that F = \r
i=1F (Ti) 6= ; and Ti(p) =
fpg ; (i = 1; 2; :::; r) for each p 2 F. Let fxng be de…ned in (1.6), and in+ in2
(i = 1; 2; :::; r) satisfying the condition (II). Then fxng converges strongly to a
common …xed point of Ti for i = 1; 2; :::; r.
Corollary 2. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X. Let Ti : E ! CB(E); (i = 1; 2; :::; r) be a …nite family of
multivalued nonexpansive mappings. Assume that F = \r
i=1F (Ti) 6= ; and Ti(p) =
fpg ; (i = 1; 2; :::; r) for each p 2 F. Let fxng be de…ned in (1.6), and in+ in2
[a; b] (0; 1) for i = 1; 2; :::; r andP1n=1 in< 1 for each i. Assume that either E is compact or one of the multivalued mappings fTi : i = 1; 2; :::; rg is semicompact.
Then fxng converges strongly to a common …xed point of Ti for i = 1; 2; :::; r.
Corollary 3. Let E be a nonempty, closed and convex subset of a uniformly convex Banach space X with the Opial property. Let Ti: E ! K(E); (i = 1; 2; :::; r) be a
…nite family of multivalued nonexpansive mappings. Assume that F = \r
i=1F (Ti) 6=
; and Ti(p) = fpg ; (i = 1; 2; :::; r) for each p 2 F. Let fxng be de…ned in (1.6),
and in+ in2 [a; b] (0; 1) for i = 1; 2; :::; r and
P1
n=1 in< 1 for each i. Then
fxng converges weakly to a common …xed point of Ti for i = 1; 2; :::; r.
Acknowledgement 1. The author would like to express their thanks to the Re-viewers and the Editors for their helpful suggestions and advices.
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Current address : Department of Mathematics, Ataturk University, Erzurum 25240, Turkey E-mail address : isayildirim@atauni.edu.tr
