D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 3 5 IS S N 1 3 0 3 –5 9 9 1
ON THE NEW MULTI-STEP ITERATION PROCESS FOR MULTI-VALUED MAPPINGS IN A COMPLETE GEODESIC
SPACE
AYNUR ¸SAHIN AND METIN BA¸SARIR
Abstract. The purpose of this paper is to prove the strong and 4-convergence theorems of the new multi-step iteration process for multi-valued quasi-nonexpansive mappings in a complete geodesic space. Our results extend and improve some results in the literature.
1. Introduction
For a real number ; a CAT( ) space is de…ned by a geodesic metric space whose geodesic triangle is su¢ ciently thinner than the corresponding comparison triangle in a model space with the curvature . The concept of this space has been studied by a large number of researchers (see [3, 5, 7, 8, 10, 14, 15]). Since any CAT( ) space is a CAT( 0) space for 0 (see [2, p.165]), all results for a CAT(0) space can immediately be applied to any CAT( ) space with 0: Moreover, a CAT( ) space with positive can be treated as a CAT(1) space by changing the scale of the space. So we are interested in a CAT(1) space.
Panyanak [11] studied the Ishikawa iteration process for multi-valued mappings in a CAT(1) space as follows.
Let K be a nonempty closed convex subset of a CAT(1) space X and f ng ; f ng
[0; 1] and T : K ! 2K be a multi-valued mapping whose values are nonempty
prox-iminal subsets of K. For each x 2 K, let PT : K ! 2K be a multi-valued mapping
de…ned by PT(x) = fu 2 T (x) : d(x; u) = d(x; T (x))g.
(A) For x12 K; the sequence of Ishikawa iteration is de…ned by
xn+1= nzn (1 n)xn; yn = nz 0 n (1 n)xn; n 1; (1.1) where zn2 T (yn) and z 0 n 2 T (xn).
Received by the editors: August 11, 2015; Accepted: Nov. 06, 2015. 2010 Mathematics Subject Classi…cation. 54E40, 54H25.
Key words and phrases. CAT( ) space, …xed point, multi-valued mapping, strong conver-gence, 4-convergence.
c 2 0 1 5 A n ka ra U n ive rsity
(B) For x1 2 K; the sequence of Ishikawa iteration is de…ned as in (1.1) where
zn2 PT(yn) and z 0
n2 PT(xn) :
Panyanak [11] proved, under some suitable assumptions, that the sequences de-…ned by (A) and (B) converge strongly to a …xed point of T in a CAT(1) space.
Gürsoy et al. [4] introduced a new multi-step iteration process for single-valued mappings in a Banach space as follows.
For an arbitrary …xed order k 2 and x12 K;
8 > > > > > > > < > > > > > > > : xn+1= (1 n)yn1+ nT (yn1); yn1= (1 1n)yn2+ 1nT (yn2); y2 n= (1 2n)yn3+ 2nT (yn3); .. . ynk 2= (1 kn 2)ynk 1+ kn 2T (ynk 1); yk 1 n = (1 k 1 n )xn+ k 1n T (xn); n 1; or, in short, 8 < : xn+1= (1 n)y1n+ nT (y1n); yi n= (1 in)yni+1+ inT (yni+1); i = 1; 2; :::; k 2; ykn 1= (1 kn 1)xn+ k 1n T (xn); n 1: (1.2) By taking k = 3 and k = 2 in (1.2), we obtain the SP-iteration process of Phuengrattana and Suantai [12] and the two-step iteration process of Thianwan [18], respectively. Recently, Ba¸sar¬r and ¸Sahin [1] studied the iteration process (1.2) for single-valued mappings in a CAT(0) space.
Now, we apply the new multi-step iteration process for multi-valued mappings in a CAT(1) space as follows.
(C) Let T be a multi-valued quasi-nonexpansive mapping from K into 2K. Then
for an arbitrary …xed order k 2 and x12 K; the sequence fxng is de…ned by
8 > > > > > > > < > > > > > > > : xn+1= (1 n)yn1 nzn1; y1 n = (1 1 n)yn2 1 nzn2; yn2 = (1 2n)yn3 2nzn3; .. . yk 2 n = (1 k 2 n )yk 1n k 2 n znk 1; yk 1 n = (1 k 1n )xn kn 1z 0 n; n 1; or, in short, 8 < : xn+1= (1 n)yn1 nzn1; yi n = (1 i n)yni+1 i nzi+1n ; i = 1; 2; :::; k 2; yk 1 n = (1 k 1n )xn kn 1z 0 n; n 1; (1.3) where zi
n2 T yni for each i = 1; 2; :::; k 1 and z 0
(D) Let T be a multi-valued quasi-nonexpansive mapping from K into P (K). Then for an arbitrary …xed order k 2 and x12 K; the sequence fxng is de…ned
as in (1.3) where zin2 PT yin for each i = 1; 2; :::; k 1 and z 0
n2 PT(xn).
In this paper, motivated by the above results, we prove some theorems related to the strong and 4-convergence of the new multi-step iteration processes de…ned by (C) and (D) for multi-valued quasi-nonexpansive mappings in a CAT(1) space.
2. Preliminaries and Lemmas
Let K be a nonempty subset of a metric space (X; d). The diameter of K is de…ned by diam(K) = sup fd(u; v) : u; v 2 Kg. The set K is called proximinal if for each x 2 X, there exists an element k 2 K such that d(x; k) = d(x; K); where d(x; K) = inf fd(x; y) : y 2 Kg. Let 2K; C(K) and P (K) denote the family
of nonempty all subsets, nonempty closed all subsets and nonempty proximinal all subsets of K, respectively. The Hausdor¤ distance on 2K is de…ned by
H(A; B) = max sup
x2A
d(x; B); sup
y2B
d(y; A) for all A; B 2 2K:
An element p 2 K is a …xed point of T if p 2 T (p). The set of all …xed points of T is denoted by F (T ).
De…nition 1. A multi-valued mapping T : K ! 2K is said to
(i) be nonexpansive if H(T (x); T (y)) d(x; y) for all x; y 2 K;
(ii) be quasi-nonexpansive if F (T ) 6= ; and H(T (x); T (p)) d(x; p) for all x 2 K and p 2 F (T );
(iii) satisfy Condition(I) if there exists a non-decreasing function f : [0; 1) ! [0; 1) with f(0) = 0 and f(r) > 0 for all r 2 (0; 1) such that
d(x; T (x)) f (d(x; F (T ))) for all x 2 K; (iv) be hemi-compact if for any sequence fxng in K such that
lim
n!1d(xn; T (xn)) = 0;
there exists a subsequence fxnkg of fxng such that limk!1xnk= p 2 K;
(v) be 4-demiclosed if for any 4-convergent sequence fxng in K; its 4-limit
belongs to F (T ) whenever limn!1d(xn; T (xn)) = 0:
It is clear that each multi-valued nonexpansive mapping with a …xed point is quasi-nonexpansive. But there exist multi-valued quasi-nonexpansive mappings that are not nonexpansive.
Example 1. [16, p.838-839] Let K = [0; +1) be endowed with the usual metric and T : K ! 2K be de…ned by T (x) = f0g if x 1; x 3 4; x 1 3 if x > 1:
Then, clearly F (T ) = f0g and for any x 2 K we have H(T (x); T (0)) jx 0j hence T is quasi-nonexpansive. However, if x = 2; y = 1 we get that H(T (x); T (y)) > jx yj = 1 and hence not nonexpansive.
Remark 1. From the proof of Lemma 5.1 in [9], it is easy to see that if T is a multi-valued nonexpansive mapping then T is 4-demiclosed.
The following lemma will be useful in this study.
Lemma 1. [11, Lemma 2.2] Let K be a nonempty subset of a metric space (X; d) and T : K ! P (K) be a multi-valued mapping. Then
(i) d(x; T (x)) = d(x; PT(x)) for all x 2 K;
(ii) x 2 F (T ) () x 2 F (PT) () PT(x) = fxg ;
(iii) F (T ) = F (PT):
Let X be a metric space with a metric d and x; y 2 X with d(x; y) = l. A geodesic path from x to y is an isometry c : [0; l] R ! X such that c(0) = x and c(l) = y. The image of c is called a geodesic segment joining x and y. A geodesic segment joining x and y is not necessarily unique in general. When it is unique, this geodesic segment is denoted by [x; y]. This means that z 2 [x; y] if and only if there exists 2 [0; 1] such that d(x; z) = (1 )d(x; y) and d(y; z) = d(x; y). In this case, we write z = x (1 )y: The space (X; d) is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be a uniquely geodesic if there is exactly one geodesic joining x to y for each x; y 2 X.
In a geodesic space (X; d), the metric d : X X ! R is convex if for any x; y; z 2 X and 2 [0; 1], one has
d(x; y (1 )z) d(x; y) + (1 )d(x; z):
Let D 2 (0; 1]: If for every x; y 2 X with d(x; y) < D, a geodesic from x to y exists, then X is said to be D-geodesic space. Moreover, if such a geodesic is unique for each pair of points then X is said to be a D-uniquely geodesic. Notice that X is a geodesic space if and only if it is a D-geodesic space.
To de…ne a CAT( ) space, we use the following concept called model space. For = 0; the two-dimensional model space M2= M2
0 is the Euclidean space R2 with
the metric induced from the Euclidean norm. For > 0; M2is the two-dimensional
sphere p1
S2 whose metric is a length of a minimal great arc joining each two points. For < 0; M2is the two-dimensional hyperbolic space p1
H2 with the
metric de…ned by a usual hyperbolic distance. The diameter of M2 is denoted by
D = p > 0;
+1 0:
A geodesic triangle 4(x; y; z) in a geodesic space (X; d) consists of three points x; y; z in X (the vertices of 4) and three geodesic segments between each pair of
vertices (the edges of 4). We write p 2 4(x; y; z) when p 2 [x; y] [ [y; z] [ [z; x]. A comparison triangle for 4(x; y; z) is a triangle 4(x; y; z) in M2 such that
d(x; y) = dM2(x; y) ; d(y; z) = dM2(y; z) and d(z; x) = dM2(z; x) :
If 0, then such a comparison triangle always exists in M2. If > 0, then such
a triangle exists whenever d(x; y) + d(y; z) + d(z; x) < 2D . A point p 2 [x; y] is called a comparison point for p 2 [x; y] if d(x; p) = dM2(x; p) :
A geodesic triangle 4(x; y; z) in X is said to satisfy the CAT( ) inequality if for any p; q 2 4(x; y; z) and for their comparison points p; q 2 4(x; y; z), one has
d(p; q) dM2(p; q):
We are ready to introduce the concept of CAT( ) space in the following de…nition taken from [2].
De…nition 2. If 0, then X is called a CAT( ) space if X is a geodesic space such that all of its geodesic triangles satisfy the CAT( ) inequality. If > 0, then X is called a CAT( ) space if it is D -geodesic and any geodesic triangle 4(x; y; z) in X with d(x; y) + d(y; z) + d(z; x) < 2D satis…es the CAT( ) inequality.
It follows from [2, p.160] that any CAT( ) space is D -uniquely geodesic. Let fxng be a bounded sequence in a metric space X. For x 2 X, we put
r(x; fxng) = lim supn!1d(x; xn). The asymptotic radius r(fxng) of fxng is de…ned
by
r (fxng) = inf
x2Xr (x; fxng) :
Further, the asymptotic center of fxng is de…ned by
A (fxng) = fx 2 X : r (x; fxng) = r (fxng)g :
Recall that the sequence fxng is 4-convergent to x 2 X if x is the unique
asymp-totic center of any subsequence of fxng.
Lemma 2. [3] Let (X; d) be a complete CAT(1) space and fxng be a sequence in X:
If r(fxng) < =2; then the following statements hold:
(i) A (fxng) consists of exactly one point,
(ii) fxng has a 4-convergent subsequence.
The following lemma is needed for our main result.
Lemma 3. [11, Lemma 2.4] If (X; d) is a CAT (1) space with diam(X) < =2, then there exist a constant K > 0 such that
d2((1 )x y; z) (1 )d2(x; z) + d2(y; z) K
2 (1 )d
2(x; y)
3. Main results We start with the following key lemmas.
Lemma 4. Let (X; d) be a CAT(1) space with convex metric, K be a nonempty, closed and convex subset of X and let T : K ! 2K be a multi-valued
quasi-nonexpansive mapping with F (T ) 6= ; and T (p) = fpg for each p 2 F (T ). Then, for the sequence fxng de…ned by (1.3), limn!1d(xn; p) exists for each p 2 F (T ).
Proof. For any p 2 F (T ), we have
d(xn+1; p) = d((1 n)yn1 nzn1; p) (1 n)d(yn1; p) + nd(zn1; p) (1 n)d(yn1; p) + nH T y1n ; T (p) (1 n)d(yn1; p) + nd(yn1; p) = d(yn1; p): Also, we obtain d(y1n; p) = d((1 1n)yn2 1nz2n; p) (1 1n)d(yn2; p) + 1nd(zn2; p) (1 1n)d(yn2; p) + 1nH T yn2 ; T (p) (1 1n)d(yn2; p) + 1nd(y2n; p) = d(yn2; p): Continuing the above process, we get
d(xn+1; p) d(yn1; p) d(yn2; p) ::: d(yk 1n ; p) d(xn; p): (3.1)
This inequality guarantees that the sequence fd(xn; p)g is non-increasing and
bound-ed below, and so limn!1d(xn; p) exists for any p 2 F (T ):
Lemma 5. Let (X; d) be a CAT(1) space with convex metric and diam(X) < =2, K be a nonempty, closed and convex subset of X and let T : K ! 2K be a
multi-valued quasi-nonexpansive mapping with F (T ) 6= ; and T (p) = fpg for each p 2 F (T ). Let fxng be the sequence de…ned by (1.3) with kn 12 [a; b] (0; 1). Then
limn!1d(xn; T (xn)) = 0.
Proof. It follows from Lemma 4 that limn!1d(xn; p) exists for each p 2 F (T ). We
(3.1), we get limn!1d(yk 1n ; p) = r: By Lemma 3, we also have d2(yk 1n ; p) = d2((1 k 1n )xn k 1n z 0 n; p) (1 k 1n )d2(xn; p) + k 1n d2(z 0 n; p) K 2 k 1 n (1 k 1n )d2(xn; z 0 n) (1 k 1n )d2(xn; p) + k 1n H2(T (xn) ; T (p)) K 2 k 1 n (1 kn 1)d2(xn; z 0 n) (1 k 1n )d2(xn; p) + k 1n d2(xn; p) K 2 k 1 n (1 k 1 n )d2(xn; z 0 n) = d2(xn; p) K 2 k 1 n (1 k 1n )d2(xn; z 0 n);
which implies that d2(xn; z 0 n) 2 a(1 b)K d 2(x n; p) d2(ynk 1; p) : Hence, limn!1d(xn; z 0 n) = 0. Since d(xn; T (xn)) d(xn; z 0 n); then we obtain lim n!1d(xn; T (xn)) = 0:
We give the 4-convergence of the iteration process de…ned by (C) in a CAT(1) space.
Theorem 1. Let X; K and fxng satisfy the hypotheses of Lemma 5, X be a
com-plete space and let T : K ! C(K) be a multi-valued quasi-nonexpansive mapping with F (T ) 6= ; and T (p) = fpg for each p 2 F (T ). If T is 4-demiclosed, then the sequence fxng is 4-convergent to a …xed point of T .
Proof. Let fung be a subsequence of fxng. Since r(fung) r(fxng) < =2,
by Lemma 2(i), there exists a unique asymptotic center u of fung : Moreover,
by Lemma 2(ii), there exists a subsequence fvng of fung such that fvng is
4-convergent to v for some v 2 X. Further, since limn!1d(vn; T (vn)) = 0 (by
Lemma 5) and T is 4-demiclosed, we have v 2 F (T ): By Lemma 4, limn!1d(xn; v)
exists. Then we can show that u = v: If not, from the uniqueness of the asymptotic center, we have
lim sup
n!1
d(un; u) < lim sup
n!1 d(un; v) = lim n!1d(xn; v) = lim sup n!1 d(vn; v) < lim sup n!1 d(vn; u) lim sup n!1 d(un; u): (3.2)
This is a contradiction. Hence we get u = v 2 F (T ): Next, we show that for any subsequence of fxng, its asymptotic center consists of the unique element. Let fung
be a subsequence of fxng with A (fung) = fug and let A (fxng) = fxg. We have
already seen that u = v: Finally, we show that x = v: If not, then the existence of limn!1d(xn; v) and the uniqueness of the asymptotic center imply that there
exists a contradiction as (3.2). Hence we get x = v 2 F (T ): Therefore, the sequence fxng is 4-convergent to a …xed point of T .
It is well known that every convex subset of a CAT(0) space, equipped with the induced metric, is a CAT(0) space (see [2]). If (X; d) is a CAT(0) space and K is a convex subset of X, then (K; d) is a CAT(0) space and hence it is a CAT( ) space with > 0. Now we give an example of such mappings which are multi-valued quasi-nonexpansive mappings as in Theorem 1.
Example 2. Let X be the real line with the usual metric and let K = [0; 1]. De…ne two mappings S; T : K ! C(K) by S(x) = h 0;x 4 i and T (x) = h 0;x 2 i :
Obviously, F (S) = F (T ) = f0g : It is proved in [6, Example 2] that both S and T are multi-valued nonexpansive mappings. Therefore, they are multi-valued quasi-nonexpansive mappings. Additionally, for 0 2 F (S) = F (T ), we have that S(0) = T (0) = f0g :
We prove the strong convergence of the iteration process de…ned by (C) in a CAT(1) space as follows.
Theorem 2. Let X; K; T and fxng be the same as in Theorem 1.
(i) If T satis…es Condition (I), then the sequence fxng is convergent strongly to
a point in F (T ).
(ii) If T is hemi-compact and continuous, then the sequence fxng is convergent
strongly to a point in F (T ).
Proof. (i) By Condition (I) and Lemma 5, we have lim
n!1 f (d (xn; F (T ))) nlim!1 d(xn; T (xn)) = 0:
That is, limn!1f (d(xn; F (T ))) = 0: Since f is a non-decreasing function satisfying
f (0) = 0 and f (r) > 0 for all r 2 (0; 1), it follows that limn!1d (xn; F (T )) = 0:
The proof of the remaining part follows the proof of Theorem 3.2 in [13], therefore we omit it.
(ii) From hemi-compactness of T and Lemma 5, there exists a subsequence fxnkg
of fxng such that limk!1xnk= q 2 K. Since T is continuous, we have
d(q; T (q)) = lim
This implies that q 2 F (T ) since T (q) is closed. Thus limn!1d(xn; q) exists by
Lemma 4. Hence the sequence fxng is convergent strongly to a …xed point q of
T .
Since every nonexpansive mapping having a …xed point is quasi-nonexpansive, then we get the following corollary.
Corollary 1. Let X; K and fxng satisfy the hypotheses of Theorem 1 and T : K !
C(K) be a multi-valued nonexpansive mapping with F (T ) 6= ; and T (p) = fpg for each p 2 F (T ). Then the sequence fxng is 4-convergent to a …xed point of T .
Moreover, if T satis…es Condition (I) or is hemi-compact, then the sequence fxng
is convergent strongly to a …xed point of T .
To avoid the restriction of T , that is, T (p) = fpg for each p 2 F (T ), we use the iteration process de…ned by (D). Using this iteration, we give the 4-convergence result in a CAT(1) space.
Theorem 3. Let (X; d) be a complete CAT(1) space with convex metric and diam(X) < =2, K be a nonempty, closed and convex subset of X and let T : K ! P (K) be a multi-valued mapping with F (T ) 6= ; and PT is quasi-nonexpansive. Let fxng be
the sequence de…ned by (D) with k 1n 2 [a; b] (0; 1). If T is 4-demiclosed, then the sequence fxng is 4-convergent to a …xed point of T .
Proof. It follows from Lemma 1 that d(x; PT(x)) = d(x; T (x)) for all x 2 K,
F (PT) = F (T ) and PT(p) = fpg for each p 2 F (PT). The 4-demiclosedness
of PT follows from the 4-demiclosedness of T . Then, applying Theorem 1 to the
mapping PT; we can conclude that the sequence fxng is 4-convergent to a point
p 2 F (PT) = F (T ).
We now present an example of a multi-valued mapping T for which PT is
quasi-nonexpansive.
Example 3. Let X and K be de…ned as in Example 2. De…ne a mapping T : K ! P (K) by T (x) = [0; x] if x 2 0; 1 2 ; 1 2 if x 2 1 2; 1 : Then, we have PT(x) = fxg if x 2 0; 1 2 ; 1 2 if x 2 1 2; 1 :
Clearly, F (T ) = F (PT) = x : 0 x 12 . It is proved in [17, Example 5] that T
is not nonexpansive and PT is quasi-nonexpansive.
We give several strong convergence results of the iteration process de…ned by (D) in a CAT(1) space.
Theorem 4. Let X; K; T and fxng be the same as in Theorem 3. If T satis…es
Proof. By following the same proof of Theorem 3.4 in [11] and using Lemma 1, we can obtain that PT satis…es Condition (I) and PT(x) is closed for any x 2 K. Then,
applying Theorem 2(i) to the mapping PT; we can conclude that the sequence fxng
is convergent strongly to a …xed point of T .
Theorem 5. Let X; K and fxng satisfy the hypotheses of Theorem 3 and T : K !
P (K) be a multi-valued hemi-compact mapping with F (T ) 6= ; and PT is
quasi-nonexpansive and continuous. Then the sequence fxng is convergent strongly to a
…xed point of T .
Proof. From the hemi-compactness of T , we can prove that PT is hemi-compact.
The conclusion follows from Theorem 2(ii).
Corollary 2. Let X; K and fxng be the same as in Theorem 3 and T : K !
P (K) be a multi-valued mapping with F (T ) 6= ; and PT is nonexpansive. Then
the sequence fxng is 4-convergent to a …xed point of T . Moreover, if T satis…es
Condition (I) or is hemi-compact, then the sequence fxng is convergent strongly to
a …xed point of T .
References
[1] Ba¸sar¬r, M. and ¸Sahin, A., On the strong and 4-convergence of new multi-step and S-iteration processes in a CAT(0) space, J. Inequal. Appl. 2013, Article ID 482, 13 pages.
[2] Bridson, M. and Hae‡iger, A., Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
[3] Espinola, R. and Fernandez-Leon, A., CAT( )-spaces, weak convergence and …xed points, J. Math. Anal. Appl. 353 (2009), 410-427.
[4] Gürsoy, F., Karakaya, V. and Rhoades, B. E., Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013, Article ID 76, 12 pages.
[5] He, J. S., Fang, D. H., López, G. and Li, C., Mann’s algorithm for nonexpansive mappings in CAT( ) spaces, Nonlinear Anal. 75 (2012), 445-452.
[6] Khan, S. H., Fukhar-ud-din, H. and Kalsoom, A., Common …xed points of two multivalued nonexpansive maps by a one-step implicit algorithm in hyperbolic spaces, Matematiµcki Vesnik 66(4) (2014), 397-409.
[7] Kimura, Y. and Nakagawa, K., Another type of Mann iterative scheme for two mappings in a complete geodesic space, J. Inequal. Appl. 2014, Article ID 72, 9 pages.
[8] Kimura, Y., Saejung, S. and Yotkaew, P., The Mann algorithm in a complete geodesic space with curvature bounded above, Fixed Point Theory Appl. 2013, Article ID 336, 13 pages. [9] Kimura, Y. and Satô, K., Halpern iteration for strongly quasinonexpansive mappings on a
geodesic space with curvature bounded above by one, Fixed Point Theory Appl. 2013, Article ID 7, 14 pages.
[10] Kirk, W. A. and Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689-3696.
[11] Panyanak, B., On the Ishikawa iteration processes for multivalued mappings in some CAT( ) spaces, Fixed Point Theory Appl. 2014, Article ID 1, 9 pages.
[12] Phuengrattana, W. and Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), 3006-3014.
[13] Puttasontiphot, T., Mann and Ishikawa iteration schemes for multi-valued mappings in CAT(0) spaces, Appl. Math. Sci. 4 (2010), 3005-3018.
[14] Salahifard, H., Vaezpour, S. M. and Dhompongsa, S., Fixed point theorems for some general-ized nonexpansive mappings in CAT(0) spaces, J. Nonlinear Anal. Optim. 4 (2013), 241-248. [15] Shabanian, S. and Vaezpour, S. M., A minimax inequality and its applications to …xed point
theorems in CAT(0) spaces, Fixed Point Theory Appl. 2011, Article ID 61, 9 pages. [16] Shahzad, N. and Zegeye, H., On Mann and Ishikawa iteration schemes for multi-valued maps
in Banach spaces, Nonlinear Anal. 71 (2009), 838-844.
[17] Song, Y. and Cho, Y. J., Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc. 48(3) (2011), 575-584.
[18] Thianwan, S., Common …xed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009), 688-695.
Current address : Aynur SAH·IN, Department of Mathematics, Faculty of Sciences and Arts, Sakarya University, Sakarya, 54050, Turkey.
E-mail address : ayuce@sakarya.edu.tr
Current address : Metin BA¸SARIR, Department of Mathematics, Faculty of Sciences and Arts, Sakarya University, Sakarya, 54050, Turkey.
