On the convergence of SP-iterative scheme for three multivalued nonexpansive mappings in CAT (κ) spaces

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1 Palestine Journal of Mathematics Vol. 4(1) (2015), Palestine Polytechnic University-PPU 2015 On the convergence of SP-iterative scheme for three multivalued nonexpansive mappings in CAT () spaces R. A. Rashwan S. M. Altwqi Communicated by Ayman Badawi MSC 2010 Classifications: Primary 20M99, 13F10; Secondary 13A15, 13M05. Keywords phrases: Common fixed points, three iterative scheme, multivalued nonexpansive mappings, CAT () spaces. Abstract In this paper strong convergence -convergence theorem is established for the SP-iterative scheme for three multivalued nonexpansive mappings in CAT () spaces for any > 0. Our results extend improve the recent ones announced by [6, 20]. 1 Introduction The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler [15]. The Banach contraction principle has been extended in different directions either by using generalized contractions for multivalued mappings hybrid pairs of single multivalued mappings, or by using more general spaces. The terminology CAT () spaces was introduced by M. Gromov to denote a distinguished class of geodesic metric spaces with curvature bounded above by R. In recent years, CAT () spaces have attracted the attention of many authors as they have played a very important role in different aspects of geometry. A very thorough discussion on these spaces the role they play in geometry can be found in the book by M.R. Bridson A. Haefliger [1]. As it was noted by W.A. Kirk in his fundamental works [8, 9], the geometry of CAT () spaces is rich enough to develop a very consistent theory on fixed point under metric conditions. These works were followed by a series of new works by different authors (see for instance [2, 4, 10, 11, 17, 18, 21]. Also, since any CAT () space is a CAT ( ) space for >, all results for CAT (0) spaces immediately apply to any CAT () with > 0. Recently B. Pia tek in [20] proved that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the complete CAT () spaces. Very recently, J.S. He et al. in [6] proved that the sequence defined by Mann s algorithm -converges to a fixed point in complete CAT () spaces. 2 Preinaries Let (E, d) be a bounded metric space, then for D, K E nonempty, set r x (D) = sup{d(x, y) : y D}, x E, rad K (D) = inf{r x (D) : x K}, diam(d) = sup{d(x, y) : x, y D}. Let x, y E. A geodesic path joining x to y (or, more briefly, a geodesic from x to y) is a map c : [0, l] E such that c(0) = x, c(l) = y, d(c(t), c(t )) = t t for all t, t [0, l]. In particular, c is an isometry between [0, l] c([0, l]), d(x, y) = l. Usually, the image c([0, l]) of c is called a geodesic segment (or metric segment) joining x y. A metric segment joining x y is not necessarily unique in general. In particular, in the case when the geodesic segment joining x y is unique, we use [x, y] to denote the unique geodesic segment joining x y, this means that z [x, y] if only if there exists t [0, 1] such that d(z, x) = (1 t)d(x, y) d(z, y) = td(x, y). In this case, we will write z = tx (1 t)y for simplicity. For fixed D (0, + ), the space (E, d) is called a D-geodesic space if any two points of E with their distance smaller than D are joined by a geodesic segment. An A -geodesic space is simply called a geodesic space. Recall that a geodesic triangle := (x, y, z) in the metric space (E, d) consists of three points in E (the vertices of ) three geodesic segments between each pair of vertices (the edges of ). For the sake of saving printing space, we write p when a point p E lies in the union of [x, y], [x, z] [y, z]. The triangle is called a comparison triangle for if d(x, y) = d(x, y), d(x, z) = d(x, z) d(z, y) = d(z, y). By [[1], Lemma 2.14, page 24], a comparison triangle for always exists

2 74 R. A. Rashwan S. M. Altwqi provided that the perimeter d(x, y) + d(y, z) + d(z, x) < 2D (where D = if > 0 otherwise). A point P [x, y] is called a comparison point for p [x, y] if d(p, x) = d(p, x). Recall that a geodesic triangle in E with perimeter less than 2D is said to satisfy the CAT () inequality if, given a comparison triangle in Mk 2 for, one has that d(p, q) d(p, q), p, q, where p Aq are respectively the comparison points of p q. The model spaces M 2 are defined as follows. Definition 2.1. Given a real number, we denote by M 2 the following metric spaces: (i) if = 0 then M 2 is Euclidean space E n ; (ii) if > 0 then M 2 is obtained from the sphere S n by multiplying the distance function by 1 the constant. (iii) if < 0 then M 2 is obtained from hyperbolic space H n by multiplying the distance function 1 by.. The metric space (E, d) is called a CAT () space if it is D -geodesic any geodesic triangle in E of perimeter less than 2D satisfies the CAT () inequality. Proposition 2.2. M 2 is a geodesic metric space. If 0, then M 2 is uniquely geodesic all balls in M 2 are convex. If > 0, then there is a unique geodesic segment joining x, y M 2 if only if d(x, y) <. If > 0, closed balls in M 2 of radius smaller than are convex. 2 Proposition 2.3. Let E be a CAT () space. Then any balls in E of radius smaller than convex. 2 are In a geodesic space E with unique metric segments, the metric d is said to be convex if for p, x, y E, t (0, 1) a point m [x, y] such that d(x, m) = td(x, y) d(y, m) = (1 t)d(x, y), then d(p, m) (1 t)d(p, x) + td(p, y). And for all x, y, z E t [0, 1] then, d((1 t)x ty, z) (1 t)d(x, z) + td(y, z) (2.1) R-trees are a particular class of CAT () spaces for any real which will be named at certain points of our exposition (see [1], pg. 167] for more details). Definition 2.4. An R-tree is a metric space E such that: (i) it is a uniquely geodesic metric space, (ii) if x, y z T are such that [y, x] [x, z] = {x}, then [y, x] [x, z] = [y, z]. Therefore the family of all closed convex subsets of a CAT () space has uniform normal structure in the usual meteric (or Banach space) sense. A subset K of E is said to be convex if K includes every geodesic segment joining any two of its points. We will denote by P (E) the family of nonempty proximinal subsets of E, by CC(E) the family of nonempty closed convex subsets of E, by KC(E) the family of nonempty compact convex subsets of E. Let H be the Hausdorff distance on CC(E), that is H(A, B) = max{sup d(x, B), sup d(y, A)}, x A y B for every A, B CC(E), where d(x, B) = inf{d(x, y) : y B} is the distance from the point x to the subset B. A multivalued mapping T : E CC(E), is said to be nonexpansive if H(T x, T y) x y, x, y E. A point x E is said to be a fixed point for a multivalued mapping T if x T x. We use the notation F (T ) sting for the set of fixed points of a mapping T. Let us recall the following definitions.

3 On the convergence of SP-iterative scheme 75 Definition 2.5. Three of multivalued nonexpansive mappings T, S, R : K CC(K), where K a subset of E, are said to satisfy condition (I) if there exists a nondecreasing function f : [0, ) [0, ) with f(0) = 0, f(r) > 0 for all r (0, ) such that d(x, T x) f(d(x, F ) or d(x, Sx) f(d(x, F ) or d(x, Rx) f(d(x, F ) for all x K, where F = F (T ) F (S) F (R), the set of all common fixed points of the mappings T, S R Definition 2.6. The mapping T : E CC(E), is called hemicompact if, for any sequence {x n } in E such that d(x n, T x n ) 0 as n, there exists a subsequence {x nr } of {x n } such that x nr p E. Proposition 2.7. The modulus of convexity for CAT () space E (of dimension 2) number r < let m denote the midpoint of the segment [x, y] joining x y define by the modulus δ r by sitting δ(r, ɛ) = inf{1 1 d(a, m)}, r where the infimum is taken over all points a, x, y E satisfying d(a, x) r, d(a, y) r ɛ d(x, y) < Next we state the following useful lemmas. Lemma 2.8. [see [13], Lemma 7]. Let (E, d, W ) be a uniformly convex hyperbolic with modulus of uniform convexity δ. For any r > 0, ɛ (0, 2), λ [0, 1] a, x, y X, if d(x, a) r, d(y, a) r d(x, y) ɛr then d((1 λ)x λy, a) (1 2λ(1 λ)δ(r, ɛ)r. Lemma 2.9. Let E be a complete CAT () space with modulus of convexity δ(r, ɛ) let x E. Suppose that δ(r, ɛ) increases with r (for a fixed ɛ) suppose {t n } is a sequence in [b, c] for some b, c (0, 1) {x n }, {y n } are sequences in E such that sup d(x n, x) r, sup d(y n, x) r d((1 t n)x n t n y n, x) = r for some r = 0. Then d(x n, y n ) = 0. Proof. The proof similar as in lemma 9 in [12]. For scaler valued case, we state the following iterative processes as the following: In 1953, W. R. Mann defined the Mann iteration [14] as where {α n } is a sequences of positive numbers in [0, 1]. In 1974, S. Ishikawa defined the Ishikawa iteration [5] as x n+1 = (1 α n )x n + α n T x n, (2.2) x n+1 = (1 α n )x n + α n T y n, y n = (1 β n )x n + β n T x n, (2.3) where {α n } {β n } are sequences of positive numbers in [0, 1]. In 2008, S. Thianwan defined the new two step iteration [22] as x n+1 = (1 α n )y n + α n T y n, y n = (1 β n )x n + β n T x n, (2.4) where {α n } {β n } are sequences of positive numbers in [0, 1]. In 2001, M. A.Noor defined the three step Noor iteration [16] as x n+1 = (1 α n )x n + α n T y n, y n = (1 β n )x n + β n T z n, (2.5) z n = (1 γ n )x n + γ n T x n, where where {α n }, {β n } {γ n } are sequences of positive numbers in [0, 1]. Recently, Phuengrattana Suantai defined the SP-iteration [19] as x n+1 = (1 α n )y n + α n T y n, y n = (1 β n )z n + β n T z n, (2.6) z n = (1 γ n )x n + γ n T x n,

4 76 R. A. Rashwan S. M. Altwqi where where {α n }, {β n } {γ n } are sequences of positive numbers in [0, 1]. clearly the Mann Ishikawa iteration are special cases of the Noor iteration. In the following definition, we extend the SP -iteration process (2.6) to the case of three multivalued nonexpansive mappings on closed convex subset of E modifying the above ones. Definition Let E be a CAT () space, K be a nonempty closed convex subset of E T, S, R : K CC(K), be three multivalued nonexpansive mappings. The sequence {x n } of the modified SP -iteration is defined by: x 1 K, x n+1 = (1 α n )y n α n u n, n N, y n = (1 β n )z n β n v n, (2.7) z n = (1 γ n )x n γ n w n, where u n T y n, v n Sz n, w n Rx n {α n }, {β n }, {γ n } [0, 1]. Remark (i) If γ n = 0 then we have x 1 K, x n+1 = (1 α n )y n α n u n, n N, (2.8) y n = (1 β n )x n β n v n, where u n T y n, v n Sx n {α n }, {β n } [0, 1]. (ii) If γ n = β n = 0 then we obtain x 1 K, where u n T x n, {α n } [0, 1]. x n+1 = (1 α n )x n α n u n, n N, (2.9) (2.10) In this paper, we study the strong convergence -convergence of SP-iterative scheme for three multivalued nonexpansive mappings in CAT () spaces for any > 0 under some conditions. Our results extend improve the some results in [6, 20]. 3 Strong convergence theorems First of all, we prove the following lemmas, which play very important rule in the latter. Lemma 3.1. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T, S, R : K CC(K) be three multivalued nonexpansive mappings {x n } be the sequence as defined in (2.7). If F T p = Sp = Rp = {p} for any p F then d(x n, p) exists for all p F. Proof. Assume that F. Let p F. Then from (2.7) we have, d(x n+1, p) = d((1 α n )y n α n u n, p) (1 α n )d(y n, p) + α n d(u n, p) (1 α n )d(y n, p) + α n H(T y n, T p) (1 α n )d(y n, p) + α n d(y n, p) = d(y n, p), (3.1) d(y n, p) = d((1 β n )z n β n v n, p) (1 β n )d(z n, p) + β n d(v n, p) (1 β n )d(z n, p) + β n H(Sz n, Sp) (1 β n )d(z n, p) + β n d(z n, p) = d(z n, p), (3.2)

5 On the convergence of SP-iterative scheme 77 From (3.2) (3.3) we obtain, From (3.3) (3.4) we have, d(z n, p) = d((1 β n )x n β n w n, p) (1 β n )d(x n, p) + β n d(w n, p) (1 β n )d(x n, p) + β n H(Rx n, Rp) (1 β n )d(x n, p) + β n d(x n, p) = d(x n, p). (3.3) d(y n, p) d(x n, p) (3.4) d(x n+1, p) d(x n, p). Thus d(x n, p) exists for each p F, hence {x n } is bounded. Lemma 3.2. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2 let T, S, R : K CC(K), be three multivalued nonexpansive mappings {x n } be the sequence as defined in (2.7). If F T p = Sp = Rp = {p} for any p F then d(x n, T y n ) = d(x n, Sz n ) = d(x n, Rx n ) = 0. Proof. Let p F. By lemma (3.1), d(x n, p) exists, {x n } is bounded. Put From (3.4) we have sup d(x n, p) = c. (3.5) also sup d(y n, p) c, (3.6) d(u n, p) d(y n, p), for all n 1. so From (3.3) we have also sup d(u n, p) c. (3.7) sup d(z n, p) c, (3.8) d(v n, p) d(z n, p), thus Further, sup d(v n, p) c. (3.9) this gives d(x n+1, p) = d((1 α n)y n α n u n, p) [(1 α n)d(y n, p) + α n d(u n, p)] [(1 α n) sup d(y n, p) + α n sup d(u n, p)] [(1 α n)c + α n c] = c, ((1 α n)d(y n, p) + α n d(u n, p)) = c. (3.10)

6 78 R. A. Rashwan S. M. Altwqi Applying lemma (2.9) we obtain Noting that which yields that then from (3.7) we have, In turn this implies that By (3.6) (3.14), we obtain Further, this gives Moreover, hence d(y n, u n ) = 0. (3.11) d(x n+1, p) = d((1 α n )y n α n u n, p) d((1 β n )y n + β n (u n, p) (1 α n )d(y n, p) + α n d(u n, p) (1 α n )d(y n, u n ) + d(u n, p), c inf d(u n, p), d(u n, p). d(v n, p) d(y n, p), c inf d(y n, p). (3.12) d(y n, p). d(v n, p) d(z n, p), sup d(v n, p) c d(y n, p) = d((1 β n)z n β n v n, p) From lemma (2.9) we have [(1 β n) sup d(z n, p) + β n sup d(v n, p)] [(1 β n)c + β n c] = c d(x n, p) = c, d((1 β n)z n β n v n, p) = c. d(z n, v n ) = 0, (3.13) d(y n, p) = d((1 β n )z n β n v n, p) (1 β n )d(z n, p) + β n d(v n, p) (1 β n )[d(z n, v n ) + d(v n, p)] + β n d(v n, p) (1 β n )d(z n, v n ) + β n d(v n, p),

7 On the convergence of SP-iterative scheme 79 this yields that c inf d(v n, p), so (3.9) gives that d(v n, p). In turn d(v n, p) d(z n, p). this yields c inf d(z n, p). (3.14) By (3.6) (3.14), we obtain d(z n, p). Moreover, d(z n, p) = d((1 γ n)x n γ n w n, p) [(1 γ n) sup d(x n, p) + γ n sup d(w n, p)] [(1 γ n)c + γ n c] = c. Thus, By lemma (2.9) we have d((1 γ n)x n γ n w n, p) = c. d(x n, w n ) = 0, (3.15) d(z n, x n ) = d((1 γ n)x n γ n w n, x n ) [(1 γ n) sup d(x n, x n ) + γ n sup d(w n, x n )], that is, d(z n, x n ) = 0 d(y n, z n ) = d((1 γ n)z n γ n v n, z n ) [(1 γ n) sup d(z n, z n ) + γ n sup d(v n, z n )], then Also then, d(y n, z n ) = 0. (3.16) d(u n, x n ) d(u n, y n ) + d(y n, x n ), (3.17) d(u n, x n ) = 0. (3.18)

8 80 R. A. Rashwan S. M. Altwqi Also that is, d(v n, x n ) d(v n, z n ) + d(z n, x n ), (3.19) Then we have d(v n, x n ) = 0. (3.20) d(x n, T y n ) d(x n, u n ) 0, as n. d(x n, Sz n ) d(x n, v n ) 0, as n. d(x n, Sx n ) d(x n, w n ) 0, as n. The following theorems gives the strong convergence of three multivalued nonexpansive mappings, with value in closed convex space, Theorem 3.3. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T, S, R : K CC(K), be three multivalued nonexpansive mappings satisfying condition (I), {x n } be the sequence as defined in (2.7). If F T p = Sp = Rp = {p} for any p F, then {x n } converges strongly to a common fixed point of T, S, R. Proof. Since T, S, R, satisfies condition (I), we have f(d(x n, F )) = 0. Thus there is a subsequence {x nr } of {x n } a sequence {p r } F such that for all r > 0. By lemma (3.1) we obtain that d(x nr, p r ) < 1 2 r, d(x nr+1, p r ) d(x nr, p r ) < 1 2 r. We now show that {p r } is a Cauchy sequence in K. Observe that d(p r+1, p r ) d(p r+1, x nr+1 ) + d(x nr+1, p r ) < < 1 2 r r 1 2 r 1. This shows that {p r } is a Cauchy sequence in K thus converges to p K. Since d(p r, T p) H(T p, T p r ) d(p, p r ), p r p as r, it follows that d(p, T p) = 0, which implies that p T p. Similarity d(p r, Sp) H(Sp, Sp r ) d(p, p r ), p r p as r, it follows that d(p, Sp) = 0, which implies that p Sp. Similarity d(p r, Rp) H(Rp, Rp r ) d(p, p r ), p r p as r, it follows that d(p, Rp) = 0, which implies that p Rp. Consequently, p F. d(x n, p) exists, we conclude that {x n } converges strongly to a common fixed point p.

9 On the convergence of SP-iterative scheme 81 Theorem 3.4. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T, S, R : K CC(K), be three hemicompact continuous multivalued nonexpansive mappings. If F T p = Sp = Rp = {p} for any p F, then {x n } be the sequence as defined in (2.7) converges strongly to a common fixed point of T, S, R. Proof. From lemma (3.2) we obtain d(x n, T y n ) = d(x n, Sz n ) = d(x n, Rx n ) = 0 T, S R are hemicompact, there is a subsequence {x nr } of {x n } such that x nr p as r for some p K. Since T, S R are continuous, we have d(p, T p) d(p, x nr ) + d(x nr, T y nr ) + H(T y nr, T p) 2d(p, x nr ) + d(x nr, T y nr ) 0 as r, d(p, Sp) d(p, x nr ) + d(x nr, Sx nr ) + H(Sx nr, Sp) 2d(p, x nr ) + d(x nr, Sx nr ) 0 as r. d(p, Rp) d(p, x nr ) + d(x nr, Rx nr ) + H(Rx nr, Rp) 2d(p, x nr ) + d(x nr, Sx nr ) 0 as r. This implies that p T p, p Sp p Rp. by lemma (3.1) d(x n, p) exists, thus p F is the strong it of the sequence {x n } itself. Corollary 3.5. Let E be a uniformly convex Banach space K a nonempty closed convex subset of E. Let T, S be two multivalued nonexpansive mappings {x n } be the sequence as defined in (2.8) T S are hemicompact continuous. If F T p = Sp = {p} for any p F, then {x n } converges strongly to a common fixed point of T S. 4 -convergence of the SP-multivalued iteration In this section, we will study the -convergence of SP-iterative process (2.7) for three multivalued nonexpansive mappings in CAT () spaces satisfying conditions (I). Let E be a complete CAT () space {x n } a bounded sequence in E. For x E, we set r(x, (x n )) = sup d(x, x n ). The asymptotic radius r({x n }) of {x n } is given by r({x n }) = inf{r(x, {x n }) : x E}, the asymptotic radius r C ({x n }) with respect to C E of {x n } is given by r C (x n ) = inf{r(x, {x n }) : x C, the asymptotic center A({x n }) of {x n } is given by the set A({x n }) = {x E : r(x, {x n }) = r({x n })}. Therefore, the following equivalence holds for any point u E: u A({x n }) sup d(u, x n ) sup d(x, x n ), x E. (4.1) A sequence {x n } in E is said to -converge to x E if x is the unique asymptotic center of {u n } for every subsequence {u n } of {x n }. In this case we write x n = x call x the -it of {x n }. we denote ω w (x n ) := A({u n }), where the union is taken over all subsequences {u n } of {x n }. Proposition 4.1. Let E be a complete CAT () space let {x n } be a sequence in E with r({x n }) < D /2. Then the following assertions hold. A({xn }) consists of exactly one point.

10 82 R. A. Rashwan S. M. Altwqi {xn } has a -convergent subsequence. Lemma 4.2. ([3], Lemma 2.7). (i) Every bounded sequence in E has a -convergent subsequence. (ii) If K is a closed convex subset of E if {x n } is a bounded sequence in K, then the asymptotic center of {x n } is in K. (iii) If K is a closed convex subset of E if T : K E is a nonexpansive mapping, then the conditions, {x n } -converges to x d(x n, f(x n )) 0, imply x K f(x) = x. Lemma 4.3. [3] If {x n } is a bounded sequence in E with A({x n }) = {x} {u n } is a subsequence of {x n } with A({u n }) = u the sequence {d(x n, u)} converges, then x = u. Lemma 4.4. Let K be a closed convex subset of E, let T : K CC(K) be a multivalued nonexpansive mapping. Suppose {x n } is a bounded sequence defined by (2.9) in K such that d(x, T x n) = 0 {d(x n, v)} converges for all v F, then w(x n ) F. Moreover, ω w (x n ) consists of exactly one point. Proof. Let u ω w (x n ), then there exists a subsequence u n of x n such that A(u n ) = {u}. By Lemma (4.2)(i) (ii) there exists a subsequence v n of {u n } such that v n = v K. By Lemma (4.2)(iii), v F (T ). By Lemma (4.3), u = v. This shows that ω w (x n ) F (T ). Next, we show that ω w (x n ) consists of exactly one point. Let u n be a subsequence of {x n } with A({u n }) = {u} let A({x n }) = {x}. Since u ω w (x n ) F (T ), {d(x n, u)} converges. By Lemma (4.3), x = u. Theorem 4.5. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T : K CC(K), satisfying condition (I). If {x n} be the sequence in K defined by (2.9) such that d(x n, T x n ) = 0 x n = v, then v T v. Proof. Let x n = v. We note that by Lemma (4.2), v K. For each n 1, we choose z n T (v) such that d(x n, z n ) = dist(x n, T (v)). By the compactness of T (v) there exists a subsequence {z nk } of {z n } such that z n k = w T (v). Since T satisfies the condition (I) we have, Note that Thus dist(x nk, T (v)) dist(x nk, T (x nk )) + d(x nk, v), (4.2) d(x nk, w) d(x nk, z nk ) + d(z nk, w) dist(x nk, T (x nk )) + d(x nk, v) + d(z nk, w). (4.3) sup d(x nk, w) sup d(x nk, v). (4.4) By the uniqueness of asymptotic centers, we have v = w T (v). Theorem 4.6. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T, S, R : K CC(K), be three multivalued nonexpansive mappings satisfying condition (I), {x n } be the sequence as defined in (2.7). If F T p = Sp = Rp = {p} for any p F, then {x n } is -converges to a common fixed point of T, S, R. Proof. It follows from Lemma (3.2), d(x n, T y n ) = d(x n, Sz n ) = d(x n, Rx n ) = 0. Now we let ω w (x n ) := A({u n }) where the union is taken over all subsequences u n of x n. We claim that ω w (x n ) F. Let u ω w (x n ), then there exists a subsequence {u n } of {x n } such that A({u n }) = {u}. By Lemmas (4.2) (4.3), there exists a subsequence {v n } of {u n } such that v n = v. Since d(x n, T y n ) = d(x n, Sz n ) = d(x n, Rx n ) = 0, by Theorem (4.5) we have v F, the d(x n, v) exists by Lemma (3.1). Hence u = v F by Lemma (4.3). This shows that ω w (x n ) F. Next we show that ω w (x n ) consists of exactly one point. Let {u n } be a subsequence of {x n } with A({u n }) = {u} let A({x n }) = {x}. Since u ω w (x n ) F d(x n, v) converges, by Lemma (4.3) we have x = u. Corollary 4.7. Let K be a nonempty closed convex subset of a complete CAT () space E with rad(k) < 2, let T, S : K CC(K), be two multivalued nonexpansive mappings satisfying condition (I), {x n } be the sequence as defined in (2.8). If F T p = Sp = {p} for any p F, then {x n } is -converges to a common fixed point of T, S.

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