A Convergence Theorem for a Finite Family of Multivalued k-strictly Pseudononspreading

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1 Thai Joural of Mathematics Volume 13 (2015) Number 3 : ISSN A Covergece Theorem for a Fiite Family of Multivalued k-strictly Pseudoospreadig Mappigs i R-Trees 1 Khaiti Samamit Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai 50200, Thailad khaiti@phrae.mju.ac.th Abstract : I this paper, we itroduce a ew m-step iterative process for fiite family of k strictly pseudoospreadig multivalued mappigs i R-trees. We obtai a strog covergece theorem of m-step iterative method to a commo fixed poit of a fiite family of those multivalued mappigs i R-trees. Our results exted may kow recet results i the literature. We close this work with the first examples of k strictly pseudoospreadig multivalued mappigs i R-trees. Keywords : fixed poit; multivalued mappig; R-tree; k strictly pseudoospreadig; covergece theorems Mathematics Subject Classificatio : 47H09; 47H10. 1 Itroductio Fixed poit theory for sigle-valued mappigs i R-trees was first studied by Kirk [1]. He proved that every cotiuous sigle-valued mappigs defied o a geodesically bouded complete R-tree always has a fixed poit. His works are followed by a series of ew works by may authors(see, e.g., [2]-[6]). It is worth 1 This research was supported by the Thai Govermet Scholarships i the Area of Sciece ad Techology, the Graduate School of Chiag Mai Uiversity ad the Thailad Research Fud uder the project RTA Copyright c 2015 by the Mathematical Associatio of Thailad. All rights reserved.

2 582 Thai J. Math. 13 (2015)/ K. Samamit metioig that fixed poit theorems i R-trees ca be applied to graph theory, biology ad computer sciece (see e.g., [7]-[10]). I 2009, Shahzad ad Zegeye [11] proved strog covergece theorems of the Ishikawa iteratio for quasi-oexpasive multivalued mappigs satisfyig the edpoit coditio i Baach spaces. Later i 2010, Puttasotiphot [12] obtaied similar results i complete CAT(0) spaces. I 2012, Samamit ad Payaak [13] itroduced a coditio o mappigs i R-trees which is weaker tha the edpoit coditio, is called the gate coditio. They proved strog covergece theorems of a modified Ishikawa iteratio for quasi-oexpasive multivalued mappigs satisfyig such coditio. I 2011, Osilike ad Isiogugu [14] itroduced a ew class of sigle-valued k- strictly pseudoospreadig mappigs i Hilbert space as follows: A mappig T : E E is called k-strictly pseudoospreadig if Tx Ty 2 x y 2 +k x Tx (y Ty) 2 +2 x Tx,y Ty for all x,y E. I a Hilbert space, the aboved iequality is equivalet to (2 k) Tx Ty 2 k x y 2 +(1 k) y Tx 2 +(1 k) x Ty 2 +k x Tx 2 +k y Ty 2 for all x,y E. They proved weak ad strog covergece theorems for those class of mappigs i Hilbert spaces. Recetly, Phuegrattaa[15] itroduced a ew class of multivalued k-strictly pseudoospreadig mappigs i R-trees. He proved strog covergece theorems of a ew two-step iterative process for two k-strictly pseudoospreadig multivalued mappigs havig the gate coditio. I this paper, motivate by the above results, we itroduce a ew m-step iterative process for fiite k-strictly pseudoospreadig multivalued mappigs i R-trees. We also obtai the strog covergece theorem for approximatig a commo fixed poit of those multivalued mappigs i R-trees by assumig the gate coditio. Fially, we close this work with the first example for class of k-strictly pseudoospreadig multivalued mappigs i R-trees. 2 Prelimiaries Let (X,d) be a metric space. A geodesic path joiig x X to y X is a map c from a closed iterval [0,l] R to X such that c(0) = x,c(l) = y, ad d(c(t),c(t )) = t t for all t,t [0,l]. I particular, c is a isometry ad d(x,y) = l. The image of c is called a geodesic segmet joiig x ad y. Whe it is uique this geodesic is deoted by [x,y]. For x,y X ad α [0,1], we deote the poit z [x,y] such that d(x,z) = αd(x,y) by (1 α)x αy. The space (X,d) is said to be a geodesic space if every two poits of X are joied by a geodesic, ad X is said to be uiquely geodesic if there is exactly oe geodesic joiig x ad y for each x,y X. A subset E of X is said to be covex if E icludes every

3 A Covergece Theorem for a Fiite Family of Multivalued geodesic segmet joiig ay two of its poits. If x X ad E X, the the distace from x to E is defied by d(x,e) = if{d(x,y) : y E}. The set E is called proximial if for each x X, there exists a elemet y E such that d(x,y) = d(x,e), ad E is said to be gated if for ay poit x / E there is a uique poit y x such that for ay z E, d(x,z) = d(x,y x )+d(y x,z). Clearly gated sets i a complete geodesic space are always closed ad covex. The poit y x is called the gate of x i E. It is easy to see that y x is also the uique earest poit of x i E. We shall deote by CB(E) the family of oempty closed bouded subsets of E, by CC(E) the family of oempty closed covex subsets of E ad by KC(E) the family of oempty compact covex subsets of E. Let H(, ) be the Hausdorff distace o CB(E), i.e., { } H(A,B) = max sup a A d(a,b),supd(b, A) b B, A,B CB(E). Let T : E CB(E) be a multivalued mappig. For each x E, we let P Tx (x) = {u Tx : d(x,u) = d(x,tx)}. I the case of P Tx (x) is a sigleto we will assume, without loss of geerality, that P Tx (x) is a poit i E. A poit x E is called a fixed poit of T if x Tx. A poit x E is called a edpoit of T if x is a fixed poit of T ad T(x) = {x}[16]. We shall deote by F(T) the set of all fixed poits of T ad by E(T) the set of all edpoits of T. We see that for each mappig T, E(T) F(T) ad the coverse is ot true i geeral. A mappig T is said to satisfies the edpoit coditio if E(T) = F(T). A R tree is a special case of a CAT(0) space. For a thorough discussio of these spaces ad their applicatios, see [17]. We ow collect some basic properties of R trees. Lemma 2.1. Let X be a complete R tree ad E be a oempty subset of X. The the followig statemets hold: (i) [18, page 1048] the gate subsets of X are precisely its closed ad covex subsets; (ii) [17, page 176] if E is closed ad covex, the for each x X, there exists a uique poit P E (x) E such that d(x,p E (x)) = d(x,e). That is, every oempty closed covex subset of a complete R tree is proximial. (iii) [17, page 176] if E is closed covex ad x belog to [x,p E (x)], the P E (x ) = P E (x);

4 584 Thai J. Math. 13 (2015)/ K. Samamit (iv) [6, Lemma 3.1] if A ad B are bouded closed covex subsets of X, the for ay u X, d(p A (u),p B (u)) H(A,B); (v) [19, Lemma 2.5] if x,y,z X ad α [0,1], the d 2 ((1 α)x αy,z) (1 α)d 2 (x,z)+αd 2 (y,z) α(1 α)d 2 (x,y); (vi) [19, Lemma 2.3] if x,y,z X, the d(x,z) +d(z,y) = d(x,y) if ad oly if z [x,y]. We state the followig coditios i R-trees: A multivalued mappig T : E CB(E) is said to satisfy coditio (I) if there exists a odecreasig fuctio f : [0, ) [0, ) with f(0) = 0 ad f(r) > 0 for all r > 0 such that d(x,tx) f(d(x,f(t))) for all x E. A fiite family of multivalued mappigs {T i } m i=1 of E ito CB(E) is said to satisfy coditio(m) if there exists a odecreasig fuctio f : [0, ) [0, ) with f(0) = 0 ad f(r) > 0 for all r > 0 such that d(x,t i x) f(d(x,f)) for some i {1,2,...,m} ad for all x E, where F = m i=1 F(T i). The followig propositio is also eeded. Propositio2.2([20]). Let (X,d) be a complete metric space ad F be aoempty closed subset of X. Let {x } be a sequece i X such that d(x +1,p) d(x,p) for all p F ad N. The {x } coverges strogly to some poit i F if ad oly if lim d(x,f) = 0. 3 Mai Results Defiitio 3.1. Let E be a oempty subset of a complete R-tree X. A multivalued mappig T : E CB(E) is called (i) ospreadig if for all x,y E. 2H 2 (Tx,Ty) d 2 (y,tx)+d 2 (x,ty) (ii) k-strictly pseudoospreadig if there exists k [0, 1) such that (2 k)h 2 (Tx,Ty) kd 2 (x,y)+(1 k)d 2 (y,tx)+(1 k)d 2 (x,ty) for all x,y E. +kd 2 (x,tx)+kd 2 (y,ty)

5 A Covergece Theorem for a Fiite Family of Multivalued It is easy to see that every ospreadig multivalued mappig is 0-strictly pseudoospreadig. Moreover, if T is k-strictly pseudoospreadig with F(T), the for all x E ad p F(T) we have H 2 (Tx,Tp) d 2 (x,p)+kd 2 (x,tx). Thus T may ot be quasi-oexpasive. It is easy to show that if T is a k-strictly pseudoospreadig multivalued mappig with F(T), the F(T) is closed. The followig result ca be foud i [15]. Lemma 3.1. Let E be a oempty closed covex subset of a complete R-tree X. Assume that T : E KC(E) is a k-strictly pseudoospreadig multivalued mappig. If {x } is a sequece i E such that x x ad d(x,tx ) 0 as, the x Tx. Now, we are ready to prove the mai theorem. Theorem 3.2. Let E be a oempty closed covex subset of a complete R-tree X. Let T 1 : E KC(E) be a k-strictly pseudoospreadig multivalued mappig ad T 2,T 3,...,T m : E KC(E) be k-strictly pseudoospreadig ad L-Lipschitzia multivalued mappigs with F = m i=1 F(T i). Suppose that T 1,T 2,...,T m satisfy the gate coditio. Let u 1,u 2,...,u m be keys of T 1,T 2,...,T m, respectively. For x 1 E, the sequece {x } geerated by y (1) = α(1) z(1) (1 α(1) )x for all N, where z (1) is the gate of u 1 i T 1 x, ad y (2) = α(2) z(2) (1 α(2) )y(1) for all N, where z (2) is the gate of u 2 i T 2 y (1), ad where z y = α z.. (1 α )y (m 2) for all N, is the gate of u m 1 i T m 1 y (m 2), ad x +1 = α (m) z (m) (1 α (m) )y for all N, where z (m) is the gate of u m i T m y. Let {α (i) } be sequeces i [0,1] such that 0 < a α (i) b < 1 k for each i {1,2,...,m}. If oe of the followig is satisfied: (i) {T i } m i=1 satisfies coditio(m), (ii) oe member of the family {T i } m i=1 is hemicompact,

6 586 Thai J. Math. 13 (2015)/ K. Samamit the {x } coverges strogly to a elemet of F. Proof. Let p F. By the gate coditio ad Lemma 2.1(v), we have d 2 (x +1,p) = d 2 (α (m) z (m) (1 α (m) )y,p) (1 α (m) )d 2 (y,p)+α (m) d 2 (z (m),p) α (m) (1 α (m) (1 α (m) )d 2 (y,p)+α (m) d 2 (P Tmy (u m),p Tmp(u m)) α (m) (1 α (m) )d 2 (y,z (m) ) (1 α (m) )d 2 (y,p)+α (m) H 2 (T my,t mp) α (m) )d 2 (y (1 α (m) (1 α (m) )d 2 (y,p)+α (m) (d 2 (y,p)+kd 2 (y,t my )) α (m) (1 α (m) )d 2 (y,z (m) ) = d 2 (y,p) α (m) (1 k α (m) )d 2 (y. (1 α,p)+α α (m) (1 k α (m) )d 2 (y (1 α α (1 α α (1 α α = d 2 (y (m 2) (1 α (1 α (1 k α,p) α,p)+α d 2 (z,z (m) ),z (m) ),p) α,z (m) ) )d 2 (y,z (m) ) (1 α )d 2 (y (m 2),z ) d 2 (P T y (m 2) (u ),P T p(u )),z ) α (m) (1 k α (m),p)+α H 2 (T y (m 2),T p),p)+α,z (1 k α α (m) (1 k α (m) )d 2 (y (d 2 (y (m 2),z (m) ) ) α (m) (1 k α (m),z )d 2 (y )d 2 (y,z (m) ),z (m) ),p)+kd 2 (y (m 2),T y (m 2) )) ) α (m) (1 k α (m) )d 2 (y (m 2),z ) )d 2 (y d 2 (x,p) α (1) (1 k α (1) )d 2 (x,z (1) ) α (2) (1 k α (2) )d 2 (y (1),z (2) )... α (1 k α,z ) α (m) (1 k α (m) )d 2 (y,z (m) ),z (m) ). (3.1) From α (i) < 1 k for each i {1,2,...,m}, we obtai d(x +1,p) d(x,p) for all N. This implies that {d(x,p)} is oicreasig ad bouded below. Hece lim d(x,p) exists for each p F. By (3.1), we have α (1) (1 k α (1) )d 2 (x,z (1) )+α (2) (1 k α (2) )d 2 (y (1),z (2) )+...+α k α d 2 (x +1,p).,z )+α (m) (1 k α (m) )d 2 (y (1,z (m) ) d 2 (x,p)

7 A Covergece Theorem for a Fiite Family of Multivalued Thus, by 0 < a α (i) b < 1 k for each i {1,2,...,m}, we have lim d(x,z (1) ) = 0 ad lim d(y(i 1) Also, with d(x,t 1 x ) d(x,z (1) ), we have By usig the defiitio of T 2, we have d(x,t 2 x ) d(x,t 2 y (1) )+H(T 2y (1),T 2x ) Thus by (3.2), we have,z (i) ) = 0 for each i {2,3,...,m}. (3.2) lim d(x,t 1 x ) = 0. (3.3) d(x,z (2) )+Ld(y (1),x ) d(x,y (1) )+d(y(1),z(2) )+Ld(y(1),x ) = (1+L)d(x,y (1) )+d(y (1),z (2) ) (1+L)(d(x,z (1) )+d(z (1),y (1) ))+d(y (1),z (2) ) = (1+L)(d(x,z (1) )+(1 α(1) )d(z(1),x ))+d(y (1),z(2) ) = (1+L)(2 α (1) )d(x,z (1) )+d(y (1),z (2) ) (1+L)(2 a)d(x,z (1) )+d(y(1),z(2) ). Similaly, by usig the defiitio of T 3, we have d(x,t 3 x ) lim d(x,t 2 x ) = 0. (3.4) (1+L 2 )(2 a)d(x,z (1) )+(1+L 1)(2 a)d(y (1),z(2) )+d(y(2),z(3) ) ad also By usig the same way, we have for all j {4,5,...,m}. lim d(x,t 3 x ) = 0. (3.5) lim d(x,t j x ) = 0 (3.6)

8 588 Thai J. Math. 13 (2015)/ K. Samamit Case (i): {T i } m i=1 satisfies coditio(m). The there exists a odecreasig fuctio f : [0, ) [0, ) with f(0) = 0 ad f(r) > 0 for all r > 0 such that d(x,t i x) f(d(x,f)) for some i {1,2,...,m} ad for all x E, where F = m i=1 F(T i). If d(x,t 1 x) f(d(x,f)) for all x E. For eaah N, we have x E. By usig (3.3), we obtai 0 = lim d(x,t 1 x ) lim (f(d(x,f))) = f( lim (d(x,f))) 0. Hece f(lim (d(x,f))) = 0, therefore lim (d(x,f)) = 0. Similarly i other cases, we ca use (3.4), (3.5) ad (3.6) to show that lim (d(x,f))= 0. By the closedess of F ad Propositio 2.2, we have {x } coverges strogly to some poit i F. Case (ii): Oe member of the family {T i } m i=1 is hemicompact. Without loss of geerality, we assume that T 1 is hemicompact. The there exists a subsequece {x l } of {x } such that {x l } coverges strogly to z E. By (3.3), (3.4), (3.5) ad (3.6), it follows by Lemma 3.1 that z F. Sice lim d(x,p) exists for each p F, it implies that {x } coverges strogly to z F. As a direct cosequece of Theorem 3.2, we obtai the followig corollary. Corollary 3.3. Let E be a oempty closed covex subset of a complete R-tree X. Let T 1 : E KC(E) be a ospreadig multivalued mappig ad T 2,T 3,...,T m : E KC(E) be ospreadig ad L-Lipschitzia multivalued mappigs with F = m i=1 F(T i). Suppose that T 1,T 2,...,T m satisfy the gate coditio. Let u 1,u 2,...,u m be keys of T 1,T 2,...,T m, respectively. For x 1 E, the sequece {x } geerated by y (1) = α (1) z (1) (1 α (1) )x for all N, where z (1) is the gate of u 1 i T 1 x, ad y (2) = α (2) z (2) (1 α (2) )y (1) for all N, where z (2) is the gate of u 2 i T 2 y (1), ad where z y = α z. (1 α )y (m 2) for all N, is the gate of u m 1 i T m 1 y (m 2), ad x +1 = α (m) z(m) (1 α (m) )y for all N,

9 A Covergece Theorem for a Fiite Family of Multivalued where z (m) is the gate of u m i T m y. Let {α (i) } be sequeces i [0,1] such that 0 < a α (i) b < 1 k for each i {1,2,...,m}. If oe of the followig is satisfied: (i) {T i } m i=1 satisfies coditio(m), (ii) oe member of the family {T i } m i=1 is hemicompact, the {x } coverges strogly to a elemet of F. Next, this is the first example of that class i the literature. Example 3.4. (For k-strictly pseudoospreadig multivalued mappigs.) (1.) Let E = [0, ),k [0,1) ad T : E KC(E) be defied by Tx = [0,( k )x] for all x E ad b 2. b The T is k strictly pseudoospreadig. (2.) Let E = [0, ),k [0,1) ad S : E KC(E) be defied by S(x) = [(( k b ) (k b )2 )x,( k )x] for all x E ad b 2. b The S is k strictly pseudoospreadig. Proof. (1.) We see that H(Tx,Ty) = ( k b ) x,y. Sice b 2, we obtai b 2 2b ad b > k. So we have 2b > 2k ad this show that b 2 2b > 2k 2k k 2. From 2k k 2 < b 2, we have This show that From (3.7), we obtai (2 k)k 2 = 2k 2 k 3 b 2 k. (2 k)( k b )2 k. (3.7) (2 k)h 2 (Tx,Ty) = (2 k) ( k b )x (k b )y 2 = (2 k)( k b )2 x y 2 k x y 2 = kd 2 (x,y) kd 2 (x,y)+(1 k)d 2 (y,tx)+(1 k)d 2 (x,ty) +kd 2 (x,tx)+kd 2 (y,ty). Hece T is k strictly pseudoospreadig. (2.) Similarly.

10 590 Thai J. Math. 13 (2015)/ K. Samamit Ackowledgemets : I would like to thak the referees for commets ad suggestios o the mauscript. This work was supported by the Thai Govermet Scholarships i the Area of Sciece ad Techology, the Graduate School of Chiag Mai Uiversity ad the Thailad Research Fud uder the project RTA Refereces [1] W. A. Kirk, Fixed poit theorems i CAT(0) spaces ad R-trees, Fixed Poit Theory Appl. (2004) [2] A. G. Aksoy, M. A. Khamsi, A selectio theorem i metric trees, Proc. Amer. Math. Soc. 134 (2006) [3] S. M. A. Aleomraiejad, Sh. Rezapour, N. Shahzad, Some fixed poit results o a metric space with a graph, Topology Appl. 159 (2012) [4] A. Amii-Haradi, A. P. Farajzadeh, Best approximatio, coicidece ad fixed poit theorems for set-valued maps i R-trees, Noliear Aal. 71(2009) [5] R. Espíola, W. A. Kirk, Fixed poit theorems i R-trees with applicatios to graph theory, Topology ad its Applicatios 153 (2006) [6] J. T. Marki, Fixed poits, selectios ad best approximatio for multivalued mappigs i R-trees, Noliear Aal. 67 (2007) [7] I. Bartolii, P. Ciaccia, M. Patella, Strig matchig with metric trees usig a approximate distace, SPIR Lecture otes i Comput. Sci. 2476, Spriger, Berli, [8] M. Bestvia, R-trees i topology, geometry, ad group theory, i: Hadbook of Geometric Topology, North-Hollad, Amsterdam, (2002) [9] W. A. Kirk, Some recet results i metric fixed poit theory, J. Fixed Poit Theory Appl. 2 (2007) [10] C. Semple, M. Steel, Phylogeetics, Oxford Lecture Ser. Math. Appl. vol. 24, Oxford Uiv. Press, Oxford, [11] N. Shahzad, H. Zegeye, O Ma ad Ishikawa iteratio schemes for multivalued maps i Baach spaces, Noliear Aal. 71 (2009) [12] T. Puttasotiphot, Ma ad Ishikawa iteratio schemes for multivalued mappigs i CAT(0) spaces, Applied Mathematical Scieces. 4 (61) (2010) [13] K. Samamit, B. Payaak, O multivalued oexpasive mappigs i R trees, J. Appl. Math. 2012, Article ID , (2012), 13 pp. [14] M. O. Osilike, F. O. Isiogugu, Weak ad strog covergece theorems for ospreadig-type mappigs i Hilbert spaces, Noliear Aal. 74 (2011)

11 A Covergece Theorem for a Fiite Family of Multivalued [15] W. Phuegrattaa, Approximatio of commo fixed poits of two strictly pseudoospreadig multivalued mappigs i R trees, Kyugpook Math. J. 55 (2015) [16] K. Wlodarczyk, D. Klim, R. Plebaiak, Existece ad uiqueess of edpoits of closed set-valued asymptotic cotractios i metric spaces, J. Math. Aal. Appl. 328 (2007) [17] M. Bridso, A. Haefliger, Metric Spaces of No-Positive Curvature, Spriger- Verlag, Berli, Heidelberg, [18] R. Espiola, W. A. Kirk, Fixed poit theorems i R trees with applicatios to graph theory, Topology Appl. 153 (2006) [19] S. Dhompogsa, B. Payaak, O covergece theorems i CAT(0) spaces, Comput. Math. Appl. 56 (2008) [20] H. H. Bauschke, P. L. Combettes, Covex Aalysis ad Mootoe Operator Theory i Hilbert Spaces, Spriger, New York, NY, USA, (Received 30 March 2015) (Accepted 14 August 2015) Thai J. Math.