ISSN 1937 - 1055

VOLUME 1, 2015

INTERNATIONAL JOURNAL OF

MATHEMATICAL COMBINATORICS

EDITED BY

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS

March, 2015

Vol.1, 2015 ISSN 1937-1055

International Journal of

Mathematical Combinatorics

Edited By

The Madis of Chinese Academy of Sciences and

Academy of Mathematical Combinatorics & Applications

March, 2015

Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sci- ences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are:

Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,---, etc.. Smarandache geometries;

Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration;

Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;

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Editor-in-Chief

Shaofei Du Linfan MAO Cepital Normal el verely, P.R.China ; : Email: dushf@mail.cnu.edu.cn Chinese Academy of Mathematics and System Science, P.R.China Baizhou He and Beijing University of Civil Engineering and Academy of Mathematical Combinatorics & Architecture, P-.R.China Applications, USA Email: hebaizhou@bucea.edu.cn

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Chinese Academy of Mathematics and System Science, P.R.China

Email: xdhu@amss.ac.cn

Deputy Editor-in-Chief

Guohua Song Beijing University of Civil Engineering and Yuangiu Huang

Architecture, P.R.China Hunan Normal University, P.R.China

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Editors Mansfield University, USA

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: ; ; Xueliang Li Deakin University

Nankai University, P.R.China

Geelong Campus at Waurn Ponds . : Email: lxl@nankai.edu.cn

Australia

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Said Broumi

Hassan IT University Mohammedia Hay El Baraka Ben M’sik Casablanca W.B.Vasantha Kandasamy B.P.7951 Morocco Indian Institute of Technology, India

. , Email: vasantha@iitm.ac.in Junliang Cai

Beijing Normal University, P-R.China Ion Patrascu Email: caijunliang@bnu.edu.cn Fratii Buzesti National College

Craiova Romania Yanxun Chang

Beijing Jiaotong University, P.R.China Han Ren Email: yxchang@center.njtu.edu.cn East China Normal University, P.R.China

: : Email: hren@math.ecnu.edu.cn Jingan Cui

Beijing University of Civil Engineering and Ovidiu-Ilie Sandru Architecture, P.R.China Politechnica University of Bucharest

Email: cuijingan@bucea.edu.cn Romania

li International Journal of Mathematical Combinatorics

Mingyao Xu Peking University, P.R.China Y. Zhang

Email: xumy@math.pku.edu.cn Department of Computer Science

Guiying Yan Georgia State University, Atlanta, USA Chinese Academy of Mathematics and System

Science, P.R.China

Email: yanguiying@yahoo.com

Famous Words: Nothing in life is to be feared. It is only to be understood.

By Marie Curie, a Polish and naturalized-French physicist and chemist.

International J.Math. Combin. Vol.1(2015), 1-18

N*C*— Smarandache Curves of Mannheim Curve Couple

According to Frenet Frame

Sileyman SENYURT and Abdussamet CALISKAN (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey)

E-mail: senyurtsuleyman@hotmail.com

Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the Mannheim curve. Besides,

we illustrate example of our main results.

Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet

invariants.

AMS(2010): 53A04

§1. Introduction

A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve ([12]). Special Smarandache curves have been studied by some authors .

Melih Turgut and Siitha Yilmaz studied a special case of such curves and called it Smaran- dache T Bz curves in the space E} ([12]). Ahmad T.Ali studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret invariants of a special case ([{1]). Muhammed Cetin , Yilmaz Tuncer and Kemal Karacan investigated special Smarandache curves according to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of Smarandache curves, also they found the centers of the osculating spheres and curvature spheres of Smarandache curves ({5]). Senyurt and Caligkan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache curves (([4]). Ozcan Bektag and Salim Yiice studied some special Smarandache curves according to Darboux Frame in ((2]). Nurten Bayrak, Ozcan Bektag and Salim Yiice studied some special Smarandache curves in E? [3]. Kemal Tasképrii, Murat Tosun studied special Smarandache curves according to Sabban frame on S? ({11]).

In this paper, special Smarandache curve belonging to a* Mannheim partner curve such as N*C™* drawn by Frenet frame are defined and some related results are given.

1Received September 8, 2014, Accepted February 12, 2015.

2 Stileyman SENYURT and Abdussamet CALISKAN

§2. Preliminaries

The Euclidean 3-space be inner product given by (,) =a] +03 + 93

where (21,22,%3) E®. Let a: I > be a unit speed curve denote by {T,N, B} the moving Frenet frame . For an arbitrary curve a E®, with first and second curvature, « and T respectively, the Frenet formulae is given by ([6], [9])

T’ =kKN N'’=-«T+7B (2.1) Bl =-—TN.

For any unit speed a : I > E3, the vector W is called Darboux vector defined by

W =7(s)T(s) + «(s) + B(s).

If consider the normalization of the Darboux C = wm” we have

cosy = WON siny = |||

we) [WII

C = sin yT(s) + cos pB(s) (2.2)

where Z(W, B) = y. Let a : I and a* : I E®? be the C?— class differentiable unit speed two curves and let {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} be the Frenet frames of the curves @ and a*, respectively. If the principal normal vector N of the curve a is linearly dependent on the binormal vector B of the curve a*, then (qa) is called a Mannheim curve and (a*) a Mannheim partner curve of (a). The pair (a, a*) is said to be Mannheim pair ([7], [8]). The relations between the Frenet frames {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} are as follows: T* = coséT sindB

N* = sin @T + cos6B (2.3) BY =N cos 9 = oor. : * dsx (2.4) sin 6 = Ar* ds

where Z(T,T*) = 0 ([8}).

Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves

in is constant.

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 3

Theorem 2.2 Let (a,a*) be a Mannheim pair curves in E?. For the curvatures and the torsions

of the Mannheim curve pair (a, a*) we have,

ds* = en g@—— K=7* sin ae (2.5) ds* = —7* g@— T T* cos ag and .. ad, K dse Nr K2 + 7? (2.6)

d T= (sin 0 r cos) S

BS |

Theorem 2.3 Let (a,a*) be a Mannheim pair curves in For the torsions T* of the

Mannheim partner curve a* we have

Theorem 2.4([{10]) Let (a,a*) be a Mannheim pair curves in E®. For the vector C* is the

direction of the Mannheim partner curve a* we have

g’ Ct = —— a Seca (2.7) ie Cae ie Ga)

where the vector C' is the direction of the Darboux vector W of the Mannheim curve a.

§3. N*C*— Smarandache Curves of Mannheim Curve Couple According to

Frenet Frame

Let (a,a*) be a Mannheim pair curves in E* and {T*N*B*} be the Frenet frame of the Mannheim partner curve a* at a*(s). In this case, N*C* - Smarandache curve can be defined by

1 * * fils) = 5(N" +"). (3.1)

Solving the above equation by substitution of N* and C* from (2.3) and (2.7), we obtain

(cos 6|| W || + sin 04/6’? + ||W||2)T + ON + (cos 04/6”? + ||W||2 sin || W|)B

a” + || W|I?

Ai(s) =

(3.2)

Stileyman SENYURT and Abdussamet CALISKAN

The derivative of this equation with respect to s is as follows,

|W 9 _ 9’ «cos é shy = \|W || ‘\ || Tp, (8) (Arr) ee cst]? + = (yetthe) i |v fils OO

2 ( Ww \ aE 4 no 4yWI) |e (eL.)'4 0/2-+4||W]|2 0 AT |W] AT||W]| /9'24\|WIl2 0’

eet wee Ww)". ee ( ts) sind] B

677+ |W? if

2 nO? +)WI2) | ( hal \e | aa ww |. WwW 2 79/2 |W? a (3.3)

In order to determine the first curvature and the principal normal of the curve (1(s), we formalize

v3 (ri cos @ +r sin 6)T + 73N + (—7 sin @ + 2 cos 6)B

Tp, (s) = Pfr, dao Um et. ey ce « ana 2m ata (=) er 4 20? +I?) satin -2(s ML) 8 6'2+||W ||? 0’ AT||W]| AT||W || Vf 0/2 +|| W|I? 0’ where

n = 2 ) (pe) Ne EA for siwie Jor iw A es ae (ena ey Ar||W'| fo? + WIP 6! forse 9 or Bt He Ww 1/9? + (IW? ie (——)’-( |) eS (iS)

6? + |W] Ar 6? + |W] o 62 + |W]

eee yy |W |

ee ee ea fee cd : eA oe K Pe

El a la)- (2 can - Ge)

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame

el ( || ) +2 6k \{( |W pee 4 a lor? + WII? XT ||W|| 62 + WII o

3 Ww 6" a | Ww chia ( 624 IW =) 9°? + ||WII2 2(=-) ( ERAT 6

x

| een Sees |

nn +

2 Ww 207 On \2 Ww yr V9? + IW IP Iw ( 92 4. IW =) - Ga) ae 6! | ( aT

= x

|| Wl]

> Gawd) (Gera OH Wye)

e+ |W 6” + |W ||?

CA Ga) Gm (Ge AT||W || 6? + ||W||2

0

/ o? + |W 4 fo? + |W ia ay

It 6”? sn) ~ Ga pee) 0! (2) (4 ye wi y_#_)

(

We) te y

(ld

——~ nS”

CNS

NS ~

0? + WIP Vo? + WIP? 6? + |W IP ( OK )I( Ww | eel| MAM for eis ver +iwie 8

aa) ($) - Gain) (B)([C ial ee)

(

(am) Ga):

6? + ||W||?

ad )-[( ial ) av) fo? + ||WII2 6? + || WI)? e

oa OK I Il y eI 6! eo tey

AT||W|| or + |W

| ( || W'|| )’ a” + ||W|I? ( 6! eeu 0k ) | 6 + WII? oY | Y dr Xr||W|

Stileyman SENYURT and Abdussamet CALISKAN

(ae

g’

————

0! OK \? Forma, - (sa)

3 Q’ 2 OK \2 Sear, - (sa) eal ||| y ey AT gre W|I og’ eysf¢__iwh yr ve +i) iwi ao = eae I( Pie | ( wi | Ww )-25)'|( Iwi__y Jo? + |IWwIP " os yy22 AT LA fo? + | y2 ik |W ‘i (or) (eI \| | ) 0! i/o? + III? At||W I] 7 Ar gre \W|/2 el ( |W g’

i

| 2

(22+

6? + ||W||? 0

=A eae = ee 4

DS

6g’ Ok 4 OK 2

6! vereT aa - 2a) K eo Maal ace eel a " 0? + || WI

fs Ky! \|W'|| 1? + (|W 7? OK \2, KF OK ce, ( ) a HG. 7) (s7) - (a)

[gr 4. Maal Xr||W'|

Iwi VO? +N IPD tw), ¢_ 8 al ( ) I )+ (sar) [( ) vor+imi Yo? + ie? SATII LS fo? 4 a al ( |W ae Maal )( 6!

0” + IW + (IW I2/ 4/6? + IP k WI ce, (

a” + ||W |?

“[( |W | yer 6! et 6 ‘i

SS

| ee uly ( \w|| ) ear a

YS DS

+

——

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame

|W]

a

Ok

recealay,

fo? + {WIP

|W]

alt

0? + ||W|I? 14/0 + ||W|l2

)

OK

14/0? + ||W|l2

6’

I (

g’

|W

fo" + |W?

Q’

)

1]

14/6!? + ||W||2

Gam) (GS) | =) ATW LN, fo? + IW ss fe? +|wie? ( | jae lel ial as “I hal ) 24+ WI) . Paywye AT US /6? + 6? + ||W|I?7? 6! 6'k \w|| 14/0? + |||? 0" | ( pres) + (serie ara a (——) +15) |(— jee Ww) (6? + WIP ATTN, fo? + || WI? @ (6? + || WI)? 0 sie Wwe yO? + IW? wi}! ( ae *)|( eae 0’ \{( aa Vf 0? + || WII? 7/ K 6'k Ok ' K OK ||W|| 6 | anand +) Gear) |( 72 = fo pt ss (oe Ok BI |W ) Sa +e. SATII! OTT 0? + WIP P+ IEP ae 6" \( oe IC hal ) ae al [9 +. ||W|l2 Ar||WI| 7 \Ar||W | Gmerye

|W] fo" + ||WII?

The first curvature is

——~

Ta a a

Ar || W'||

: | qt

+ ||WI/?

K

ATW]

V2(V ri? + 72? + 737) Ke, = 7 ( | ) ee 4 5(O7+I|W |?) 0/2-+|| Wl? go’ AT||W I

>(

|

g!

||)

2 ll) Vor+wies

8 Stileyman SENYURT and Abdussamet CALISKAN

The principal normal vector field and the binormal vector field are respectively given by (7 cosO + fr sin 8)T + 73N + (—%4 sind + 2 cos0)B 8,5 Se (3.4) Ty + 727 + T3

_fip, &y 48 Ba (s) = ET + SN + BB. (3.5)

where

622 asd Ww Ald a ie A WI ) & = re c080( et | a F2 cos 0 E 1 ENT a

/ , = 73 (lL) + r3( Ain) | sine [9/2 +||W ||2 T||W I

by =r ( Iw ) - 0! /9!2 4.\|W ||? AT ||W

Sea ( Ww )2 eee are pee "( ial ) l 3 [9°24 Wl? 0’ AT AT /9'2 4 Wl? 6’ / y —r3 |W || af P3 OK cos 0 /9'2 +||W||2 AT ||W

2 = 2 S Ww Jor+ WI? : a 9) «(0/2 4|| WII? (71? + Fo? + 737) (te) Yee | t+ (a? 4 re? + 732) 22 4M I) <t- ) a 2( WI \e AT |W] /9'2+|| WI? go’ ,

In order to calculate the torsion of the curve (3, we differentiate

pe Ff[ooo(( ye 0? + WIP i/o? + Iwi gt te tin) fo? +lWIP fo? + we Ari vsino(( Ors ){C |W pees 6! Kk

ATW LN, fo? yp Jers ive Oe ! 0”? + ||WI|I? "Kk \2 K\2 ( ial ye In a |

versie, % tafe? yay SATII Oe

*)-[( Thad! yr) Thal

Oe meaner

fo? + (WIP

6g’

(lu

fo" + |W?

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame

aa SH) Is -sno([( ty eo

(tween) wi ae \, ( ) | (=)

Zara | eC Coed de / 1/0? + ||W ||? / +o Ok IC |W || ) | ) 60 kK

a a AIL fo? + IW IP ow? 47

(lu

( hal aa WI zi 6 )*-)|).

Q/? i \|W||2 0 Q/? 4 |W | AT||W|| AT and thus : :

my (ti cos @ + to sind + t3)T + t3N + (t2 cos 6 ty sin@ + ts)T

1 /2 where

( Iw yp cael 0! l( Iwi__y

a a, = WE fo" + ||WII? : Vo" + (|W? fo? + ||WII?

wr \w|| ) desley WI

g’

fo" + |W?

g!

fo? + |W? 7 |

-|( hal peta 6! 6k ‘i

owe 8 + we SATII

( OK al |W’ a 6! 6K ves Xr||W| 6? + ||W||? Q 6? + |W]? Ar||WI|7 Ar

( |W | plead |W | at 6K: ee 6k ( Ky? /o'? + WI? o /o? + |W)? Ar ||W| Ar ||WI| 7S Ar ae 2( Ok, ){C |W || yew oy 23 Ok ) Ar||W|| /o'? + |W)? ve /o'? + ||W||? Ar||W|| ( ial a, al 3 6k )( 6k ) /o'? + |W]? a’ /o'? + ||W|? Ar |W] 7 \Ar||W]|

10 Siileyman SENYURT and Abdussamet CALISKAN

= OK ){C \|W’|| eas 6’ Ky IW fer simi o+|w2 47

( WI yeaa ial al +2()|( Iwi__y Vo? + WIP ‘i 6? + WIP

ver simi 7

uae La \|w| +2( maa ee “Feri fer seimie

0’ RK K i

Yersime

0 = (De [ea Ee dae ee, At||W]| Ar [gir Ww 0 ly? wie + WIP

vere (_aw_y ve" jee ecs

\/ 0? + || W|I? oe fo? + III \r

( |W yr Al |W ee

6? + WI)? f 6? + |W)? A

eWay VES EIMET e SINUN O ate ‘eae 7] | 9 LIWIP (<a) oe =.) te

The torsion is then given by

vp, Heth 8 BH TEAR”

Qy

TB, = 208

where

wy2¢__ IW I, f__ ky WP é'n_\? O = ~24(55)"(Feaes) +a ((ja ae) | a2 ~ 8 (Seq) (SO) (See + |( 4 ) wr 0”

——S—= ——_—~ te Vor? + |W? Ar||W'| Vo? + WIP 0’ 0”? + ||W||?

6’ AT

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 11

|W 2 6'K |W / On \2 WII A tian + (Fae mine? | (Sav) 7 (ae ts (<3 ie (Sere) ts(57)

(SF 8) 6S) - (4S) | (YY Stale) IW (2

oe + W]e, AT? NAT] 0? + WIP 0’ e+ Wz SATII

a|(s wil eee) of nt |( Iw el _y ew of Vo? + ||WIP

vo? + WIP e Vr + (WIP ¢ Vo? + {WIP +|( IW yer Al +(e Al ah is 1, o" )’ At L\ fo? + WIP oY Jor +wi? LS /o? +] WIP Ar 6 Ar||W'|

(Ly (s,s _I_y Jerre 8 OriWy/ or er aqme! 8 Or Or WT ee = [( ol) eta) Gh) + to ( lt) ie

———— t —_— t a fe | pe

+05 ((Feroep) | tat) GS) +6 Gere) ae) ae

Ok tek Ok, K \2 0' ks K \3 -4( WT) (55) + Gaps) +6(sourD) Gos

_({# |W Als. ||| Jer + We]? Iwi OK \? = (E(t) sl ee Ga)

Iwi JOTI] oe y2¢__IWI__yIWI), ¢_O’®_)2 wy | pw y2) (_) ee 2) (see) ot Ga) E+")

f(y Vem FWY ey [¢__w_y Ver FWP |wil__y’ te =| GF Five) |-Ga (ae) a ||(Geare) vere | -( OK: ) 6! ( OK. y|C WI peaeaial

0 At||WIl/ for + wy? SATII Jo? + WIP 0 -+|( ||| y ee) -( Ok JI ( ||| ) Vo? + (WIP eer ||| AT LS /0? + ||W||? oY AT||WII7 LN /6? + [WIP VO" + |W) 2 Kf OK \! |W 1/0? + |W]? 1? 0K Kar 0K -37 (sq) rt (om aa) 0" + (sq) Ge)? a (saw) ( |W yi er) g (& (= ||| _)' Comal ||| V6? + |W? e Vo? + ae Ge 6? + WIP oY Vo? + |W —2/ 0'k )'|( ||| )’ veer) ea, Wiens ||| vl __y' wor Ar ||W|| Jo? + WIP 0’ Cre Jor + |W? 6

2 6 OK \3 Ok K\2 K\ K |W || ' Jo? + |WIP = (WT) : (seq) oe) - (x7) x5 (79 a a) Example 3.1 Let us consider the unit speed Mannheim curve and Mannheim partner curve: a(s) = —=(-—coss,—sins,s), a*(s) = —=(—2coss,—2sins,s)- W)= Fl ), a°(s) = Fel |

The Frenet invariants of the partner curve, a*(s) are given as following

T*(s) = —=(2sins,—2coss,1), (s)

S|-

12 Siileyman SENYURT and Abdussamet CALISKAN

1 N*(s) = —s(sins,coss,—2)

V5 B*(s) = (coss,sins,0)

2 2 2 2 1 C*(s) = (=sins+—<coss,—=coss + —ssins, =)

5 V5 5 VB 22

ga

ee

5

In terms of definitions, we obtain special Smarandache curve, see Figure 1.

i Figure 1 3; = —=((5 + 2V5) sins + 10coss, (5 2V5) coss + 10sins, —9V5)

5V5

References

[1] Ali A.T., Special Smarandache curves in the Euclidean space, International Journal of

Mathematical Combinatorics, Vol.2, 2010, 30-36.

[2] Bektas O. and Yiice S., Special Smarandache curves according to Dardoux frame in Eu-

clidean 3-space, Romanian Journal of Mathematics and Computer science, Vol.3, 1(2013),

48-59. [3] Bayrak N., Bektag O. and Yiice S., Special Smarandache curves in

13, International Con-

ference on Applied Analysis and Algebra, 20-24 June 2012, Yildiz Techinical University, pp.

209, Istanbul.

[4] Calgkan A., Senyurt S., Smarandache curves in terms of Sabban frame of spherical indi-

catrix curves, XI, Geometry Symposium, 01-05 July 2013, Ordu University, Ordu.

[10

(11 [12

[13

N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 13

Cetin M., Tuncer Y. and Karacan M.K.,Smarandache curves according to bishop frame in Euclidean 3-space, arxiv:1106.3202, vl [math.DG], 2011.

Hacisalihoglu H.H., Differential Geometry, Inénii University, Malatya, Mat. no.7, 1983. Liu H. and Wang F.,Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2(2008), 120-126(7).

Orbay K. and Kasap E., On mannheim partner curves, International Journal of Physical Sciences, Vol. 4 (5)(2009), 261-264.

Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006.

Senyurt S. Natural lifts and the geodesic sprays for the spherical indicatrices of the mannheim partner curves in E?, International Journal of the Physical Sciences, vol.7, No.16, 2012, 2414-2421.

Task6prii K. and Tosun M., Smarandache curves according to Sabban frame on $?, Boletim da Sociedade parananse de Mathemtica, 3 srie, Vol.32, No.1(2014), 51-59 ssn-0037-8712. Turgut M., Yilmaz S., Smarandache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, Vol.3(2008), pp.51-55.

Wang, F. and Liu, H., Mannheim partner curves in 3-space, Proceedings of The Eleventh International Workshop on Diff. Geom., 2007, 25-31.

International J.Math. Combin. Vol.1(2015), 14-28

Fixed Point Theorems of Two-Step Iterations for Generalized 7-Type Condition in CAT(0) Spaces

G.S.Saluja (Department of Mathematics, Govt. Nagarjuna P.G. College of Science, Raipur - 492010 (C.G.), India) E-mail: salujal963@gmail.com

Abstract: In this paper, we establish some strong convergence theorems of modified two- step iterations for generalized Z-type condition in the setting of CAT(0) spaces. Our results extend and improve the corresponding results of [3, 6, 28] and many others from the current

existing literature.

Key Words: Strong convergence, modified two-step iteration scheme, fixed point, CAT(0)

space.

AMS(2010): 54H25, 54E40

§1. Introduction

A metric space X is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having non-positive sectional curvature is a CAT(0) space. Fixed point theory in a CAT(0) space was first studied by Kirk (see [19, 20]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since, then the fixed point theory for single-valued and multi-valued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2], [9], [11]-[13], [17]-[18], [21]-[22], [24]- [26] and references therein). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT(k) space with k < 0 since any CAT(k) space is a CAT(m) space for every m > k (see [7).

Let (X,d) be a metric space. A geodesic path joining x X to y X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,/] C R to X such that c(0) = a, c(l) = y and d(c(t),c(t’)) = |t t’| for all t,t’ [0,/]. In particular, c is an isometry, and d(x,y) =1. The image a of c is called a geodesic (or metric) segment joining x and y. We say X is (i) a geodesic space if any two points of X are joined by a geodesic and (ii) a uniquely geodesic if there is exactly one geodesic joining x and y for each x,y X, which we will denoted by [x,y], called the segment joining x to y.

A geodesic triangle A(a1, 22,23) in a geodesic metric space (X,d) consists of three points

1Received July 16, 2014, Accepted February 16, 2015.

Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 15

in X (the vertices of A) and a geodesic segment between each pair of vertices (the edges of A). A comparison triangle for geodesic triangle A(x1, 72,23) in (X,d) is a triangle A(x, v2, 73) := A (%q,%2,%3) in R* such that dp2(%j,7j) = d(x;,x;) for i,j {1,2,3}. Such a triangle always exists (see [7]).

1.1 CAT(0) Space

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following CAT (0) comparison axiom.

Let A be a geodesic triangle in X, and let A C R? be a comparison triangle for A. Then A is said to satisfy the CAT (0) inequality if for all 2, y A and all comparison points 7,9 A,

Complete CAT(0) spaces are often called Hadamard spaces (see [16]). If x,y1, y2 are points of a CAT(0) space and yo is the mid point of the segment [yi, y2] which we will denote by (y1 @ y2)/2, then the CAT (0) inequality implies

® 1 1 1 P (0, A) <5 Pam) +5 Pew) 5 Hvry). (1.2)

The inequality (1.2) is the (CN) inequality of Bruhat and Tits [8]. The above inequality was extended in [12] as

@(z,ax@(l—a)y) < ad?(z,x) +(1—a)d?(z,y) —a(1 a)d? (x,y) (1.3)

for any a [0,1] and z,y,z€ X.

Let us recall that a geodesic metric space is a CAT (0) space if and only if it satisfies the (CN) inequality (see [7, page 163]). Moreover, if X is a CAT (0) metric space and x,y X, then for any a [0,1], there exists a unique point ax © (1 a)y [x,y] such that

d(z,axz @ (1—a)y) < ad(z,x) + (1— a)d(z,y), (1.4)

for any z X and [x,y] = {ax 6 (1—a)y: a€ (0, 1]}. A subset C of a CAT (0) space X is convex if for any x,y C, we have [x,y] C C. We recall the following definitions in a metric space (X,d). A mapping T: X X is called an a-contraction if d(Tx,Ty) <ad(x,y) for allx, y X, (1.5)

where a (0,1). The mapping T is called Kannan mapping [15] if there exists b (0,4) such that

d(Tx,Ty) < b[d(a,Tx) + d(y,Ty)| (1.6)

for all a,y X.

16 G.S.Saluja

The mapping T is called Chatterjea mapping [10] if there exists c (0,4) such that d(Tx,Ty) < c[d(x, Ty) + dy, Tx)] (1.7)

for all a,y X. In 1972, Zamfirescu [29] proved the following important result.

Theorem Z Let (X,d) be a complete metric space and T: X X a mapping for which there exists the real number a, b and c satisfying a (0,1), b, c (0,4) such that for any pair zx, y€ X, at least one of the following conditions holds:

(a1) d(Tx,Ty) < ad(x,y);

(22) d(L'x,Ty) < b|d(x, Tx) + dy, Ty)};

(23) d(L'x,Ty) < cld(x, Ty) + dy, Tx)}.

Then T has a unique fixed point p and the Picard iteration {x,}°° defined by

En+1 = Lary, n= OFT 2 ee: converges to p for any arbitrary but fixed xp X.

An operator T' which satisfies at least one of the contractive conditions (z1), (z2) and (z3) is called a Zamfirescu operator or a Z-operator.

In 2004, Berinde [5] proved the strong convergence of Ishikawa iterative process defined by: for xo C, the sequence {z,,}°° given by

Ungi = (L—an)an + OnT Yn,

Yn = (1 _ Bn )&n + PnT%y, n=O, (1.8)

to approximate fixed points of Zamfirescu operator in an arbitrary Banach space E. While proving the theorem, he made use of the condition,

Px —Tyl| <6 |jx— yl] + 25 lw To (1.9)

which holds for any z, y E where0< 6 < 1. In 1953, W.R. Mann defined the Mann iteration [23] as

Un+1 = (1 = An)Un + anT Un, (1.10)

where {a,,} is a sequence of positive numbers in [0,1]. In 1974, S.Ishikawa defined the Ishikawa iteration [14] as

Sn41 = (1— dn)$n + OnTtn,

th = (1— bn) Sn + bnT Sn, (1.11)

where {a,,} and {b,} are sequences of positive numbers in [0,1].

Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 17

In 2008, S.Thianwan defined the new two step iteration [27] as Unt = (1—an)Wn + OnT wn,

Wn = (1 = bn) + OnT Vn, (1.12) where {a,,} and {b,} are sequences of positive numbers in [0,1]. Recently, Agarwal et al. [1] introduced the S-iteration process defined as

Engi = (1L—an)T Ln + OnT Yn,

Yn = (1 bn) &n + bnT an, (1.13)

where {a,,} and {b,,} are sequences of positive numbers in (0,1).

In this paper, inspired and motivated [5, 29], we employ a condition introduced in [6] which is more general than condition (1.9) and establish fixed point theorems of S- iteration scheme in the framework of CAT(0) spaces. The condition is defined as follows:

Let C be a nonempty, closed, convex subset of a CAT(0) space X and T: C C a self map of C. There exists a constant L > 0 such that for all x, y C’, we have

d(Tx,Ty) < e& UT) [6 d(x,y) + 26d(x,Tx)], (1.14)

where 0 < 6 < 1 and denotes the exponential function of « C. Throughout this paper, we call this condition as generalized Z-type condition.

Remark 1.1 If ZL =0, in the above condition, we obtain d(Tx,Ty) < dd(a,y) + 26 d(a,T x),

which is the Zamfirescu condition used by Berinde [5] where b Cc 6 max {a, [3° —}, 0<6<1,

while constants a, b and c are as defined in Theorem Z.

Example 1.2 Let X be the real line with the usual norm |].|| and suppose C' = [0,1]. Define T:C > C by Tx = xit for all x,y C. Obviously T is self-mapping with a unique fixed point 1. Now we check that condition (1.14) is true. If v,y [0,1], then ||[Ta—Ty]| < ef llz@—Tall§ || yl] + 26 ||a Ta||] where 0 < 6 < 1. In fact

|T2—Tyl| =

t—y 2

and

-—1

eblleTel 5 a yll + 26 lle Ta] = e* lel [51a yl +5 Ie 1].

18 G.S.Saluja

Clearly, if we chose x = 0 and y = 1, then contractive condition (??) is satisfied since

1 Tx —Ty|| = == [Tx - Ty 7

t—y 2

and for L > 0, we chose L = 0, then eb lle-Tel | § 1a yl] + 26 le Ta] ] = e* lll [5 Ja yl +5 a 111]

= e9(1/2)(26) = 25, where 0<6 <1.

Therefore Px Tyl| < oP !#- 72H] |] yl| + 26 x Tal].

Hence T is a self mapping with unique fixed point satisfying the contractive condition (1.14).

Example 1.3 Let X be the real line with the usual norm |].|| and suppose K = {0,1, 2,3}. Define T: K K by Tzr=2, if «=0

= 3, otherwise.

Let us take x = 0, y= 1 and L =0. Then from condition (1.14), we have

1

IA

e°(?)15(1) + 26(2)] < 1(56) = 56

which implies 6 > ¢. Now if we take 0 < 6 < 1, then condition (1.14) is satisfied and 3 is of course a unique fixed point of T.

1.2 Modified Two-Step Iteration Schemes in CAT(0) Space

Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let T: C > C be a contractive operator. Then for a given x1 = x C, compute the sequence {z,,} by the iterative scheme as follows:

Ung = (1L—an)T Ln B OnT Yn,

Yn = (1 = bn) &n O bn» T In, (1.15)

where {a,,} and {b,} are sequences of positive numbers in (0,1). Iteration scheme (1.15) is called modified S-iteration scheme in CAT(0) space.

Vnti1 = (1—an)Wn 8 OnT wn,

Wn = (1 = bn)in 8 OnT Vn, (1.16)

where {a,,} and {b,,} are sequences of positive numbers in [0,1]. Iteration scheme (1.16) is called

Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 19

modified $.Thianwan iteration scheme in CAT(0) space. Sn41 = (1— dn)Sn ® AnTtn,

th = (1— bn) Sn ® bnT Sn, (1.17)

where {a,,} and {b,,} are sequences of positive numbers in [0,1]. Iteration scheme (1.17) is called modified Ishikawa iteration scheme in CAT(0) space.

We need the following useful lemmas to prove our main results in this paper. Lemma 1.4([24]) Let X be a CAT(0) space. (i) Fora, y€X andt [0,1], there exists a unique point z [x, y] such that d(x, 2) =td(a, y) anddly, 2) = (1—#)d{a, y). (4)

We use the notation (1—t)x @ ty for the unique point z satisfying (A).

(ii) Fora, ye X andt [0,1], we have

d((1 —t)a @ ty, z) < (1—t)d(a, z) + td(y, z).

Lemma 1.5([4]) Let {pn }°o, {an}%o, {rn} eo be sequences of nonnegative numbers satisfying

the following condition: Pnti <(1—Sn)Pn tanttn, Vn>0,

where tate COO AY df Sag ga = 00, lit, 355Gn = Og) and >) arn = tO, hen

§2. Strong Convergence Theorems in CAT(0) Space

In this section, we establish some strong convergence theorems of modified two-step iterations

to converge to a fixed point of generalized Z-type condition in the framework of CAT(0) spaces.

Theorem 2.1 Let C be a nonempty closed convex subset of a complete CAT(0) space X and let T: C > C be a self mapping satisfying generalized Z-type condition given by (1.14) with F(T) #0. For any xo C, let {an}°29 be the sequence defined by (1.15). If 07-4 an = 00 and pete Anbn = 00, then {rn }°2_ converges strongly to the unique fixed point of T.

Proof From the assumption F(T) 4 @, it follows that T has a fixed point in C, say uw. Since T satisfies generalized Z-type condition given by (1.14), then from (1.14), taking x = u

20 G.S.Saluja

and y = tn, we have Wien) eter) (6 d(u, @,) + 26 d(u, Tu) = eb d