Fixed Point Theorems for (, )-Uniformly Locally Generalized Contractions

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1 Global Joral o Pre ad Applied Mahemaics. ISSN Volme 4 Nmber 9 (208) pp Research Idia Pblicaios hp:// Fied Poi Theorems or ( -Uiormly Locally Geeralized Coracios G. Sdhaamsh Moha Reddy Facly o Sciece ad Techology ICFAI Fodaio or Higher Edcaio oaapalli Shakarpalli Road Hyderabad Idia Absrac I his paper we deie a class called ( -iormly locally geeralized coracios ad esablish a ied poi heorem or sch coracios Mahemaics Sbjec Classiicaio. 54H25. 47H0. Key words ad phrases. Fied poi Theorem Coracio Covergece. Irodcio:. eiiio: A selmap o a -meric space (X ) is called a (.) ( -iormly locally geeralized coracio i here is a mber q wih 0q ad a posiive cosa sch ha or all We ow prove Mai Theorem: ( y z) q. ( y y) r. ( ) s. ( y y y). y X wih y y ( y y) ( y ) Sp where q r s 2 y X 2. Theorem: Sppose is a ( -iormly locally geeralized coracio o a -meric space (X ) ad X is -orbially complee. The or every X eiher.

2 78 G Sdhaamsh Moha Reddy or all iegers s 0 s s s (2.) or coverges o which is a ied poi o. Also here is o oher ied poi v X wih v v. (2.2) he seqece cosider s s Proo: For ay X s s 0. The we have eiher each o he erm i his seqece is greaer ha or eqal o or or some erm i i is less ha. Le or some ieger s0 I he irs case he aleraive o (2.) o he hypohesis holds. s s 0 s0 s0. Sice is a ( - s s0 s0 iormly locally geeralized coracio ad 0 we ge mbers q r s ad (all depedig o ad y) sch ha s0 s0 2 s0 2 s0 s0 0 s Thereore s0 s0 s0 s0 s0 s0 q. r. s0 s0 2 s0 2 s. s0 s0 2 s0 2 s0 s0 s0 s0 s0 s0 s0 s0 s0 q. r. s0 s0 2 s0 2 s. s0 s0 s0 s0 s0 2 s0 2 s0 s0 s0 q r. s0 s0 2 s0 2 s. s s0 2 s0 2 s0 s0 s0 s q r 0..

3 Fied Poi Theorems or -Uiormly Locally Geeralized Coracios 79 This implies ha q r s0 s0 2 s0 2 s0 s0 s0. s. s0 s0 s0 Also we ge by repeaed se o he above ieqaliy ha s0 p s0 p s0 p s0 p s0 p s0 p. 2. s0 p2 s0 p s0 p s0 s0 s0 p. s Tha is 0 p s0 p s0 p or every ieger p ad hece or s0 we have p p 2 2 p p p... s 0 s 0 s... 0 p s 0 s 0 s 0 s 0 s 0 s... 0 p s... 0 s 0 s 0 s0 p p s0 s0 0 s 0 as Ths he seqece is a Cachy seqece i a -orbially complee - meric space (X ) ad hece here eiss X sch ha lim s0 p lim

4 80 G Sdhaamsh Moha Reddy Thereore here is a ieger 0 0 s sch ha or all 0 Now s r q s r r q r s r q which gives

5 Fied Poi Theorems or -Uiormly Locally Geeralized Coracios 8 Thereore Now leig i ollows ha 0 which implies ha showig ha he seqece coverges o some poi o X. To prove he iqeess o ied poi o sppose ha v X ad v v. The v v v v q v v r s v v v v v v q v v r s v v v v v v q 2 v v. v v which implies ha v v 0 v v or some sice ad hece v provig he secod par o (2.2). 2.2 Corollary: Sppose is a ( -iormly locally geeralized coracio o a -meric space (X ) ad X is -orbially complee. I or every X here is a ieger () sch ha ( ) ( ) ( ) (2.2.) The has a iqe ied poi provided ay wo ied pois v o are sch ha v v. Also he seqece or ay X coverges o he iqe ied poi o. Proo: Follows rom Theorem..

6 82 G Sdhaamsh Moha Reddy ACKNOWLEGEMENT The ahor wold like o hak Proessor S Chary ad he reviewers or heir valable commes ad sggesios o improve he qaliy o his paper. REFERENCES [] B. Ahmad M. Ashra ad B.E. Rhoades Fied poi heorems or epasive mappigs i -meric spaces Idia J. Pre ad Applied Mahemaics Vol- 30 No.0 pp (200). [2] Lj. B. Ciric Geeralized coracios ad ied poi heorem Pbl. Is. Mah. 2 (26) (97) pp.9-26 [3] Lj. B. Ciric A geeralizaio o Baach coracio priciple Proc. Amer. Mah. Soc. 45 (974) pp [4] B. C. hage Geeralized meric spaces ad mappigs wih ied poi Bllei o Calca Mah. Soc. Vol-84 No. 4 pp (992) [5] B. C. hage Commo ied poi priciple i -meric spaces Bllei o Calca Mah. Soc. Vol-9 No. 6 pp (999) [6] B. C. hage A. M. Paha ad B. E. Rhoades A geeral eesio priciple or ied poi heorems i -meric spaces I. J. Mah. & Mah. Sci. Vol.23 pp (2000) [7] B. Fisher Qasi-coracios o meric spaces Proc. Amer. Mah. Soc. 75 (979) pp 5-54 [8] Gerald Jgck Compaible mappigs ad commo ied pois Iere J. Mah. Mah. Sci 9 (986); [9] R. Kaa O cerai ses ad ied poi heorems Rom. Mah. Pre Appl. 4 (969) 5-54 [0] R. Kaa Some resls o ied pois- II America Mah. Mohly 76 (969) p (406) [] Solomo Leader Fied poi or geeral coracio i meric spaces Mah. Japaica 24 (979) pp 7-24 [2] S. V. R. Naid K.P.R. Rao ad N. Sriivasa Rao O he opology o - meric spaces ad geeraio o -meric spaces I. J. Mah. Mah. Sci. 5 (2004) pp [3] S. V. R. Naid K.P.R. Rao ad N. Sriivasa Rao O coceps o balls i - meric spaces I. J. Mah. Mah. Sci. (2005) pp [4] S. V. R. Naid K.P.R. Rao ad N. Sriivasa Rao O coverge seqeces ad ied poi heorems i -meric spaces I. J. Mah. Mah. Sci. (2005) pp

7 Fied Poi Theorems or -Uiormly Locally Geeralized Coracios 83 [5] B.E. Rhoades A ied poi heorem or geeralized meric spaces I. J. Mah. Sci. (2005). [6] B. Sigh ad R.K. Sharma Commo ied pois via compaible maps i - meric space Radovi Maemeicki (2002); pp [7] Shaba Sedghi Nabi Shobe ad Haiy Zho A commo ied poi heorem i -meric spaces Joral o ied poi heory ad applicaios Haidavi Pblishig corporaio Volme 2007 Aricle I page. [8] G Sdhaamsh Moha Reddy A Commo Fied Poi heorem o complee G-meric spaces Ieraioal Joral o Pre ad Applied Mahemaics Volme 8 2 (208); pp

8 84 G Sdhaamsh Moha Reddy