J. Adv. Math. Stud. Vol. 7(2014), No. 2, 20-27 http://journal.fairpartners.ro SOME EXISTENCE THEOREMS UNDER NONEXPANSIVE MAPPINGS IN NONCOMPLETE METRIC SPACE RAVINDRA K. BISHT Abstract. The aim of the present paper is two fold. Firstly, we investigate the existence of common fixed points for a pair of self-mappings satisfying non-expansive condition but without assuming the completeness of the space or containment of the ranges of the involved mappings. Secondly, we generalize the result obtained in first part for sequence of mappings, wherein some mappings need not require to satisfy any noncommuting conditions. REFERENCES [1] M. Aamri and D. El. Moutawakil: Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270(2002), 181-188. [2] R. K. Bisht: Common fixed points of generalized Meir-Keeler type condition and nonexpansive mappings, Internat. J. Math. Math. Sci., Volume 2012, Article ID 786814, 12 pages. [3] G. Jungck: Commuting mappings and fixed point, Amer. Math. Monthly, 83(1976), 261-263. [4] G. Jungck: Compatible mappings and common fixed points, Internat. J. Math. Math Sci., 9(1986), 771-779. [5] G. Jungck: Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci., 4(1996), 199-215. [6] M. A. Al-Thagafi and Naseer Shahzad: Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sinica, 24(2008), 867-876. [7] R. P. Pant: Common fixed points of noncommuting mappings, J. Math. Anal.Appl., 188(1994), 436-440. [8] R. P. Pant: Discontinuity and fixed points, J. Math. Anal. Appl., 240(1999), 284-289. [9] R. P. Pant: Common fixed points of Lipschitz type mapping pairs, J. Math. Anal. Appl., 240(1999), 280-283. [10] R. P. Pant: Fixed points of nonexpansive mapping and a generalized notion of compactness, Bull. Cal. Math. Soc., 99(2007), No. 1, 45-52. [11] Vyomesh Pant and R. P. Pant: Common fixed points of conditionally commuting maps, Fixed Point Theory, 11(2011), 113-118. [12] M. Chandra, S. N. Misra, S. L. Singh and B. E. Rhoades: Coincidences and fixed points of nonexpansive type multivalued and single valued maps, Indian J. Pure Appl. Math., 26(1995), 393- 401. [13] K. P. R. Sastry and S. R. Krishna Murthy: Common fixed points of two partially commuting tangential selfmaps on a metric space, J. Math. Anal. Appl., 250(2000), 731-734. [14] S. Sessa: On a weak commutativity condition in fixed point considerations, Publ. Inst. Math. (Beograd) (NS), 34(46)(1982) 149-153. [15] Wutiphol Sintunavarat and Poom Kumam: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math., Volume 2011, Article ID 637958, 14 pages, doi:10.1155/2011/ 637958. Received: 20 July, 2013. Revised: 28 February, 2014. 2010 Mathematics Subject Classification: 47H10, 54H25. Key words and phrases: Nonexpansive mappings, common fixed point, (CLRg ) property and conditional commutativity. c 2014 Fair Partners Team for the Promotion of Science & Fair Partners Publishers 20 Some existence theorems under nonexpansive mappings in noncomplete metric space 21 [16] Kadelburg, S. Radenovi´c and N. Shahzad: A note on various classes of compatible-type pairs of mappings and common fixed point theorems, Abstr. Appl. Anal., 2013, Article ID 697151, 6 pages. [17] M. Imdad and S. Chauhan: Employing common limit range property to prove unified metrical common fixed point theorems, Intern. J. Analysis 2013, Article ID 763261, 16 pages. [18] M. Imdad, S. Chauhan and Z. Kadelburg: Fixed point theorems for mappings with common limit range property satisfying generalized (φ − ϕ) weak contractive conditions, Math. Sciences 7:16 (2013), doi:10.1186/2251- 7456-7-16. [19] M. Imdad, B.D. Pant and S. Chauhan: Fixed point theorems in Menger spaces using the (CLRST ) property and applications, J. Nonlinear Anal. Optim., 3(2012), No. 2, 225-237. Bipin Tripahti Kumaon Institute of Technology, Department of Mathematics-Applied Sciences and Humanities, PDwarahat-262553 Almora, India E-mail address: [email protected]
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