IJPAM: Volume 117, No. 3 (2017)
Title
EDELSTEIN'S THEOREM FOR NON-SELFMULTIVALUED CONTRACTIVE MAPPINGS
Authors
V. Sankar Raj, S. Jamal FathimaDepartment of Mathematics
Manonmaniam Sundaranar University
Tirunelveli 627 012, Tamilnadu, INDIA
Abstract
Let us consider two nonempty subsets and of a metric space and an upper semicontinuous multivalued non-self mapping , where denotes the set of all nonempty subsets of . It is worth mentioning that the notion of iterated sequence is meaningless since the mapping is a non-self mapping. In this article, we introduce a new type of Picard's iteration-like sequence for a non-self multivalued mapping and provided sufficient conditions for the existence of a point in for which the distance between the point and the set is optimum. Using this notion, we obtain a generalized Edelstein's theorem for non-self multivalued contractive mappings.History
Received: 2017-01-28
Revised: 2017-08-09
Published: January 11, 2018
AMS Classification, Key Words
AMS Subject Classification: 47H10, 47J25, 54H25
Key Words and Phrases: multivalued contractive, property, upper semicontinuous, best proximity point, fixed point, Picard's iteration-like sequence, diminishing sequence
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Bibliography
- 1
- M. A. Al-Thagafi, N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Analysis, 70, No. 3 (2009), 1209-1216, doi: https://doi.org/10.1016/j.na.2008.02.004.
- 2
- A. Amini-Harandi, Best proximity points for proximal generalized contractions in metric spaces, Optim. Lett., 7, No. 5 (2013), 913-921, doi: https://doi.org/10.1007/s11590-012-0470-z.
- 3
- N. Bilgili, E. Karapinar, K. Sadarangani, A generalization for the best proximity point of Geraghty-contractions, J. Inequal. Appl., (2013), 9, doi: https://doi.org/10.1186/1029-242X-2013-286.
- 4
- M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc., 12, (1961), 7-10, doi: https://doi.org/10.1090/S0002-9939-1961-0120625-6.
- 5
- A.A. Eldred, P. Veeramani, On best proximity pair solutions with applications to differential equations, The Indian Math. Soc., Special Centenary Volume(1907-2007) (2008), 51-62.
- 6
- M. Gabeleh, Best proximity point theorems for single- and set-valued non-self mappings, Acta Math. Sci. Ser. B Engl. Ed., 34, No. 5 (2014), 1661-1669, doi: https://doi.org/10.1016/S0252-9602(14)60112-0.
- 7
- M. Gabeleh, Best proximity point theorems via proximal non-self mappings, J. Optim. Theory Appl., 164, No. 2 (2015), 565-576, doi: https://doi.org/10.1007/s10957-014-0585-8.
- 8
- M. Gabeleh, Best proximity points: global minimization of multivalued non-self mappings, Optim. Lett., 8, No. 3 (2014), 1101-1112, doi: https://doi.org/10.1007/s11590-013-0628-3.
- 9
- N. Hussain, A. Latif, P. Salimi, Best proximity point results for modified Suzuki --proximal contractions, Fixed Point Theory and Applications, 2014 2014:10, (2014), 16, doi: https://doi.org/10.1186/1687-1812-2014-10.
- 10
- W.K. Kim, K.H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl., 316, No. 2 (2006), 433-446, doi: https://doi.org/10.1016/j.jmaa.2005.04.053.
- 11
- W.K. Kim, Sangho Kum, K.H. Lee, On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear Analysis, 68, No. 8 (2008), 2216-2227, doi: https://doi.org/10.1016/j. na.2007.01.057.
- 12
- W. A. Kirk, S. Reich, P.Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24, No. 7-8 (2003), 851-862, doi: https://doi.org/10.1081/NFA-120026380.
- 13
- S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30, (1969), 475-488, http://projecteuclid.org/euclid.pjm/1102978504.
- 14
- B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226, (1977), 257-290, doi: https://doi.org/10.1090/S0002-9947-1977-0433430-4.
- 15
- S. Sadiq Basha, P. Veeramani, Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103, No. 1 (2000), 119-129, doi: https://doi.org/10.1006/jath.1999.3415.
- 16
- V. Sankar Raj, A best proximity point theorem for weakly contractive non-self mappings, Nonlinear Analysis, 74, (2011), 4804-4808, doi: https://doi.org/10.1016/j.na.2011.04.052.
- 17
- V. Sankar Raj, A.A. Eldred, A characterization of strictly convex spaces and applications, J. Optim. Theory Appl., 160, No. 2 (2014), 703-710, doi: https://doi.org/10.1007/s10957-013-0357-x.
- 18
- N. Shahzad, S. Sadiq Basha, R. Jeyaraj, Common best proximity points: global optimal solutions, J. Optim. Theory Appl., 148, No. 1 (2011), 69-78, doi: https://doi.org/s10957-010-9745-7.
- 19
- R. E. Smithson, Fixed points for contractive multifunctions, Proc. Amer. Math. Soc., 27, (1971), 192-194, doi: https://doi.org/10.1090/S0002-9939-1971-0267564-4.
- 20
- P. S. Srinivasan, P. Veeramani, On existence of equilibrium pair for constrained generalized games, Fixed Point Theory Appl., No. 1 (2004) , 21-29, doi: https://doi.org/10.1155/S1687182004308132.
- 21
- C. Vetro, P. Salimi, Best proximity point results in non-Archimedean fuzzy metric spaces, Fuzzy Inf. Eng., 5, No. 4 (2013), 417-429, doi: https://doi.org/10.1007/s12543-013-0155.
How to Cite?
DOI: 10.12732/ijpam.v117i3.2 How to cite this paper?Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 117
Issue: 3
Pages: 375 - 382
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