Weak contractions on chains in a generalized metric space with a partial order
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Weak contraction mapping principle is a generalization of the Banach contraction mapping principle. Weakly contractive mappings are intermediate to contraction mappings and nonexpansive mappings. They have been studied in several contexts. Metric fixed point theory in partially ordered spaces have rapidly developed in recent times. In this paper we extend the concept of weak contraction to subset of a partially ordered generalized metric space which are chains by themselves. It is noted that this weak contraction is different from weak contraction on the whole space. We prove here that under certain assumptions the weakly contractive mapping on certain chains will have a fixed point. Two illustrative examples are given.
Keywords\(G\)-metric space Partially ordered set Weak contraction Fixed point Orbit Monotone property
Mathematics Subject Classification (2000)54H25
The work is supported by the Council of Scientific and Industrial Research, Government of India, under Research Project No - 25(0168)/09/EMR-II. The support is gratefully acknowledged. The authors acknowledge the suggestions of the learned referees.
- 3.Mustafa, Z., Obiedat, H., Awawdeh, F.: Some of Fixed Point Theorem for Mapping on Complete G-Metric Spaces. Fixed Point Theory Appl. 2008 (2008), Article ID 189870.Google Scholar
- 6.Mustafa, Z., Shatanawi, W., Bataineh, M.: Existence of Fixed Point Result in G-Metric Spaces. Int. J. Math. Math. Sci. 2009 (2009) Article ID 283028.Google Scholar
- 7.Mustafa, Z., Sims, B.: Fixed Point Theorems for Contractive Mappings in Complete G-metric Space. Fixed Point Theory Appl. 2009 (2009) Article ID 917175.Google Scholar
- 9.Chugh, R., Kadian, T., Rani, A., Rhoades, B.E.: Property P in G-Metric Spaces. Fixed Point Theory Appl. 2010 (2010), Article ID 401684.Google Scholar
- 11.Shatanawi, W.: Fixed Point Theory for Contractive Mappings satisfying \(\phi \)-Maps in G-metric Spaces. Fixed Point Theory Appl. 2010 (2010) Article ID 181650.Google Scholar
- 12.Mustafa, Z., Aydi, H., Karapinar, E.: On common fixed points in G-metric spaces using (E.A) property. Comput. Math. Appl. doi: 10.1016/j.camwa.2012.03.051.
- 21.Alber, Ya.I., Guerre-Delabriere, S.: Principles of weakly contractive maps in Hilbert spaces, in:I. Gohberg, Yu. Lyubich(Eds.), New Results in Operator Theory. In : Advances and Appl. 98, Birkhäuser, Basel, 7–22 (1997)Google Scholar
- 27.Dutta, P. N., Choudhury, B.S.: A Generalisation of Contraction Principle in Metric Spaces. Fixed Point Theory Appl. 2008 (2008), Article ID 406368, p. 8Google Scholar
- 30.Ciric, L., Cakic, N., Rajovic, M., Ume, J. S.: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008 (2008), Article ID 131294.Google Scholar