International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
ISSN (Print): 2279-0020 ISSN (Online): 2279-0039
International Journal of Engineering, Business and Enterprise Applications (IJEBEA) www.iasir.net A Fixed Point Theorem of đ?œş −chainable Intuitionistic Fuzzy Metric Space Syed Shahnawaz Ali1 , Jainendra Jain2 , P.L. Sanodia3 , and Shilpi Jain4 Corporate Institute of Science & Technology, Hataikheda, Patel Nagar, Raisen Road, Bhopal, M.P., India. 2 O.P. Jindal Institute of Technology, Knowledge Park, Gharghoda Road, Punjipathra, Raigarh, C.G., India. 3 Institute for Excellence in Higher Education, Kaliyasot Dam, Kolar Road, Bhopal, M.P., India. 4 Govt. Motilal Vigyan Mahavidyalaya, Jehangirabad Road, Bhopal, M.P. India. 1
Abstract: Fuzzy Mathematics has seen an enormous growth since the introduction of notion of fuzzy sets by Zadeh in 1965. Kramosil and Michalek introduced the notion of fuzzy metric spaces which was later modified by George and Veeramani and others. The notion of intuitionistic fuzzy metric spaces was introduced by Park in 2004. The intuitionistic fuzzy fixed point theory has become an area of interest for specialists in fixed point theory as intuitionistic fuzzy mathematics has covered new possibilities for fixed point theorists. In this paper, we give some conditions of which four self mappings of Îľ-chainable Intuitionistic fuzzy metric spaces have a unique common fixed point. Keywords:Fixed Points, Fuzzy sets, Fuzzy Metric Spaces, Intuitionistic Fuzzy Sets, Intuitionistic Fuzzy Metric Spaces. I. Introduction The Fuzzy Mathematics commenced with the introduction of the notion of fuzzy sets by Zadeh ([1], 1965), as a new way to represent the vagueness in everyday life. In mathematical programming, problems are expressed as optimizing some goal function given certain constraints, and there are real life problems that consider multiple objectives. Generally, it is very difficult to get a feasible solution that brings us to the optimum of all objective functions. A possible method of resolution, that is quite useful, is the one using fuzzy sets by D.Turkoglu and B.E.Rhoades ([2], 2005). Atanassov([3], 1986), introduced the notion of intuitionistic fuzzy sets by generalizing the notion of fuzzy set by treating membership as a fuzzy logical value rather than a single truth value. For an intuitionistic set the logical value has to be consistent (in the senseđ?›žđ??´ (đ?‘Ľ) + đ?œ‡đ??´ (đ?‘Ľ) ≼ 1).đ?›žđ??´ (đ?‘Ľ) and đ?œ‡đ??´ (đ?‘Ľ) degree of membership and degree of non-membership, respectively. All result which holds of fuzzy sets can be transformed intuitionistic fuzzy sets but converse need not be true. Intuitionistic fuzzy set can be viewed in the context as a proper tool for representing hesitancy concerning both membership and non-membership of an element to a set. To be more precise, a basic assumption of fuzzy set theory that if we specify the degree of membership of an element in a fuzzy set as a real number from [0,1], say đ?‘Ž, then the degree of its nonmembership is automatically determined as (1 − đ?‘Ž), need not hold for intuitionistic fuzzy sets. In intuitionistic fuzzy set theory it is assumed that non-membership should not be more than (1 − đ?‘Ž). The application of intuitionistic fuzzy set instead of fuzzy sets means the introduction of another degree of freedom into a set description.Fixed point and common fixed point properties for mappings defined on fuzzy metric spaces , intuitionistic fuzzy metric spaces have been studied by many authors like Coker.([4], 1997), S. Sharma ([5], 2002), V. Gregori and A. Sapena ([7], 2002), C. Alacaetal. ([6], 2006). Most of the properties which provide the existence of fixed points and common fixed points are of linear contractive type conditions. Saadati and park ([8], 2006) studied the concept of intuitionistic fuzzy metric space and its applications. Further, they introduced the notion of Cauchy sequences in an intuitionistic fuzzy metric space and proved the well-known fixed point theorem of Banach and Edelstein extended to intuitionistic fuzzy metric space with the help of Grabiec([2], 1988) gave a generalization of Jungck’s common fixed point theorem Jungck G.([14], 1986) to intuitionistic fuzzy metric spaces. They first formulated the definition of weakly commuting and R-weakly commuting mappings in intuitionistic fuzzy metric spaces and proved the intuitionistic fuzzy version of Pant’s theorem. In this Paper, we give some conditions of which four self-mappings of Îľ −chainable Intuitionistic fuzzy metric spaces have a unique common fixed point. II. Preliminaries Definition 2.1.: (Schweizer B. and Sklar A., 1960)A binary operation ∗: [0,1] Ă— [0,1] â&#x;ś [0,1] is a continuous đ?‘Ą −norm if ∗ is satisfying the following conditions (a) ∗ is commutative and associative ; (b) ∗ is continuous ; (c) đ?‘Ž ∗ 1 = đ?‘Ž for all đ?‘Ž ∈ [0,1]; (d) đ?‘Ž ∗ đ?‘? ≤ đ?‘? ∗ đ?‘‘ whenever đ?‘Ž ≤ đ?‘? and đ?‘? ≤ đ?‘‘, for all đ?‘Ž, đ?‘?, đ?‘?, đ?‘‘ ∈ [0,1].
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