Ijebea15 445

Page 1

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].

IJEBEA 15-445; Š 2015, IJEBEA All Rights Reserved

Page 40

Syed Shahnawaz Aliet al., International Journal of Engineering, Business and Enterprise Applications, 14(1), September-November, 2015, pp. 40-45

Definition 2.2.: (Schweizer B. and A. , 1960)A binary operation â—Š: [0,1] Ă— [0,1] â&#x;ś [0,1] is a continuous đ?‘Ą −norm if đ?‘? is satisfying the following conditions (a) â—Š is commutative and associative ; (b) â—Š is continuous ; (c) đ?‘Ž â—Š 0 = đ?‘Ž for all đ?‘Ž ∈ [0,1]; (d) đ?‘Ž â—Š đ?‘? ≤ đ?‘? â—Š đ?‘‘ whenever đ?‘Ž ≤ đ?‘? and đ?‘? ≤ đ?‘‘, for all đ?‘Ž, đ?‘?, đ?‘?, đ?‘‘ ∈ [0,1]. Definition 2.3.: A 5-tuple (đ?‘‹, â„ł,∗,â—Š) is said to be an intuitionistic fuzzy metric space if đ?‘‹ is an arbitrary set, ∗ is a continuous đ?‘Ą − norm, is a continuous đ?‘Ą −conorm and â„ł, đ?‘ are fuzzy set on đ?‘‹ 2 Ă— (0, ∞) satisfying the following conditions: (i) â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) + đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) ≤ 1 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹ and đ?‘Ą > 0; (ii) â„ł(đ?‘Ľ, đ?‘Ś, 0) = 0 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹; (iii) â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 1 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹andđ?‘Ą > 0 if and only if đ?‘Ľ = đ?‘Ś; (iv) â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) = â„ł(đ?‘Ś, đ?‘Ľ, đ?‘Ą) for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹andđ?‘Ą > 0; (v) â„ł (đ?‘Ľ, đ?‘Ś, đ?‘Ą) ∗ â„ł(đ?‘Ś, đ?‘§, đ?‘ ) ≤ â„ł(đ?‘Ľ, đ?‘§, đ?‘Ą + đ?‘ ) for all đ?‘Ľ, đ?‘Ś, đ?‘§ ∈ đ?‘‹ and s, đ?‘Ą > 0; (vi) â„ł(đ?‘Ľ, đ?‘Ś, . ): (0, ∞) â&#x;ś [0,1] is left continuous for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹; (vii) lim â„ł (đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 1 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹andđ?‘Ą > 0; đ?‘›â†’∞

(viii) đ?‘ (đ?‘Ľ, đ?‘Ś, 0) = 1 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹; (ix) đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 0 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹and đ?‘Ą > 0 if and only if đ?‘Ľ = đ?‘Ś; (x) đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) = đ?‘ (đ?‘Ś, đ?‘Ľ, đ?‘Ą) for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹andđ?‘Ą > 0; (xi) đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) â—Š đ?‘ (đ?‘Ś, đ?‘§, đ?‘ ) ≼ đ?‘ (đ?‘Ľ, đ?‘§, đ?‘Ą + đ?‘ ) for all đ?‘Ľ, đ?‘Ś, đ?‘§ ∈ đ?‘‹ and s, đ?‘Ą > 0; (xii) đ?‘ (đ?‘Ľ, đ?‘Ś, . ): (0, ∞) â&#x;ś [0,1] is right continuous for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹; (xiii) lim đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 0 for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹; đ?‘›â†’∞

Then (â„ł, đ?‘ ) is called an intuitionistic fuzzy metric on đ?‘‹. The functions â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą)denote the degree of nearness between đ?‘Ľ and đ?‘Ś with respect to đ?‘Ą, respectively. Example1 Let đ?‘‹ = â„•.Define đ?‘Ž ∗ đ?‘? = đ?‘šđ?‘Žđ?‘Ľ{0, đ?‘Ž + đ?‘? − 1}and đ?‘Ž â—Š đ?‘? = đ?‘Ž + đ?‘? − đ?‘Žđ?‘?for allđ?‘Ž, đ?‘? ∈ [0,1] and let â„ł and đ?‘ be fuzzy sets on đ?‘‹ 2 Ă— (0, ∞) defined as follows: đ?‘Ľ , đ?‘–đ?‘“ đ?‘Ľ ≤ đ?‘Ś đ?‘Ś â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą ) = {đ?‘Ś , đ?‘–đ?‘“ đ?‘Ś ≼ đ?‘Ľ đ?‘Ľ đ?‘Śâˆ’đ?‘Ľ , đ?‘–đ?‘“ đ?‘Ľ ≤ đ?‘Ś đ?‘Ś đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą ) = {đ?‘Ľ − đ?‘Ś , đ?‘–đ?‘“ đ?‘Ś ≼ đ?‘Ľ đ?‘Ľ for all đ?‘Ľ, đ?‘Ś đ?œ– đ?‘‹ and đ?‘Ą > 0. Then (đ?‘‹, â„ł,∗,â—Š)is an intuitionistic fuzzy metric space. Definition 2.4.: (Atanassov K.T., 1986)An intuitionistic fuzzy set đ?’œđ?œ ,đ?œ‚ in a universe đ?‘ˆ is an object đ?’œđ?œ ,đ?œ‚ = {(đ?œ đ?’œ(đ?‘˘), đ?œ‚đ?’œ(đ?‘˘))|đ?‘˘ ∈ đ?‘ˆ}, where, for all đ?‘˘ ∈ đ?‘ˆ, đ?œ đ?’œ(đ?‘˘) ∈ [0, 1] and đ?œ‚đ?’œ(đ?‘˘) ∈ [0, 1] are called the membershipdegree and the non-membership degree respectively of đ?‘˘in đ?’œđ?œ ,đ?œ‚ and furthermore they satisfy đ?œ đ?’œ(đ?‘˘) + đ?œ‚đ?’œ(đ?‘˘) ≤ 1. Remark 2.5.:(Alaca C. ,Turkoglu D. and Aitun I, 2008) An intuitionistic fuzzy metric spaces with continuous đ?‘Ą −norm ∗ and continuous đ?‘Ą − conorm â—Š defined by đ?‘Ž ∗ đ?‘Ž ≼ đ?‘Ž and (1 − đ?‘Ž) â—Š (1 − đ?‘Ž) ≤ (1 − đ?‘Ž) for all đ?‘Ž ∈ [0,1]. Then for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹, â„ł(đ?‘Ľ, đ?‘Ś,∗) is non decreasing and đ?‘ (đ?‘Ľ, đ?‘Ś,â—Š) is non increasing. Remark 2.6(Alaca, Turkoglu and Yildiz,2006) ,Inintuitionistic fuzzy metric space đ?‘‹, â„ł(đ?‘Ľ, đ?‘Ś, . ) is non decreasing and đ?‘ (đ?‘Ľ, đ?‘Ś, . ) is non-increasing for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹. (Alaca, Turkoglu and Yildiz,2006), introduced the following notions: Definition 2.7.: (Alaca, Turkoglu and Yildiz,2006) Let (đ?‘‹, â„ł,∗,â—Š) be an intuitionistic fuzzy metric space.Then (i) A sequence {đ?‘Ľđ?‘› } in đ?‘‹ is said to be Cauchy sequence if for all đ?‘Ą > 0, and

đ?‘? > 0,

lim â„ł (đ?‘Ľđ?‘›+đ?‘? , đ?‘Ľđ?‘› , đ?‘Ą) = 1, lim đ?‘ (đ?‘Ľđ?‘›+đ?‘? , đ?‘Ľđ?‘› , đ?‘Ą) = 0.

đ?‘›â†’∞

đ?‘›â†’∞

(ii) A sequence {đ?‘Ľđ?‘› } in đ?‘‹ is said to be convergent to a point đ?‘Ľ ∈ đ?‘‹ if, for all đ?‘Ą > 0, lim â„ł (đ?‘Ľđ?‘› , đ?‘Ľ, đ?‘Ą) = 1, lim đ?‘ (đ?‘Ľđ?‘› , đ?‘Ľ, đ?‘Ą) = 0. đ?‘›â†’∞

đ?‘›â†’∞

Since ∗ and â—Š are continuous, the limit is uniquely determined from (v) and (xi) of definition 2.3, respectively. Definition 2.8.: An intuitionistic fuzzy metric space (đ?‘‹, â„ł,∗,â—Š) is said to be complete if and only if every Cauchy sequence in đ?‘‹ is convergent.

IJEBEA 15-445; Š 2015, IJEBEA All Rights Reserved

Page 41

Syed Shahnawaz Aliet al., International Journal of Engineering, Business and Enterprise Applications, 14(1), September-November, 2015, pp. 40-45

Definition 2.9.: An intuitionistic fuzzy metric space (đ?‘‹, â„ł,∗,â—Š) is said to be compact if every sequence in đ?‘‹ contains a convergent subsequence. Lemma 2.10.:.Let (đ?‘‹, â„ł,∗,â—Š)be an intuitionistic fuzzy metric space and for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‹, đ?‘Ą > 0 and if for a numberđ?‘˜ ďƒŽ (0, 1), â„ł(đ?‘Ľ, đ?‘Ś, đ?‘˜đ?‘Ą) ≼ â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą)and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘˜đ?‘Ą) ≤ đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) then đ?‘Ľ = đ?‘Ś. Proof.Since â„ł(đ?‘Ľ, đ?‘Ś, đ?‘˜đ?‘Ą) ≼ â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą), and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘˜đ?‘Ą) ≤ đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą), then using results of (Sharma, 2002) , we đ?‘Ą đ?‘Ą have â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) ≼ â„ł(đ?‘Ľ, đ?‘Ś, ) and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) ≤ đ?‘ (đ?‘Ľ, đ?‘Ś, ). By repeated application of above inequalities, we đ?‘˜ đ?‘Ą

đ?‘Ą

đ?‘˜

đ?‘˜

have ℳ(�, �, �) ≼ ℳ(�, �, ) ≼ ℳ(�, �, �

and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) ≤ đ?‘ (đ?‘Ľ, đ?‘Ś, ) ≤ đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘˜

đ?‘Ą đ?‘˜2

đ?‘˜

2 ) ≼. . . . ≼ ℳ(�, �,

) ≤. . . . ≤ đ?‘ (đ?‘Ľ, đ?‘Ś,

đ?‘Ą đ?‘˜đ?‘›

đ?‘Ą đ?‘˜đ?‘›

) ≼⋯

) ≤⋯

For đ?‘› ďƒŽ â„• , which tends to and 0 as đ?‘› → ∞, respectively. Thus â„ł(đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 1 and đ?‘ (đ?‘Ľ, đ?‘Ś, đ?‘Ą) = 0 for all đ?‘Ą > 0 and we get đ?‘Ľ = đ?‘Ś. Lemma2.11.:. Let (đ?‘‹, â„ł,∗,â—Š)be an intuitionistic fuzzy metric space and {đ?‘Śđ?‘› } be a sequence in đ?‘‹. If there exists a number đ?‘˜ďƒŽ (0, 1)such that â„ł(đ?‘Śđ?‘›+2 , đ?‘Śđ?‘›+1 , đ?‘˜đ?‘Ą) ≼ â„ł(đ?‘Śđ?‘›+1 , đ?‘Śđ?‘› , đ?‘Ą), đ?‘ (đ?‘Śđ?‘›+2 , đ?‘Śđ?‘›+1 , đ?‘˜đ?‘Ą) ≤ đ?‘ (đ?‘Śđ?‘›+1 , đ?‘Śđ?‘› , đ?‘Ą) (1) for all đ?‘Ą > 0and đ?‘› = 1, 2, . . ., then {đ?‘Śđ?‘› } is a Cauchy sequence in đ?‘‹. Proof. By simple induction with the condition (1) with the help of (C. Alaca etal.,2006), we have for all đ?‘Ą > 0 and đ?‘› = 0, 1, 2, â‹Ż, đ?‘Ą â„ł(đ?‘Śđ?‘›+1 , đ?‘Śđ?‘›+2 , đ?‘Ą) ≼ â„ł (đ?‘Ś1 , đ?‘Ś2 , đ?‘› ), đ?‘˜ đ?‘Ą đ?‘ (đ?‘Śđ?‘›+1 , đ?‘Śđ?‘›+2 , đ?‘Ą) ≤ đ?‘ (đ?‘Ś1 , đ?‘Ś2 , đ?‘› ) (2) đ?‘˜ Thus by (2) and using definition 2.3((v) and (xi) ),for any positive integer đ?‘?and real number đ?‘Ą > 0, we have đ?‘Ą đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ đ?‘Ą â„ł(đ?‘Śđ?‘› , đ?‘Śđ?‘›+đ?‘? , đ?‘Ą) ≼ â„ł (đ?‘Śđ?‘› , đ?‘Śđ?‘›+1 , ) ∗ â‹Ż ∗ â„ł(đ?‘Śđ?‘›+đ?‘?−1 đ?‘Śđ?‘›+đ?‘? , ) đ?‘? đ?‘? đ?‘Ą đ?‘Ą đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ ≼ â„ł(đ?‘Ś1 , đ?‘Ś2 , đ?‘›âˆ’1 ) ∗ â‹Ż ∗ â„ł(đ?‘Ś1 , đ?‘Ś2 , đ?‘›+đ?‘?−2 ) đ?‘?đ?‘˜ đ?‘?đ?‘˜ đ?‘Ą đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘

And đ?‘ (đ?‘Śđ?‘› , đ?‘Śđ?‘›+đ?‘? , đ?‘Ą) ≤ đ?‘ (đ?‘Śđ?‘› , đ?‘Śđ?‘›+1 , ) đ?‘?

đ?‘Ą

∗ â‹Ż ∗ đ?‘ (đ?‘Śđ?‘›+đ?‘?−1 đ?‘Śđ?‘›+đ?‘? , ) đ?‘?

đ?‘Ą đ?‘Ą )đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ ∗ â‹Ż ∗ đ?‘ (đ?‘Ś1 , đ?‘Ś2 , đ?‘›+đ?‘?−2 ). đ?‘›âˆ’1 đ?‘?đ?‘˜ đ?‘?đ?‘˜ Therefore, by definition 2.3((vii) and (xiii) ), we have lim â„ł (đ?‘Śđ?‘› , đ?‘Śđ?‘›+đ?‘? , đ?‘Ą) ≼ 1đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ ∗ â‹Ż ∗ 1 ≼ 1. ≤ đ?‘ (đ?‘Ś1 , đ?‘Ś2 ,

đ?‘›â†’∞

And lim N (đ?‘Śđ?‘› , đ?‘Śđ?‘›+đ?‘? , đ?‘Ą) ≤ 0đ?‘?−đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ â—Š â‹Ż â—Š 0 ≤ 0.

đ?‘›â†’∞

Which implies that {đ?‘Śđ?‘› }is a Cauchy sequence in đ?‘‹. This completes the proof. (Alaca Turkoglu and Yildiz, 2008) introduced the notions of compatible mappings in intuitionistic fuzzy metric space, akin to the concept of compatible mappings introduced by (Jungck, G., 1986) in metric spaces as follows: Definition2.12.: A pair of self mappings (đ?‘“, đ?‘”) of an intuitionistic fuzzy metric space (đ?‘‹, â„ł,∗,â—Š) is said to be compatible if lim â„ł (đ?‘“đ?‘”đ?‘Ľđ?‘› , đ?‘”đ?‘“đ?‘Ľđ?‘› , đ?‘Ą) = 1, and lim đ?‘ (đ?‘“đ?‘”đ?‘Ľđ?‘› , đ?‘”đ?‘“đ?‘Ľđ?‘› , đ?‘Ą) = 0 for every đ?‘Ą > 0, whenever {đ?‘Ľđ?‘› } is a đ?‘›â†’∞

đ?‘›â†’∞

sequence in đ?‘‹ such that

lim đ?‘“đ?‘Ľđ?‘› = lim đ?‘”đ?‘Ľđ?‘› = đ?‘§ for some đ?‘§ ∈ đ?‘‹.

đ?‘›â†’∞

đ?‘›â†’∞

Definition2.13.: A pair of self mappings (đ?‘“, đ?‘”) of an intuitionistic fuzzy metric space (đ?‘‹, â„ł,∗,â—Š) is said to be non compatible if lim â„ł (đ?‘“đ?‘”đ?‘Ľđ?‘› , đ?‘”đ?‘“đ?‘Ľđ?‘› , đ?‘Ą) ≠1, or non-existent and lim đ?‘ (đ?‘“đ?‘”đ?‘Ľđ?‘› , đ?‘”đ?‘“đ?‘Ľđ?‘› , đ?‘Ą) ≠0 or non-existent đ?‘›â†’∞

đ?‘›â†’∞

for every đ?‘Ą > 0, whenever {đ?‘Ľđ?‘› } is a sequence in đ?‘‹ such that lim đ?‘“đ?‘Ľđ?‘› = lim đ?‘”đ?‘Ľđ?‘› = đ?‘§ for some đ?‘§ ∈ đ?‘‹. đ?‘›â†’∞

đ?‘›â†’∞

(Jungck, G. and Rhoades, 1998) introduced the concept of weakly compatible maps as follows: Definition2.14.: Two self maps đ?‘“ and đ?‘” are said to be weakly compatible if they commute at coincidence points. Definition2.15.: Let (đ?‘‹, â„ł,∗,â—Š)be an intuitionistic fuzzy metric space. A finite sequence đ?‘Ľ = đ?‘Ľ0 , đ?‘Ľ1 , đ?‘Ľ2 , â‹Ż , đ?‘Ľđ?‘› = đ?‘Ś is called ∈ −chain from đ?‘Ľ to đ?‘Ś if there exists a positive number ∈ > 0 such that â„ł(đ?‘Ľđ?‘– , đ?‘Ľđ?‘–−1 , đ?‘Ą) > 1−∈ and đ?‘ (đ?‘Ľđ?‘– , đ?‘Ľđ?‘–−1 , đ?‘Ą) > 1−∈ for all đ?‘Ą > 0 and đ?‘– = 1,2, â‹Ż , đ?‘›. An intuitionistic fuzzy metric space(đ?‘‹, â„ł,∗,â—Š) is called ∈ −chainable if for any đ?‘Ľ, đ?‘Ś ∈ đ?‘‹, there exists an

IJEBEA 15-445; Š 2015, IJEBEA All Rights Reserved

Page 42

Syed Shahnawaz Aliet al., International Journal of Engineering, Business and Enterprise Applications, 14(1), September-November, 2015, pp. 40-45

∈ −chain from đ?‘Ľ tođ?‘Ś. III. Main Result Theorem3.1. Let đ??´, đ??ľ, đ?‘†, and đ?‘‡ be self maps of a complete ∈ − chainable intuitionistic fuzzy metric spaces(đ?‘‹, â„ł,∗,â—Š) with continuous đ?‘Ą −norm ∗ and continuous đ?‘Ą −conorm â—Š defined by đ?‘Ž ∗ đ?‘Ž ≼ đ?‘Ž and (1 − đ?‘Ž) â—Š (1 − đ?‘Ž) ≤ (1 − đ?‘Ž) for all đ?‘Ž ∈ [0,1] Satisfying the following condition: 1. đ??´(đ?‘‹) ⊆ đ?‘‡(đ?‘‹) and đ??ľ(đ?‘‹) ⊆ đ?‘†(đ?‘‹), 2. đ??´ and đ?‘† are continuous , 3. The pairs (đ??´, đ?‘†) and (đ??ľ, đ?‘‡) are weakly compatible, 4. There exist đ?‘ž ∈ (0,1) such that 1 1 â„ł(đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘žđ?‘Ą) ≼ {â„ł(đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) ∗ â„ł(đ?‘†đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą) ∗ [â„ł(đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) + â„ł(đ??´đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)] ∗ [â„ł(đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą) + 2

2

â„ł(đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą)] ∗ â„ł(đ??ľđ?‘Ś, đ?‘‡đ?‘Ś, đ?‘Ą) ∗ â„ł(đ??´đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) ∗ â„ł(đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą)} and đ?‘ (đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘žđ?‘Ą) ≤ {đ?‘ (đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) â—Š 1

1

đ?‘ (đ?‘†đ?‘Ľ, đ??´đ?‘Ľ, đ?‘Ą) â—Š [đ?‘ (đ?‘†đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) + đ?‘ (đ??´đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą)] â—Š [đ?‘ (đ??´đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą) + đ?‘ (đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą)] â—Š đ?‘ (đ??ľđ?‘Ś, đ?‘‡đ?‘Ś, đ?‘Ą) â—Š 2

2

đ?‘ (đ??´đ?‘Ľ, đ?‘‡đ?‘Ś, đ?‘Ą) â—Š đ?‘ (đ?‘†đ?‘Ľ, đ??ľđ?‘Ś, đ?‘Ą)}. For every đ?‘Ľ, đ?‘Ś ∈ đ?‘‹ and đ?‘Ą > 0. Then đ??´, đ??ľ, đ?‘† and đ?‘‡ have a unique common fixed point in đ?‘‹. Proof. As đ??´(đ?‘‹) ⊆ đ?‘‡(đ?‘‹), for any đ?‘Ľ0 ∈ đ?‘‹, there exist a point đ?‘Ľ1 ∈ đ?‘‹ such that đ??´đ?‘Ľ0 = đ?‘‡đ?‘Ľ1 . Since đ??ľ(đ?‘‹) ⊆ đ?‘†(đ?‘‹), for this point đ?‘Ľ1 , we can choose a point đ?‘Ľ2 ∈ đ?‘‹ such that đ??ľđ?‘Ľ1 = đ?‘†đ?‘Ľ2 . Inductively, we can find a sequence {đ?‘Śđ?‘› } in đ?‘‹ as follows: đ?‘Ś2đ?‘›âˆ’1 = đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 = đ??´đ?‘Ľ2đ?‘›âˆ’2 and đ?‘Ś2đ?‘› = đ?‘†đ?‘Ľ2đ?‘› = đ??ľđ?‘Ľ2đ?‘›âˆ’1 for đ?‘› = 1,2, â‹Ż By Theorem of (Alaca et al. ,2006) , we can conclude that {đ?‘Śđ?‘› } in đ?‘‹. Since đ?‘‹ is complete, therefore sequence {đ?‘Śđ?‘› } in đ?‘‹ converges to đ?‘§ for some đ?‘§ in đ?‘‹ and so the sequences {đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 }, {đ??´đ?‘Ľ2đ?‘›âˆ’2 }, {đ?‘†đ?‘Ľ2đ?‘› } and {đ??ľđ?‘Ľ2đ?‘›âˆ’1 } also converges to đ?‘§. Since đ?‘‹ is ∈ −chainable, there exists ∈ −chain from đ?‘Ľđ?‘› to đ?‘Ľđ?‘›+1 , that is there exists a finite sequence đ?‘Ľđ?‘› = đ?‘Ś1 , đ?‘Ś2 , â‹Ż , đ?‘Śđ?‘™ = đ?‘Ľđ?‘›+1 such that â„ł(đ?‘Śđ?‘– , đ?‘Śđ?‘–−1 , đ?‘Ą) > 1−∈ and đ?‘ (đ?‘Śđ?‘– , đ?‘Śđ?‘–−1 , đ?‘Ą) < 1−∈ for all đ?‘Ą > 0 and đ?‘– = 1,2, â‹Ż , đ?‘™. Thus we have đ?‘Ą đ?‘Ą â„ł(đ?‘Ľđ?‘› , đ?‘Ľđ?‘›+1 , đ?‘Ą) ≼ â„ł (đ?‘Ś1 , đ?‘Ś2 , ) ∗ â„ł (đ?‘Ś2 , đ?‘Ś3 , ) ∗ â‹Ż ∗ đ?‘™ đ?‘™ đ?‘Ą đ?‘Ą đ?‘Ą â„ł (đ?‘Śđ?‘™âˆ’1 , đ?‘Śđ?‘™ , ) > (1−∈) ∗ (1−∈) ∗ â‹Ż ∗ (1−∈) ≼ (1−∈) and đ?‘ (đ?‘Ľđ?‘› , đ?‘Ľđ?‘›+1 , đ?‘Ą) ≤ đ?‘ (đ?‘Ś1 , đ?‘Ś2 , ) â—Š đ?‘ (đ?‘Ś2 , đ?‘Ś3 , ) â—Š đ?‘™ đ?‘™ đ?‘™ â‹Żâ—Š đ?‘Ą đ?‘ (đ?‘Śđ?‘™âˆ’1 , đ?‘Śđ?‘™ , ) < (1−∈) â—Š (1−∈) â—Š â‹Ż â—Š (1−∈) ≤ (1−∈) đ?‘™ For đ?‘š > đ?‘›, đ?‘Ą đ?‘Ą â„ł (đ?‘Ľđ?‘› , đ?‘Ľđ?‘š , đ?‘Ą) ≼ â„ł(đ?‘Ľđ?‘› , đ?‘Ľđ?‘›+1 , ) ∗ â„ł(đ?‘Ľđ?‘›+1 , đ?‘Ľđ?‘›+2 , ) đ?‘šâˆ’đ?‘› đ?‘šâˆ’đ?‘› đ?‘Ą ∗ ¡ ¡ ¡ ∗ â„ł(đ?‘Ľđ?‘šâˆ’1 , đ?‘Ľđ?‘š , ) > (1−∈) ∗ (1−∈) ∗ â‹Ż ∗ (1−∈) ≼ (1−∈) and đ?‘šâˆ’đ?‘› đ?‘Ą đ?‘Ą đ?‘ (đ?‘Ľđ?‘› , đ?‘Ľđ?‘š , đ?‘Ą) ≤ đ?‘ (đ?‘Ľđ?‘› , đ?‘Ľđ?‘›+1 , ) â—Š đ?‘ (đ?‘Ľđ?‘›+1 , đ?‘Ľđ?‘›+2 , ) đ?‘šâˆ’đ?‘› đ?‘šâˆ’đ?‘› đ?‘Ą â—Š ¡ ¡ ¡ â—Š â„ł(đ?‘Ľđ?‘šâˆ’1 , đ?‘Ľđ?‘š , ) < (1−∈) â—Š (1−∈) â—Š â‹Ż â—Š (1−∈) ≤ (1−∈) đ?‘šâˆ’đ?‘› Therefore {đ?‘Ľđ?‘› } is a Cauchy sequence in đ?‘‹ and hence there exists đ?‘Ľ in đ?‘‹ such that đ?‘Ľđ?‘› → đ?‘Ľ. from condition (2) đ??´đ?‘Ľ2đ?‘›âˆ’2 → đ??´đ?‘Ľ, đ?‘†đ?‘Ľ2đ?‘› → đ?‘†đ?‘Ľ as limit đ?‘› → ∞. By uniqueness of limits, we have đ??´đ?‘Ľ = đ?‘§ = đ?‘†đ?‘Ľ. Since pair (đ??´, đ?‘†) is weakly compatible, therefore, đ??´đ?‘†đ?‘Ľ = đ?‘†đ??´đ?‘Ľ and so đ??´đ?‘§ = đ?‘†đ?‘§. from condition (2) we have đ??´đ?‘†đ?‘Ľ2đ?‘› → đ??´đ?‘†đ?‘Ľ and therefore, đ??´đ?‘†đ?‘Ľ2đ?‘› → đ?‘†đ?‘§. Also from continuity of đ?‘†, we have đ?‘†đ?‘†đ?‘Ľ2đ?‘› → đ?‘†đ?‘§. from condition (4), we get 1 â„ł(đ??´đ?‘†đ?‘Ľ2đ?‘› , đ??ľđ?‘Ľ2đ?‘›âˆ’1 , đ?‘žđ?‘Ą) ≼ {â„ł(đ?‘†đ?‘†đ?‘Ľ2đ?‘› , đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą) ∗ â„ł(đ?‘†đ?‘†đ?‘Ľ2đ?‘› , đ??´đ?‘†đ?‘Ľ2đ?‘› , đ?‘Ą) ∗ [â„ł(đ?‘†đ?‘†đ?‘Ľ2đ?‘› , đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą) + 1

2

â„ł(đ??´đ?‘†đ?‘Ľ2đ?‘› , đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą)] ∗ [â„ł(đ??´đ?‘†đ?‘Ľ2đ?‘› , đ??ľđ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą) + â„ł(đ?‘†đ?‘†đ?‘Ľ2đ?‘› , đ??ľđ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą)] ∗ â„ł(đ??ľđ?‘Ľ2đ?‘›âˆ’1 , đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą) ∗ 2

â„ł(đ??´đ?‘†đ?‘Ľ2đ?‘› , đ?‘‡đ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą) ∗ â„ł(đ?‘†đ?‘†đ?‘Ľ2đ?‘› , đ??ľđ?‘Ľ2đ?‘›âˆ’1 , đ?‘Ą)}and

IJEBEA 15-445; Š 2015, IJEBEA All Rights Reserved

Page 43

Syed Shahnawaz Aliet al., International Journal of Engineering, Business and Enterprise Applications, 14(1), September-November, 2015, pp. 40-45

𝑁(𝐴𝑆𝑥2𝑛 , 𝐵𝑥2𝑛−1 , 𝑞𝑡) ≤ {𝑁(𝑆𝑆𝑥2𝑛 , 𝑇𝑥2𝑛−1 , 𝑡) ◊ 𝑁(𝑆𝑆𝑥2𝑛 , 𝐴𝑆𝑥2𝑛 , 𝑡) 1 ◊ [𝑁(𝑆𝑆𝑥2𝑛 , 𝑇𝑥2𝑛−1 , 𝑡) + 𝑁(𝐴𝑆𝑥2𝑛 , 𝑇𝑥2𝑛−1 , 𝑡)] 2 1 ◊ [𝑁(𝐴𝑆𝑥2𝑛 , 𝐵𝑥2𝑛−1 , 𝑡) + 𝑁(𝑆𝑆𝑥2𝑛 , 𝐵𝑥2𝑛−1 , 𝑡)] ◊ 𝑁(𝐵𝑥2𝑛−1 , 𝑇𝑥2𝑛−1 , 𝑡) 2 ◊ 𝑁(𝐴𝑆𝑥2𝑛 , 𝑇𝑥2𝑛−1 , 𝑡) ◊ 𝑁(𝑆𝑆𝑥2𝑛 , 𝐵𝑥2𝑛−1 , 𝑡)}. Proceeding limit as 𝑛 → ∞, we have 1 1 ℳ(𝑆𝑧, 𝑧, 𝑞𝑡) ≥ {ℳ(𝑆𝑧, 𝑧, 𝑡) ∗ ℳ(𝑆𝑧, 𝑆𝑧, 𝑡) ∗ [ℳ(𝑆𝑧, 𝑧, 𝑡) + ℳ(𝑆𝑧, 𝑧, 𝑡)] ∗ [ℳ(𝑆𝑧, 𝑧, 𝑡) + ℳ(𝑆𝑧, 𝑧, 𝑡)] ∗ 2

2

ℳ(𝑧, 𝑧, 𝑡) ∗ ℳ(𝑆𝑧, 𝑧, 𝑡) ∗ ℳ(𝑆𝑧, 𝑧, 𝑡)}and 1

1

2

2

𝑁(𝑆𝑧, 𝑧, 𝑞𝑡) ≤ {𝑁(𝑆𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑆𝑧, 𝑆𝑧, 𝑡) ◊ [𝑁(𝑆𝑧, 𝑧, 𝑡) + 𝑁(𝑆𝑧, 𝑧, 𝑡)] ◊ [𝑁(𝑆𝑧, 𝑧, 𝑡) + 𝑁(𝑆𝑧, 𝑧, 𝑡)] ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑆𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑆𝑧, 𝑧, 𝑡)}.From lemma 2.10, we get 𝑆𝑧 = 𝑧, and hence 𝐴𝑧 = 𝑆𝑧 = 𝑧. Since 𝐴(𝑋) ⊆ 𝑇(𝑋), there exists 𝑣 in 𝑋 such that 𝑇𝑣 = 𝐴𝑧 = 𝑧. from condition (4), we haveℳ(𝐴𝑥2𝑛 , 𝐵𝑣, 𝑞𝑡) ≥ 1 1 {ℳ(𝑆𝑥2𝑛 , 𝑇𝑣, 𝑡) ∗ ℳ(𝑆𝑥2𝑛 , 𝐴𝑥2𝑛 , 𝑡) ∗ [ℳ(𝑆𝑥2𝑛 , 𝑇𝑣, 𝑡) + ℳ(𝐴𝑥2𝑛 , 𝑇𝑣, 𝑡)] ∗ [ℳ(𝐴𝑥2𝑛 , 𝐵𝑣, 𝑡) + 2

2

ℳ(𝑆𝑥2𝑛 , 𝐵𝑣, 𝑡)] ∗ ℳ(𝐵𝑣, 𝑇𝑣, 𝑡) ∗ ℳ(𝐴𝑥2𝑛 , 𝑇𝑣, 𝑡) ∗ ℳ(𝑆𝑥2𝑛 , 𝐵𝑣, 𝑡)} and

𝑁(𝐴𝑥2𝑛 , 𝐵𝑣, 𝑞𝑡) ≤

1

1

2

2

{𝑁(𝑆𝑥2𝑛 , 𝑇𝑣, 𝑡) ◊ 𝑁(𝑆𝑥2𝑛 , 𝐴𝑥2𝑛 , 𝑡) ◊ [𝑁(𝑆𝑥2𝑛 , 𝑇𝑣, 𝑡) + 𝑁(𝐴𝑥2𝑛 , 𝑇𝑣, 𝑡)] ◊ [𝑁(𝐴𝑥2𝑛 , 𝐵𝑣, 𝑡) + 𝑁(𝑆𝑥2𝑛 , 𝐵𝑣, 𝑡)] ◊ 𝑁(𝐵𝑣, 𝑇𝑣, 𝑡) ◊ 𝑁(𝐴𝑥2𝑛 , 𝑇𝑣, 𝑡) ◊ 𝑁(𝑆𝑥2𝑛 , 𝐵𝑣, 𝑡)}. 1

Letting 𝑛 → ∞, we have ℳ(𝑧, 𝐵𝑣, 𝑞𝑡) ≥ {ℳ(𝑧, 𝑇𝑣, 𝑡) ∗ ℳ(𝑧, 𝑧, 𝑡) ∗ [ℳ(𝑧, 𝑇𝑣, 𝑡) + ℳ(𝑧, 𝑇𝑣, 𝑡)] ∗ 2

1 2 1 2

[ℳ(𝑧, 𝐵𝑣, 𝑡) + ℳ(𝑧, 𝐵𝑣, 𝑡)] ∗ ℳ(𝐵𝑣, 𝑇𝑣, 𝑡) ∗ ℳ(𝑧, 𝑇𝑣, 𝑡) ∗ ℳ(𝑧, 𝐵𝑣, 𝑡)} = {ℳ(𝑧, 𝑧, 𝑡) ∗ ℳ(𝑧, 𝑧, 𝑡) ∗ 1

[ℳ(𝑧, 𝑧, 𝑡) + ℳ(𝑧, 𝑧, 𝑡)] ∗ [ℳ(𝑧, 𝐵𝑣, 𝑡) + ℳ(𝑧, 𝐵𝑣, 𝑡)] ∗ ℳ(𝐵𝑣, 𝑧, 𝑡) ∗ ℳ(𝑧, 𝑧, 𝑡) ∗ ℳ(𝑧, 𝐵𝑣, 𝑡)} ≥ 2

1

1

2

2

ℳ(𝐵𝑣, 𝑧, 𝑡)and𝑁(𝑧, 𝐵𝑣, 𝑞𝑡) ≤ {𝑁(𝑧, 𝑇𝑣, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ [𝑁(𝑧, 𝑇𝑣, 𝑡) + 𝑁(𝑧, 𝑇𝑣, 𝑡)] ◊ [𝑁(𝑧, 𝐵𝑣, 𝑡) + 𝑁(𝑧, 𝐵𝑣, 𝑡)] ◊ 𝑁(𝐵𝑣, 𝑇𝑣, 𝑡) ◊ 𝑁(𝑧, 𝑇𝑣, 𝑡) ◊ 𝑁(𝑧, 𝐵𝑣, 𝑡)} 1 1 = {𝑁(𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ [𝑁(𝑧, 𝑧, 𝑡) + 𝑁(𝑧, 𝑧, 𝑡)] ◊ [𝑁(𝑧, 𝐵𝑣, 𝑡) + 𝑁(𝑧, 𝐵𝑣, 𝑡)] ◊ 𝑁(𝐵𝑣, 𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) 2 2 ◊ 𝑁(𝑧, 𝐵𝑣, 𝑡)} ≤ 𝑁(𝐵𝑣, 𝑧, 𝑡). By lemma 2.10, we have 𝐵𝑣 = 𝑧, and therefore, we have 𝑇𝑣 = 𝐵𝑣 = 𝑧. Since (𝐵, 𝑇) is weakly compatible, therefore, 𝑇𝐵𝑣 = 𝐵𝑇𝑣 and hence 𝑇𝑧 = 𝐵𝑧. from condition (4), we have 1 ℳ(𝐴𝑥2𝑛 , 𝐵𝑧, 𝑞𝑡) ≥ {ℳ(𝑆𝑥2𝑛 , 𝑇𝑧, 𝑡) ∗ ℳ(𝑆𝑥2𝑛 , 𝐴𝑥2𝑛 , 𝑡) ∗ [ℳ(𝑆𝑥2𝑛 , 𝑇𝑧, 𝑡) + ℳ(𝐴𝑥2𝑛 , 𝑇𝑧, 𝑡)] ∗ 1 2

2

[ℳ(𝐴𝑥2𝑛 , 𝐵𝑧, 𝑡) + ℳ(𝑆𝑥2𝑛 , 𝐵𝑧, 𝑡)] ∗ ℳ(𝐵𝑧, 𝑇𝑧, 𝑡) ∗ ℳ(𝐴𝑥2𝑛 , 𝑇𝑧, 𝑡) ∗ ℳ(𝑆𝑥2𝑛 , 𝐵𝑧, 𝑡)}and 1

𝑁(𝐴𝑥2𝑛 , 𝐵𝑧, 𝑞𝑡) ≤ {𝑁(𝑆𝑥2𝑛 , 𝑇𝑧, 𝑡) ◊ 𝑁(𝑆𝑥2𝑛 , 𝐴𝑥2𝑛 , 𝑡) ◊ [𝑁(𝑆𝑥2𝑛 , 𝑇𝑧, 𝑡) + 𝑁(𝐴𝑥2𝑛 , 𝑇𝑧, 𝑡)] ◊ 1 2

2

[𝑁(𝐴𝑥2𝑛 , 𝐵𝑧, 𝑡) + 𝑁(𝑆𝑥2𝑛 , 𝐵𝑧, 𝑡)] ◊ 𝑁(𝐵𝑧, 𝑇𝑧, 𝑡) ◊ 𝑁(𝐴𝑥2𝑛 , 𝑇𝑧, 𝑡) ◊ 𝑁(𝑆𝑥2𝑛 , 𝐵𝑧, 𝑡)}.

Letting 𝑛 → ∞, we have 1 1 ℳ(𝑧, 𝐵𝑧, 𝑞𝑡) ≥ {ℳ(𝑧, 𝑇𝑧, 𝑡) ∗ ℳ(𝑧, 𝑧, 𝑡) ∗ [ℳ(𝑧, 𝑇𝑧, 𝑡) + ℳ(𝑧, 𝑇𝑧, 𝑡)] ∗ [ℳ(𝑧, 𝐵𝑧, 𝑡) + ℳ(𝑧, 𝐵𝑧, 𝑡)] ∗ 2

2

ℳ(𝐵𝑧, 𝑇𝑧, 𝑡) ∗ ℳ(𝑧, 𝑇𝑧, 𝑡) ∗ ℳ(𝑧, 𝐵𝑧, 𝑡)} 1

1

2

2

= {ℳ(𝑧, 𝑧, 𝑡) ∗ ℳ(𝑧, 𝑧, 𝑡) ∗ [ℳ(𝑧, 𝑧, 𝑡) + ℳ(𝑧, 𝑧, 𝑡)] ∗ [ℳ(𝑧, 𝐵𝑧, 𝑡) + ℳ(𝑧, 𝐵𝑧, 𝑡)] ∗ ℳ(𝐵𝑧, 𝑧, 𝑡) ∗ 1

ℳ(𝑧, 𝑧, 𝑡) ∗ ℳ(𝑧, 𝐵𝑧, 𝑡) ≥ ℳ(𝐵𝑧, 𝑧, 𝑡)}and 𝑁(𝑧, 𝐵𝑧, 𝑞𝑡) ≤ {𝑁(𝑧, 𝑇𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ [𝑁(𝑧, 𝑇𝑧, 𝑡) + 2

1

𝑁(𝑧, 𝑇𝑧, 𝑡)] ◊ [𝑁(𝑧, 𝐵𝑧, 𝑡) + 𝑁(𝑧, 𝐵𝑧, 𝑡)] ◊ 𝑁(𝐵𝑧, 𝑇𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑇𝑧, 𝑡) ◊ 𝑁(𝑧, 𝐵𝑧, 𝑡)} = {𝑁(𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ 1 2

2

1

[𝑁(𝑧, 𝑧, 𝑡) + 𝑁(𝑧, 𝑧, 𝑡)] ◊ [𝑁(𝑧, 𝐵𝑧, 𝑡) + 𝑁(𝑧, 𝐵𝑧, 𝑡)] ◊ 𝑁(𝐵𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑧, 𝑧, 𝑡) ◊ 𝑁(𝑧, 𝐵𝑧, 𝑡)} ≤ 2

𝑁(𝐵𝑧, 𝑧, 𝑡).Which implies that 𝐵𝑧 = 𝑧. Therefore, 𝐴𝑧 = 𝑆𝑧 = 𝐵𝑧 = 𝑇𝑧 = 𝑧. Hence 𝐴, 𝐵, 𝑆 and 𝑇 have a unique common fixed point in 𝑋. For uniqueness, let 𝑤 be another common fixed point of 𝐴, 𝐵, 𝑆 and 𝑇. Then from condition (4), we have

IJEBEA 15-445; © 2015, IJEBEA All Rights Reserved

Page 44

Syed Shahnawaz Aliet al., International Journal of Engineering, Business and Enterprise Applications, 14(1), September-November, 2015, pp. 40-45 1

â„ł(đ?‘§, đ?‘¤, đ?‘žđ?‘Ą) = â„ł(đ??´đ?‘§, đ??ľđ?‘¤, đ?‘žđ?‘Ą) ≼ {â„ł(đ?‘†đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) ∗ â„ł(đ?‘†đ?‘§, đ??´đ?‘§, đ?‘Ą) ∗ [â„ł(đ?‘†đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) + â„ł(đ??´đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą)] ∗ 1 2

2

[â„ł(đ??´đ?‘§, đ??ľđ?‘¤, đ?‘Ą) + â„ł(đ?‘†đ?‘§, đ??ľđ?‘¤, đ?‘Ą)] ∗ â„ł(đ??ľđ?‘¤, đ?‘‡đ?‘¤, đ?‘Ą) ∗ â„ł(đ??´đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) ∗ â„ł(đ?‘†đ?‘§, đ??ľđ?‘¤, đ?‘Ą)} ≼ â„ł(đ?‘§, đ?‘¤, đ?‘Ą)and 1

đ?‘ (đ?‘§, đ?‘¤, đ?‘žđ?‘Ą) = đ?‘ (đ??´đ?‘§, đ??ľđ?‘¤, đ?‘žđ?‘Ą) ≤ { đ?‘ (đ?‘†đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) â—Š đ?‘ (đ?‘†đ?‘§, đ??´đ?‘§, đ?‘Ą) â—Š [ đ?‘ (đ?‘†đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) + đ?‘ (đ??´đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą)] â—Š 2

1

[ đ?‘ (đ??´đ?‘§, đ??ľđ?‘¤, đ?‘Ą) + đ?‘ (đ?‘†đ?‘§, đ??ľđ?‘¤, đ?‘Ą)] â—Š đ?‘ (đ??ľđ?‘¤, đ?‘‡đ?‘¤, đ?‘Ą) â—Š đ?‘ (đ??´đ?‘§, đ?‘‡đ?‘¤, đ?‘Ą) â—Š đ?‘ (đ?‘†đ?‘§, đ??ľđ?‘¤, đ?‘Ą)} ≤ đ?‘ (đ?‘§, đ?‘¤, đ?‘Ą).From lemma 2 2.10, we conclude that đ?‘§ = đ?‘¤. Hence đ??´, đ??ľ, đ?‘† and đ?‘‡ have a unique common fixed point in đ?‘‹. IV. Conclusions In this paper, we give some conditions of which four self-mappings of ∈ −chainable Intuitionistic fuzzy metric spaces have a unique common fixed point.This work can be easily extended by increasing the number of selfmappings and establishing the fixed point theorems in more generalized settings. V.

References

[1] [2]

Zadeh L.A., Fuzzy Sets, Inform and Control, vol. 8, pp. 338 – 353, (1965). Turkoglu D. and Rhoades B.E, A Fixed Fuzzy Point for Fuzzy Mapping in Complete Metric Spaces, Math. Communications, vol. 10 (2), pp. 115 121, (2005).

[3] [4] [5] [6]

Atanassov K.T., Intuitionistic Fuzzy Sets, Fuzzy sets and Systems vol. 20, pp. 87–96, (1986). Coker D., An Introduction to Intuitionistic Fuzzy Topological Spaces, Fuzzy sets and Systems, vol. 88 (1), pp. 81–89, (1997). Sharma S, Common Fixed Point Theorems in Fuzzy Metric Spaces, Fuzzy sets and Systems, vol. 127, pp. 345–352, (2002). Alaca C., Turkoglu D. and Yildiz C, Fixed Points in Intuitionistic Fuzzy Metric Spaces, Chaos, Solitons and Fractals, vol. 29, pp. 1073 –1078, (2006). Gregori V. and Sapena A, On Fixed Point Theorems in Fuzzy Metric Spaces, Fuzzy sets and Systems, vol. 125, pp. 245–253, (2002). Saadati R. and Park J.H, On the Intuitionistic Fuzzy Topological Spaces, Chaos, Solitons and Fractals, vol. 27, pp. 331 –344, (2006). Schweizer B. and Sklar A., Statistical Metric Spaces, Pacific Journal of Mathematics, vol. 10, pp. 314-334, (1960). Alaca C., Turkoglu D. and Altun I., On Compatible Mappings of Type (I) and Type (II) in Intuitionistic Fuzzy Metric Spaces, Korean Mathematical Society, vol.23 (3), pp. 427 –446, (2008). Sharma, S. and Desphande, B., On Compatible Mappings Satisfying an Implicit Relation in Common Fixed Point Consideration, Tamkang Journal of Mathematics, vol. 33, pp 245–252, (2002). Alaca C., Turkoglu, D. and Yildiz C., Compatible Maps and Compatible Maps of Types (Îą) and (β) in Intuitionistic Fuzzy Metric Spaces, Demonstration Mathematical, vol. 39 (3), pp. 671 –684, (2008). Jungck G. and Rhoades B. E Fixed Point for Set Valued Functions Without Continuity, Indian Journal of Pure & Applied Mathematics, vol. 29, pp. 227 – 238, (1998). Jungck G., Compatible Mappings and Common Fixed Point, International Journal of Mathematics and Mathematical Sciences, vol. 9 (4), pp.771-779, (1986). Deschrijver G. and Kerre E.E., On the Relationship Between Some Extensions of Fuzzy Set Theory, Fuzzy sets and Systems, vol. 23, pp. 227–235, (2003). Deschrijver G., Corenelis C. and Kerre E.E., On the Representation of Intuitionistic Fuzzy đ?’• −norms and đ?’• −conorms, IEEE Transactions on Fuzzy Systems, vol. 12, pp. 45–61, (2004). Park J.H., Intuitionistic Fuzzy Metric Spaces, Chaos, Solitons and Fractals, vol. 22, pp. 1039 – 1046, (2004). Atanassov K.T., Intuitionistic Fuzzy Sets, Fuzzy sets and Systems, vol. 20, pp. 87–96, (1986). Pant R.P., Common Fixed Points of Non Commuting Mappings, Journal of Mathematical Analysis and Gregori V., Romaguera S. and Veereamani P., A Note on Intuitionistic Fuzzy Metric Spaces, Chaos, Solitons and Fractals, vol. 28, pp. 902 – 905, (2006). M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy sets and Systems, 27, 245-252 (1988).

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

IJEBEA 15-445; Š 2015, IJEBEA All Rights Reserved

Page 45

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.