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# Common fixed point of weakly compatible

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### Common fixed point of weakly compatible

1. 1. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012 COMMON FIXED POINT OF WEAKLY COMPATIBLE MAPPINGS UNDER A NEW PROPERTY IN FUZZY METRIC SPACES Rajinder Sharma Department of General Requirements (Mathematics Section) College of Applied Sciences, Sohar, Sultanate of Oman E-mail : rajind.math@gmail.comAbstract: In this paper, we prove some common fixed point theorems for weakly compatible mappingsunder a new property in fuzzy metric spaces. We prove a new result under (S-B) property defined bySharma and Bamboria.Keywords: Fixed point; Fuzzy metric space; (S-B) property.Introduction:The foundation of fuzzy mathematics was laid down by Zadeh [25] with the evolution of the concept offuzzy sets in 1965. The proven result becomes an asset for an applied mathematician due to its enormousapplications in various branches of mathematics which includes differential equations, integral equationetc. and other areas of science involving mathematics especially in logic programming and electronicengineering. It was developed extensively by many authors and used in various fields. Especially, Deng[7], Erceg [8], and Kramosil and Michalek [17] have introduced the concepts of fuzzy metric spaces indifferent ways. To use this concept in topology and analysis, several researchers have studied fixed pointtheory in fuzzy metric spaces and fuzzy mappings [2],[3],[4],[5], [10],[14], [15] and many others.Recently, George and Veeramani [12],[13] modified the concept of fuzzy metric spaces introduced byKramosil and Michalek and defined the Hausdoff topology of fuzzy metric spaces. Theyshowed also that every metric induces a fuzzy metric.Grabiec [11] extended the well known fixed pointtheorem of Banach [1] and Edelstein [9] to fuzzy metric spaces in the sense of Kramosil and Michalek 32
2. 2. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012[15].Moreover, it appears that the study of Kramosil and Michalek [17] of fuzzy metric spacespaves the way for developing a soothing machinery in the field of fixed point theorems in particular,for the study of contractive type maps. In this paper we prove a new result under (S-B) property definedby Sharma and Bamboria [22].Preliminaries *Definition 1 : [20] A binary operation is called a continuous t-norm if * is an Abelian topological monoid with the unit 1 such that * * whenever for all are in Examples of are * *Definition 2 : [17] The 3-tuple * is called a fuzzy metric space (shortly FM-space) if is an arbitrary set, * is a continuous t-norm and M is a fuzzy set in satisfying the following conditions for all *In what follows, will denote a fuzzy metric space. Note that can be thought as the degree of nearness between and with 33
3. 3. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012 respect to . We identify with for all and with and we can find some topological properties and examples of fuzzy metric spaces in (George and Veeramani [12]).Example 1: [12] Let be a metric space. Define * * andfor all ,Then * is a fuzzy metric space. We call this fuzzy metric induced by the metric thestandard fuzzy metric.In the following example, we know that every metric induces a fuzzy metric.Lemma 1 : [11] For all is non-decreasing.Definition 3 : [11] Let * be a fuzzy metric space :(1) A sequence is said to be convergent to a point , if ,for all(2) A sequence called a Cauchy sequence if , for all and . 34
4. 4. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012(3) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.Remark 1 : Since * is continuous, it follows from that the limit of the sequence in space is uniquely determined. Let * be a fuzzy metric space with the following condition: for all .Lemma 2 : [6],[18] If for all and for a number k ∈ (0,1), thenLemma 3 : [6],[18] Let be a sequence in a fuzzy metric space * with the condition If there exists a number such thatfor all and then is a Cauchy sequence in X.Definition 4 : [18] Let and be mappings from a fuzzy metric space * into itself. Themappings and are said to be compatible iffor all whenever is a sequence in such that 35
5. 5. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012Definition 5 : [6] Let and be mappings from a fuzzy metric space into itself. Themappings and are said to be compatible of type if,for all , whenever is a sequence in such thatDefinition 6 : [16] A pair of mappings S and T is called weakly compatible pair in fuzzy metricspace if they commute at coincidence points; i.e. , if for some thenDefinition 7 : Let and be two self mappings of a fuzzy metric space * We say that and satisfy the property (S-B) if there exists a sequence in suchthatExample 2 : [22] Let byConsider the sequence , clearlyThen and satisfy 36
6. 6. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012Example 3 : [22] LetSuppose property holds; then there exists in a sequence satisfyingThereforeThen which is a contradiction since . Hence and do not satisfy the propertyRemark 2 : It is clear from the definition of Mishra et al. [18] and Sharma and Deshpande [23] that twoself mappings S and T of a fuzzy metric space * will be non-compatible if there exists at leastone sequence { } in X such thatbut is either not equal to 1 or non-existent. Thereforetwo noncompatible selfmappings of a fuzzy metric space * satisfy the propertyIt is easy to see that if and are compatible , then they are weakly compatible and the converse is nottrue in general.Example 2 : Let . Define and by : 37
7. 7. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012AsTherefore are weakly compatible.Turkoglu, Kutukcu and Yildiz [24] prove the following :Theorem A : Let * be a complete fuzzy metric space with * and let be maps from into itself such that(i)(ii) there exists a constant such thatfor all and , where , such that(iii) the pairs and are compatible of type(iv) and are continuous and .Then have a unique common fixed point.Sharma, Pathak and Tiwari [21] improved Theorem A and proved the following . 38
8. 8. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012Theorem B : * be a complete fuzzy metric space with and let and be maps from into itself such that(1.1)(1.2) there exists a constant such thatfor all where such that and(1.3) the pairs are weak compatible.Then have a unique common fixed point.Rawal [19] proved the following :Theorem C : Let * be a complete fuzzy metric space with . Let be mappings of into itself such that(1.1)(1.2) there exists a constant k ∈ (0, 1) such that 39
9. 9. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012(1.3) If the pairs are weakly compatible, then have a unique common fixed point inMain ResultsWe prove Theorem C under a new property [22] in the following way.Theorem 1 : Let * be a fuzzy metric space with for all and condition Let be mappings of into itself such that (1.1) (1.2) satisfies the property(1.3) there exists a constant such that 40
10. 10. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012(1.4) if the pairs are weakly compatible,(1.5) one of is closed subset of then have a unique common fixed point inProof : Suppose that satisfies the property Then there exists a sequence } insuch that for someSince there exists in a sequence such that . Hence . Letus show that . Indeed, in view of (1.3) for we have 41
11. 11. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012Thus it follows that Since the t-norm * is continuous and is continuous, letting we have 42
12. 12. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012It follows thatand we deduce thatSuppose S(X) is a closed subset of X. Then z = Su for some u ∈ X. Subsequently, we have = .By (1.3) with we haveTaking the limn → ∞ , we have 43
13. 13. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012Thus .This gives .Therefore by Lemma 2, we have . The weak compatibility of and implies that On the other hand, sincethere exists a point such that . We claim that using (1.3) with , we haveThus 44
14. 14. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012It follows thatTherefore by Lemma 2, we haveThus The weak compatibility of and impliesthat and Let us show that is a common fixedpoint of in view of (1.3) with we have This gives 45
15. 15. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012 Therefore by Lemma 2, we have and is a common fixed point ofSimilarly, we can prove that is a common fixed point of and Since we concludethat is a common fixed point ofIf then by (1.3) with wehaveThis gives Thus 46
16. 16. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012By Lemma 2, we have and the common fixed point is a unique. This completes the proof of thetheorem.If we take in Theorem 1, we have the following.Corollary 1 : Let *) be a fuzzy metric space with and condition Let be mappings of into itself such that (1.3) (1.4) satisfies the property (1.3) there exists a constant such thatfor all such that (1.4) if thepairs and are weakly compatible,(1.5) one of is closed subset of , then have a unique common fixed point in X.References:[1] Banach, S. : Surles operations danseles ensembles abstracts at leur applications aux equations integrals. Fund.Math., 3(1922), 133-181.[2] Bose, B. K. and Sahani, D. : Fuzzy mappings and fixed point theorems, Fuzzy Sets and Systems,21(1987),53-58. [3] Butnariu, D.: Fixed points for fuzzy mappings , Fuzzy Sets and Systems, 7(1982),191-207. 47
17. 17. Network and Complex Systems www.iiste.orgISSN 2224-610X (Paper) ISSN 2225-0603 (Online)Vol 2, No.4, 2012 [4] Chang, S. S.: Fixed point theorem for fuzzy mappings, Fuzzy Sets and Systems, 17(1985),181-187.[5] Chang, S.S., Cho, Y.J., Lee, B.S. and Lee, G.M. : Fixed degree and fixed point theorems for fuzzy mappings,Fuzzy Sets and Systems,87(3)(1997),325-334.[6] Cho, Y. J. : Fixed points in fuzzy metric spaces, J. Fuzzy Math., 5(4)(1997), 949- 962.[7] Deng, Z. K. : Fuzzy pseudo metric spaces , J. Math. Anal. Appl., 86(1982), 74-95.[8] Erceg, M. A.: Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69(1979), 205-230.[9] Edelstein, M. : On fixed point and periodic points under contractive mappings, J. London Math. Soc., 37(1962),74-79.[10] Frigon, M. and O’ Regan, D. : Fuzzy contractive maps and fuzzy fixed points, Fuzzy Sets and Systems,129(2002),39-45. [11] Grabiec, M. : Fixed point in fuzzy metric spaces, Fuzzy Sets and Systems, 27(1988), 385-389.[12] George, A. and Veeramani, P. : On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(1994),395-399.[13] George, A. and Veeramani, P. : On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems,90(1997), 365-368. [14] Gregori, V. and Romaguera, S. : Fixed point theorems for fuzzy mappings in quasi-metric spaces, Fuzzy Sets andSystems, 115(2000),477-483.[15] Heilpern, S. : Fuzzy mappings and fixed point theorems, J. Math.Anal. Appl.,83(1981),566-569.[16] Jungck G., and Rhoades, B. E. : Fixed point for set valued functions without continuity, Ind. J. Pure Appl.Maths., 29(3)(1998), 227-238.[17] Kramosil, I. and Michalek, J. : Fuzzy metric and statistical metric spaces, Kybernetika, 11(1975), 336-344.[18] Mishra, S. N., Sharma, N. and Singh, S. L. : Common fixed points of maps in fuzzy metric spaces. Internat. J.Math. Math. Sci., 17(1994), 253-258.[19] Rawal, M. : On common fixed point theorems in metric spaces and other spaces, Ph.D. thesis, Vikram Univ.,Ch.5(2009),72-84.[20] Schweizer, B. and Sklar, A. : Statistical metric spaces, Pacific. J. Math., 10(1960), 313-334.[21] Sharma, Sushil, Pathak, A. and Tiwari, R. : Common fixed point of weakly compatible maps without continuityin fuzzy metric space, Internat. J. Appl. Math.Vol.20,No.4 (2007),495-506.[22] Sharma, Sushil and Bamboria, D. : Some new common fixed point theorems in fuzzymetric space under strict contractive conditions, J. Fuzzy Math., Vol. 14 No.2 (2006),1-11.[23] Sharma, Sushil and Deshpande, B. : Common fixed point theorems for finite number of mappings withoutcontinuity and compatibility on fuzzy metric spaces, Mathematica Moravica, Vol.12(2008),1-19.[24] Turkoglu, D., Kutukcu, S. and Yildiz, C. : Common fixed point of compatible maps of type (α) on fuzzy metricspaces, Internat. J. Appl. Math., 18(2)(2005),189-202.[25] Zadeh, L.A. : Fuzzy Sets, Inform Contr., 8(1965), 338-353. 48