Fuzzy metric space has been defined using fuzzy numbers by many researchers. Recently Xia and Guo [20] have introduced fuzzy metric space using fuzzy scalars which is similar to the classical metric space. In this work we have generalized a fuzzy metric space to a partial fuzzy metric space using fuzzy scalars and its fixed point theorem have been established.Publisher's Versio

Kumar, Vijay

Tehlan, Aanchal

TWMS Journal of Applied and Engineering Mathematics

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TWMS J. App. and Eng. Math. V.12, N.3, 2022, pp. 888-894FIXED POINT THEOREMS OF FUZZY PARTIAL METRIC SPACESAANCHAL TEHLAN1, VIJAY KUMAR1, §Abstract. Fuzzy metric space has been defined using fuzzy numbers by many re-searchers. Recently Xia and Guo [20] have introduced fuzzy metric space using fuzzyscalars which is similar to the classical metric space. In this work we have generalized afuzzy metric space to a partial fuzzy metric space using fuzzy scalars and its fixed pointtheorem have been established.Keywords: Fuzzy metric space, Fuzzy partial metric space, fixed point theorem.AMS Subject Classification: 47H10, 03E72.1. IntroductionMetric is a function which defines distance between two points of the set and metricspace is a set together with a metric on that set. It generalized the notion of distancefunction in Euclidean n-dimensional space. Mathews [13] developed the theory of par-tial metric spaces, which helps in the generalization of metric spaces. Many researchershave used fuzzy set theory in various disciplines of research. Michalek and Kramosil [11]introduced the concept of fuzzy metric spaces, which are closely related to the class ofprobabilistic metric spaces. Kaleva [9] and Dia [4] used fuzzy numbers to define metricspaces for measuring distances in fuzzy sets. Felbin [5] investigated various fixed pointtheorems, some useful results and properties of these spaces were investigated by George[6] who utilized fuzzy numbers to define fuzzy metric space. Chaudhri [3], Diamond [4]and Boxer [2] used similar type of approaches in their work under different kind of fuzzyenvironment, which leaves unique impact in the development of the theory. Among variousapproaches to define these spaces, Xia [20] used fuzzy scalars to define fuzzy metric space,which became a stronger form in fuzzy topological spaces as fuzzy metric spaces. Relatedwork in this direction is seen in the works of Mazaheri [14] and Khalik [10]. Recently,Wu [19] and Gregori [7] used fuzzy numbers to define partial metric space. In this work,Banach’s fixed point theorem has been proved for fuzzy partial metric space by using fuzzyscalars.1 Department of Mathematics, Manav Rachna International Institute of Research and Studies,Faridabad-121004, Haryana, India.e-mail: tehlanaanchal@gmail.com; ORCID: https://orcid.org/0000-0001-9762-8416.e-mail: drvijaykumarsudan@gmail.com; ORCID: https://orcid.org/0000-0002-6905-5160.§ Manuscript received: April 22, 2020; accepted: October 12, 2020.TWMS Journal of Applied and Engineering Mathematics, Vol.12, No.3 © Işık University, Departmentof Mathematics, 2022; all rights reserved.888A. TEHLAN, V. KUMAR: FIXED POINT THEOREMS OF FUZZY PARTIAL METRIC SPACES 8892. PreliminariesDefinition 2.1 (Fuzzy point). Xia [20] defined fuzzy point. In this paper, we denote(x, λ) as fuzzy points and a set containing all the fuzzy points of X as PF (X). Also, whenX = {x : x is a real number} are called fuzzy scalars, and is written as SF (R). Fornon-negative fuzzy scalars, set is defined as S+F (R).Definition 2.2 (Fuzzy Scalars). According to Singh [17], consider, (x, λ), (y, γ) as anytwo fuzzy scalars. Then:(2.2.1) (a, λ) ≥ (b, γ), if a  b or (a, λ) = (b, γ).(2.2.2) (a, λ) is said to be not less than (b, γ) if a ≥ b, then (a, λ)  (b, γ or (b, γ) ≺ (a, λ)(2.2.3) (a, λ) is non-negative if a > 0.Definition 2.3 (Partial Fuzzy Metric Space). Let X be a non-empty set and let a mappingpF : PF (X)×PF (X)→ S+F (R), then (PF (X), pf ) is said to be partial fuzzy metric space, iffor every fuzzy scalars (x, λ), (y, γ), (z, ρ) ⊂ (PF (X, )pF ) satisfies the following conditions:(i) 0 ≤ pF ((x, λ), (x, λ)) ≤ pF ((x, λ), (y, γ))(ii) pF ((x, λ), (x, λ)) = pF ((x, λ), (y, γ)) = pF ((y, γ), (y, γ)) then x = y and λ = γ = 1(iii) pF ((x, λ)(y, γ)) = pF ((y, γ), (x, λ))(iv) pF ((x, λ), (z, ρ)) ≤ pF ((x, λ)(y, γ)) + pF ((y, γ), (z, ρ))− pF ((y, γ), (y, γ))Example 2.1. Suppose (X, p) is a partial metric space, then the distance of (x, λ), (y, γ)in PF (X) is defined aspF ((x, λ), (y, γ)) = {p(x, y),min(λ, γ)}where p(x, y) defines the distance between points x and y.Here, we shall show that pF satisfies all the conditions, given in (2.3).Given that p(x, y) is a partial metric and p(x, y) ≥ 0.We have pF ((x, λ)(y, γ)) = {p(x, y),min(λ, γ)}, which is non-negative as 0 ≤ p(x, x) ≤p(x, y).Symmetric. Consider, (x, λ)(y, γ) ⊂ PF (X) we have:pF ((x, λ)(y, γ)) = {p(x, y),min(λ, γ)}pF ((x, λ)(y, γ)) = {p(y, x) = min(γ, λ)}which gives,pF ((x, λ)(y, γ)) = pF ((y, γ), (y, γ))Triangularity. Consider, (x, λ), (y, γ), (z, ρ) ⊂ PF (X), hencepF (x, λ), (z, ρ) = {p(x, z),min(λ, ρ)}pF (x, λ), (z, ρ) ≺ {p((x, y) + (y, z)− (y, y),min(λ, γ, ρ))}= {p((x, y),min(λ, γ)}+ {p((y, z),min(γ, ρ)} − {p((y, y),min(γ, γ)}= pF (x, λ), (y, γ) + pF (y, γ), (z, ρ)− pF (y, γ), (y, γ)Definition 2.4. Consider any sequence (xn, λn) in some fuzzy partial metric space (x, λ) ∈PF (X). Then, (xn, λn) will converge to any fuzzy point (x, λ) ∈ X if limn→∞pF ((xn, λn), (x, λ)) =0λ ∀ γ ∈ (0, 1], where λ ≤ γ, (x, λ) is the limit of (xn, λn) given as limn→∞(xn, λn) = (x, λ).Definition 2.5. Any sequence defined using fuzzy points (xn, λn) ∈ (PF (X), pF ) is cauchyif for any λ ∈ (0, 1], limn→∞pF ((xm+n, λm+n), (xn, λn)) = 0λ for all m ∈ N .890 TWMS J. APP. AND ENG. MATH. V.12, N.3, 2022Example 2.2. Consider, X = [0,∞) and (PF (X), pF ) = {p(x, y),min(λ, γ)} wherep(x, y) = max(x, y) which means (PF (X), pF ) is a partial fuzzy metric. Consider, {xn} ={1, 2, 1, 2, . . .}. Then sequence will converge for all x ≥ 2, whenlimn→∞pF ((xn, λn), (x, γ)) = pF ((x, λ), (x, γ))therefore L(xn) = {xn ∈ X : xn → x} = [2,∞).But limn→∞pF ((xm+n, λm+n), (xn, λn)) does not exist and hence (xn, λn) is not Cauchy.Lemma 2.1 (Masihaa [12]). A partial fuzzy metric space is complete if for some cauchysequence belonging to it there will exist some unique limit in the space.Moreover, limn→∞(xn, λn) = 0 iff pF ((x, λ), (x, λ)) = limn→∞pF (xn, λn), (x, λ) = limn→∞pF (xn, λn), (xm, λm)Since every cauchy sequence which has fuzzy points will have some unique limit point asfuzzy point, as in the classical case.Definition 2.6 (Masihaa [12]). Let φ : [0, 1]→ [0, 1] satisfying;(i) φ(0) = 0 and φ(t) > 0 for each t > 0.(ii) φ is right lower semi-continuous(iii) For any sequence {rn} with limn→∞rn = 0, there exists, a ∈ (0, 1) and no ∈ N suchthat φ(rn) ≥ arn for each n ≥ no.3. Main ResultsTheorem 3.1. Suppose we have (X,≤) as some set which is partially ordered and pF onPF (X) is a fuzzy partial metric space which is complete. Let T : PF (X) → PF (X) be amap which is continuous and non-decreasing :pF (T (x, λ), T (y, γ)) ≤ pF ((x, λ), (y, γ))− ϕ{pF ((x, λ), (y, γ))} (3.1.1)∀ x, y ∈ X where x, y are comparable and φ satisfies Definition 2.6.Whenever, we have an (x0, y0) ∈ PF (X) so that (x0, y0) ≤ T (x0, y0), then T will have aunique fixed point which means there will exist (x, λ) ∈ PF (X) and T (x, λ) = (x, λ).Proof. Let T (x0, λ0) = (x0, λ0) then there is nothing to prove.Let (x0, λ0) 6= T (x0, λ0). Also, let (xn, λn) = T ((xn−1, λn−1).We need to show that (xn0 , λn0) = (xn0+1, λn0+1) which gives that (xn0 , λn0) is a fixedpoint of T .Therefore, let (xn+1, λn+1) 6= (xn, λn).As, (x0, λ0) ≤ T ((x0, λ0)), and T is non-decreasing for every n ∈ N . Therefore(xn0 , λn0) ≤ (x1, λ1) ≤ (x2, λ2) ≤ · · · ≤ (xn, λn) ≤ ((xn+1, λn+1))As (xn−1, λn−1) ≤ (xn, λn) using inequality (3.1.1). ThereforepF (((xn+1, λn+1), (xn, λn)) = pF (T (xn, λn)), T ((xn−1, λn−1))≤ pF (xn, λn), (xn−1, λn−1)))− φ(pF (xn, λn), ((xn−1, λn−1)))< pF (xn, λn)(xn−1, λn−1))Therefore, pfn = {(pF (xn−1, λn−1), (xn,λn)} is a non-negative and decreasing sequence.Therefore, it has a limit p. Let p > 0 then ∃ n0 ∈ N such that φ(pfn) ≥ φ(p) > 0 for alln > no and pfn ≤ pfn−1 − φ(pfn−1) ≤ pfn−1 − φ(p).Let n→∞ we get:p ≤ p− φ(p) ≤ pwhich gives a contradiction that pF = 0.A. TEHLAN, V. KUMAR: FIXED POINT THEOREMS OF FUZZY PARTIAL METRIC SPACES 891Now, we need to show that (xn, λn) is a cauchy sequence.Since, limn→∞pF ((xn, λn), (xn−1, λn−1) = 0 from (iii) of φ there exists 0 < a < 1 and n0 ∈ Nsuch that φ(pF ((xn, λn), (xn−1, λn−1)) ≥ a(xn+1, λn+1), (xn, λn), for all n > n0.Also,pF ((xn+1, λn+1), (xn, λn) ≤ pF ((xn, λn), (xn−1, λn−1)− φ(pF ((xn, λn), (xn−1, λn−1))≤ (1− a)pF ((xn, λn), (xn−1, λn−1) (3.1.2)By this we get:pF ((xn+1, λn+1), (xn, λn) ≤ (1− a)pF ((xn+1, λn+1), (xn, λn) ≤ . . .≤ (1− a)npF ((x1, λ1), (x0, λ0). (3.1.3)We fix K = (1− a), therefore,pF ((xn+1, λn+1), (xn, λn) = 2pF ((xn+1, λn+1), (xn, λn)− pF ((xn+1, λn+1), (xn+1, λn+1)+ pF ((xn, λn), (xn, λn)≤ 4KnpF ((x1, λ1), (x0, λ0))Therefore, limn→∞pF (xn+1, λn+1), ((xn,λn)) = 0 therefore (xn, λn) is cauchy and hencelimn→∞pF (xm, λm), (xn, λn) = 0.Using Lemma 2.1 we get:pF ((x, λ), (x, λ)) = limn→∞pF (xn, λn), (x, λ) = limn→∞pF (xn, λn), (xm, λm) = 0 (3.1.4)Hence, (xn, λn) is Cauchy.Also, as limn→∞pF ((xn+1, λn+1)), T ((x, λ))) = pF ((T (x, λ), T (x, λ)), andpF ((x, λ), T (x, λ)) ≤ pF ((x, λ), (xn+1, λn+1)) + pF ((xn+1, λn+1)), T (x, λ))− pF ((xn+1, λn+1)), ((xn+1, λn+1)))≤ pF ((xn+1, λn+1)), (x, λ)) + pF ((xn+1, λn+1)), T (x, λ)).Letting n→∞ we get:pF ((x, λ), T (x, λ)) ≤ pF ((T (x, λ), T (x, λ))≤ pF (x, λ), (x, λ))− φ(pF (x, λ), (x, λ)) = 0.Therefore pF ((x, λ), T (x, λ)) = 0.Hence, T (x, λ) = (x, λ). Theorem 3.2. Suppose (X,≤) and (PF (X), pF ) is the same as in Theorem 3.1. LetT : PF (X)→ PF (X) be a non-decreasing map as below:pF (T (x, λ), (y, γ)) ≤ pF ((x, λ), (y, γ))− φ{pF ((x, λ), (y, γ))}∀ x, y ∈ X where x, y are comparable and φ satisfies Definition 2.6.Let {(xn, λn)} be increasing such that (xn, λn)→ (x, λ) in PF (X) which implies (xn, λn) ≤ (x, λ)∀ n. If ∃ (x0, λ0) ∈ PF (X) with (x0, λ0) ≤ T (x0, λ0) such that (x, λ) = T (x, λ)Proof. As in Theorem 3.1, we will be able to have a sequence (xn0 , λn0) in PF (X) by usingxn = T (xn−1) such that(x0, λ0) ≤ (x1, λ1) ≤ (x1, λ1) ≤ (x2,λ2) ≤ · · · ≤ (xn,λn) ≤ (xn+1, λn+1)Also, we will prove that {(xn, λn)} is cauchy (PF (X), pF ) and ∃ x ∈ X givingpF ((x, λ), (x, λ)) = limn→∞pF ((xn, λn), (x, λ)) = limn→∞pF ((xn, λn), (xm, λm) = 0892 TWMS J. APP. AND ENG. MATH. V.12, N.3, 2022We claim that T (x, λ) = (x, λ). Suppose pF ((x, λ), T (x, λ)) > 0 by conditions of Theo-rem 3.1:pF ((x, λ), T (x, λ)) ≤ pF ((x, λ), (xn+1, λn+1))) + pF ((xn+1, λn+1)), T (x, λ))− pF ((xn+1, λn+1)), (xn+1, λn+1))≤ pF ((xn+1, λn+1)), (x, λ)) + pF ((xn+1, λn+1)), T (x, λ)))Letting n→∞ we have, pF ((x, λ), T (x, λ)) = 0. We will now give a condition which sufficiently proves the uniqueness of fixed point theo-rems in both Theorem 3.1 and 3.2. The condition is stated below:for every (x, λ), (y, γ) ∈ pF (X) ∃ either some lower bound or upper bound. (3.2.1)By [12] this condition is same as the following condition:∀ x, y ∈ X there shall exist some z ∈ X comparable to (x, λ) and (y, γ). (3.2.2)Theorem 3.3. We now prove the uniqueness of T in Theorems 3.1 and 3.2 using thecondition (3.2.2).Proof. Let there exist (z, ρ) and (y, γ) ∈ pF (X) as the two different fixed points of T .Therefore, pF ((z, ρ), (y, γ)) > 0. Now, we will be considering two cases:Case I: If (z, ρ) and (y, γ) are comparable thenTn(z, ρ) = (z, ρ) and Tn(y, γ)) = (y, γ) are comparable for n = 0, 1, 2By condition (3.1.1), we havepF ((z, ρ), (y, γ)) ≤ pF (Tn(z, ρ), Tn(y, γ))≤ pF (Tn−1(z, ρ), Tn−1(y, γ))− φ(Tn−1(z, ρ), Tn−1(y, γ))= pF ((z, ρ), (y, γ))− φ(pF ((z, ρ), (y, γ))< pF ((z, ρ), (y, γ)) which is a contradiction.Case II: If (z, ρ) and (y, γ) are comparable then ∃ x ∈ X comparable to z and y.Since, T is increasing therefore (Tn(x) is comparable to Tn(z, ρ) = (z, ρ) andTn(y, γ) = (y, γ) for n = 0, 1, 2ThereforepF ((z, ρ), Tn(x)) = pF (Tn(z, ρ), Tn(x, λ))≤ pF (Tn−1(z, ρ), Tn−1(x, λ))− φ(pF (Tn−1(z, ρ), Tn−1(x, λ))= pF ((z, ρ) , Tn−1((x, λ)))− φpF ((z, ρ), Tn−1((x, λ)))< pF ((z, ρ), Tn−1(x, λ))Therefore, we have got that pF ((z, ρ), Tn(x, λ)) is a sequence which is non-negative andnon-decreasing, having a limit let it be b > 0. By last inequality we get that:b ≤ b− φ(b) < bTherefore, a = 0, In the same manner we can show that limn→∞pF ((y, γ), Tn−1(x, λ)) = 0.pF (z, ρ), (y, γ) ≤ pF (z, ρ), Tn(x, λ)− pF (Tn(x, λ), Tn(x, λ)≤ pF (z, ρ), Tn(x, λ) + pF (Tn(x, λ), Tn(x, λ)taking limit n→∞ gives pF (z, ρ), (y, γ) = 0 which is a contradiction as pF (z, ρ), (y, γ) > 0.Hence, proved. A. TEHLAN, V. KUMAR: FIXED POINT THEOREMS OF FUZZY PARTIAL METRIC SPACES 8934. ConclusionWe have generalized a fuzzy metric space to a partial fuzzy metric space using fuzzy scalarswhich have many practical applications.5. AcknowledgementThe contribution of each author is on an equal basis in this work. Also the final manuscripthave been read and approved by all authors. No conflict of interest is declared by theauthors.References[1] Bag, T., Samanta, S. K., (2003), Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11,(3), 687–705.[2] Boxer, L., (1997), On Hausdirff-like metirc for fuzzy sets, Pattern Recognition Letters, 18, 115–118.[3] Chaudhary, B. B., Rosenfeld, A., (1996), On a metric distance between fuzzy sets, Pattern RecognitionLetters, 17, 1157–1160.[4] Diamond, P., Kloden, P., (1990), Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 35, 241–249.[5] Felbin, C., (1992), Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48, 239–248.[6] George, A., Veeramani, P. V., (1994), On some results of fuzzy metric spaces, Fuzzy Sets and System,64, 395–399.[7] Gregori, V., Minana, J. J., Miravet, D., (2019), Fuzzy partial metric spaces, International Journal ofGeneral Systems, 48, 260–279.[8] Gregori, V., Romaguera, S., (2000), Some properties of fuzzy metric spaces, Fuzzy Sets and Systems,115, 485–489[9] Kaleva, O., Seikkala, (1984), On fuzzy metric space, Fuzzy Sets and Systems, 12, 215–229[10] Khalik, M. A., Almyaly, A. H., (2015), Construction Smooth Hausdroff Space from Fuzzy MetricSpace, Journal of Mathematics (IOSR-JM), (Mar-April), 32–33.[11] Kramosil, I., Michalek, J., (1975), Fuzzy metric and statistical metric spaces, Kybernetika, 11, 326–334.[12] Masihaa, H. P., Sabetghadama, F., Shahzadb, N., (2013), Fixed point theorems in partial metricspaces with an application, 27, (4), 617–624.[13] Matthews, S., (1994), Partial Metric Topology, Proceedings of the 8th Summer Conference on Topol-ogy and its Applications, Annals of The New York Academy of Sciences, 728, 183–197.[14] Mazaheri, H., Sarikhani, R., (2014), λ-Fuzzy fixed points in fuzzy metric spaces, Journal of Mathe-matics and Computer Science, 9, 332–341.[15] Pu, B. M., Liu, Y. M., (1980), Fuzzy Topology I: Neighbourhood structure of a fuzzy poiny andMoore-Smith convergance, J. Math. Anal. Appl., 76, 517–599.[16] Rudin, W., (1976), Principles of Mathematical Analysis, McGraw-Hill Kogakusha Ltd., 3rd edition.[17] Singh, R. K., Alam, M. Z., (2005), Fixed point theorem in fuzzy metric spaces, IOSR Journal ofMathematics, 13, 55–59.[18] Valero, O., (2005), On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol., 2,229–240.[19] Wu, J., Yue, Y., (2017), Formal balls in fuzzy partial metric spaces, Iranian Journal of Fuzzy Systems,14, 155–164[20] Xia, Z., Guo, F. F. (2004), Fuzzy metric spaces, J. Appl. Math. & Computing, 16, 371–381.894 TWMS J. APP. AND ENG. MATH. V.12, N.3, 2022 Aanchal Tehlan is a research scholar in the Department of Applied Sciences (Mathe-matics), Faculty of Engineering & Technology, Manav Rachna International Instituteof Research & Studies, Faridabad. She is doing her research in fuzzy sets and itsgeneralizations.                        Dr. Vijay Kumar is working as a professor in the Department of Applied Sciences(Mathematics), Faculty of Engineering & Technology, Manav Rachna InternationalInstitute of Research & Studies, Faridabad. His research interest includes fuzzy setsand its generalizations, decision making.

Fixed point theorems of fuzzy partial metric spaces

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Fixed point theorems of fuzzy partial metric spaces

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