# Best Proximity Point Theorems for Single and Multivalued Mappings in Fuzzy Multiplicative Metric Space

Authors: Misbah Farheen; Tayyab Kamran; Azhar Hussain
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce fuzzy multiplicative metric space and prove some best proximity point theorems for single-valued and multivalued proximal contractions on the newly introduced space. As corollaries of our results, we prove some fixed-point theorems. Also, we present best proximity point theorems for Feng-Liu-type multivalued proximal contraction in fuzzy metric space. Moreover, we illustrate our results with some interesting examples.


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## Body

## 1. Introduction and Preliminaries

Best proximity point is the generalization of fixed point and is useful when contraction map is not a self-map that isT:A⟶B where A∩B=ϕ. A point μ∈A is known as best proximity point if dμ,Tμ=dA,B. Fan [1] presented best approximation theorem which is stated as follows: “If K is a nonempty compact convex subset of a Hausdorff locally convex topological vector space E and T:K⟶E is a continuous non-self-mapping, then there exists an element μ in K such a way that dμ,Tμ=dTμ,K.” A best proximity point theorem is more applicable than best approximation theorem, as it provides optimal approximate solution. Therefore, best proximity point theory seeks attention of authors such as [2–7]. Many research works done on multivalued non-self-maps use Nadler’s approach [8]. Nadler’s theorem is stated as follows: “Let M,d be a complete metric space and T be a mapping from M into CBM, where CBM is the collection of all closed and bounded subsets of M, such that for all μ,ν∈M, HTμ,Tν≤λdμ,ν where 0<λ<1. Then, T has a fixed point.” Another way of defining multivalued contraction is approached by Feng and Liu [9]. They proved a fixed-point theorem for newly defined multivalued contraction which is stated as follows: “Let M,d be a complete metric space, T:M⟶CM, where CM is the collection of all closed subsets of M, be a multivalued mapping. If there exists a constant c∈0,1 such that for any μ∈M, there is ν∈Ibμ (where Ibμ=ν∈Tμ∣bdμ,ν≤dμ,Tμ⊂M for some b∈0,1) satisfying dν,Tν≤cdμ,ν. Then, T has a fixed point in M provided that c<b and fμ=dμ,Tμ is lower semicontinuous”. With the help of example, in the same article, they also have shown that Feng-Liu-type multivalued contraction is more general than Nadler’s multivalued contraction. Recently, Sahin et al. [10] proved best proximity point theorem for Feng-Liu-type multivalued map.On the other hand, fuzzy metric space was firstly defined by Kramosil and Michalek [11] and then modified by George and Veeramani [12]. The modified definition is given as follows.Definition 1 (see [12]).
A 3-tupleM,FM,⋆ is called fuzzy metric space if M is an arbitrary set, ⋆ is continuous t-norm, and FM is a fuzzy set on M×M×0,∞ satisfying the following conditions for all μ,ν,ρ∈M and t,s>0:
FM1:FMμ,ν,t>0
FM2:FMμ,ν,t=1 if and only if μ=ν
FM3:FMμ,ν,t=FMν,μ,t
FM4:FMμ,ρ,t+s≥FMμ,ν,t⋆FMν,ρ,s
FM5:FMμ,ν,.:0,∞⟶0,1 is continuousThet−norm is defined as follows.Definition 2 (see [12]).
A continuoust-norm is a binary operation ⋆:0,12⟶0,1 if the pair 0,1,⋆ is a topological monoid, that is,
(1)
⋆ satisfies associative and commutative laws(2)
⋆ is continuous(3)
a⋆1=a, ∀a∈0,1(4)
for everya,b,c,d∈0,1, a⋆b≤c⋆d whenever a≤c and b≤d
Some known examples of a continuoust-norm are a⋆1b=mina,b,a⋆2b=ab/maxa,b,λfor0<λ<1,a⋆3b=ab,a⋆4b=maxa+b−1,0.Many researches have been produced on fixed-point theory in fuzzy metric spaces [4, 13–19]. Vetro and Salimi [20] proved best proximity point theorem in fuzzy metric spaces. Due to the development of new calculus by Grossman and Katz [21], known as multiplicative calculus, a metric was introduced by Bashirov et al. [22] called multiplicative metric defined as follows.Definition 3 (see [22]).
Assume a nonempty setM. Regard multiplicative metric as a mapping d:M×M⟶ℝ obeying the following assertions:
M1:dμ,ν>1 for all μ,ν∈M and dμ,ν=1 if and only if μ=ν
M2:dμ,ν=dν,μ
M3:dμ,ρ≤dμ,ν·dν,μ for all μ,ν,ρ∈MGetting inspiration from all the work mentioned above, we firstly introduce fuzzy multiplicative metric space and prove some of its topological properties. Moreover, we obtain some best proximity point theorems for Feng-Liu-type multivalued non-self-maps on fuzzy multiplicative metric space.


## 2. Fuzzy Multiplicative Metric Spaces

This section introduces a new type of metric space which is fuzzy analogy of multiplicative metric space. We give an example to show the existence of such space.Definition 4.
A tripletM,FMM,⋆ is termed as fuzzy multiplicative metric space if ⋆ is continuous t-norm, M is arbitrary set, and FMM is fuzzy set on M×M×1,∞ fulfilling the accompanying conditions for all μ,ν,ρ∈M and t,s>1.
FMM1:FMMμ,ν,t>0
FMM2:FMMμ,ν,t=1 if and only if μ=ν
FMM3:FMMμ,ν,t=FMMν,μ,t
FMM4:FMMμ,ρ,t.s≥FMMμ,ν,t⋆FMMν,ρ,s
FMM5:FMMμ,ν,.:1,∞⟶0,1 is continuousHere, we have an example of fuzzy multiplicative metric which cannot be fuzzy metric.Example 5.
LetM=ℝ+ and FMMμ,ν,t=t+1/t+μ/ν∗, consider a continuous t-norm ⋆:0,1×0,1⟶0,1 as μ⋆ν=μν. Then, M is fuzzy multiplicative metric space.Remark 6.
(1)
LetM,FMM,⋆ be a fuzzy multiplicative metric space. Whenever FMMμ,ν,t>1−ε for μ,ν∈M and t>1, 0<ε<1, we can find t0, 1<t0<t such that FMMμ,ν,t0>1−ε(2)
Letε1,ε2,ε3,ε4,ε5∈0,1. For any ε1>ε2, we can find ε3 such that ε1⋆ε3≥ε2, and for any ε4, we can find ε5 such that ε5⋆ε5≥ε4We now discuss some topological properties of fuzzy multiplicative metric space.Definition 7.
LetM,FMM,⋆ be a fuzzy multiplicative metric space and 0<ε<1, t>1; then, an open ball having center μ and radius ε is defined as
(1)Bμ,ε,t=ν∈M:FMMμ,ν,t>1−ε.Proposition 8.
Every open ball is an open set in fuzzy multiplicative metric space.Proof.
Consider an open ballBμ,ε,t and let ν∈Bμ,ε,t. This implies that FMMμ,ν,t>1−ε. Since FMMμ,ν,t>1−ε, using Remark 6, we can find t0, 1<t0<t, such that FMMμ,ν,t0>1−ε. Let ε0=FMMμ,ν,t0>1−ε. Since ε0>1−ε, therefore by using Remark 6, we can find ε1, 0<ε1<1, such that ε0>1−ε1>1−ε. Now, for a given ε0 and ε1 such that ε0>1−ε1, we can find ε2,0<ε2<1 such that ε0⋆ε2≥1−ε1. Now, consider the ball Bν,1−ε2,t/t0. We claim that Bν,1−ε2,t/t0⊂Bμ,ε,t.
Now,ρ∈Bν,1−ε2,t/t0 implies that FMMν,ρ,t/t0>ε2. Therefore,
(2)FMMμ,ρ,t≥FMMμ,ν,t0⋆FMMν,ρ,tt0≥ε0⋆ε2≥1−ε1>1−ε.
Therefore,ρ∈Bμ,ε,t, and hence,
(3)Bν,1−ε2,tt0⊂Bμ,ε,t.☐Proposition 9.
LetM,FMM,⋆ be a fuzzy multiplicative metric space. Define τ=A⊂M:μ∈Aifandonlyifthereexistt>1andε,0<ε<1suchthatBμ,ε,t⊂A.
Then,τ is a topology on M.Theorem 10.
Every fuzzy multiplicative metric space is Hausdorff.Proof.
Assume thatM,FMM,⋆ is a given fuzzy multiplicative metric space. Let μ,ν be two distinct points of M, and then, 0<FMMμ,ν,t<1. Let FMMμ,ν,t=ε, 0<ε<1. For each ε0, ε<ε0<1, using Remark 6, we can find ε1 such that ε1⋆ε1≥ε0. Now, consider the open balls Bμ,1−ε1,t1/2 and Bν,1−ε1,t1/2. Clearly,
(4)Bμ,1−ε1,t1/2∩Bν,1−ε1,t1/2=ϕ.
For if there exists(5)ρ∈Bμ,1−ε1,t1/2∩Bν,1−ε1,t1/2.
Then,(6)ε=FMMμ,ν,t≥FMMμ,ρ,t1/2⋆FMMρ,ν,t1/2≥ε1⋆ε1≥ε0>ε,
which is a contradiction. Therefore,M,FMM,⋆ is Hausdorff.☐Definition 11.
In a fuzzy multiplicative metric spaceM,FMM,⋆, a sequence μa is a convergent sequence which converges to μ if and only if there exist a1∈ℕ with FMMμa,μ,t>1−ε, for all a≥a1 and for each ε>0,t>1.Theorem 12.
LetM,FMM,⋆ be a fuzzy multiplicative metric space, μ∈M and μa be a sequence in M. Then, μa converges to μ if and only if FMMμa,μ,t⟶1 as a⟶∞ for each t>1.Proof.
Suppose thatμa⟶μ. Then, for each t>1 and ε∈0,1, there exists a natural number a1 such that FMMμa,μ,t>1−ε for all a≥a1. We have 1−FMMμa,μ,t<ε. Hence, FMMμa,μ,t⟶1 as a⟶∞.
Conversely, suppose thatFMMμa,μ,t⟶1 as a⟶∞. Then, for each t>1 and ε∈0,1, there exist a natural number a1 such that 1−FMMμa,μ,t<ε for all a≥a1. In that case, FMMμa,μ,t>1−ε. Hence, μa⟶μ as a⟶∞.☐Definition 13.
Consider a sequenceμa in a fuzzy multiplicative metric space M,FMM,⋆. If for each ε>0,t>1, there exist a1∈ℕ such that FMMμa,μb,t>1−ε for all a,b≥a1, and then, μa is termed as Cauchy sequence in M.Theorem 14.
LetM,FMM,⋆ be a fuzzy multiplicative metric space, μ∈M and μa be a sequence in M. Then, μa is Cauchy if and only if FMMμa,μb,t⟶1 as a,b⟶∞ for each t>1.Proof.
Suppose thatμa is a Cauchy sequence in M. Then, for each t>1 and ε∈0,1, there exists a natural number a1 such that FMMμa,μb,t>1−ε for all a,b≥a1. We have 1−FMMμa,μb,t<ε. Hence, FMMμa,μb,t⟶1 as a,b⟶∞.
Conversely, suppose thatFMMμa,μb,t⟶1 as a,b⟶∞. Then, for each t>1 and ε∈0,1, there exists a natural number a1 such that 1−FMMμa,μb,t<ε for all a,b≥a1. In that case, FMMμa,μb,t>1−ε. Hence, μa is a Cauchy sequence.☐Proposition 15.
In a fuzzy multiplicative metric spaceM,FMM,⋆, if a sequence μa converges in M, then μa is Cauchy.Proof.
Letε and t be real numbers with ε∈0,1,t>1. Since ε∈0,1, there is some ε0∈0,1 such that 1−ε0⋆1−ε0>1−ε. Also, suppose that μa converges in M, say it converges to μ∈M. Then, there exists a0∈ℕ such that for each a≥a0,
(7)FMMμa,μ,t1/2>1−ε0.
Thus, fora>b≥a0, we have
(8)FMMμa,μb,t≥FMMμa,μ,t1/2⋆FMMμb,μ,t1/2>1−ε0⋆1−ε0>1−ε.☐That isμa is a Cauchy sequence.Definition 16.
A fuzzy multiplicative metric spaceM,FMM,⋆ is termed as complete if and only if every sequence in M which is Cauchy must converge in M.Definition 17.
LetM,FMM,⋆ be a fuzzy multiplicative metric space. A subset A of M is closed if for each sequence μa in A which is convergent with μa⟶μ, we have μ∈A.Remark 18.
LetM,FMM,⋆ be a complete fuzzy multiplicative metric space. A subset A of M is closed if and only if A,FMM,⋆ is complete.The following lemma is the analogue of Kiany’s lemma [16] in the setting of newly defined space.Lemma 19.
LetM,FMM,⋆ be a fuzzy multiplicative metric space such that for μ,ν∈M,t>1 and h>1(9)lima⟶∞⋆i=a∞FMMμ,ν,thi=1.
Supposeμa is a sequence in M such that for all a∈ℕ(10)FMMμa,μa+1,tα≥FMMμa−1,μa,t,where 0<α<1. Then, μa is a Cauchy sequence.Proof.
For eacha∈ℕ and t>1, we have
(11)FMMμa,μa+1,t≥FMMμa−1,μa,t1/α≥FMMμa−2,μa−1,t1/α2≥⋯≥FMMμ0,μ1,t1/αa−1.
Thus, for eacha∈ℕ, we get
(12)FMMμa,μa+1,t≥FMMμ0,μ1,t1/αa−1.☐Choosing constantsh>1 and l∈ℕ such that hα<1 and ∑i=l∞1/hi=1/hl/1−1/h<1. Therefore, for b≥a,
(13)FMMμa,μb,t≥FMMμa,μb,t1/hl+1/hl+1+⋯+1/hl+b≥FMMμa,μa+1,t1/hl⋆FMMμa+1,μa+2,t1/hl+1nonumber,⋆⋯⋆FMMμb−1,μb,t1/hl+b.Using (12) in above inequality, we have
(14)FMMμa,μb,t≥FMMμ0,μ1,t1/αa−1hl⋆FMMμ0,μ1,t1/αahl+1⋆⋯⋆FMMμ0,μ1,t1/αb−2hl+b−a−2.That is(15)FMMμa,μb,t≥FMMμ0,μ1,t1/αha−1⋆FMMμ0,μ1,t1/αha⋆⋯⋆FMMμ0,μ1,t1/αhb−2.The above expression can be simplified as(16)FMMμa,μb,t≥⋆i=a∞FMMμ0,μ1,t1/αhi−1.Then, from the above, we have(17)lima,b⟶∞FMMμa,μb,t≥lima⟶∞⋆i=a∞FMMμ0,μ1,t1/αhi−1=1,for eacht>1. Hence, for each t>1,
(18)lima,b⟶∞FMMμa,μb,t=1,which shows thatμa is a Cauchy sequence.Definition 20.
Consider a fuzzy multiplicative metric spaceM,FMM,⋆ and A,B⊂M; then, for all t>1,
(19)A0=μ∈A:FMMμ,ν,t=FMMA,B,tforsomeν∈B,B0=ν∈B:FMMμ,ν,t=FMMA,B,tforsomeμ∈A,where
(20)FMMA,B,t=SupFMMμ,ν,t,μ∈A,ν∈B,
for allt>1.Definition 21.
LetM,FMM,⋆ be a fuzzy multiplicative metric space and A,B⊂M. If every sequence μa of A, fulfilling the condition that FMMν,μa,t⟶FMMν,A,t for some ν in B and for all t>1, has a convergent subsequence, then A is termed as approximatively compact with respect to B.


## 3. Best Proximity Point Theorems in Fuzzy Multiplicative Metric Spaces

In the present section, we prove some best proximity point theorems for single-valued and multivalued proximal contractions. First, we define the analogous of proximal contractions in the setting of fuzzy multiplicative metric space and then proceed to the main results.Definition 22.
LetM,FMM,⋆ be a fuzzy multiplicative metric space and A,B⊂M. A mapping T:A⟶B is named as multiplicative contraction of first kind if there exists α∈0,1, such that for all u,v,μ,ν∈A(21)FMMu,Tμ,t=FMMA,B,t,(22)FMMv,Tν,t=FMMA,B,t⇒FMMu,v,tα≥FMMμ,ν,t.Theorem 23.
LetM,FMM,⋆ be a complete fuzzy multiplicative metric space and A,B⊂M such that B is approximatively compact with respect to A. Assume that limt⟶∞FMMμ,ν,t=1, T:A⟶B be multiplicative contraction of first kind and TA0⊂B0. Then, T possesses best proximity point.Proof.
Letμ0∈A0 then for Tμ0∈TA0⊂B0, there exist μ1∈A0 such that
(23)FMMμ1,Tμ0,t=FMMA,B,t.
Further, sinceTμ1∈TA0⊂B0, there exist μ2∈A0 such that
(24)FMMμ2,Tμ1,t=FMMA,B,t.
Similarly, forTμ2∈TA0⊂B0, there exist μ3∈A0 such that
(25)FMMμ3,Tμ2,t=FMMA,B,t.
By continuing the similar steps, we get(26)FMMμa+1,Tμa,t=FMMA,B,tforalla∈ℕ.
By successive application of fuzzy multiplicative contraction, we have for alla∈ℕ∪0(27)FMMμa,μa+1,tα≥FMMμa−1,μa,t≥FMMμa−2,μa−1,t1/α≥FMMμa−3,μa−2,t1/α2≥⋯≥FMMμ0,μ1,t1/αa−1.
For anyq∈ℕ,
(28)FMMμa,μa+q,t≥FMMμa,μa+1,t1/q⋆FMMμa+1,μa+2,t1/q⋆⋯⋆FMMμa+q−1,μa+q,t1/q.
Using (27) in above inequality, we obtain
(29)FMMμa,μa+q,t≥FMMμ0,μ1,t1/qαa⋆FMMμ0,μ1,t1/qαa+1⋆⋯⋆FMMμ0,μ1,t1/qαa+q−1.
By assumption,limt⟶∞FMMμ,ν,t=1, we get that
(30)lima⟶∞FMMμa,μa+q,t=1⋆1⋆⋯⋆1=1.
Hence,μa is a Cauchy sequence. The completeness of fuzzy multiplicative metric space M,FMM,⋆ implies that μa converges to μ∗∈A, that is,
(31)lima⟶∞FMMμa,μ∗,t=1,
for allt>1. Notice that
(32)FMMμ,B,t≥FMMμ,Tμa,t≥FMMμ,μa+1,t1/2⋆FMMμa+1,Tμa,t1/2=FMMμ,μa+1,t1/2⋆FMMA,B,t≥FMMμ,μa+1,t1/2⋆FMMμ,B,t.☐Therefore,FMMμ,Tμa,t⟶FMMμ,B,t as a⟶∞. Since B is approximatively compact with respect to A, so Tμa has a convergent subsequence Tμak converging to some ρ∈B. Further, for each k∈ℕ, we have
(33)FMMA,B,t≥FMMμ,ρ,t≥FMMμ,μak+1,t1/3⋆FMMμak+1,Tμak,t1/3⋆FMMTμak,ρ,t1/3=FMMμ,μak+1,t1/3⋆FMMA,B,t1/3⋆FMMTμak,ρ,t1/3.Lettingk⟶∞, we get FMMμ,ρ,t=FMMA,B,t, which implies that μ∈A0 and TA0⊆B0 implies that Tμ∈B0, there exist μ∗∈A, such that FMMμ∗,Tμ,t=FMMA,B,t. From this and equation (26) implies that
(34)FMMμa+1,μ∗,t≥FMMμa,μ,t1/α.Applying limita⟶∞ to above inequality gives FMMμ,μ∗,t=1 which implies that μ=μ∗. Hence, FMMμ,Tμ,t=FMMA,B,t, which shows that T possesses best proximity point μ.Example 24.
LetM=ℝ+×ℝ+ and FMMμ,ν,t=t+1/t+dμ,ν where dμ,ν=μ1/ν1∗.μ2/ν2∗ for μ=μ1,μ2 and ν=ν1,ν2. Then, M,FMM,⋆ is complete fuzzy multiplicative metric space with ⋆:0,12⟶0,1 defined as a⋆b=ab. Let A=1,μ:μ∈ℝ+ and B=2,ν:ν∈ℝ+ then A and B are closed subsets of M and FMMA,B,t=t+1/t+2, A0=A,B0=B. Define T:A⟶B as
(35)T1,μ=2,μ22.Letμ=1,μ,ν=1,ν∈A and then u=1,μ2/2 and v=1,ν2/2∈A such that FMMu,Tμ,t=FMMA,B,t=FMMv,Tν,t. It can be easily checked that T is proximal contraction in fuzzy multiplicative metric space M with α=2/3. Also, the condition limt⟶∞FMMμ,ν,t=1 holds.Since all statements of Theorem23 hold, therefore T possesses best proximity point. We can see that 1,2 is best proximity point of T.IfA=B=M in Theorem 23, then we obtain the following corollary which is the fixed-point theorem for fuzzy multiplicative contraction in fuzzy multiplicative metric space.Corollary 25.
LetM,FMM,⋆ be a complete fuzzy multiplicative metric space. A mapping T:M⟶M satisfying FMMμ,ν,tα≥FMMμ,ν,t has fixed point provided that limt⟶∞FMMμ,ν,t=1.Now, we prove a best proximity theorem for Feng-Liu-type multivalued contraction in fuzzy multiplicative metric space.Theorem 26.
LetM,FMM,⋆ be complete fuzzy multiplicative metric space. A,B⊆M be two nonempty closed subsets of M having P-property and A0≠ϕ. Let T:A⟶CB be a mapping such that TA0⊆B0 and for all μ∈A0 and ν∈Tμ, there exist ρ∈A0 satisfying
(36)FMMν,ρ,t=FMMA,B,tandFMMν,Tρ,tc≥FMMμ,ρ,t,
for somec∈0,1 and t>1. Assume that M,FMM,⋆ satisfy
(37)lima⟶∞⋆i=a∞FMMμ,ν,thi=1,
for everyμ,ν∈M,t>1 and h>1. Then, T has best proximity point in A provided that fμ,ν=FMMν,Tμ,t is upper semicontinuous.Proof.
Letμ0∈A0 be arbitrary point. Choose ν0∈Tμ0. Then, by assumption, there exist μ1∈A0 such that
(38)FMMν0,μ1,t=FMMA,B,t,FMMν0,Tμ1,tc≥FMMμ0,μ1,t.
Presently, letb∈c,1, and then, we can discover ν1∈Tμ1 such that
(39)FMMν0,ν1,t≥FMMν0,Tμ1,tb.
Again by assumption, there existμ2∈A0 such that
(40)FMMν1,μ2,t=FMMA,B,t,FMMν1,Tμ2,tc≥FMMμ1,μ2,t.
Also, we can findν2∈Tμ2 such that
(41)FMMν1,ν2,t≥FMMν1,Tμ2,tb.☐Proceeding in similar manner, we develop two sequencesμa and νa in A and B, respectively, with μa∈A0, νa∈Tμa and
(42)FMMνa,μa+1,t=FMMA,B,t,(43)FMMνa,Tμa+1,tc≥FMMμa,μa+1,t,(44)FMMνa,νa+1,t≥FMMνa,Tμa+1,t,for alla∈ℕ and t>1. Then again, since A and B have P-property, so from inequality (43), we get
(45)FMMμa,μa+1,t=FMMνa−1,νa,t.Therefore, from inequality (44), we have
(46)FMMμa,μa+1,t=FMMνa−1,νa,t≥FMMνa−1,Tμa,tb.From inequality (44), we have
(47)FMMνa−1,Tμa,t≥FMMμa−1,μa,t1/c.Combining inequalities (46) and (47), we get
(48)FMMμa,μa+1,t≥FMMμa−1,μa,tb/c,for alla≥1 and t>1.Letk=c/b and then 0<k<1. The inequality (48) gives
(49)FMMμa,μa+1,tk≥FMMμa−1,μa,t,for0<k<1 and t>1. By our assumption (37) and Lemma 19, μa is Cauchy sequence.Now, from inequalities (44) and (46), we have
(50)FMMνa,Tμa+1,tc≥FMMμa,μa+1,t≥FMMνa−1,Tμa,tb⇒FMMνa,Tμa+1,t≥FMMνa−1,Tμa,tb/c.Also, from inequalities (44) and (56), we have
(51)FMMνa,νa+1,t1/b≥FMMνa,Tμa+1,t≥FMMνa−1,Tμa,tb/c⇒FMMνa,νa+1,tc≥FMMνa−1,Tμa,t,for0<c<1 and t>1. Hence, νa is Cauchy sequence.As subsetsA and B are closed in M, therefore μa,νa converges to points of A and B, respectively. Thus, there exist μ∗∈A and ν∗∈B such that μa⟶μ∗ and νa⟶ν∗ as a⟶∞.Lettinga⟶∞ in inequality (43), we have
(52)FMMμ∗,ν∗,t=FMMA,B,t,fort>1. The inequality (56) shows that the sequence fμa,νa=FMMνa,Tμa,t is increasing and it converges to 1. Since fμ,ν is upper semicontinuous, so
(53)1=limsupa⟶∞fμa,νa≤fμ∗,ν∗≤1implies to the fact that fμ∗,ν∗=1, that is, FMMν∗,Tμ∗,t=1, and hence, ν∗∈Tμ∗. Therefore,
(FM1)
(54)FMMA,B,t≥FMMμ∗,Tμ∗,t≥FMMμ∗,ν∗,t=FMMA,B,t,that is,FMMμ∗,Tμ∗,t=FMMA,B,t. This shows that T possesses best proximity point μ∗.IfA=B=M in Theorem 26, then we obtain the following corollary which is the fixed-point theorem for Feng-Liu-type contraction in fuzzy multiplicative metric space.Corollary 27.
LetM,FMM,⋆ be complete fuzzy multiplicative metric space. Let T:M⟶CM be a mapping, for all μ∈M and ν∈Ibμ (where Ibμ=ν∈Tμ∣FMMμ,ν,t≥FMMμ,Tμ,tb⊂M for some b∈0,1) satisfying
(55)FMMν,Tν,tc≥FMMμ,ν,t,
for somec∈0,1 and t>1. Then, T possesses fixed point provided that c<b and fμ=FMMμ,Tμ,t is upper semicontinuous.


## 4. Best Proximity Point Theorems of Feng-Liu-Type Mappings in Fuzzy Metric Space

Getting motivation from the work of Sahin et al. [10], we prove the following result.Theorem 28.
LetM,FM,⋆ be complete fuzzy metric space. A,B⊆M be closed and nonempty having P-property and A0≠ϕ. Let T:A⟶CB be a mapping such that TA0⊆B0 and for all μ∈A0 and ν∈Tμ there exist ρ∈A0 satisfying
(56)FMν,ρ,t=FMA,B,t,FMν,Tρ,ct≥FMμ,ρ,t,
for somec∈0,1 and t>0. Assume that M,FM,⋆ satisfy
(57)lima⟶∞∗i=a∞FMμ,ν,thi=1,
for everyt>0,h>1 and μ,ν∈M. Then, T possesses best proximity point in A provided that fμ,ν=FMν,Tμ,t is upper semicontinuous.Proof.
Letμ0∈A0 be arbitrary point. Choose ν0∈Tμ0. Then, by assumption, there exist μ1∈A0 such that FMν0,μ1,t=FMA,B,t and FMν0,Tμ1,ct≥FMμ0,μ1,t.
Presently, letb∈c,1, then we can discover ν1∈Tμ1 such that
(58)FMν0,ν1,t≥FMν0,Tμ1,bt.
Again by assumption, there existμ2∈A0 such that FMν1,μ2,t=FMA,B,t and FMν1,Tμ2,ct≥FMμ1,μ2,t.
Also, we can findν2∈Tμ2 such that
(59)FMν1,ν2,t≥FMν1,Tμ2,bt.☐Proceeding in a similar manner, we develop two sequencesμa and νa in A and B, respectively, with μa∈A0, νa∈Tμa and
(60)FMνa,μa+1,t=FMA,B,t,(61)FMνa,Tμa+1,ct≥FMμa,μa+1,t,(62)FMνa,νa+1,t≥FMνa,Tμa+1,t,for alla∈ℕ and t>0. Then again, since A and B have P-property, so from inequality (61), we get
(63)FMμa,μa+1,t=FMνa−1,νa,t.Therefore, from inequality (62), we have
(64)FMμa,μa+1,t=FMνa−1,νa,t≥FMνa−1,Tμa,bt.From inequality (62), we have
(65)FMνa−1,Tμa,t≥FMμa−1,μa,1ct.Combining inequalities (64) and (65), we get
(66)FMμa,μa+1,t≥FMμa−1,μa,bct,for alla≥1 and t>0.Letk=c/b and then 0<k<1. The inequality (66) gives
(67)FMμa,μa+1,kt≥FMμa−1,μa,t,for0<k<1 and t>0. By our assumption (57) and Lemma 19, μa is Cauchy sequence.Now, from inequalities (62) and (64), we have
(68)FMνa,Tμa+1,ct≥FMμa,μa+1,t≥FMνa−1,Tμa,bt⇒FMνa,Tμa+1,t≥FMνa−1,Tμa,bct.Also, from inequalities (62) and (68), we have
(69)FMνa,νa+1,1bt≥FMνa,Tμa+1,t≥FMνa−1,Tμa,bct⇒FMνa,νa+1,ct≥FMνa−1,Tμa,t,for0<c<1 and t>0. Hence, νa is Cauchy sequence.As subsetsA and B are closed in M, so μa,νa converges to points of A and B, respectively. Thus, there is some μ∗∈A and ν∗∈B such that μa⟶μ∗ and νa⟶ν∗ as a⟶∞.Lettinga⟶∞ in equation (61), we have
(70)FMμ∗,ν∗,t=FMA,B,t,for t>0. The inequality (68) shows that the sequence fμa,νa=FMνa,Tμa,t is increasing sequence, so it converges to 1. Since fμ,ν is upper semicontinuous, so
(71)1=limsupa⟶∞fμa,νa≤fμ∗,ν∗≤1implies to the fact that fμ∗,ν∗=1, that is, FMν∗,Tμ∗,t=1, and hence, ν∗∈Tμ∗.Therefore,(72)FMA,B,t≥FMμ∗,Tμ∗,t≥FMμ∗,ν∗,t=FMA,B,t,that is,FMμ∗,Tμ∗,t=FMA,B,t. This shows that T possesses best proximity point μ∗.Example 29.
LetJ=0,1∪1/2a:a∈ℕ and M=J×J, FMμ,ν,t=t/t+dμ,ν and dμ,ν=μ1−ν1+μ2−ν2 for μ=μ1,μ2 and ν=ν1,ν2∈M. Then, M,FM,⋆ is complete fuzzy metric space where ⋆:0,12⟶0,1 defined by a⋆b=ab. Let A=0,1/2a:a∈ℕ∪0,0,0,1 and B=1,1/2a:a∈ℕ∪1,0,1,1. Then, A0=A, B0=B and FMA,B,t=t/t+1. Define T:A⟶CB as
(73)T1,μ=0,12a+1,0,1ifμ=12a,a=0,1,2,⋯,0,0,0,12ifμ=0.
For allμ,ν∈M,lima⟶∞⋆i=a∞FMμ,ν,thi=1 which implies that M satisfies 16. Let μ=1,1/2a∈A0 and ν=0,1/2a1∈Tμ=1,1/2a; then, for ρ=1,1/2a1∈A0, we have FMν,ρ,t=FMA,B,t and FMν,Tρ,t=1≥FMμ,ν,t.
Also,(74)fμ,ν=FMν,Tμ,t=tt+dν,Tμ=tt+1/2a+1forμ=1,12a1forμ=1,0,1,1
is continuous. Since all conditions of Theorem28 are satisfied, so best proximity point for T exists. Furthermore, for u=1,1/2a,v=1,0∈A0HFMT1,1/2a,T1,0,ct=ct/ct+1/2 and FM1,1/2a,1,0,t=t/t+1/2a.
Assume that forc∈0,1, HFMT1,1/2a,T1,0,ct≥FM1,1/2a,1,0,t. That is
(75)ctct+1/2≥tt+1/2a,
which implies thatc≥2a−1 for a∈ℕ which is a contradiction. This shows that T does not satisfy the contraction condition of Nadler’s multivalued mapping.As corollary of Theorem28, we obtain a result which was proved in [23]. We get the corollary by taking A=B=M.Corollary 30.
LetM,FM,⋆ be complete fuzzy metric space. Let T:M⟶CM be a mapping, for all μ∈M and ν∈Ibμ (where Ibμ=ν∈Tμ∣FMμ,ν,t≥FMμ,Tμ,bt⊂M for some b∈0,1) satisfying
(76)FMν,Tν,ct≥FMμ,ν,t,for some c∈0,1 and t>1. Then, T possesses fixed point provided that c<b and fμ=FMμ,Tμ,t is upper semicontinuous.


## 5. Conclusion

Zadeh [24] introduced the notion of fuzzy logic to cope with the problem of uncertainty that occurs essentially while studying real-life problem. Many researchers found easiness to study the phenomenon of different fields that were too complex to be analyzed by conventional techniques. Fuzzy metric introduced by Kaleva and Seikkala [25] measures the imprecision of distance between elements. Fuzzy metric has been applied in variety of applications like color image filtering [26] and in engineering methods [15]. Multiplicative calculus has its great applications in various fields, few of which are in biomedical image analysis [27] and contour detection in images [28]. In this paper, we introduced fuzzy multiplicative metric space and proved some best proximity point and fixed-point results in this new framework. The above discussion shows the possible applications in this framework in the future.



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