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SOME NOVEL FIXED POINT RESULTS FOR (Ω, ∆)-WEAK CONTRACTION CONDITION IN COMPLETE FUZZY METRIC SPACES

ABSTRACT

In the present article, some fixed point theorems are investigated for two pairs of weakly compatible maps through (Ω, ∆)-type weak contractive maps in the framework of fuzzy metric spaces. The results studied in this workpiece are generalizations of some recent results existing in literature. Also, some illustrative examples are presented in last section to check the authenticity of our results.

Keywords:
fuzzy metric space; fixed points; (Ω; ∆)-weak contractive map; altering distance function

1 INTRODUCTION

Contraction principle given by Banach (1922BANACH S. 1922. Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae, 3: 133-181.) is the most eminent result in the era of metrical fixed point theory. Though this principle requires the continuity of the mapping, still it works as the back-bone even for the recent results in different metric spaces. An open question on the continuity of the mapping in Banach principle is answered by many authors. In 1968, Kannan (1968KANNAN R. 1968. Some result on fixed points. Bulletin of Calcutta Mathematical Society, 6: 71-78.) settled this problem in a robust way by introducing the following inequality:

d ^ ( T ρ , T σ ) β [ d ^ ( ρ , T ρ ) + d ^ ( σ , T σ ) ] for all ρ , σ U and β 0 , 1 2 . (1)

Later on, Rakotch (1962RAKOTCH A. 1962. A note on contraction mappings. Proceedings of American Mathematical Society , 13: 459-462.), Boyd & Wong (1969BOYD DW & WONG TSW. 1969. On nonlinear contraction. Proceedings of American Mathematical Society, 20: 458-464.) extended the contraction inequality due to Banach (1922BANACH S. 1922. Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae, 3: 133-181.) by characterizing a control function stated below:

d ^ ( T ρ , T σ ) α ( ρ , σ ) d ^ ( ρ , σ ) for all ρ , σ U and α : [ 0 , ] [ 0 , 1 ]

and

d ^ ( T ρ , T σ ) ϕ ( d ^ ( ρ , σ ) ) for all ρ , σ U ,

where ϕ:[0,][0,] is a non-decreasing continuous function such that ϕ(t) vanishes at t = 0. On the other hand, Alber & Guerre-Delabriere (1997ALBER YI & GUERRE-DELABRIERE S. 1997. Principles of weakly contractive maps in Hilbert spaces. New Results in Operator Theory in Advances and Appl., 98: 7-22.) introduced a modified contractive condition in Hilbert spaces which was further elaborated by Rhoades (2001RHOADES BE. 2001. Some theorems on weakly contractive maps. Nonlinear Analysis, 47: 2683-2693.) as follows:

If a mapping T: UU satisfies the following condition:

d ^ ( T ρ , T σ ) d ^ ( ρ , σ ) - Δ ( d ^ ( ρ , σ ) ) for all ρ , σ U ,

then T possesses a fixed point.

Zhang & Song (2009ZHANG Q & SONG Y. 2009. Fixed point theory for generalized-weak contractions. Applied Mathematics Letters , 22: 75-78.) proved unique common fixed point results for hybrid generalized ∆-weak contractive mappings in complete metric spaces whereas Doric (2009DORIC D. 2009. Common fixed points for generalized (ψ, ϕ) weak contraction. Applied Mathematics Letters, 22: 1896-1900.) established some related theorems using control functions. This work was an extension to the results due to Zhang & Song (2009ZHANG Q & SONG Y. 2009. Fixed point theory for generalized-weak contractions. Applied Mathematics Letters , 22: 75-78.). Then, Murthy et al. (2015MURTHY PP, TAS K & PATEL UD. 2015. Common fixed point theorems for generalized (ψ, ϕ) weak contraction condition in complete metric spaces. Journal of Inequalities and Applications Applied Mathematics, 139: 14 pages.) proved some results using weak contractive condition on two pairs of discontinuous weakly compatible mappings.

In 1975, the concept of fuzzy metric space is initiated by Kramosil & Michalek (1975KRAMOSIL I & MICHALEK J. 1975. Fuzzy metric and statistical metric spaces. Kybernetica, 11: 336-344.) with the concept of t-norm. Later on, George & Veeramani (1994GEORGE A & VEERAMANI P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64: 395-399.) extended the notion of fuzzy metric space by defining the Hausdorff topology in this framework. After that, Mihet (2008MIHET D. 2008. Fuzzy ψ-contractive mapping in non-Archimedean fuzzy metric spaces. Fuzzy sets Syst., 159: 739-744.) introduced the fuzzy version of Banach’s result and introduced fuzzy ψ-contractive type mapping in non-Archimedean fuzzy environment. A key distinction between a fuzzy metric and a classical metric is that the latter contains a parameter in its definition. This concept has been used successfully in engineering applications including colour picture filtering and perceived colour disparities. (For details, one can refer to the study of Camarena et al. (2008CAMARENA J, GREGORI V, MORILLAS S & SAPENA A. 2008. Fast detection and removal of impulsive noise using peer groups and fuzzy metrics. J. Visual Commun. Image Representation, 19: 20-29.), Camarena et al. (2010CAMARENA J, GREGORI V, MORILLAS S & SAPENA A. 2010. Two-step fuzzy logic-based method for impulse noise detection in colour images. Pattern Recognition Letters, 31: 1842-1849.), Morillas et al. (2009MORILLAS S, GREGORI V & HERVAS A. 2009. Fuzzy peer groups for reducing mixed gaussian impulse noise from color images. IEEE Transactions on Image Processing, 18: 1452-1466.), Morillas et al. (2007MORILLAS S, GREGORI V & PERIS-FAJARNES G. 2007. New adaptative vector filter using fuzzy metrics. J. Electron. Imaging, 16: 1-15.), Morillas et al. (2005MORILLAS S, GREGORI V, PERIS-FAJARNES G & LATORRE P. 2005. A fast impulsive noise color image filter using fuzzy metrics. Real-Time Imaging, 11: 417-428.), Morillas et al. (2008aMORILLAS S, GREGORI V, PERIS-FAJARNES G & LATORRE P. 2008a. Isolating impulsive noise color images by peer group techniques. Computer Vision and Image Understanding, 110: 102-116.), Morillas et al. (2008bMORILLAS S, GREGORI V, PERIS-FAJARNES G & SAPENA A. 2008b. Local self-adaptative fuzzy filter for impulsive noise removal in color image. Signal Process, 8: 390-398.)).

It has been demonstrated, in particular, that the class of topological spaces that are fuzzy metrizable matches with the class of topological spaces that may be metrized and then some traditional metric completeness and compactness theorems have been modified for fuzzy metric spaces. (See George & Veeramani (1995GEORGE A & VEERAMANI P. 1995. Some theorems in fuzzy metric spaces. J. Fuzzy Math., 3: 933-940.), Gregori & Romaguera (2000GREGORI V & ROMAGUERA S. 2000. Some properties of fuzzy metric spaces. Fuzzy Sets and Systems , 115: 485-489.)). However, compared to the traditional theories of metric completion, the theory of fuzzy metric completion is significantly distinct. In actuality, some fuzzy metric spaces are non-completable (See Gregori (2002GREGORI V. 2002. On completion of fuzzy metric spaces. Fuzzy Sets and Systems , 130: 399-404.)). The example below demonstrates the existence of a fuzzy metric space that forbids fuzzy metric completion.

Example 1 (Gregori (2002GREGORI V. 2002. On completion of fuzzy metric spaces. Fuzzy Sets and Systems , 130: 399-404.)).The continuous t-norm defined on [0, 1] × [0, 1] is indicated by the symboland is defined as

l * m = m a x { 0 , l + m - 1 }

for each l , m [ 0 , 1 ] .

Now, let { u m } m = 3 and { v m } m = 3 be two arbitrary sequences of non-identical points such that U V = ϕ , where U = { u m : m 3 } and V = { v m : m 3 } .

PutW=UV. Let M be a real-valued function defined on W × W × (0, ∞) as:

M ( u m , u n , t ) = M ( v m , v n , t ) = 1 - 1 m n - 1 m n M ( u m , v n , t ) = M ( v n , u m , t ) = 1 m + 1 n ,

for every m, n ≥ 3.

We firstly claim that (M, ∗) is a fuzzy metric on W.

Observe that the first four characteristics are nearly evident. (for m, n ≥ 3):

  1. 0<M(u,v,t)1for allu,vW, t>0;

  2. M(u,v,t)=1if and only if u = v;

  3. M(u,v,t)=M(v,u,t)for allu,vW, t>0;

  4. For every u, vW; M(u, v, .) is a continuous function on (0, ∞).

Also, a straightforward calculation reveals that, for every m, n, p ≥ 3 and s, t > 0;

M ( u m , u n , s ) * M ( u n , u p , t ) M ( u m , u p , s + t )

and

M ( v m , v n , s ) * M ( v n , v p , t ) M ( v m , v p , s + t ) .

Finally, the relationships listed below are simple:

M ( u m , u n , s ) * M ( u n , v p , t ) M ( u m , v p , s + t )

Similarly,

M ( u m , v n , s ) * M ( v n , v p , t ) M ( u m , v p , s + t )

and

M ( u m , v n , s ) * M ( v n , u p , t ) M ( u m , u p , s + t )

Thus, for every u, v, wW and s, t > 0, we have

M ( u , v , s ) * M ( v , w , t ) M ( u , w , s + t ) .

Hence, (M, ∗) is a fuzzy metric on W.

Next, we assert that in the fuzzy metric space (W, M, ∗); {um}m=3is a Cauchy sequence.

For fixed ε ∈ (0, 1) and s > 0, there exists m 0 ≥ 3 such that|1m-1n|<εfor all m, nm 0 .

Let nm. Then,

M ( u m , u n , s ) = 1 - 1 m - 1 n > 1 - ε

for m, nm 0 . Thus,{um}m=3is a Cauchy sequence in (W, M, ∗). Similarly,{vm}m=3is also a Cauchy sequence in (W, M, ∗).

Although,{um}m=3and{vm}m=3do not converge in W w.r.t. the topology ςMinduced by (M, ∗). Actually, ς M is the discrete topology on W as for every m ≥ 3 and each s > 0, we have forBM(u,ε,s)={uW:M(u,v,s)>1-ε}(where B M is the base with family of open sets of the form{BM(u,ε,s):uW,0<ε<1,t>0})

B M u m , 1 m ( m + 1 ) , s = { u m } a n d B M v m , 1 m ( m + 1 ) , s = { v m } .

To demonstrate the two prior equality claims, it is sufficient to observe that for m, n ≥ 3, with m ≠ n, and s > 0, we have

M ( u m , u n , s ) = 1 - 1 m n - 1 m n 1 - 1 m - 1 m + 1 = 1 - 1 m ( m + 1 ) .

Similarly,

M ( v m , v n , s ) = 1 - 1 m ( m + 1 )

and for m, n ≥ 3 and s > 0, we have

M ( u m , v n , s ) = 1 m + 1 n 1 - 1 m - 1 m + 1 .

Similarly,

M ( v m , u n , s ) = 1 - 1 m - 1 m + 1 .

Hence, (W, M, ∗) is not complete.

The main intent of our work is to extend and generalize (∆, Ω)-weak contraction due to Murthy et al. (2015MURTHY PP, TAS K & PATEL UD. 2015. Common fixed point theorems for generalized (ψ, ϕ) weak contraction condition in complete metric spaces. Journal of Inequalities and Applications Applied Mathematics, 139: 14 pages.) to fuzzy metric spaces. The authenticity of the results is further verified with some illustrative examples.

Theorem 1.[Murthy et al. (2015MURTHY PP, TAS K & PATEL UD. 2015. Common fixed point theorems for generalized (ψ, ϕ) weak contraction condition in complete metric spaces. Journal of Inequalities and Applications Applied Mathematics, 139: 14 pages.)] Let (U, d) be a metric space equipped with completeness, and C, D, E and T be the self mappings defined on U satisfying

Ω ( d ^ ( C ρ , D σ ) ) Ω ( M ( ρ , σ ) ) - Δ ( N ( ρ , σ ) )

for all ρ, σU, with ρ ≠ σ and

M ( ρ , σ ) = max d ^ ( E ρ , T σ ) , 1 2 ( d ^ ( E ρ , C ρ ) + d ^ ( T σ , D σ ) ) , 1 2 ( d ^ ( E ρ , D σ ) + d ^ ( T σ , C ρ ) )

and

N ( ρ , σ ) = min d ^ ( E ρ , T σ ) , 1 2 ( d ^ ( E ρ , C ρ ) + d ^ ( T σ , D σ ) ) , 1 2 ( d ^ ( E ρ , D σ ) + d ^ ( T σ , C ρ ) ) ,

C(U)T(U)andD(U)E(U), (C, E) and (D, T) are weakly compatible pairs.

Δ:[0,][0,)is such that ∆(t) > 0, which is lower semi-continuous for all t > 0 andis discontinuous at t = 0 with ∆(0) = 0,Ω:(0,)[0,)is an altering distance. Then C, D, E and T have a unique common fixed point in U.

2 PRELIMINARIES

Definition 1 (George & Veeramani (1994GEORGE A & VEERAMANI P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64: 395-399.)).Let*:[0,1]×[0,1][0,1]be a binary operation.is a continuous t-norm if it satisfies the postulates stated below:

  1. is commutative as well as associative;

  2. is a continuous binary operation;

  3. a*1=a a[0,1];

  4. a*bc*dprovided ac andbd a,b,c,d[0,1].

Definition 2 (George & Veeramani (1994GEORGE A & VEERAMANI P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64: 395-399.)). (U, M, ∗) is named a fuzzy metric space if U is any non-empty set, M is a fuzzy set onU2×[0,)and ‘’ is a continuous t-norm, satisfying the following axioms σ1,σ2,σ3Uand t, s > 0:

  1. M ( σ 1 , σ 2 , t ) is positive;

  2. M(σ1,σ2,t)=1 t>0σ1=σ2;

  3. M(σ1,σ2,t)=M(σ2,σ1,t);

  4. M(σ1,σ2,t)*M(σ2,σ3,s)M(σ1,σ3,t+s);

  5. M ( σ 1 , σ 2 , t , · ) : [ 0 , ) [ 0 , 1 ] is left continuous.

Here, M(σ1, σ2, t) signifies the degree of nearness between two elements σ1and σ2w.r.t. t. These spaces are referred as GV-spaces.

Lemma 1 (George & Veeramani (1994GEORGE A & VEERAMANI P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64: 395-399.)).if (U, M, ∗) is a fuzzy metric space (FMS), then M(ρ, σ, ·) is non-decreasing ρ,σU.

Definition 3 (George & Veeramani (1994GEORGE A & VEERAMANI P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64: 395-399.)).Let (U, M, ∗) be a FMS. Then,

  1. any sequence {ρ n } in U is convergent to a point t>0,limnM(ρn,ρ,t)=1.

  2. any sequence {ρ n } in U is named a Cauchy sequence if t>0and for eachϵ ]0,1),n0Nsuch thatM(ρn,ρm,t)>1-ϵ n,mn0.

  3. A fuzzy metric space in which every Cauchy sequence convergent in it, is named as complete fuzzy metric space.

3 MAIN RESULTS

Theorem 2.Let (U, M, ∗) be a complete fuzzy metric space, and let Θ, D, E and T : U → U be four mappings satisfying

Ω ( M ( Θ ρ , D σ , t ) ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) (2)

for all ρ, σ ∈ U, with ρ ≠ σ and

κ 1 ( ρ , σ , t ) = min { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T y , Θ ρ , t ) } ,

Θ ( U ) T ( U ) and D ( U ) E ( U ) , (3)

( Θ , E ) and ( D , T ) are weakly compatible pairs, (4)

where

Δ : [ 0 , 1 ] [ 0 , 1 ] is an upper semi-continuous mapping and Δ ( t ) is less than 1 t < 1 ,

Δ is discontinuous at t = 1 with Δ ( 1 ) = 0 , (5)

Ω : [ 0 , 1 ] [ 0 , 1 ] is a non-decreasing and continuous function with Ω ( t ) = 1 t = 1 . (6)

Then Θ, D, E and T possess a unique common fixed point.

Proof. Let ρ0 be any arbitrary point in U. As Θ(U)T(U) and D(U)E(U), therefore, there exists another point ρ1U for which Θρ0=Tρ1, and in the similar manner, for the point ρ1U, there exists a point ρ2U for which Dρ1=Eρ2. Following the same pattern, we can set up a sequence {σn } such that

σ 2 n + 1 = Θ ρ 2 n = T ρ 2 n + 1 , σ 2 n + 2 = D ρ 2 n + 1 = E ρ 2 n + 2 , for n = 0 , 1 , 2 ,

Let us suppose that for all nN{0},

σ 2 n σ 2 n + 1 . (7)

Now, we show that M(σ2n,σ2n+1,t)1 as n tends to nN{0}. Assume that ρ=ρ2n and σ=σ2n+1 in (2).

Ω ( M ( Θ ρ 2 n , D ρ 2 n + 1 , t ) ) = Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) Ω ( κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) ) + Δ ( κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) ) , (8)

where

κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) = min { M ( σ 2 n , σ 2 n + 1 , t ) , M ( σ 2 n , σ 2 n + 1 , t ) * M ( σ 2 n + 1 , σ 2 n + 2 , t ) , M ( σ 2 n , σ 2 n + 1 , t ) * M ( σ 2 n + 1 , σ 2 n + 1 , t ) }

and

κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) = max { M ( σ 2 n , σ 2 n + 1 , t ) , M ( σ 2 n , σ 2 n + 1 , t ) * M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) , M ( σ 2 n , σ 2 n + 2 , t ) * M ( σ 2 n + 1 , σ 2 n + 1 , t ) } .

Then by triangle inequality, we have

κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) min { M ( σ 2 n , σ 2 n + 1 , t ) , M ( σ 2 n , σ 2 n + 1 , t ) * M ( σ 2 n + 1 , σ 2 n + 2 , t ) M σ 2 n , σ 2 n + 1 , t 2 * M σ 2 n + 1 , σ 2 n + 2 , t 2 * 1 }

and

κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) = max { M ( σ 2 n , σ 2 n + 1 , t ) , M ( σ 2 n , σ 2 n + 1 , t ) * M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) , M ( σ 2 n , σ 2 n + 2 , t ) * M ( σ 2 n + 1 , σ 2 n + 1 , t ) } .

If

M ( σ 2 n , σ 2 n + 1 , t ) M ( σ 2 n + 1 , σ 2 n + 2 , t ) , (9)

then we obtain

κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) M ( σ 2 n + 1 , σ 2 n + 2 , t ) (10)

and (8) implies

Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) Ω ( κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) ) + Δ ( κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) ) ,

and so,

Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) Ω ( κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) ) .

Using monotonically increasing property of ∆ and Ω functions, we have

M ( σ 2 n + 1 , σ 2 n + 2 , t ) κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) . (11)

From (10) and (11), we get

κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) = M ( σ 2 n + 2 , σ 2 n + 1 , t ) . (12)

Since

1 ( M σ 2 n + 1 , σ 2 n + 2 , t 2 * M σ 2 n , σ 2 n + 1 , t 2 ) M ( σ 2 n + 2 , σ 2 n , t ) , (13)

we have, κ2(ρ2n,ρ2n+1,t)<1, then from (8), (12) and the properties of ∆ and Ω functions, one can get,

Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) + Δ ( κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) > Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) ,

this is a contradiction, thus we have

M ( σ 2 n + 1 , σ 2 n + 2 , t ) M ( σ 2 n , σ 2 n + 1 , t ) . (14)

So, we obtain the following

κ 1 ( ρ 2 n , ρ 2 n + 1 , t ) = M ( σ 2 n , σ 2 n + 1 , t ) , (15)

κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) = M ( σ 2 n , σ 2 n + 2 , t ) . (16)

Now putting (15) and (16) in (8), we have

Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) Ω ( M ( σ 2 n , σ 2 n + 1 , t ) ) + Δ ( M ( σ 2 n , σ 2 n + 2 , t ) ) , (17)

Ω ( M ( σ 2 n , σ 2 n + 1 , t ) ) , (18)

As Ω is a non-decreasing function, therefore, we get,

M ( σ 2 n + 1 , σ 2 n + 2 , t ) M ( σ 2 n , σ 2 n + 1 , t ) .

This shows that M(σ2n,σ2n+1,t) is a non-decreasing sequence, so there exists r > 0 such that

lim n M ( σ 2 n , σ 2 n + 1 , t ) = r . (19)

By (7) and (13), it follows that κ2(ρ2n,ρ2n+1,t)<1.

Taking limit n tends to ∞ in (18) and using (19), we get

lim n Ω ( M ( σ 2 n + 1 , σ 2 n + 2 , t ) ) lim n Ω ( M ( σ 2 n , σ 2 n + 1 , t ) + lim n Δ ( κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) ) ,

which gives,

Ω ( r ) Ω ( r ) + lim n Δ ( κ 2 ( ρ 2 n , ρ 2 n + 1 , t ) ) .

This is impossible with ∆ function, therefore

lim n M ( σ 2 n , σ 2 n + 1 , t ) = 1 .

Thus, nN{0}, we have

lim n M ( σ 2 n + 1 , σ 2 n + 2 , t ) = 1 ,

that is,

lim n M ( σ n , σ n + 1 , t ) = 1 . (20)

Next, we claim that the sequence {σn } is Cauchy.

For this, it is sufficient to prove that the sub-sequence {σ2n } of the sequence {σn } is Cauchy. Let us assume in a contrary manner that {σ2n } is not a Cauchy sequence. Consider the sequences {2n(k)} and {2m(k)} such that 2n(k) > 2m(k) > 2k for kN and

M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) 1 - ϵ . (21)

Choose 2n(k) to be the smallest index in such a way that (21) holds true.

Then,

M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) - 1 , t ) > 1 - ϵ for all k N . (22)

Putting ρ=ρ2m(k)-1 and σ=ρ2n(k)-1 in (2),

Ω ( M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) ) Ω ( κ 1 ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) ) + Δ ( κ 2 ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) ) , (23)

where

M ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) = min { M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) - 1 , t ) , M ( σ 2 m ( k ) - 1 , σ 2 m ( k ) , t ) * M ( σ 2 n ( k ) - 1 , σ 2 n ( k ) , t ) , M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) , t ) * M ( σ 2 n ( k ) - 1 , σ 2 m ( k ) , t ) } .

By triangle inequality, we obtain,

M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) M σ 2 m ( k ) , σ 2 n ( k ) - 1 , t 2 * M σ 2 n ( k ) - 1 , σ 2 n ( k ) , t 2 ,

taking k → ∞, we get

lim k M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) = 1 - ϵ . (24)

Now, for every k, we have

M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) - 1 , t ) M σ 2 m ( k ) , σ 2 m ( k ) - 1 , t 3 * M σ 2 m ( k ) , σ 2 n ( k ) , t 3 * M σ 2 n ( k ) - 1 , σ 2 n ( k ) , t 3 , M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) M σ 2 m ( k ) , σ 2 m ( k ) - 1 , t 3 * M σ 2 m ( k ) - 1 , σ 2 n ( k ) - 1 , t 3 * M σ 2 n ( k ) - 1 , σ 2 n ( k ) , t 3 .

Letting limit k → ∞ and using (20)-(24), we get

lim k M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) - 1 , t ) = 1 - ϵ . (25)

Also, for each positive value of k, we get

M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) , t ) M σ 2 m ( k ) - 1 , σ 2 m ( k ) , t 2 * M σ 2 m ( k ) , σ 2 n ( k ) , t 2 , M ( σ 2 m ( k ) , σ 2 n ( k ) , t ) M σ 2 m ( k ) , σ 2 m ( k ) - 1 , t 2 * M σ 2 m ( k ) - 1 , σ 2 n ( k ) , t 2 .

Taking k → ∞ and using (20)-(25), we get

lim k M ( σ 2 m ( k ) - 1 , σ 2 n ( k ) , t ) = 1 - ϵ . (26)

Again, for each positive value of k, we get

M ( σ 2 n ( k ) - 1 , σ 2 m ( k ) , t ) M σ 2 n ( k ) - 1 , σ 2 n ( k ) , t 2 * M σ 2 n ( k ) , σ 2 m ( k ) , t 2 , M ( σ 2 n ( k ) , σ 2 m ( k ) , t ) M σ 2 n ( k ) , σ 2 n ( k ) - 1 , t 2 * M σ 2 n ( k ) - 1 , σ 2 m ( k ) , t 2 .

Taking limit k → ∞ and using (20)-(26), we get

lim k M ( σ 2 n ( k ) - 1 , σ 2 m ( k ) , t ) = 1 - ϵ . (27)

From (23)-(27), one obtains

lim k M ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) = 1 - ϵ (28)

and

lim k κ 2 ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) = 1 .

Taking k → ∞ in (23), we get

Ω ( 1 - ϵ ) Ω ( 1 - ϵ ) + lim k Δ ( κ 2 ( ρ 2 m ( k ) - 1 , ρ 2 n ( k ) - 1 , t ) ) . (29)

As ∆ is discontinuous at t = 1 where ∆(t) = 0 and Δ(t)<1 t<1, the last term in (29) vanishes, which eventually lead to a contradiction.

Thus, {σn } is a Cauchy sequence. By the property of completeness, this sequence converges to some point ζ (say) in U. Consequently, its sub-sequences also converges to ζ in U i.e.

Θ ρ 2 n ζ , T ρ 2 n + 1 ζ , D ρ 2 n + 1 ζ , E ρ 2 n ζ .

Since D(U)E(U), there exists V such that ζ=E.

Let M(ζ,Θ,t)1 putting ρ= and y=ρ2n+1 in (2), we get

Ω ( M ( Θ , D ρ 2 n + 1 , t ) ) Ω ( κ 1 ( , ρ 2 n + 1 , t ) ) + Δ ( κ 2 ( , ρ 2 n + 1 , t ) ) , (30)

where

κ 1 ( , ρ 2 n + 1 , t ) = min { M ( E , T ρ 2 n + 1 , t ) , M ( E , Θ , t ) * M ( T ρ 2 n + 1 , D ρ 2 n + 1 , t ) , M ( E , D ρ 2 n + 1 , t ) * M ( T ρ 2 n + 1 , Θ , t ) }

and

κ 2 ( , ρ 2 n + 1 , t ) = max { M ( E , T ρ 2 n + 1 , t ) , M ( E , Θ , t ) * M ( T ρ 2 n + 1 , D ρ 2 n + 1 , t ) , M ( E , D ρ 2 n + 1 , t ) * M ( T ρ 2 n + 1 , Θ , t ) } .

Taking n → ∞ and using ζ=E, we have

M ( , ζ , t ) = max { M ( E , ζ , t ) , M ( E , Θ , t ) * M ( ζ , ζ , t ) , M ( E , ζ , t ) * M ( ζ , Θ , t ) } = M ( ζ , Θ , t ) .

Also, we have

Ω ( M ( Θ , ζ , t ) ) Ω ( M ( ζ , Θ , t ) ) + lim n Δ ( κ 2 ( , ρ 2 n + 1 , t ) ) .

As ∆ is discontinuous at t = 1 and ∆(t) = 0, we notice that

Ω ( M ( Θ , ζ , t ) ) Ω ( M ( ζ , Θ , t ) ) .

Consequently, we reach a contradiction with Ω function. Thus,

M ( ζ , Θ , t ) = 1 Θ = ζ Θ = ζ = E .

As (Θ, E) is a weakly compatible pair, it commutes at its coincidence point ℏ, i.e.ΘE=EΘΘζ=Eζ.

Next, we claim that Θζ = Eζ = ζ.

For this, putting ρ = ζ and σ = ρ2n+1 in (2), we get

Ω ( M ( Θ ζ , D ρ 2 n + 1 , t ) ) Ω ( κ 1 ( ζ , ρ 2 n + 1 , t ) ) + Δ ( κ 2 ( ζ , ρ 2 n + 1 , t ) ) , (31)

where

κ 1 ( ζ , ρ 2 n + 1 , t ) = min { M ( E ζ , T ρ 2 n + 1 , t ) , M ( E ζ , Θ ζ , t ) * M ( T ρ 2 n + 1 , D ρ 2 n + 1 , t ) , M ( E ζ , D ρ 2 n + 1 , t ) * M ( T ρ 2 n + 1 , Θ ζ , t ) }

and

κ 2 ( ζ , ρ 2 n + 1 , t ) = max { M ( E ζ , T ρ 2 n + 1 , t ) , M ( E ζ , Θ ζ , t ) * M ( T ρ 2 n + 1 , D ρ 2 n + 1 , t ) * M ( E ζ , D ρ 2 n + 1 , t ) * M ( T ρ 2 n + 1 , Θ ζ , t ) .

Taking n → ∞ and using Θζ = Eζ, we get

κ 1 ( ζ , ζ , t ) = M ( E ζ , ζ , t ) .

Now, (30) implies that

Ω ( M ( E ζ , ζ , t ) ) Ω ( M ( E ζ , ζ , t ) ) + lim n Δ ( κ ( ζ , ρ 2 n + 1 , t ) ) .

As ∆ is discontinuous at t = 1, we get ∆(t) = 0, which implies

Ω ( M ( E ζ , ζ , t ) ) > Ω ( M ( E ζ , ζ , t ) ) ,

but it is a contradiction. Therefore M(Eζ, ζ, t) = 1, implies

E ζ = ζ E ζ = Θ ζ = ζ .

Likewise, we can demonstrate that Tζ = Dζ = ζ.

Hence, Eζ = Θζ = Tζ = Dζ = ζ.

We now assert that ζ is the unique common fixed point of Θ, D, E and T. To show this, let w be another fixed point of Θ, D, E and T.

Now put ρ = ζ and σ = w in (2), we obtain

Ω ( M ( ζ , w , t ) ) Ω ( M ( ζ , w , t ) ) + Δ ( M ( ζ , w , t ) ) ,

which contradicts itself. Thus, M(ζ,w,t)=1ζ=w.

Hence Θ, D, E and T possess an unrepeated common fixed point in U. □

Assuming E = T = I (identity map), we deduce the following result:

Theorem 3.Let (U, M, ∗) be a fuzzy metric space equipped with completeness. Let Θ, D: UU be two self-mappings which satisfy the following inequality:

Ω ( M ( Θ ρ , D σ , t ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) , (32)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) } ;

and

  1. ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

  2. Ω : [0, 1] → [0, 1] is an altering distance function.

Then Θ and D possess a unique fixed point in U.

The result below is obtained by taking Ω = I(identity function):

Corollary 1.Let (U, M, ∗) be a fuzzy metric space equipped with completeness property. Let Θ, D, E and T : U → U be self-mappings holding following inequality:

M ( Θ ρ , D σ , t ) κ 1 ( ρ , σ , t ) + Δ ( κ 2 ( ρ , σ , t ) ) , (33)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) } ;

  1. Θ(U) ⊂ T(U) and D(U) ⊂ E(U),

  2. (Θ, E) and (D, T) are weakly compatible pairs,

  3. ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

Then Θ, D, E and T possess a unique common fixed point in U.

Corollary 2.Let (U, M, ∗) be a fuzzy metric space equipped with completeness property. Let Θ and D : U → U be self-mappings satisfying the following inequality:

M ( Θ ρ , D σ , t ) κ 1 ( ρ , σ , t ) + Δ ( κ 2 ( ρ , σ , t ) ) , (34)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) } ;

∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

Then Θ and D possess a unique fixed point in U.

If the aforementioned condition

κ 2 ( ρ , σ , t ) = max { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) }

is changed to

κ 2 ( ρ , σ , t ) = max { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) } ;

another result will be deduced as follows:

Theorem 4.Let (U, M, ∗) be a fuzzy metric space equipped with completeness. Let Θ, D, E and T be self-mappings defined on U such that they satisfy the following inequality:

Ω ( M ( Θ ρ , D σ , t ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) , (35)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) } ;

  1. Θ(U)T(U)andD(U)E(U),

  2. (Θ, E) and (D, T) are weakly compatible pairs,

  3. ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

  4. Ω : [0, 1] → [0, 1] is a strictly monotonically increasing altering distance function.

Then Θ, D, E and T possess a unique common fixed point in U.

On the same lines, the above theorem is easily demonstrable as Theorem 3.1.

Theorem 5.Let (U, M, ∗) be a fuzzy metric space equipped with completeness. Let Θ and D be self-mappings defined on U such that they satisfy the following inequality:

Ω ( M ( Θ ρ , D σ , t ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) , (36)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) , M ( ρ , D σ , t ) * M ( σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( ρ , σ , t ) , M ( ρ , Θ ρ , t ) * M ( σ , D σ , t ) } ;

  1. ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0,

  2. Ω : [0, 1] → [0, 1] is a strictly monotonically increasing altering distance function.

Then Θ, D possess a unique common fixed point in U.

Following are a few corollaries that arise from the results stated above:

Corollary 3.Let (U, M, ∗) be a fuzzy metric space equipped with completeness. Let Θ, D, E and T be self-mappings defined on U such that they satisfy the following inequality:

M ( Θ ρ , D σ , t ) κ 1 ( ρ , σ , t ) + Δ ( κ 2 ( ρ , σ , t ) ) , (37)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) } ;

  1. Θ(U)T(U)andD(U)E(U),

  2. (Θ, E) and (D, T) are weakly compatible pairs,

  3. ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

Then Θ, D, E and T possess a unique common fixed point in U.

Corollary 4.Let (U, M, ∗) be a fuzzy metric space equipped with completeness. Let Θ and D be self-mappings defined on U such that they satisfy the following inequality:

M ( Θ ρ , D σ , t ) κ 1 ( ρ , σ , t ) + Δ ( κ 2 ( ρ , σ , t ) ) , (38)

where ρ, σ ∈ U, ρ ≠ σ,

κ 1 ( ρ , σ , t ) = min { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) }

and

κ 2 ( ρ , σ , t ) = max { M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) } ;

Here, ∆ : [0, 1] → [0, 1] with ∆(t) < 1 is upper semi-continuous for each t ∈ (0, 1) andis discontinuous at the point t = 1 with ∆(t) = 0.

Then Θ, D possess a unique common fixed point in U.

Example 2.Let U = [0, 2] be equipped with the (usual) metricd^(ρ,σ)=|ρσ|and (U, M, ∗) be a fuzzy metric space. Let Θ, D, E and T be self mappings defined on U as

Θ ( ρ ) = { 0 if ρ = 0 ρ 7 + 1 otherwise E ( ρ ) = { 0 if ρ = 0 3 ρ 7 + 1 otherwise D ( ρ ) = { 0 if ρ = 0 2 ρ 7 + 1 otherwise T ( ρ ) = { 0 if ρ = 0 4 ρ 7 + 1 otherwise ,

where ρ , σ U , Θ ( U ) = [ 0 , 8 7 ] , E ( U ) = [ 0 , 9 7 ] , D ( U ) = [ 0 , 10 7 ] , T ( U ) = [ 0 , 11 7 ]

Here, Θ(U)T(U)andD(U)E(U), and (Θ, E) and (D, T) are weakly compatible maps at ρ = 0.

Let Ω(t) = t andΔ(t)={t2, if t10,t=1

Now, we examine Theorem 3.1’s inequality in several cases.

Case I. If ρ = 0 and σ = 0

Ω ( M ( Θ ρ , D σ , t ) ) = Ω ( t t + | Θ ρ D σ | ) = Ω ( 1 ) = 1 κ 1 ( ρ , σ , t ) = 1 Ω ( κ 1 ( ρ , σ , t ) ) = 1 κ 2 ( ρ , σ , t ) = 1 Δ ( κ 2 ( ρ , σ , t ) ) = 0 .

Hence,

Ω ( M ( Θ ρ , D σ , t ) ) = Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) .

Case II. If ρ = 0 and σ ≠ 0,

Ω ( M ( Θ ρ , D σ , t ) ) = Ω ( t t + | Θ ρ D σ | ) = Ω ( t t + | 0 ( 2 σ 7 + 1 ) | ) = t t + | 1 + 2 σ 7 | .

Now,

{ M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) } = { t t + | E ρ T σ | , t t + | E ρ Θ ρ | * t t + | T σ D σ | , t t + | E ρ D σ | * t t + | T σ Θ ρ | } = { t t + | 4 σ 7 + 1 | , 1 * t t + | 4 σ 7 + 1 1 2 σ 7 | , t t + | 0 ( 2 σ 7 + 1 ) | * t t + | 4 σ 7 + 1 0 | } = { t t + | 4 σ 7 + 1 | , t t + | 2 σ 7 | , t t + | 1 + 2 σ 7 | * t t + | 4 σ 7 + 1 | } ,

implies,

κ 1 ( ρ , σ , t ) = t t + | 4 σ 7 + 1 | κ 2 ( ρ , σ , t ) = t t + | 2 σ 7 | .

Therefore,

Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) = Ω ( t t + | 4 σ 7 + 1 | ) + Δ ( t t + | 2 σ 7 | ) = t t + | 4 σ 7 + 1 | + 1 20 ( t t + | 2 σ 7 | ) t t + | 2 σ 7 + 1 | = Ω ( M ( Θ ρ , D σ , t ) ) .

Hence,

Ω ( M ( Θ ρ , D σ , t ) ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) .

Case III. If ρ ≠ 0 and σ = 0.

Ω ( M ( Θ ρ , D σ , t ) ) = Ω ( t t + | Θ ρ D σ | ) = Ω ( t t + | ρ 7 + 1 | ) = t t + | ρ 7 + 1 | .

Now,

{ M ( E ρ , T σ , t ) , M ( E ρ , Θ ρ , t ) * M ( T σ , D σ , t ) , M ( E ρ , D σ , t ) * M ( T σ , Θ ρ , t ) } = { t t + | 3 ρ 7 + 1 | , t t + | 3 ρ 7 + 1 ρ 7 1 | * 1 , t t + | 3 ρ 7 + 1 | * t t + | ρ 7 + 1 | } = { t t + | 3 ρ 7 + 1 | , t t + | 2 ρ 7 | , t t + | 3 ρ 7 + 1 | * t t + | ρ 7 + 1 | } ,

By definition of κ1 and κ2 in Theorem 2, we get

κ 1 ( ρ , σ , t ) = t t + | 3 ρ 7 + 1 | , κ 2 ( ρ , σ , t ) = t t + | 2 ρ 7 | , Ω ( κ 1 ( ρ , σ , t ) ) + Ω ( κ 2 ( ρ , σ , t ) ) = t t + | 3 ρ 7 + 1 | + 1 20 ( t t + | 2 ρ 7 | ) t t + | ρ 7 + 1 | .

Hence,

Ω ( M ( Θ ρ , D σ , t ) ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) .

Case IV. If ρ ≠ 0 and σ ≠ 0.

Ω ( M ( Θ ρ , D σ , t ) ) = Ω ( t t + | Θ ρ D σ | ) = Ω ( t t + | ρ 7 + 1 2 u 7 1 | ) = Ω ( t t + ρ 7 ) .

Now,

{ t t + | E ρ T σ | , t t + | E ρ Θ ρ | * t t + | T σ D σ | , t t + | E ρ D σ | * t t + | T σ Θ ρ | } = { t t + ρ 7 , t ( t + 2 ρ 7 ) * t ( t + 2 σ 7 ) , t t + | 3 ρ 7 2 σ 7 | * t t + | 4 σ 7 ρ 7 | } ,

and

κ 1 ( ρ , σ , t ) = { t t + 2 ρ 7 , σ < 2 ρ 7 t t + 2 σ 7 , σ > 2 ρ 7 κ 2 ( ρ , t , σ ) = t t + | 3 ρ 7 2 σ 7 | ,

this implies,

Ω ( Θ ρ , D σ , t ) Ω ( κ 1 ( ρ , σ , t ) ) + Δ ( κ 2 ( ρ , σ , t ) ) .

The inequality is therefore true in each instance. As a result, Theorem 3.1’s criteria are all fulfilled and therefore Θ, D, E and T possess a unique common fixed point. Here, ρ = 0 is the unique common fixed point in U.

4 CONCLUSION

In this work, control functions are well used to locate fixed point for pairs of discontinuous maps in the setting of fuzzy environment. This is a fruitful strategy to broaden and generalize the literature’s findings in the direction of fuzzy metric space.

Acknowledgements

The authors declare that they have no competing interests.

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Publication Dates

  • Publication in this collection
    09 Oct 2023
  • Date of issue
    2023

History

  • Received
    15 Mar 2023
  • Accepted
    29 Apr 2023
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