New approach of Lebesgue integral in revised fuzzy cone metric spaces via unique coupled fixed point theorems Текст научной статьи по специальности «Математика»

New approach of Lebesgue integral in revised fuzzy cone metric spaces via unique coupled fixed point theorems

Ravichandhiran Thangathamizha, Angamuthu Muralirajb, Periyasamy Shanmugavelc

a Jeppiaar Institute of Technology (Autonomous), Department of Mathematics, Kanchipuram, Tamil Nadu, Republic of India, e-mail: [email protected], corresponding author,

ORCID Ю: https://orcid.Org/0000-0002-2449-5103

b Barathidasan University, Urumu Dhanalakshmi College,

PG & Research Department of Mathematics,

Trichy, Tamilnadu, Republic of India, e-mail: [email protected],

ORCID Ю: https://orcid.org/0009-0007-6696-6683

c Periyar University, Selvamm Arts and Science College (Autonomous), Department of Mathematics, Namakkal, Tamil Nadu, Republic of India, e-mail: [email protected],

ORCID Ю: https://orcid.org/0009-0004-8783-7088

doi https://doi.Org/10.5937/vojtehg72-48816

FIELD: mathematics

ARTICLE TYPE: original scientific paper

Abstract:

Introduction/purpose: This article introduces the concept of revised fuzzy cone contraction by using the concept of a traiangular conorm and Revised Fuzzy Cone contractive conditions.

Methods: This article established new Revised Fuzzy Cone Contraction (RFC-C) type unique coupled Fixed Point theorems (FP theorems) in revised fuzzy cone metric spaces (RFCMS) by using the triangular property of RFCMS.

Results: The obtained results on fixed points in revised fuzzy cone metric spaces generalize some known results in the litrature and present illustrative examples to support the main work.

Conclusion: The RFC contractive conditions generalize some important contraction types and examine the existence of a fixed point in revised fuzzy cone metric spaces. In addition, the Lebesgue integral type mapping is applied to get the existence result of a unique coupled fixed point in RFCMS to validate the main work.

Key words: revised fuzzy metric, revised fuzzy cone, fixed point.

Thangathamizh, R. et al, New approach of Lebesgue integral in revised fuzzy cone metric spaces via unique coupled fixed point theorems, pp.1029-1045

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Introduction

In the year 1965, Zadeh (Zadeh, 1965) introduced the concept of fuzzy sets which permit the gradual assessment of the membership of elements in a set. To use this concept in topology, Kramosil & Michalek (1975) introduced the class of fuzzy metric spaces [FMS]. After that, George & Veeramani (1994) modified the concept of fuzzy metric spaces and defined a Hausdorff topology on this fuzzy space. After that In 2015, the notion of fuzzy cone metric space (FCM space) was introduced by Oner et al. (2015). Grabiec (1988), gave the well-known Banach contraction principle in the case of fuzzy metric spaces, in the sense of Kramosil and Michalek.

Indeed, Huang & Zhang (2007) rediscovered the idea of a Banachvalued metric space. Indeed, many mathematicians proposed it, but it became popular after Huangand Zhang’s study. By adopting the theory that the underlying cone is normal, they demonstrated the convergence properties and some FP-theorems. In 2015, Oner et al. (2015) gave the idea of a fuzzy cone metric space (FCM-space), and they also presented some fundamental properties and “a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy.” After that, Li et al. (2021) settled some generalized fuzzy cone contractive type FP-results neglecting that “all the sequences are Cauchy” in a complete FCM-space. And later, Jabeen et al. (2020) presented some common FP theorems for three self-mappings, by taking into consideration the idea of weakly compatible in FCM-spaces with an integral type application.

Chen et al. (2020), gave the idea of coupled fuzzy cone contractive-type mappings. They proved “some coupled FP-theorems in FCM-spaces with non-linear integral type application.” Latterly Rehman & Aydi (2021) presented the concept of rational type fuzzy cone contraction mappings in FCM-spaces. They used “the triangular property of fuzzy metric” as a fundamental tool and proved some common FP-theorems and give an application. Guo & Lakshmikantham (1987) proved “coupled FP results for the nonlinear operator with applications”. Later, Bhaskar & Lakshmikantham (2006) present some coupled FP-theorems in the context of partially ordered metric spaces, and this work is also presented by Lakshmikantham & Ciric (2009). The concept of a cone metric space is introduced by Huang & Zhang (2007) and they also proved FP results. Some more fixed point results in a cone metric space and Fuzzy Metric spaces can be found in (Jankovic et al, 2010; Javed et al, 2021;

Kadelburg et al, 2011; Karapinar, 2010; Rezapour & Hamlbarani, 2008; Shamas et al, 2021) and the references therein.

Alexander Sostak (2018) additionally represented the idea of George-Veeramani Fuzzy Metrics Revised [RFMS]. Presently, Olga Grigorenko et al. (2020) introduced “On t-conorm primarily based Fuzzy (Pseudo) metrics”. In 2023, Muraliraj et al. (2023) proved some common coupled FP-results for commuting mappings in FMS. Muraliraj & Thangathamizh (2021b) introduced the concept of a Revised fuzzy modular metric space [RFMMS]. Moreover, Muraliraj & Thangathamizh (2023a) tend to prove that a Revised fuzzy cone topological space is pre-compact if and providing each sequence in it is a Cauchy subsequence. Further, we tend to show that Xt xX2 may be a complete Revised fuzzy cone topological space if and providing X1 and X2 are complete Revised fuzzy cone metric areas. Finally, it is tried that each divisible Revised fuzzy cone topological space is second calculable and a mathematical space of a separable Revised fuzzy cone topological space is separable. Some more t-conorm results in various metric spaces can be found in (Kider, 2020, 2021; Muraliraj & Thangathamizh, 2021a, 2022, 2023b; Oner & Sostak, 2020; Parakath Nisha Bagam et al, 2024; Muraliraj et al, 2024) and the references therein.

This paper presents some unique coupled FP findings in RFCMS by taking the idea of Guo & Lakshmikantham (1987) and Chen et al. (2020). Furthermore, we have also presented an application of the two Lebesgue Integral Equations (LIE) for a common solution to uphold our work. This paper is organized as follows: Section 2 consists of preliminaries. Section 3 establishes some unique coupled FP-results in RFCMS with illustrative examples. Section 4 presents an application of Lebesgue integral mapping to get the existence result of unique coupled FP in RFCMS to hold up the main work. In Section 5, we discuss the conclusion of our work presenting the objectives and hypotheses of the research or intervention.

Priliemeries

Some fundamental definitions and lemmas are given in this section. Definition 1 (George & Veeramani, 1994).

An operation О : [0, 1]2 ^ [0, 1] is called a continuous t-conorm if

(i) Ois associative, commutative, and continuous

(ii) 0 Oqi = 4i and qtOq2 ^ q3 O?4, whenever qt< q3 and

q2 < q4, V qt, q2, q3, q4 G [0, 1]

(iii) The maximum; q4Oq2 = max iqi, q2)

(iv) The product; qtOq2 = qi + q2~qiq2

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(vi) The Lukasiewicz; q30q2 = min {q1 + q2 ,1}.

Definition 2 (Huang & Zhang, 2007).

Let E be a real Banach space and 9 be the zero element of E, and p is a subset of E. Then, p is called a cone if,

(i) p is closed and nonempty, and рФ {#}

(ii) a1, a2 E R, at, a2> 0 and V a, bEp, then ata + a2bE p

(iii) both aEp and —a E pand then a = d

A partial ordering on a given cone pc E is defined by a<b^b — aEp. a< b stands for a<b and аФЬ, while a«b stands for b — a E int{p}. In this paper, all cones have nonempty interior.

Definition 3 (Sostak, 2018).

Let U be a set and O: [0,1]2 ^ [0,1] is a continuous t-conorm. A RFMS, on the set U is a pair (N0,O) or simply N0, where the mapping N0: U2 ^ [0, 1] satisfying the following conditions,

(RIF 1) N0{bltb2,t) < 1 and N0{bltb2, t) = 0 <^bl = b2 (RF 2) N0(b1, b2, t)=N0(b2, b3, t)

(RF 3) N0(b1, b2, t) QN0(b2, b3, s) >N0(b1, b3, t + s)

(RF 4) N0(b1, b2, -): (0, от) ^ 0, 1) is right continuous V b1, b2, b3E U andt,s> 0. Then, (N0,O) is said to be a RFM on Ж.

Definition 4 (Muraliraj & Thangathamizh, 2023).

A 3-tuple (U, N0,0) is said to be RFCMS if p is a cone of E, U is an arbitrary set, О is a continuous t-conorm and (N0,0)be a RFCM on U 2 x int (p) satisfying the following conditions; V bx, b2, b3 e U and t, s Eint(p).

i. N0(b3, b2, t) < 1 and N0{b1, b2, t) = 0 ^b3 = b2

ii. N0(bt, b2, t) =N0(b2, b3, t)

iii. N0(b1, b2, t) ON0(b2, b3, s) >N0(b1, b3, t + s)

iv. N0(b1, b2, -): int (p) ^ [0,1] is continuous.

Definition 5 (Muraliraj & Thangathamizh, 2023)

Let (U, N0,O) is a RFCMS, 3 E U and {bj} be any sequence in U.

(i) [bj] converges to b3 if for any ce (0, 1), t >>0, and 3 j3 eN such that

N0(bj,b3,t)<c, for j > \л. This can be written as limb: = 0 , or b,- ^b3 as ™.

V J ' J J

(ii) (bj) is Cauchy if for any c e (0, 1), t >0 and 3 jtEN such that N0(bj, b3, t) < c for j, k> j\.

(iii) (U,N0,0) is complete if every Cauchy sequence is convergent in U

(iv) [bj] is RFC contractive if 3 a(0,1) so that

for t >>в, j > 1.

(1)

Lemma 6

Let (U,N0,0)is a RFCMS and a sequence

bj U N0{bj,b3, t)^ 0 as j^<x for each t »в.

Definition 7

Let (U,N0,0)is a RFCMS. The RFCM N0 is triangular if

N0(b1, b3, t) <N0(b1, b2, t) + N0(b2, b3, t), V b3, b2, b3eU, t» в, (2)

Definition 8

Let (U, N0,0)is a RFCMS and A: U^U. Then, A is called RFC contractive if there is ae (0,1) so that

N0(Ab1, Ab2, t) <a(N0(b1, b2, t)), V b3, b2 eU and t» в, (3)

Definition 9

Let {Xj, X2) eU 2. Then, it is said to be coupled FP of a mapping A : U 2 if

A(Jэ2, b3) = b2,

A{b3, b2) = b3. (4)

As a follow-up to our original work, we now prove a few special pair FP theorems in RFCMS with examples. Additionally, we offer a Lebesgue integral contractive type application.

Main results

Now, the first main result is presented.

Theorem 10.

Let A: U2 is a mapping on complete RFCMS (U, N0,O) in which N0 is triangular and satisfies the following inequality

No(.A(a, b), A($, 0, t) <l(N0(a, f, t))+m(M(A(a, b), A($, 0,0) where

MiA^ b), A&0, t) = {+^о(а,щ, a t)+No(f' A(a,

V a, b, %, ^eU,t> в, l e 0,1), and m> 0 with (1 + Am) < 1.

Then, A has a unique coupled FP in U.

Proof:

N0(a, A(a, b), t)+N0($, Atf, 0, t) |

i, b), t))

(5)

(6)

Any a0, bQ eU; we define sequences {a,-} and {bj} in U such that

A(aj, bj) = aj+i, A{bj, aj) = bj+i, for j> 0. (7)

From (5) for t> 0,one gets

N0(aj, aj+1, t) = N0{A{aj_1, bj_-L), A{aj, bj), t)

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<1 (^N0{aj_1, aj, t)) + m [м{А{а]_1, bj_1), A(aj, bj), t)), where

м(д(п h (N°(aj~i,A(aJ~i,bi~i),t) + No(aj, A(aj,bj),t)\ (9)

M(^(a/'“1, ^ ^,bj),l) = I +N0{aj_1, A(a,, bj), t) ) (9)

= (w0(a/_1, aj, t), N0(aj, aj+1, t), N0{aj_1, aj+1, t))

< 2 ^N0{aj_1, aj, t) + N0{aj, aj+1, t)"j From (8) and (9), for t» d,

^0(^7, ^7 + 1, 0 — l (^o(^/-l, ^7, О) 2Ш^Л^о(я^_1, Ctj, t) + Nq(Uj, Q-j + i, t)) (10)

After simplification, one obtains

N0(aj, aJ+1, t)<A (N0{aj_1, aj, t)) for t»d, (11)

where A= ( l+2m ) < 1.

(1-2 m)

Similarly,

N0{a.j_1, aj, t) < A^N0(aj_2, aj_x, t)j for t» d.

From (11) and (12), by induction, for t» d,

N0(aj, aj+1, t)< A ^N0{aj_1, aj, tjj <A2 ^N0{aj_1, a}, tj]

< — <A2(^Nq(q.q,&]_, t))^ 0, as M.

The above shows that [aj] be a RFC-C; therefore, limN0(aj, a,+1, t) = 0, for t» d.

Now for i > j and for t» ,then

N0(aj + a]+1, t) < (N0(aj + a]+1, t) + N0{aj+1 + aj+2, t) + ••• + A2{N0(al_1 + at, t))) <AJ(N0(cl0 + аг, t))+ A^+1{N0(a0 + a±, t)) + —h V 1{N0(a0 + аг, t)) = {А-* + A-,+1 + — + A^ 1)(j{N0{a0 + Я]_, t)) ^ as <x>.

= 771 ((^oOo +at, t)) ) as j^<x. (15)

Hence, the sequence {a,-} is Cauchy. Now for the sequence {bj} and from (5), for t» tf,there is

No (bj, bj+1, t) = N0{A{bj_1, bj_t), A(bj, a.j), t)

<1 (N0{bj_1, bj, t)) +m [м{а{Ь]_1, aj_1), {bj, aj), t)), (16)

where

M(A(b, „a, ,),ft,,a,).t) = ("АЬ-»0+ «.(<*■ -»(V«ДO'

1 1 } 1 +N0{bj-1, A{bj, aj), t)

= bj, t)+N0{bj, bj+L t) + N0{bj_1, bj+L t))

(12)

(13)

(14)

= 2 (N0(bJ_1, bj, t) + N0(bj, bj+1, t))

Now, from (16) and (17), for t>> в,

N0{bj, bj+1, t) < l (N0(bj_1, bj, t)) + 2m(N0(bj_ll bj, t) + N0(bj, b,-+1, t))

one gets, after simplification,

N0{bj, bj+1, t)<X (N0(bJ_1, bj, t)), for t>> d,

where X = {l+2m{ < 1.

(1-2 m)

Similarly,

N0(bj_1, bj, t)<X (N0(bj_2, bJ_1, t)), for t>> .

Now, from (19) and (20) and by induction, for t>> d,

(17)

(18)

(19)

(20)

N0 ( b j ,b,+1 ,t ) ^ ( N0 ( bj-1 ,bj ,t)) ^ ^ ( N0 ( b,-2,bj -, t))

<■•■ <Xi{N0(b0, bx, t)) (21)

It shows that the sequence {bj} is a RFC-C; therefore,

limN0(bj, bi+1, t} = 0, for t>> d. (22)

K J J J

Now, for i > j and for t>> ,there is

N0{bj, bj+1, t) < (N0(bj, bj+1, t)) + (N0(bj+1, bj+2, t)) + - + X2{N0(bi_1, bt, t)) <XJ(N0(b0, b±, t)) + XJ+1(N0(b0, b±, t)) + --- + XJ~1(N0(b0, b±, t))

= {XJ +XJ+1 + - + Xj-1){N0(b0, b±, t)) as ».

= ttj {N0(b0, bt, t)) as j^«>. (23)

Hence, the sequence [bj] is Cauchy. Since A is complete and {a,}, [bj] are Cauchy sequences in A, so 3 a, b e A such that aj and bj ^b as

j^<x or this can be written as aj = a and bj = b.

Therefore,

lim.j^m N0{aj, a, t) = 0, lim.j^m N0{bj, b, t) = 0, for t>> d. (24)

Hence,

limj^ aj+1 — limj^ A(a.j, bj} — A{limj^„ aj, limj^ bj}^A(a, b) — a. (25) Similarly,

limj^m bj+1 = limj^m A{bj, aj)- A{limj^m bj, limj^m aj)^ A(b, a) = b.(26) Regarding its uniqueness, suppose (a±, b±) and (_b±, a±) are another coupled FP pairs in U2 such that A(a±,b±) = a± and A(b±,a±) = b±. Now, from (5), for t>> ,there exists

N0(a, a1, t) =N0(A(a, b), A(au b1), t)

<l(N0(a, a1, t))+m(M(A(a, b), (%, b±), t)), (27)

where

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меж»,«,(<>.. ь,).() = (^/)

+N0 (а. Л (а!. ЬД t) +N0 (%. Л (а. fo). t)

= (Л/0(а. а. t) +^0(а1. с^. t) + Л/0(а. с^. t) + ^0(а1. а. t))

= 2(W0(a.a1.t)) . (28)

Now, from (27) and for t» fl,

МоСа.аД) <Z(JV0(a.a1.t)) + 2m(W0(a.%. t)) = (/ + 2m)(W0(a.a1.t))

= (/ + 2m) < (/ + 2m)2(W0(a. t))

<•••<(/ + 2m)J'(W0(a.a1.t))^ 0, as M, (29)

where (/ + 2m) < 1.

Hence, there exists л/0(а.t) = 0 for a = and t» fl.

Similarly, again from (4), t» fl,there is

WoCfo.foi.t) =^0(Л(Ь.а).Л(Ь1.а1)д)

<l(N0(b.b1.t)) + m(M(A(b.a).(b1.a1).t)), (30)

where

м(л(ь.а).(ь1.а1).с) = (У?:/'Д/ Д

+N0 (fo. Л (fei. аД t) +W0(fo1. Л (fo. a). t)

= (iV0(fo. fo. t) + W0(fo1. fei. t) + N0(fo. fei. t) + W0(fo1. fo. t))

= 2(ВДА.О) (31)

Now, from (30) and for t» fl,

WoCfo.bai.t) </(iV0(fo.ba1.t)) + 2m(N0(fo.t)) = (/ + 2m)(N0(fo.t))

= (/ + 2m) < (/ + 2m)2(N0(fo.t))

<•••<(/ + 2m);(W0(fo. Ьг. t))^ 0, as y'^ M. (32)

Hence, there exists N0(fo.t) = 0 /or fo = and t» fl.

Corollary 11

Let Л : U2 be a mapping on complete RFCMS (U.N0.O) in which N0 is triangular and satisfies

w.«. **. о. о ^{+m[„o(a.,(:(:j;:;;i';(o)(,,(,f).t)]} <33>

v аДДДЕ U,t> в, /Е 0.1), and m> 0 with (/ + 2m) < 1. Then, Л has a unique coupled FP in U.

Corollary 12

Let Л : U 2 be a mapping on complete RFCMS (U. N0.O) in which N0 is triangular and satisfies

Л/0(Л(а. fo). Л(^. О. t)<( г(^о(а. f. 0) | (34)

+m[N0(a. Л(£. О. t) + N0(|'. Л(а. fo). t)]

v a, fo, (etf, t>e, le 0,1), and m> 0 with (1 + 2m) < 1. Then, A has a unique coupled FP in U.

Example 13

A = (0,от), ©is a t-conorm, and A : U 2 x (0, °°) ^ [0,1]is defined as

(35)

Л/0(а, fo, t) = ^ , d(a, fo) = ja — foj,

t+d(a,b)

v a, be Uand t>e. Then, it is easy to verify that N0 is triangular and (U, No,0)is a complete RFCM-space. We define

^, ifa, be [0,1]

b) = ) 2a + 2b — 2

- , ifa, be [1, «).

Now from (5), for t> 0,one obtains

-^-(W0(a, f, t))

(36)

N0(A(a,fo),A(£,0,t) = I , 12 ' ' I, for t>9.

0 +±(M(A(a, fo), A(Z, O, t))‘

It is easy to verify that conditions of Theorem 10 are satisfied with

l = m = —. Then, A has unique coupled FP for a = 2 and b = 2.

A(a, fo) =Л(2,2) = 2(2)+2(2)~2 = 2 (2,2) = 2.

(37)

(38)

N0(A(a,fo),Л(£, О, t) <

(39)

Theorem 14

Let A : U2 be a mapping on complete RFCMS (U, N0,O) in which N0 is triangular and satisfies the inequality

l(N0(a, t))

+m(N0(a, A(a, fo), t) +W0(f, A(%, 0, t))

,+4Wo(f,Л(а,fo), t) OW0((,Л£, 0,0), v a, fo, %, ^eU,t>6, le 0,1), and m, n>0 with (/ + 2m, я) < 1. Then, A has a unique coupled FP in U.

Corollary 15

Let A: U2 be a mapping on complete RFCMS (U, N0,O) in which N0 is triangular and satisfies the inequality

l(N0(a, %, t))

+n(N0(^, A(a, fo), t) OW0([, A$, 0, t))J v a, fo, %, ^eU,t>6, le 0,1), and m, n>0 with (/ + 2m, я) < 1. Then, A has a unique coupled FP in U.

Example 16

A = (0,от), o is a t-conorm, and A : U 2 x (0, ^ [0,1] is defined as

N0(A(a,fo),A{$, 0, t) <

(40)

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w°(a’b- t)=;SS) ’d(a-b) = la~bl (41)

v a, be Uand t> e.Then, it is easy to verify that N0 is triangular and (U,N0,0)is a complete RFCM-space. We define

^, ifa, be [0,1]

K) = ^2a + 2b-3 ,c ,Г1 л (42)

- , ifa, be [1, »).

Now from (39), for t> 0,one obtains, for t> в,

i

12

— (N0(a, f, t))

N0(A(a, b), Atf, 0, t) = | ^ 12

+ - (N0(a, A(a, b), t)+N0tf, A(£, Q, t))

It is easy to verify that conditions of Theorem 13 are satisfied with

l = m = - and n = 0.

8

Then, A has unique coupled FP for a = 2 and b = 2.

A(a, b) =Л(3,3) = 2(3)+2(3)~2 = 2 ^Л(3,3) = 3.

(43)

(44)

Application

In this section, we present an application of Lebesgue integral (LI) mapping to support our main work. In 2002, Branciari (Branciari, 2002) proved the following result on a complete metric space for a unique FP.

Theorem 17

Let (U, d) be a complete metric space, ae (0,1), and A: U^U a mapping such that v a, beU,

A (Aa, Ab) d (a,b)

I (pAs)ds <a I (pAs)ds , (45)

0 0

where y: (0, m) ^ (0, m) is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of (0, m) and for each r> 0,

dA a,

| pAs)ds.

(46)

Then, A has a unique FP ueU such that for any aeU, limAJa = u. Now,

we are in the position to use the above concept and to prove a unique coupled FP-theorem in FCM-spaces.

Theorem 18

Let A: U2 be a mapping on complete RFCMS (U, N0lO) in which N0 is triangular and satisfies the inequality

jV(ssjds < lS)ds + )ds

0 0 0

M(A(a, b), Atf, 0, t) = {

N0(a, A(a, b), t)+N0Q, A(%, (), t) j +N0 (а, Л(^, О, t) + N0 (^, Л (a, fo), t)J

(47)

v a, b, %, (e U,t>6, le 0,1), and m> 0 with (7 + 4m) < 1 and where q>:0,<x>) ^ 0, <x) is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of 0, m) and for each

t> 0,

d (a,b)

j (p(s)ds (48)

0

Then, A has a unique coupled FP in U.

Conclusion

This article introduced the idea of coupled FP-results in RFCMS and used "the triangular property of RFCMS" to demonstrate certain special coupled FPT under the revised contractive type requirements. Some examples are provided that supported our conclusions as well. Furthermore, a Lebesgue integral mapping application is provided to enhance our primary findings. With the aid of this novel idea, it is possible to demonstrate more modified and universal contractive type coupled FP results with various integral contractive type of conditions and applications throughout the whole RFCMS.

References

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Nuevo enfoque de la integral de Llebesgue en espacios metricos de conos difusos revisados mediante teoremas de punto fijo acoplados Lnicos

Ravichandhiran Thangathamizha, Angamuthu Muralirajb,

Periyasamy Shanmugavelc

a Instituto de Tecnologla Jeppiaar (Autonomo), Departamento de Matematicas, Kanchipuram, Tamil Nadu, Republica de la India, autor de correspondencia

b Universidad Barathidasan, Universidad Urumu Dhanalakshmi,

PG y Departamento de Investigacion de Matematicas,

Trichy, Tamilnadu, Republica de la India,

c Universidad de Periyar, Facultad de Artes y Ciencias de Selvamm (autonoma), Departamento de Matematicas,

Namakkal, Tamil Nadu, Republica de la India,

CAMPO: matematicas

TIPO DE ARTICULO: articulo cientifico original Resumen:

Introduccion/objetivo: Este articulo presenta el concepto de contraccion revisada del cono difuso utilizando el concepto de conorma triangular y las condiciones contractivas revisadas del cono difuso.

Metodos: Este articulo establecio nuevos teoremas de punto fijo acoplados unicos (teoremas FP) del tipo de contraccion difusa revisada (RFC-C) en espacios metricos de cono difuso revisados (RFCMS) mediante el uso de la propiedad triangular de RFCMS.

Resultados: Los resultados obtenidos en puntos fijos en espacios metricos de cono difuso revisados generalizan algunos resultados conocidos en la literatura y presentan ejemplos ilustrativos para respaldar el trabajo principal.

Conclusion: Las condiciones contractivas de RFC generalizan algunos tipos de contraccion importantes y examinan la existencia de un punto fijo en espacios metricos de cono difuso revisados. Ademas, se aplica el mapeo de tipo integral de Lebesgue para obtener el resultado de existencia de un punto fijo acoplado unico en RFCMS para validar el trabajo principal.

Palabras claves: metrica difusa revisada, cono difuso revisado, punto fijo.

Новый подход к интегралу Лебега в пересмотренных нечетких конусообразных метрических пространствах с помощью единой связанной неподвижной точки

Равичандиран Тангатамиж3, Ангамуту Муралираджб,

Периясами Шанмугавилв

a Технологический институт Джеппиара (автономный), математический факультет, Канчипурам, Тамилнад, Республика Индия, корреспондент

б Университет Баратидасан, колледж Дханалакшми в Уруму, Математический факультет и научно-исследовательский центр,

Тричи, Тамилнад, Республика Индия

в Университет Перияр, Салемский колледж искусств и естественных наук (автономный), математический факультет,

Намаккал, Тамилнад, Республика Индия

РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций,

27.39.15 Линейные пространства, снабженные

топологией, порядком и другими структурами ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: В данной статье представлена концепция пересмотренного сокращения нечеткого конуса с использованием концепции треугольника и пересмотренных условий сокращения нечеткого конуса.

Методы: В статье представлены новые пересмотренные теоремы об единой связанной неподвижной точке с типом сжатия нечеткого конуса (RFC-C) (теоремы FP) в

пересмотренных метрических пространствах нечеткого конуса (RFCMS), используя свойство треугольности RFCMS. Результаты: Полученные результаты по неподвижным точкам в пересмотренных нечетких конусообразных метрических пространствах обобщают некоторые известные результаты в литературе и представляют иллюстративные примеры в поддержку основной работы.

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Выводы: Условия сжатия RFC обобщают некоторые важные типы сжатия и исследуют существование неподвижной точки в пересмотренных нечетких конусообразных метрических пространствах. Помимо того, отображение интегрального типа Лебега применяется для получения результата существования единой связанной фиксированной точки в RFCMS для подтверждения основной работы.

Ключевые слова: пересмотренная нечеткая метрика,

пересмотренный нечеткий конус, неподвижная точка.

Нови приступ Лебеговом интегралу у прера^еним фази конусним метричким просторима помогу теорема ]единствене спрегнуте непокретне тачке

Равичандиран Тангатамиж!\ Ангамуту Муралирад* б,

Пери]асами Шанмугавилв

a Институт за технологи|у Цепиар, Оде^еже за математику,

Канчипурам, Тамил Наду, Република Инди]а, аутор за преписку

б Универзитет Баратидасан, Уруну Даналакшми колед,

Одележе математике за последипломске и истраживачке студне, Тиручирапали, Тамилнаду, Република Инди]а в Универзитет Периар, Колед уметности и науке Селвам (аутономни), катедра математике, Намакал, Тамил Наду, Република Инди]а

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:

Увод/циж Оваj рад уводи по}ам прера^ене фазне конусне контракцце помоПу концепта троугаоне конорме и прера^еног фазног конусног контрактивног услова.

Методе: Представ^ене су нове теореме ]единствене спрегнуте непокретне тачке типа RFC-C (revised fuzzy cone contraction -прере^ене фази конусне контракцие) у прера^еним фази конусним метричким просторима (RFCMS - revised fuzzy cone metric spaces) коришПе^ем своства троугла ще поседуу RFCMS.

Резултати: Добуени резултати на непокретним тачкама у прера^еним фази конусним метричким просторима генерализуу неке познате резултате из литературе и представъа}у илустративне примере щи подржава}у основу овог рада.

Закъучак: Контрактивни услови RFC генерализуу неке важне типове контракци'а и испитуу посто]аъе непокретне тачке у прера^еним фази конусним метричким просторима. Применено}е и пресликаваъе типа Лебеговог интеграла за добцаъе резултата

]единствене спрегнуте непокретне тачке у RFCMS за валидаццу овог рада.

К^учне речи: прера^ена фази метрика, прера^ен фази конус, непокретна тачка.

Paper received on: 20.01.2024.

Manuscript corrections submitted on: 22.09.2024. Paper accepted for publishing on: 23.09.2024.

© 2024 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).

Thangathamizh, R. et al, New approach of Lebesgue integral in revised fuzzy cone metric spaces via unique coupled fixed point theorems, pp.1029-1045