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8
Fixed point results in controlled revised fuzzy metric spaces with an application to the transformation of solar energy to electric power
Ravichandran Thangathamizha, Abdelhamid Moussaouib, Tatjana Dosenovicc, Stojan Radenovicd
a Jeppiaar Institute of Technology (Autonomous), Department of Mathematics, Kanchipuram, Tamil Nadu, Republic of India, e-mail: [email protected], corresponding author, ORCID ID: https://orcid.org/0000-0002-2449-5103 b Sultan Moulay Slimane University, Faculty of Sciences and Technics, LMACS, Beni Mellal, Kingdom of Morocco, e-mail: [email protected], ORCID ID: https://orcid.org/0000-0003-4897-1132 c University of Novi Sad, Faculty of Technology, Novi Sad, Republic of Serbia, e-mail: [email protected], ORCID ID: https://orcid.Org/0000-0002-3236-4410 d University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia, e-mail: [email protected],
ORCID ID: https://orcid.org/0000-0001-8254-6688
doi https://d0i.0rg/l 0.5937/vojtehg72-49064
FIELD: mathematics
ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: This study establishes sufficient conditions for a sequence to be Cauchy within the framework of controlled revised fuzzy metric spaces. It also generalizes the concept of Banach's contraction principle by introducing several new contraction conditions. The aim is to derive various fixed-point results that enhance the understanding of these mathematical structures.
Methods: The researchers employ rigorous mathematical techniques to develop their findings. By defining a set of novel contraction mappings and utilizing properties of controlled revised fuzzy metric spaces, they analyze the implications for sequence convergence. The methodology includes constructing specific examples to illustrate the theoretical results. Results: The study presents several fixed-point theorems derived from the generalized contraction conditions. Additionally, it provides a number of non-trivial examples that substantiate the claims and demonstrate the applicability of the results in practical scenarios. A significant application is
w
further research in both theoretical mathematics and its practical applications, particularly in the field of renewable energy.
•f explored regarding the conversion of solar energy into electric power,
| utilizing differential equations to highlight this connection.
" Conclusion: The findings deepen the understanding of Cauchy sequences
in fuzzy metric spaces and offer a broader perspective on the application of 5 the fixed-point theory in real-world scenarios. The results pave the way for
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t£ Key words: fixed point theorems, revised fuzzy metric space (RFMS),
y contraction principles(CP), Green's function, differential equation.
8 Introduction and preliminaries
o The existence of a unique fixed point (UFP) for self-mappings under
x suitable contraction conditions over complete metric spaces(CMS) is ¡^ guaranteed by Banach's fixed point theory (BSP) (also known as "the contraction mapping theorem"), one of the most significant sources of dc existence and uniqueness theorems in numerous areas of analysis. New extensions and generalizations of fixed point results are important because they increase our understanding of mathematical systems, enable the solution of specific problems, extend current theorems, and lead to the development of new theories and applications. They are an < important aspect of mathematical study and have far-reaching o ramifications in a variety of fields.
Fuzzy logic (FL) was established by Zadeh (1965). Unlike the theory of traditional logic, some numbers are not contained within the w stand. FL afflation of the numbers in the set defines an element within
the interval [0,1]. Zadeh has learned theories of fuzzy sets to be art, the problem of indefinity with the aid of uncertainty, the essential part of genuine difficulty.
& The theory is seen as a fixed point (FP) in the fuzzy metric space
(FMS) for various processes, one of them utilizing fuzzy logic. Later on, following Zadeh's outcomes, Heilpern (1981) established the fuzzy mapping (FM) notion and a theorem on an FP for fuzzy contraction mapping (FCM) in linear MS, expressing a fuzzy general form of BC-theory. In the definition of FMSs provided by Kaleva & Saikkala (1984), the imprecision is introduced if the distance between the elements is not a precise integer. After the first by Kramosil & Michalek (1975) and further work by George & Veeramani (1994), the notation of an FMS was introduced. After that, Klement et al. (2004) presented some problems on trigonomtric terms and releted operators.
Branga & Olaru (2022) proved several fixed point results for self-mappings by utilizing generalized contractive conditions in the context
of altered MS. Al-Khaleel et al. (2023) used cyclic contractive mappings of Kannan and Chatterjea type in generalized metric spaces. Czerwik (1993) found the solution of the well-known BFPT in the context of b-metric spaces (b-MS). Mlaiki et al. (2018) defined controlled MS as a generalization of b-MS by utilizing a control function a: S2 ^[1, m) of the otherside of the b-triangular inequality. The relation between b-MS and FMS has been discussed by Hassanzadeh & Sedghi (2018). Li et al. (2022) used Kaleva-Seikkala's type FbMSs and proved several fixed point results by using contraction mappings. Furthermore, Sedghi & Shobe (2012) and Sedghi & Shobkolaei (2014) proved various common FPTs for R-weakly commuting maps in the frame work of FbMSs. Sezen (2021) introduced controlled FMS as a generalization of FMS and FbMS by applying a control function a: S2 ^[1,m) in a triangular inequality of the form:
M(b, w,t + s) > M (b, z, ———-) + M (z, w,—;--) ,for all b, w,z £: and s,t
\ a(b,z)J \ a(z, w)J
> 0 .
If we take a(b,z) = a(z, w) = 1 then its an FMS and for a(b, z) = a(z, w) > s with s > 1 it is then an FbMS.
Ishtiaq et al. (2022, 2023) established the theory of double-controlled intuitionistic fuzzy metric-like spaces by "considering the case where the self-distance is not zero"; if the metric's value is 0, afterwards it must be a self-distance and also an established FP theorem for contraction my mappings. See (Schweizer & Sklar, 1960) for triangular conorm (TCN), | continuous triangular conorm (CTCN) (Kaleva & Seikkala, 1984), and TN of H-type (Hadzic, 1979; Hassanzadeh & Sedghi, 2018). In (Hussain et al, 2022), the authors worked on CFMSs by utilizing orthogonality and pentagonal CFMSs and proved several FPRs for contraction mappings. Rakic et al. (2020) proved a fuzzy version of BFPT by using Ciric-quasi-contraction in the context of FbMSs. Younis et al. (2022) proved several FPRs in the context of dislocated b-metric spaces and solved the turning in circuit problem. iop d
In 2018, Sostak (2018) explored the idea of Revised Fuzzy Metric Spaces (shortly, RFM-Space) as a new idea to express a revised fuzzy set. Then Grigorenko et al. (2020) made several proposals based on RFM-Spaces. Building on the work of Sostak (2018), Muraliraj et al. (2023) defined the notion of retinal-type revised fuzzy contraction and proved a fixed point theorem of RFM spaces. Muraliraj & Thangathamizh (Muraliraj & Thangathamizh, 2021ab) partially came up with the idea of fixed point results for RFM in revised fuzzy contraction (shortly, RFM contractions). The following authors (Adabitabar Firozja & Firouzian, 2015; Kider, 2020,
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2021; Moussaoui et al, 2022; Muraliraj & Thangathamizh, 2021a, 2023) provided many concepts of RFMS and they proved to be quite useful in this study. Gregori & Minana (2021) introduced a new version of contraction principles in the context of fuzzy metric spaces. It is defined by the means of t-conorms. After that Muraliraj & Thangathamizh (2022), Thangathamizh et al. (2024) established new revised fuzzy cone contraction type unique coupled fixed point theorems in revised fuzzy cone metric spaces by using the property of triangular.
In this article,
- we prove that a sequence must be Cauchy in the CRFMS under some conditions;
- we prove a fixed point result by using new Ciric-quasi-contraction and generalize the Banach contraction principle by utilizing several new contraction conditions;
- we provide several non-trivial examples to show the validity of the main results;
- we discuss an application concerning the transformation of solar energy to electric power.
Now, we provide several definitions and results that are helpful to understand the main section.
Definition 1 (Sostak, 2018). A binary operation r: [0,1]2 ^[0,1] is a CTCN if it verifies the conditions below:
(C 1) r is commutative and associative,
(C 2) r is continuous,
(c 3) r(K,0) = k for all kg[0,1] ,
(c 4) r(K, p) <r(c, d) for k, p,c, d e [0,1] such that k< c and p< d.
The examples of CTCN are rp(a,b) = a + b - a .b, rmax(a,b) = max{a, b}, and rL(a, b) = min{a + b, 1}.
Definition 2 (Hadzic, 1979). Suppose that r is a TCN and suppose that rT: [0,1] ^ [0,1], xe N , express the process given below:
^0) = r(b,b),rT+1(b) = r(TT(b),b), xeN,b e[0,1].
Then, TCN r is H-type if the family {rT(b)}TeN is equi-continuous at b = 1.
A TCN of H-type is rmax and see [9, 18] for examples.
Each t-conorm can be generalized in a different way to an n-ary process via associativity, taking (b1,...,bn)e [0,1]n for the values
r1i=1bi = bi, r1i=1bi = r(r1i=1bi,bT) = r(bi.....bT).
Example 1. An n-ary generalization of the TCNrmin, rL, and rP are:
rmax (bi,...,bx) = max{bi,...,bT}
rL(bi.....bT) = min{Ei=1 bi - (x- 1) ,1}, TpCbi.....bT)= n?=i b;
A TCN r can be extended to accountable infinite operation, for any sequence (bT)TeN considering from [0,1] the value r^ = lim r^ b
The sequence {ri=1T(b)}TGN is non-increasing and bounded below and so the limit ri=1m(bi) exists. In the FP theory (Hadzic & Pap, 2001; Hadzic, 1979), it might be interesting to look at the category of TCNr and sequence (bT) in the range [0,1] such that lim bT = 0 and
lim ri=1m(bi) = lim ri=1m(bT+i) (1)
Proposition 1. Suppose (bT)TGN is a series of numbers with the range [0,1] such that lim bT = 0 and assume r to be a TCN of H-type. Then,
lim ri=1mbT+i. Throughout this study, we utilize H2;HxS.
Definition 3 (Sostak, 2018). A 3-tuple (S,M,r) is known as an RFMS if S be a some (nonempty) set, r is a CTCN, and M is an RFM on S2 x(0, ot) and satisfies the following conditions, for all b, w,z £s, and t,s > 0:
(Rfm 1) M(b, w,t) < 1, (Rfm 2) M(b, w,t) = 0 iff b = w, (Rfm 3) M (b, w,t) = M (w,b, t), (Rfm 4) r(M(b, w,t),M(w,z, s)) > M(b, z, t + s), (Rfm 5) M(b,w,.): (0,ot) ^ [0,1] is continuous. Definition 4. A 3-tuple (S,M,F) is called an RFbMS if S is a random (non-empty) set, r is a CTCN, and M is an RFM on S2 x(0, ot) and e satisfies the following conditions for all b, w,z GS, t,s > 0, and p> 1 as a | real number:
(Rb 1) M(b, w,t) < 1, (Rb 2) M(b, w,t) = 0 iff b = w, (Rb 3) M (b, w,t) = M (w,b, t), (Rb 4) r(M(b, w,t),M(w,z,s))> M(b,z, p(t + s)), (Rb 5) M(b, w, -): (0, ot) ^[0,1] is continuous.
The following fixed point theorem is using a new Ciric-quasi- tino contraction in the context of RFbMSs.
Theorem 1. Suppose that (S,M,rmin) is a complete RFbMS, assume that /:S^S. If for some KG (0,1), such that
(M« (b, w, -) ,Ma (fb, b, -) ,Ma (fw, w, -),) Ma(fb,fw,t)< max] V m « \ - k b, wG
[ Ma(fb, w, 7J ,Ma^b,fw, -J J
S,t > 0. Then, f has a UFP in S.
Lemma 1. Let M(b, w, -) be an RFbMS. Then, M(b, w,t) is b-non-decreasing with respect to for all b, wGS.
Si
Definition 5. Let S be a non-empty set, a: S2 ^[1, œ), r is a CTCN, | and Ma is an RFM on S2 x[0, œ) and satisfies the following conditions for
all b, w,z GS,s, and t > 0:
(ERFMa 1) Ma(b, w,0) < 1, (ERFMa 2) Ma(b, w,t) = 0 iff b = w, (ERFMa 3) Ma(b, w,t) = Ma(w,b,t),
(ERFMa 4) Ma (b, z, a(b, z)(t + s))< r(Ma (b, w,t),Ma (w,z, s)), (ERFMa 4) M(b, w,.): (0, ^[0,1] is right continuous. E Then, the triple (S,Ma, r) is said to be an extended revised fuzzy b-
o metric space and Ma is said to be a controlled RFM on S.
Theorem 2. Suppose that (S,M,F) is a complete RFMS with /: S2 ^
0 [1,ro), assume that lim Ma(b, w,t) = 0, for all b, wGS. Iff : S^S satisfies
1 the following for some KG (0,1), such that
<
M
((fb,fw,t)< Ma (b, w,for all b, wGS,t > 0.
dc Also suppose that for arbitrary b0 GS and n,q g N, there is
a(bn,bn+q) — ^
where bn = f nb0.Then, f has a UFP in S.
Definition 6. Let S be a non-empty set, a: S2 ^[1, ot), r is a CTCN, w and Ma is an RFM on S2 x[0, m) and satisfies the following conditions for all b, w,z gS, s, and t > 0:
s2 (RFMa1)Ma(b, w,0)< 1,
° (RFMa2)Ma(b, w,t) = 0 iff b = w,
(RFMa3)Ma(b, w,t) = Ma(w,b,t),
(^FMa4)Ma(b,z,t + s) <r (Ma (b, W,,Ma (w,z, ,
(RFMa5)M(b, a),.): (0,oo) [0,1] is right continuous. & Then, the triple Ma, T) is said to be a CRFMS and Ma is said to be
a controlled RFM on S.
Muraliraj and Thangathamizh, (2021a) proved the following Banach contraction principle in the context of CRFMS.
Theorem 3. Suppose that (S,M, F) is a complete CRFMS with a: S2 ^[1,ot), assume that lim Ma(b, w,t) = 0, for all b, wGS.
If /: S^S satisfies the following for some Kg(0, 1), such that
Ma(fb,fw,t)< Ma (b, w,£), for all b, wGS,t > 0.
Also, suppose that for all b GS, we obtain lim Ma(bn, w) and
lim Ma(w,bn) which exist and are finite. Then, f has a UFP in S.
Definition 7. Suppose M(b, w,t) is a CRFMS. For t > 0, the open ball B(b,l,t) with the center b £2 and the radius 0 <Z < 1 is expressed as a sequence {bT}:
G-Convergent to b if M(bT,b,t)^ 0 as x^^ or for every t > 0.We write lim bT = b.
is said to be a Cauchy sequence (CS) if for all 0 <e < 1 and t > 0 there exist satisfying x0 £ N such that M(bT,bm,t) <e for all x,m >x0.
The CRFMS(2,M, Q is a G-complete if every Cauchy sequence is convergent in 2.
Main results
In this part, we discuss several new results in the context of CRFMSs.
Lemma 2. Suppose {bT} is a sequence in a CRFMS(2,Ma, r). Let tt£ (0,1) exist such that
M«(bT,bT+1,t) < Ma (bT_!,bT, 3 (2)
and bT £2, and v £(0,1) exist such that
lim M(b, w,t) = 0, t > 0 (3)
Then, {bT} is a Cauchy sequence.
Proof. Suppose k£ (N,1) and the total E^K is convergent, x0 £ N exists such that E^K < 1 for every x>x0. Let x>m > x0. Since Ma b-non-decreasing, by (RFMa 4), for every t > 0, one can obtain
( ET:Fm"1Ki M«(bT,bT+m,t)< Ma (bT,bT+m^^-
a(bT,bT+m) <
r M„ (bT,bт+1, a(bT,bT+i)a(bT,bT+m)) ,Ma (^lA^ «(b^^Mb^^)))
/ Ma (bт,bт+l, a(bT,bT+1)«(bT,bT+m)) , \
<r
M A b _tKT+1_\
aV т+1, т+2, a(bx+1,bx+2)a(bx+1,bx+m)a(bx,bx+m)J ,
\Ma (bт+2,bт+m, a(bx+2,bx+m)a(bx+1,bx+m)a(bx,bx+m))/
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( M„ (bт,bт+1, a(bT,bT+i)a(bT,bT+m)) ,
M A b _tKT+1_N
Ma (bт+2,bт+3, Ma ^'^S^t^^
x(bx+2,bx+3)a(bx+2,bx+m)a(bx+1,bx+m)a(bx,bx+m)>'
a(bx+3,bx+4)a(bx+2,bx+3)a(bx+2,bx+m)a(bx+1,bx+m)a(bx,bx+m)y
Ma (b
'T+m-2,bT+m-l,
tKT
V
M
^bx+m-l,'
«(bt+m-z.bt+m-i) niJx Z«(bi,bT+m)
)
^ a(bT+m_2,bT+m_1) n^?1-1 «(bi,bT+m)J / From inequality (2), one deduces Ma(bT,bT+1,t) < Ma (b0,b1, K7) and
since x> m and a: S2 ^[1, ot), one can obtain
Ma(bT,bT+m,t) ^
i M„fb„.b,.-—-j, \
<r
M
M«(b0,b1,
(bo,bl, a(bT,bT+1)a(bT,bT+m)KT txT
a(bT+l,bT+2)a(bT+l,bT+m)a(bx ,bT+m)K
T+l
M«(b0,b1,
tK
.T+m-2
a(bT+m_2,bT+m_1)ni=+Tm"' a(bi,bT+m)K
jT+m-l
V
Malbcbi,-
tKT
/
As x^ot and by utilizing (3),one obtains Ma(bT,bT+m,t) <r(0,0,— ,0) = 0. Hence,{bT} is a Cauchy sequence in S.
Corollary 1. Suppose {bT} is a sequence in CRFMS (S,Ma, Q and r is H-type. If KG (0,1) exists such that
M«(bT,bT+1,t) < Ma(bT_1,bT, K), xGN,t > 0. (4)
Then,{bT} is a continuous Sequence. Lemma 3. If for b, wg S, and some Kg(0,1),
Ma(b, w,t)< Ma(b, w, K) ,t > 0. (5)
Then, b = w.
Proof. Inequality (5) implies that
,(b, w,t)< Ma (b, w, —),iGM,t > 0.
Mn
r
Now,
Ma(b, w,t) < lim Ma (b, w,—) = 0,t > 0.
And by (RFMa 2), it is easy to see that b = w. Theorem 4. Suppose that (2,M,F) is a complete CRFMS and suppose that f: 2^2. Let them exist KG (0,1) such that
'mq (b, w, 0 ,Ma (b, fw, 0 ,Ma (fb, w, ^
Mq(b,f^,K)+ Mg(fb,^) 2
Ma(b,f^)+ Mg(fb,^)
1+Ma(b,«,i) J
Ma(fb,fw,t) < min
,b, w£
:,t > 0.
and b, w£2 such that
lim Ma(b, w,t ) = 0,t > 0
t^ 00
(6) (7)
Then, f has a UFP in 2.
Proof. Suppose b0 G2 and bT+1 = fbT, iGi Consider b = bT an dw = bT_! in (6), then one can obtain
Ma(bT,bT+1,t) < Ma(fbT_1,fbT,t ), <
min
M
(bT-i,bT, 3, r (mq (bT_!,bT, ,Ma (bT,bT+1,3/ Ma(bT,bT, i),
< min ¡Ma (bT_!,bT,,Ma (bT,bT+1,£)}
If Ma(bT,bT+1,t ) < Ma(bT,bT+1,,x£N,t > 0.
Then by Lemma 3 such that bT = bT+1,t£ N, there is
Ma(bT,bT+1,t )< Ma(bT_1,bT,,x£N,Ma(bT,bT+1,t ) and by
Lemma 2 it follows that {bT} is a CS. Since, (2,M,F) is complete, b £2 exist such that lim bT = b and
lim Ma(b,bT,t) = 0,t > 0. (8)
By utilizing (6) and (RFMa4), it is easy to see that b is a FP for /. Suppose k1 £(N, 1) and k2 = 1 -k by (6); one can obtain
tK,
Ma(fb, b,t) ^W Ma (fb,bT-l, ,Ma (^V. J^)
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M„fb,bT_1, ) ,Mafb,bT, ,tKl , )
" V T 1 2a(fb,bT)N/ a V ' T' 2a(fb,bT)Ky
r| M
a (fb, b,
a\ u, ut-1,
tKi
(2)2a(fb,b)a(fb,bT)K
tKi
't-1,
(2)2a(b,bT_1)a(fb,bT)K
Mafb.b,
tKi
2a(fb,bT)K,
+rl
Ma(fb'b'(2)2a(fb,b)a(fb,bx)Kj'
Mg(b,bT_1,(2)2a(b,bTt^^i1)a(fb,bT)-)y
m ihh tKi ^ J a( ' '(2)2a(fb,b)a(fb,bx)Kj'
MalB'B^'2a(fb,bT)Kj.1l M (, , tKl_N
_V al^pT-1,(2)2a(b,bT_1)a(fb,bT)K;
l + Ma(b,bT_1,2a(fbtbT)K)
for all t > 0. By (8) and as x^ot, one obtains
/ ( 0,0
<r
min
(mq (fb,b, (fb,b,
r
1 + r
tKl
(2)2a(fb,b)a(fb,bT)K
tKi
(2)2a(fb,b)a(fb,bT)K/' 1
0. r
(fb, b, (2)2a(fb, b)1a(fb, bT)—), j
1 + 0 0
Suppose that b and w are two different FP for /. Then, by applying (6), one can obtain
Ma(b, w,t) = Ma( f b,fw,t)
1 Ma (b, w, - ) ,Ma (b, fw, ,m„ (fb, w, - ),
< min ■
M
(b, fw, K) + Ma (fb, w, K)
M
(b, fw, -) .Ma(fb, w, -) 1 + Ma (b, w, K)
r
= min
M,
(b, w,£) ,Ma (b, w,£) ,Ma (b, w,£) ,
Ma(b,M,£).Ma(b,M,£)
and by Lemma 3, it is easy to see that b =w. Remark 1. If one takes
Ma (b, w,,Ma (b, fw,^ ,Ma (fb, w,^ , Ma(b,fw, |) + Ma(fb, w, !)
= Ma (b, w, J) ,
t > 0.
min
Ma (b,fw, £) .Ma(fb, w, I)
= Ma(b, w, II )
v 1 + Ma(b, w, ,
in the above theorem, one then obtains a revised fuzzy version of the
Banach contraction principle.
Example 2. Let ~={0,1,3}, Ma(b, w,t) = e"
(b-M)2
(b-M)2
e t
, and
r = rp. Then, (2,M,r) is a complete CRFMS with a(b, w) = b + w + 1. Define the function /: 2^2 such that /(0)=/(1) = 1 and f (3) = 0. Observe that if b = w or wG {0,1}, then Ma( fb,fw,t) = 0 and t > 0 and
condition (6) is fulfilled. Suppose b = 1 and w = 3. Then, KG Q,i) and one obtains
i
e st + e st e st e st | min^e"st,e st,e st,-2-,-
x 1 + e"st
Now, suppose b = 1 and w = 3, then, choosing K from (i, one deduces
Ma(fb,fw,t) = e t - et j < mil
Ma(fb,
,fw,t) = e t - et J < mi
-A
_± __± e it + e st e st.e st 1 min \ e st,e st,e st,-,-
1 + e st
Similarly, if b = 3 and w = 1 as well as b = 3 and w = 1, one
establishes that for se Q,i) condition (6) is fulfilled for all b,we~,
and t > 0. Hence, all the conditions of Theorem 4 are satisfied with a UFP b = 1.
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Corollary 2. Supposing that (2,M,r) is a complete CRFMS with 2 ^[1,ot), assume that lim Ma(b, w,t) = 0. For all b, wG2. If/: 2^2
satisfies the following, for some Kg(0, 1), such that
Ma(fb,fw,t) < Ma (b, w, 0 ,for all b, wG2, t > 0. Then, f has a UFP i
in
Example 3. 2 = A u B, where A= [0,1],B = -, and Ma: ~2 x (0, œ)
[0,1], is defined by
Ma(b, w,t) =
t +t è
0 if b = w if b e B and we A
if b e A and we B
otherwise
It + max{b, w}
Then, (2,M, r) is CRFMS with r(b, w) = b. w and a controlled function
a:,
[0, œ) defined by
0 if b = w
a(b w) = i 0 i ^ , J (min{b, w} otherwise It is easy to see that all the conditions of Corollary 2 are satisfied. Consider the triangular inequality (RFEMa4) of the revised fuzzy extended b-metric space defined in Definition 5 as
Ma (b, z, a(b, z)(t + s)) < r(Ma (b, w,t),Ma (w,z, s))
Let b, z e A, and we B, the, na(b, z) = 0. Assume t = s = 2,b = -,z =
1, and
w = 40. There is 1
t + s +
min{b, z}
<r
t + s +■
1 , 1
t + — s +—) ww
max{b, z}
One obtains 0.8 > 0.975, which is a contradiction. Hence, Ma is not an extended revised fuzzy b-metric space. Now, consider the triangular inequality (b 4) of RFbMS defined in Definition 4 as
Ma(b, z, p(t + s)) < r(Ma(b, w,t),Ma(w,z,s)). One obtains
p(s + t^ + min{b,z} I t s
<r
P(s +1) +
max{b, z}
1 , 1
t+ - s + — i ww
For p£[1,9], the above inequality is not satisfied. Theorem 5. Supposing that (S,Ma, r) is a complete CRFMS assuming that /: S^S, then K£(0,1) exists
Ma (fb, fw,t) < max [m„ (b, w, ,Ma (fb, w, ,Ma (b, fw, -)} (9)
For all b, wGS, t > 0 such that
lim Ma(b, w,t) = 0 (10)
For all t > 0.Then, f has a UFP in S.
Proof. Suppose b0 eS,bT+1 = fbT, and tgN from (9) with b = bT and w = bT_!, for every tgN and t > 0, one can obtain
Ma(bT+1,bT,t) < max{m„ (bT,bT_!,,Ma (bT+1,bT,,Ma (bT,bT_!, < max {m„ (bT,bT_!, ,Ma (bT+1,bT, If Ma(bT+1,bT,t) < Ma(bT+1,bT,i£N, t> 0. Then, Lemma 3
implies that bT,bT+1,xG N, such that Ma(bT+1,bT,t) < Ma (bT,bT_1,xg N, t > 0.
Moreover, by Lemma 2 {bT} is a Continuous Sequence. Hence, b GS exists such that limbT = b and
limMa(b,bT,t) = 0, t > 0. (11)
Now, it is shown that b is an FP for/. Letting k1 g(-, 1) and k2 = 1 -k-l by (9),one can obtain g-
Ma fb, fa ,M« i^Sd)
rl max
a V , т, a(b,bT)/ , a V , , a(b,fb)J I M / tK2 ^
E
1=
Taking x^^ and utilizing (11), one deduces a.
Ma (fb, b, t)<r (max {0, Ma (b, fb, , 0})
^M^,^) ,0)=M«(b,fb, 9 ,t> 0.
where v = a(b[^fb)- g(0,1), one has Ma(fb,b,t) < Ma(fb,b,t > 0,
and by Lemma 3, one has /6 = b. Suppose that b and are to different FP for /, that is, /6 = b and /w = o». By (9), one deduces
Ma(fb, fw,t) < max [m„ (b, w,,Ma (b, fb, ,Ma (w,fw, -)}
LU
O O
<
X
=max
(mo (b, w, 0,0,0} = Ma (b, w, -) = Ma (fb, fw, 3
0
J» For t > 0, and by utilizing the Lemma 3, one has = /w, which gives b = w.
cn
^ Remark 2. If one takes
"o
>
■«r
C\l
o
C\l
max Jm„ (b, w, N) ,Ma (fb, b, N) ,Ma (fw, w, N)j= Ma (b, w, N) in the above theorem, then one obtains a revised fuzzy version of the
cc Banach contraction principle.
(b-M)2 / (b-M)2
% Example 4. Suppose £ = (0,2), Ma(b, w,t) = e t , (1 - e t
and r = rp Then,(~,M, r)is a complete CFMS with a(b, w) = b + w + 2. Let
¿5 (2 - b, b g(0,1)
«H2 z!
b G[1,2)
ft Part 1: If b, we [1,2), then Ma(fb,fw,t) = 0,t > 0 and (9) are trivially
£ verified.
< Part 2: If b e[1,2) and we (0,1), such that Ne Q, i), one can obtain
(l-«)2 / (l~«)2\ ( 4N(l-«)2
Ma(fb,fw,t) = e t 11 - e t ) = I1 - e t „ Ma(fw, w,,t > 0.
o Part 3: As in the preceding section, for Ne Q,i), one obtains
| Ma(fb,fw,t) < Ma(fb,b, nN) ,b e(0,1), we [1,2),t > 0.
uu Part 4: If b,we (0,1), then for Ne Q,^, one has
(l-M)2 / (l-«)2\ 4N(l-«)2 / 4N(l-«)2
Ma(fb,fw,t) = e t (1 - e t 1 = e t (1 - e t
& = Ma (fa), où,0,b > a), t > 0 and Ma(fb,f(jû,t) < Ma (foo, go, , b >
w, t> 0
So, condition (9) is fulfilled for all b,we,t > 0 and by Theorem 5 it follows that b = 1 is a UFP for /. A new Ciric-quasi-contraction is analyzed in the following theorem.
Theorem 6. Supposing that (£,M,rmax) is a complete CRFMS, assume that /: If for some -g (0,1), such that
( Ma(b, w, -) ,Ma(fb,b, -), )
M«(fb,fw,t)< max j ^ m - \ [ ,b,w G
(Ma (fw, w ,-J ,Ma [fb, w ,7J ,Ma [b, fw , -JJ
S,t > 0. (12)
Then, f has a UFP in
Proof. Suppose b0 GS and bT+1 = fbT,tg N. By utilizing condition (12) with b = bT, w = bT_!, utilizing (FMa4), and the assumption that r = rmax, one can obtain
Ma(bT+1,bT,t) <
'Ma (bT,bT_!, £) ,Ma (bT+1,bT, -) ,Ma (bT,bT_!,
max <
M «¿¿M?) -M« (b-b-1- «^b»)
M,
<
max {M„ (bT,bT_1, ,M„ (^A-!, a(bJbT_i}J} , TGN,t > 0.
By Lemma 3 and Corollary 2, it is possible to demonstrate Theorem 5 such that
M«(bT+1,bT,t) < Ma(bT,bT_1, a(bT,btT i)-), tgN, t > 0.and {bT} is a
CS. So, b GS exists such that lim bT = b,
lim Ma(b, bT,t) = 0, t > 0 (13)
Suppose k-l g(—, 1) and k2 = 1 -k^ By (12) and (FMa4), one deduces
M«(fb,b,t) < max {M«(fb,fb,, ^K^,^)}
<
max
max
tKi
't+1,
max
(m„ ffb, b,-^-) ,M„ fb, bT,-^-))
I a V ' ' a(b,fb)a(fb,bT)N/ ; a V ; T; a(b,bT)a(fb,bT)N/J
M
a(b,l
tKi
't + 1, ,
a(b,bT+1)-
For all tgN and t > 0. Taking t^ot and utilizing (13), one obtains
0,M«(b,fb,
Ma(fb,b,t) < max
Îmax < lr
•,o:
' max ÎM„ (fb, b, ——-,o), <
k. I a V a(b,fb)a(fb,bT)Ky ' J '
= Ma fib, b, ——7-7-,t > 0.
a V ' ' a(b,fb)a(fb,bT)Ky '
and by Lemma 3 with v = a(b,fb)a(fb,bT)x g (0,1)such that f b = b. By
ki
condition (12), for two different FPs b = f band w = fw, one can obtain
E
Si
Ma(fb,fw,t) <
I ( Ma(b, w, ,Ma(fb,b, ,Ma(fw, w, ,
cm max
Is- I max
15 >
g = max
<n I max
DC
yy
q:
o o
fM« (fb,b, ^) ,M« (b,w ,M« (b,fw, 3 J
Ma (b, w, ^ ,1,1,
{1,M«(b,w, «¿J»)} ,M«(b,fw, ^
M«(b,w,
= Ma (fb,fw, ,t > 0. and by Lemma 3, it follows that b = w.
< The next theorem aims to establish a new contractive condition with
° the weaker TCN. g Example 5. Suppose S = A u B where A =[0,1],B = N\1 and
m Ma: S2 x (0, ot) —>[0,1] is a revised fuzzy metric defined by
0 if b = w __i_
e if b e B and wG A
e bt if b G A and wG B
e t 11 - e t )
otherwise
LU
M(b, w,t) =
1" e_r~)
0 Then, (S,M,F) is a CRFMS with r(b, w) = b. w and a controlled
s2 function a: S2 — [0, ot) defined by | rb a _i 0 if b = w
, (min{b + w, 1} otherwise It is easy to see that all the conditions of Theorem 6 are satisfied. Let b = j, w = 40, z = 1, and t = s = 1. Then, it does not satisfy the
triangle inequality (EMa4) of Definition 5. Hence, it is not an extended & revised fuzzy b-metric space. Now, show that it is not an RFbMS. Considering the triangle inequality (b4) of Definition 4, there is
min{b,m} / min{b,m}\ , 1 i % { 1 N
e p(t+s) |1 _ e p(t+s) )<r(e~^,e~^M ^ e_2P I 1 - e2P I
< e"20 - e20 j = e"To - eio j
It is clear that the above inequality is not satisfied for p = 2. Hence, it is not RFbMS.
Theorem 7. Supposing that (S,M, r), r>rp is a complete CRFMS, assume that /: S — S. For some xe(0,1), let
(
Ma(fb,fw,t) < max
M
(b, w, -) ,Ma (fb, b, -) ,Ma (fw, w, -)
,b, wG
|Ma (fb, w, ,Ma(b,fw,-)
S,t > 0. ' (14)
And b, wGS exists such that lim Ma(b, w,t) = 0,t > 0. (15)
Then, f has a UFP in
Proof. Let b0 gS and bT+1 = fbT,tg N.Taking b = bT and w = b^ in (14), by (RFMa4) and r> rp, one can obtain
Ma(bT+1,bT,t) <
' Ma(bT,bT+1, -) ,Ma(fb,b, -), ^
ma^i Ma(bт+l,bт,-) ,M«(bx,bт-l, -)
Ma (ft, w, 2) . M„ (bT,bT_1, ,M„ (bX,bX, -) ,
For all b, wG S,t > 0. Since Ma(b, w,t) is a b-non-decreasing in t and
^k. p = max{K, p}, one deduces
Ma(bT+1,bT,t) < max i " Qr all tGN, t > 0.
Mn
By Lemmas 2 and 3, one can obtain,
M« (bT+1,bT,t) < Ma (bT,bT_!, a(bJTi)J, TGN, t > 0.
Hence, {bT} is a CS. Since (S,M,r) is complete, b GS exist such that lim bT = b and lim Ma(b,bT,t) = 0,t > 0 (16)
Supposing k1 g(N ,1) and k2 = 1 -k by (14) and (FMa4) one can obtain
M«(fb,b,t) — r ( M« f fb„ «sg^) ,M« Kb, ^J^)
<
max
b _^_i
jMa (fb, b a(fb,b)a(fb,bT) J .Ma (b, ^ a(fb,bT)a(b,bT) J M«(b,b^+1, «fit»)
E
¡r
1= &
r
0 tn
m
CN
"o
>
■«r
C\l
0 C\l
01 yy
01 ID
o
o <
o
X
0
LU
h>
01 <
w <
o
SZ >o
X LU
0
<
max i
max
m„ fb,bT, ftK\) ,m„ fb,fb, ,tK\ ) ,m„ fbT,bT+1, , tKl , Y
aV a(b,bT)N/ aV a(b,fb)N/ aV T' T+1' a(bT,bT+1)-/ {Ma (fb,b, a(fb,b)a(fb,bj ,M« , a^totj)}
For all tgN and t > 0.Taking t^ot and utilizing (16), there is
0,M«(b,fb, ^) ,0
/
Ma(fb,b,t) <r| max<
max
M« (fb> b' a(fb,b)a(fb,bT)J '
(m„ fib, b,-—-) ,o) ,1
I a V ' ' a(fb,b)a(fb,bT)N/, J ,
t > 0.
And by Lemma 3 with v = a(fb,b)a(fb,bT)x e(0,l) such that /i = b.
Kl
Let b and w are two different FPs for f. By (2.13), one obtains
Ma (b, w, £) ,Ma (fb, b, J) ,Ma (fw, w, J)
Ma(fb,fw,t) <rl
Mn
(fb, b, _ \ ) .M„ (b, w, _ \ ) ,M„ (b, w,-)
<r (mh (b, w, £) ,0,0, max {0, Ma (b, w, ,Ma (b, w, ^
= M„ fb, w, , \ ) = Ma ffb, fw, ——-——1 ,t > 0. And thus, by Lemma 3,one obtains b = w.
(b-m)2 / (b-M)2
Example 6. Suppose S = {0,1,3},Ma(b, w,t) = e t 11 - e t
and r = rp. Then,(S,M,r) is a complete CRFMS with a(b, w) = b + w + 1. Define the function f: S ^ S such that f 0 = f 1 = 1 and f3 = 0. Observe that if b = w or wG {0,1}, then, Ma(fb,fw,t) = 0,t > 0 and (14) is fulfilled. Suppose b = 1 and w = 3.Then, -g (i,one obtains
Ma(fb,fw,t) = e t(1 - etj< max je t ,e t,e t ,e t,0|
Suppose b = 1 and w = 3.Then, by choosing -g (i,one has
Ma(fb,fw,t) = e th - e tj< max je t ,0,e t ,e t ,e t|
r
Similarly, if b = 3 and w = 1 as well as b = 3 and w = 1, then for xe (i,condition (14) is met for all b, w eS, and t > 0. As a result, Theorem 7 is satisfied with a UFP b = 1.
An application to the transformation of solar energy to electric power
Sun-based boards are currently being distributed and shown widely to reduce people's reliance on petroleum derivatives which are less environmentally friendly. Nearly 19 trillion kilowatts of power were transported internationally in 2007. In comparison, the amount of day light that enters the Earth's surface in a single hour is enough to illuminate the entire planet for a full year. The question is: how do those dazzling and warm beams of light obtain power? A numerical model of the electric flow in an RLC equal circuit, often known as a "tuning" circuit, can be presented with a basic understanding of how light is converted into power. In the fields of radio and communication engineering, this circuit has several uses. The version that is being presented can be used to calculate the production of electric power, provide tools to improve building performance, and can be used as a decision-making tool when designing a hybrid renewable electricity system based on solar power. Every aspect of this system is mathematically expressed as a differential equation in (Younis et al, 2022) using the following equation
|2| = n(8,tt(8))_?2| (17)
( e(0) = 0, r(0) = m
where fi: [0,1] is a continuous function that is condition (17)
to the integral equation to which it is equivalent.
e(«) = ft N(fi,l)fi(l,e(l))dl,fie [0,1] (18)
where the Green's function N(fi,b), it follows:
N(fi,b) = i(fi-l)en(M(b^)(fi-1))0 * c < l < 1 (19) |
v 7 I 0,0 <fi< l < 1 v '
where fi(M(b, w)) > 0 is a constant, as determined by the values of ^ and fi, mentioned in (3.1).
Let S = C([0, fi], be the set of all real continuous positive functions that are expressed on the set [0,c]. Let S be endowed with the CRFMS given by the following
( 0 if t = 0
M(b, w ,t) =j sup min{b'"}+t otherwise for all b, w G S (20)
(.tG[0,l] maxib,«}+t
E
¡r
One can verify that (~,M,r)is a complete CRFMS with a controlled function a: ~2 ^[0, OT),defined by a(b, oo) = b + 00 + 1.
It is obvious thatb*is a solution of integral Equation (18), and as a result, a solution of differential equation (17) which governs the system of converting solar energy into electric power if and only if b* is an FP of /. It 3 is installed as a guarantee of the existence of FP of /.
Theorem 8. Assume the following problem fulfills: a: f: [0,S]2 is a continuous function;
EE there exists a continuous function N: [0,S]2 ^ R+ such that
o sup J® N(a,l)> 1
° ae[o,fi] u
< max{f(a,l,b(l),f(a,l, oo(l))} > N(a,b)max{D(b(l), oo(l))} and
min{f(a,l,b(l),f(a,l,w(l))} > N(a,b)min{D(b(l),oo(l))} for all a,l,G
x
o [0,1], b, 00 g #+and KG (0,1) exists such that
LU
1-
cr <
w <
o
X
(b(l), o(l))
rMa (b(l), o(l),j) ,Ma (b(l),fo(l), j) ,Ma (fb(l), o(l),j) Ma (b(l),fo(l), j) + Ma (fb(l),o(l),j)
= min
Ma (b(l),fo(l), j) + Ma (fb(l),o(l),j)
1 + Ma (b(l), o(l), j)
Differential equation (17) that represents the solar energy problem 2 has a solution as a result and integral equation (18) also has a solution. Proof. For b,00 G-, by use of assumptions (I) to (III), one has
min{/0®N(fi,l)n(l,b(l))dl,/0i,N(fi,l)n(l,«(l))dl}+t
^ M(fb,f(0 ,t) = sup -^a-9-\—
^ te[0,l] max{/o N(fl,l))il(l,b(l))dl,/^ N(fl,l)il(l,o)(l))dlj+t'
X® min{N(S,l)n(l,b(l)), N(S,l)H(l, w(l))} dl +t
= sup —^-
tG[o,i] £max{N(S,l))H(l,b(l)), N(S,l)H(l, 0(l))} +t'
X® N(S,l)min{H(l, b(l))dl H(l, w(l))dl}+t
= sup —=-
tG[o,i] J"* N(S,l)max{H(l, b(l))dl H(l, w(l))dl}+t'
Xf N(S,l) min{D(b(l), 0(l))} dl +t
< sup -—---= M(D(b,w ,t))
tG[o,i] N(S,l) max{D(b(l), w(l))} dl +t'
Thus, all conditions of Theorem 4 are fulfilled, i.e., the operator f has an FP which is the solution to differential equation (17) regulating the conversion of solar energy to electrical power.
Open Problems 1. The following open problem is provided for further applications of the findings in this article:
Optional appliance renewal is one of the most basic concerns in management science and engineering economics. Corporations periodically purchase new appliances and sell old ones in order to operate the equipment permanently. If 5(t,z) is the efficiency of the appliance at time period T and 5(T) is the cost at the purchasing time, then,
e-^S(T) = JTA_1 e^z[S(T,z) -S(a(z),z)]du, -oo <T <o.
where z is the usage time of the machine and n is the constant of the industry wide discount rate.
Can the results established in this note or their variants be applied to solve the aforementioned integral equation?
Can the results derived in this article be controlled in graphical revised fuzzy metric spaces? Can one demonstrate the aforementioned findings for multi-valued mappings?
Conclusions
In the perspective of controlled revised fuzzy metric spaces, this manuscript contains a number of fixed point theorems and a sufficient condition for a sequence to be Cauchy. As a result, the well-known contraction requirements with controlled revised fuzzy metric spaces have been combined to simplify the proofs of several fixed point theorems. Furthermore, an application to transform solar energy to electric power has been discussed. In the future, these results will be enhanced in the framework of tripled controlled revised fuzzy metric spaces and pentagonal controlled revised fuzzy metrics spaces.
References
Adabitabar Firozja, A. & Firouzian, S. 2015. Definition of fuzzy metric space K with t-conorm. Annals of Fuzzy Mathematics and Informatics, 10(4), pp.649-655 [online]. Available at: http://www.afmi.or.kr [Accessed: 02 February 2024].
Al-Khaleel, M., Al-Sharif, S. & AlAhmad, R. 2023. On Cyclic Contractive Mappings of Kannan and Chatterjea Type in Generalized Metric Spaces. Mathematics, 11(4), art.number:890. Available at:
https://doi.org/10.3390/math11040890.
Branga, A.N. & Olaru, I.M. 2022. Generalized Contractions and Fixed Point Results in Spaces with Altering Metrics. Mathematics, 10(21), art.number:4083. Available at: https://doi.org/10.3390/math10214083.
E
o
^ Czerwik, S. 1993. Contraction mappings in b-metricspaces. Acta
I Mathematica et Informatica Universitatis Ostraviensis, 1(1), pp.5-11 [online]. w Available at: https://dml.cz/handle/10338.dmlcz/120469 [Accessed: 02 February 2024].
George, A. & Veeramani, P. 1994. On some results in fuzzy metric spaces. > Fuzzy Sets and Systems, 64(3), pp. 395-399. Available at: S https://doi.org/10.1016/0165-0114(94)90162-7.
° Gregori, V. & Miñana, J.-J. 2021. A Banach contraction principle in fuzzy
oc metric spaces defined by means of t-conorms. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115, g art.number:129. Available at: https://doi.org/10.1007/s13398-021-01068-6.
0 Grigorenko, O., Miñana, J.J., Sostak, A. & Valero, O. 2020. On t-Conorm < Based Fuzzy (Pseudo)metrics. Axioms, 9(3), art.number:78. Available at: y https://doi.org/10.3390/axioms9030078.
1 Hadzic, O. 1979. A fixed point theorem in Menger spaces. Publications De l¡j L'institute Mathématique, Nouvelle serie, 20(40), pp.107-112 [online]. Available
at: http://elib.mi.sanu.ac.rs/files/journals/publ/46/17.pdf [Accessed: 02 February dc 2024].
Hadzic, Q. & Pap, E. 2001. Probabilistic metric spaces. In: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications, 536, pp.47-94. Dordrecht: Springer. Available at: https://doi.org/10.1007/978-94-017-1560-7_2.
w Hassanzadeh, Z. & Sedghi, S. 2018. Relation betweenb-metric and fuzzy
^ metric spaces. Mathematica Moravica, 22(1), pp.55-63. Available at: " https://doi.org/10.5937/MatMor1801055H.
o Heilpern, S. 1981. Fuzzy mappings and fixed point theorem. Journal of
x
Mathematical Analysis and Applications, 81(2), pp.566-569. Available at: □ https://doi.org/10.1016/0022-247X(81)90141-4.
Hussain, A., Ishtiaq, U., Ahmed, K. & Al-Sulami, H. 2022. Qn Pentagonal Controlled Fuzzy Metric Spaces with an Application to Dynamic Market Equilibrium. Journal of Function Spaces, 2022(1), art.number: 5301293. Available ^ at: https://doi.org/10.1155/2022/5301293.
Ishtiaq, U., Kattan, D.A., Ahmad, K., Sessa, S. & Ali, F. 2023. Fixed Point Results in Controlled Fuzzy Metric Spaces with an Application to the Transformation of Solar Energy to Electric Power. Mathematics, 11(15), art.number:3435. Available at: https://doi.org/10.3390/math11153435.
Ishtiaq, U., Saleem, N., Uddin, F., Sessa, S., Ahmad, K. & di Martino, F. 2022. Graphical Views of Intuitionistic Fuzzy Double-Controlled Metric-Like Spaces and Certain Fixed-Point Results with Application. Symmetry, 14(11), art.number:2364. Available at: https://doi.org/10.3390/sym14112364.
Kaleva, Q. & Seikkala, S. 1984. Qn fuzzy metric spaces. Fuzzy Sets and Systems, 12(3), pp.215-229. Available at: https://doi.org/10.1016/0165-0114(84)90069-1.
Kider, J.R. 2020. Some Properties of Algebra Fuzzy Metric Space. Journal of Al-Qadisiyah for Computer Science and Mathematics, 12(2), pp.43-56. Available at: https://doi.org/10.29304/jqcm.2020.12.2.695.
Kider, J.R. 2021. Application of Fixed Point in Algebra Fuzzy Normed Spaces. Journal of Physics: Conference Series, 1879, art.number:022099. Available at: https://doi.org/10.1088/1742-6596/1879/2/022099.
Klement, E.P., Mesiar, R. & Pap, E. 2004. Problems on triangular norms and related operators. Fuzzy Sets and Systems, 145(3), pp.471-479. Available at: https://doi.org/10.1016/S0165-0114(03)00303-8.
Kramosil, I. & Michalek, J. 1975. Fuzzy metrics and statistical metric spaces. Kybernetika, 11(5), pp.336-344 [online] . Available at: https://www.kybemetika.cz/content/1975/5Z336 [Accessed: 19 January 2024].
Li, S.-F., He, F. & Lu, S.-M. 2022. Kaleva-Seikkala'sType Fuzzy b-Metric Spaces and Several Contraction Mappings. Journal of Function Spaces, 2022(1), art.number: 2714912. Available at: https://doi.org/10.1155/2022/2714912.
Mlaiki, N., Aydi, H., Souayah, N. & Abdeljawad, T. 2018. Controlled Metric Type Spaces and the Related Contraction Principle. Mathematics, 6(10), art.number:194. Available at: https://doi.org/10.3390/math6100194.
Moussaoui, A., Hussain, N., Melliani, S., Nasr, H. & Imdad, M. 2022. Fixed point results via extended FZ-simulation functions in fuzzy metric spaces. Journal of Inequalities and Applications, art.number:69. Available at: https://doi.org/10.1186/s13660-022-02806-z.
Muraliraj, A. & Thangathamizh, R. 2021a. Fixed point theorems in revised fuzzy metric space. Advances in Fuzzy Sets and Systems, 26(2), pp.103-115. Available at: https://doi.org/10.17654/FS026020103. _
Muraliraj, A. & Thangathamizh, R. 2021b. Introduction on Revised fuzzy ^ modular spaces. Global Journal of Pure and Applied Mathematics, 17(2), pp.303317. Available at: https://doi.org/10.37622/GJPAM/17.2.2021.303-317.
Muraliraj, A. & Thangathamizh, R. 2022. Relation-Theoretic Revised Fuzzy Banach Contraction Principle and Revised Fuzzy Eldestein Contraction Theorem. JMSCM Journal of Mathematical Sciences & Computational Mathematics, 3(2), pp.197-207. Available at: https://doi.org/10.15864/jmscm.3205.
Muraliraj, A. & Thangathamizh, R. 2023. Some topological properties of revised fuzzy cone metric spaces. Ratio Mathematica, 47, pp.42-51. Available at: https://doi.org/10.23755/rm.v47i0.734. f
Muraliraj, A., Thangathamizh, R., Popovic, N., Savic, A. & Radenovic, S. 2023. The First Rational Type Revisd Fuzzy-Contractions in Revisd Fuzzy Metric Spaces with an Applications. Mathematics, 11(10), art.number:2244. Available at: Available at: https://doi.org/10.3390/math11102244.
Rakic, D., Mukheimer, A., Dosenovic, T., Mitrovic, Z. & Radenovic, S. 2020. On some new fixed point results in fuzzy b-metric spaces". Journal of Inequalities and Applications, art.number:99. Available at: https://doi.org/10.1186/s13660-020-02371-3.
Sedghi, S. & Shobe, N. 2012. Common fixed point theorem in b-fuzzy metric space. Nonlinear Functional Analysis and Applications (NFAA), 17(3), pp.349-
E
o
^ 359 [online]. Available at: http://nfaa.kyungnam.ac.kr/journal-§ nfaa/index.php/NFAA/article/view/38 [Accessed: 02 February 2024]. tn Sedghi, S. & Shobkolaei, N. 2014. Common fixed point theorem for R-weakly
commuting maps in b-fuzzy metric spaces. Nonlinear Functional Analysis and Applications (NFAA), 19(2) pp.285-295 [online]. Available at: > http://nfaa.kyungnam.ac.kr/journal-nfaa/index.php/NFAA/article/view/238 24 [Accessed: 02 February 2024].
° Sezen, M.S. 2021. Controlled fuzzy metric spaces and some related fixed
oL point results. Numerical Methods for Partial Differential Equations, 37(1), pp.583593. Available at: https://doi.org/10.1002/num.22541. g Schweizer, B. & Sklar, A. 1960. Statistical metric spaces. Pacific Journal of
0 Mathematics, 10(1), pp.313-334. Available at: ^ https://doi.org/10.2140/pjm.1960.10.313.
^ Sostak, A. 2018. George-Veeramani Fuzzy Metrics Revised. Axioms, 7(3),
1 art.numner:60. Available at: https://doi.org/10.3390/axioms7030060.
ft Thangathamizh, R., Muraliraj, A. & Shanmugavel, P. 2024. New approach of
Lebesgue integral in revised fuzzy cone metric spaces via unique coupled fixed in point theorems. Vojnotehnicki glasnik/Military Technical Courier, 72(3), pp.1029-^ 1045. Available at: https://doi.org/10.5937/vojtehg72-48816.
Younis, M., Singh, D. & Abdou, A.A.N. 2022. A fixed point approach for tuning circuit problem in dislocated b-metric spaces. Mathematical Methods in the Applied Science, 45(4), pp.2234-2253. Available at: w https://doi.org/10.1002/mma.7922.
Zadeh, L.A. 1965. Fuzzy Sets. Information and Control, 8(3), pp.338-353.
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Available at: https://doi.org/10.1016/S0019-9958(65)90241-X.
UU El punto fijo da como resultado espacios métricos difusos revisados y
controlados, con una aplicación a la transformación de energía solar en energía eléctrica
Ravichandhiran Thangathamizh3, Abdelhamid Moussaouib, Tatjana Dosenovicc, Stojan Radenovicd
a Instituto de Tecnología Jeppiaar (Autónomo), Departamento de Matemáticas, Kanchipuram, Tamil Nadu, República de la India, autor de correspondencia
b Universidad Sultán Moulay Slimane, Facultad de Ciencias y Técnicas,
LMACS, Beni Mellal, Reino de Marruecos c Universidad de Novi Sad, Facultad de Tecnología,
Novi Sad, República de Serbia d Universidad de Belgrado, Facultad de Ingeniería Mecánica, Belgrado, República de Serbia
CAMPO: matemáticas
TIPO DE ARTÍCULO: artículo científico original
Resumen:
Introducción/objetivo: Este estudio establece condiciones suficientes para que una secuencia sea Cauchy dentro del marco de espacios métricos difusos revisados y controlados. También generaliza el concepto del principio de contracción de Banach al introducir varias condiciones nuevas de nuevas. El objetivo es derivar varios resultados de punto fijo que mejoren la comprensión de estas estructuras matemáticas. Métodos: Los investigadores emplean técnicas matemáticas rigurosas para desarrollar sus hallazgos. Al definir un conjunto de asignaciones de contracción novedosas y utilizar propiedades de espacios métricos difusos revisados y controlados, analizan las implicaciones para la convergencia de secuencias. La metodología incluye la construcción de ejemplos específicos para ilustrar los resultados teóricos.
Resultados: El estudio presenta varios teoremas de punto fijo derivados de las condiciones de contracción generalizada. Además, proporciona una serie de ejemplos no triviales que fundamentan las afirmaciones y demuestran la aplicabilidad de los resultados en escenarios prácticos. Se explora una aplicación importante con respecto a la conversión de energía solar en energía eléctrica, utilizando ecuaciones diferenciales para resaltar esta conexión.
Conclusión: Los hallazgos profundizan la comprensión de las secuencias de Cauchy en espacios métricos difusos y ofrecen una perspectiva más amplia sobre la aplicación de la teoría del punto fijo en escenarios del mundo real. Los resultados allanaron el camino para futuras investigaciones tanto en matemáticas teóricas como en sus aplicaciones | prácticas, particularmente en el campo de las energías renovables.
Palabras claves: teoremas del punto fijo, espacio métrico difuso revisado (RFMS), principios de contracción (CP), función de Green, ecuación diferencial.
Результаты неподвижной точки в управляемых пересмотренных фазовых метрических пространствах, применяемых для преобразования солнечной энергии
Равичандиран Тангатамижа, Абделхамид Мусауиб, Татьяна Дошеновичв, Стоян Раденовичг a Технологический институт Джеппиара (автономный), математический факультет, Канчипурам, Тамилнад, Республика Индия, корреспондент
б Университет Султана Мулай Слимана, факультет естественных наук и
техники, MACS, Бени-Меллаль, Королевство Марокко в Нови-Садский университет, технологический факультет,
г. Нови-Сад, Республика Сербия г Белградский университет, машиностроительный факультет, г. Белград, Республика Сербия
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РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций, з 27.39.15 Линейные пространства, снабженные
ю топологией, порядком и другими структурами
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ВИД СТАТЬИ: оригинальная научная статья Резюме:
Введение/цель: В данном исследовании установлены условия для
о того, чтобы последовательность Коши находилась в рамках
2
контролируемых пересмотренных нечетких метрических ш пространств. В статье также обобщается концепция принципа
^ сжатия Банаха, вводя несколько новых условий сжатия. Цель
о статьи заключается в получении различных результатов с
" фиксированной точкой, которые улучшат понимание этих
^ математических структур.
Методы: В исследовании применялись строгие математические о методы для представления открытий. Определяя набор новых
сокращающихся отображений и используя свойства контролируемых пересмотренных нечетких метрических < пространств, были проанализированы импликации для
сходимости последовательностей. Методология включает в себя разработку конкретных примеров, иллюстрирующих теоретические результаты.
м Результаты: В исследовании представлено несколько теорем о
неподвижной точке, полученных из обобщенных условий сжатия. ^ Помимо того, приводится ряд нетривиальных примеров,
о которые обосновывают утверждения и демонстрируют
применимость результатов в практических сценариях. ш Рассматривается важная сфера применения, связанная с
преобразованием солнечной энергии в электрическую с использованием дифференциальных уравнений.
Выводы: Полученные результаты углубляют понимание последовательностей Коши в фазовых метрических пространствах и раскрывают более широкую перспективу для применения теории фиксированной точки в реальных сценариях. Результаты прокладывают путь для дальнейших исследований как в области теоретической математики, так и в области ее практического применения, в частности, в области возобновляемых источников энергии.
Ключевые слова: теоремы о неподвижной точке, пересмотренное нечеткое метрическое пространство (RFMS), принципы сжатия (CP), функция Грина, дифференциальное уравнение.
Резултати непомичне тачке у контролисаним ревидираним фази метричким просторима примежени на претвараже соларне енерги]е у електричну
Равичандиран Тангатамижа, Абделхамид Мусауи6, Татjана ДошеновиЬ8, Стоjан Раденови1йг
а Институт за технолог^у Цепиар (аутономни), Оде^еже за математику, Канчипурам, Тамил Наду, Република Инди]а, аутор за преписку 6 Универзитет „Султан Мула] Слиме]н", Факултет природних и техничких наука, 1_МАСв, Бени Мелал, Кра^евина Мароко
в Универзитет у Новом Саду, Технолошки факултет,
Нови Сад, Република Ср6и]а г Универзитет у Београду, Машински факултет, Београд, Република Ср6и]а
ОБЛАСТ: математика
КАТЕГОРИJА (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:
Увод/циъ: У студии се успоставъа}у довоъни услови да секвенца буде Кошцева у оквиру контролисаних ревидираних фази метричких простора. Тако^е, генерализуе се концепт Банаховог принципа контракцце уво^еъем неколико нових услова контракцце. Циъ ]е да се изведу различити резултати непомичне тачке щи доводе до боъег разумеваъа ове математичке структуре. ^
Методе: Аутори разви]а]у сво}а откриПа коришПеъем ригорозних математичких техника. Дефинисаъем скупа нових пресликаваъа контракцца и коришПеъем сво]ства контролисаних ревидираних фази метричких простора анализиранесу импликацце за конвергенциу секвенце. Методологща укъучуе конструисаъе конкретних примера за илустраццу теорцских резултата. Резултати: Студща представъа неколико теорема непомичне тачке изведених из генерализованих услова контракцце. Поред | тога, наводи бро]не нетривцалне примере щи поткрепъуу тврдъе и демонстрира}у применъивост резултата у практичним сценари'има. Приказана ¡е важна примена у области претвараъа соларне енергце у електричну енергцу помоПу диференццалне Уедначине.
Закъучак: Налази продубъуу разумеваъе Кошцевих секвенци у фази метричким просторима и нуде ширу перспективу примене теори е непокретне тачке у сценари има из реалног живота. Резултати отвара]у пут за даъа истраживаъа, како у теорцсщ
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математици, тако и у ъеним практичним применама, посебно у области обнов^иве енерги'е.
К^учне речи: теореме непокретне тачке, прера^ени фази метрички простор (RFMS), принципи контракци'е (CP), Гринова функци'а, диференци'ална ]едначина.
Paper received on: 03.02.2024.
Manuscript corrections submitted on: 16.11.2024.
Paper accepted for publishing on: 18.11.2024.
© 2024 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, BTr.M0.ynp.cp6). 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/).
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