CC BY
0
Exploring multivalued probabilistic s
^-contractions with orbits in b-Menger spaces
Youssef Achtouna, Stojan Radenovicb, Ismail Tahiric, Mohammed Lamarti Sefian
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a Abdelmalek Essaadi University, Normal Higher School, Department of Mathematics and Computer Science, Tetouan, Kingdom of Morocco, e-mail: achtoun44@outlook.fr, corresponding author, ORCID iD: ©https://orcid.org/0009-0005-5334-2383 b University of Belgrade, Faculty of Mechanical Engineering, js
Belgrade, Republic of Serbia, e-mail: radens@beotel.rs,
ORCID iD: ©https://orcid.org/0000-0001-8254-6688 c Abdelmalek Essaadi University, Normal Higher School, Department of Mathematics and Computer Science, Tetouan, Kingdom of Morocco, e-mail: istahiri@uae.ac.ma,
ORCID iD: ©https://orcid.org/0000-0002-7723-3721 d Abdelmalek Essaadi University, Normal Higher School, Department of Mathematics and Computer Science, Tetouan, Kingdom of Morocco, £ e-mail: lamarti.mohammed.sefian@uae.ac.ma ORCID iD: ©https://orcid.org/0000-0001-8270-2660
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doi https://doi.org/10.5937/vojtehg72-49063
FIELD: mathematics ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: The paper presents a novel approach to certain well-established fixed point theorems for multivalued probabilistic contractions in b-Menger spaces, leveraging the boundedness of the orbits. w The aim was to generalize and enhance the results previously derived ^ by Fang and Hadzic.
Methods: The boundedness of orbits in b-Menger spaces is used to establish their approach for multivalued probabilistic contractions. Results: The findings of the study not only generalized the existing fixed ^ point theorems but also enhanced them significantly. The effectiveness of the approach in extending the results originally proposed by Fang and Hadzic was showcased. Moreover, the applicability of the coincidence fixed point theorem in fuzzy b-metric spaces was demonstrated. Conclusions: The study presented a novel perspective on fixed point theorems in multivalued probabilistic contractions within b-Menger spaces. By leveraging boundedness and introducing a coincidence fixed point
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theorem for fuzzy b-metric spaces, the work contributed to the advancement in this field.
Key words: fixed point, b-Menger spaces, multivalued ^-contraction, fuzzy b-metric space.
Introduction
In 1981, Vul'pe et al. (Berinde & Pacurar, 2022) introduced the concept of b-metric space as a generalization of metric spaces, a framework later utilized by Bakhtin and Czerwik (Bakhtin, 1989; Czerwik, 1993) to establish the well-known Banach fixed point theorem in these spaces (Banach, 1922). The significance of the fixed point theory resonates across various branches of pure and applied mathematics due to its broad range of applications.
The concept of probabilistic metric spaces, introduced by K. Menger in 1942 (Menger, 2003), constitutes a crucial extension of metric spaces. The exploration of the fixed point theory in Menger spaces, concerning both multivalued and single-valued contractions, has become an integral part of probabilistic analysis, attracting the attention of numerous mathematicians (Achtoun et al., 2023; Mbarki & Oubrahim, 2017; Huang et al., 2023; Mihet, 2005; Patle et al., 2019). Recently, Mbarki and Oubrahim (Mbarki & Oubrahim, 2017) introduced the b-metric version of probabilistic metric spaces, termed a b-Menger space, which stands as the most general concept among those mentioned earlier. Notably, numerous fixed point results have been derived within this type of space (Mbarki & Oubrahim, 2017; Mihet, 2005). A parallel idea emerged in the realm of fuzzy metric spaces, where Nadaban (Nadaban, 2016) introduced the concept of a fuzzy b-metric space, generalizing the notion put forth by Kramosil and Michalek (Kramosil & Michalek, 1975).
The concept of multivalued contractions in metric spaces was pioneered by Nadler (Nadler Jr, 1969), and Hadzic (Hadzic, 1989) later extended this notion to multivalued ^-contractions in probabilistic metric spaces. She established a fixed point theorem employing the concept of probabilistic function of non-compactness. Building on this foundation, Fang (Fang, 1992) presented a generalization of Hadzic's results by substituting the condition of a continuous t-norm with a t-norm of H-type.
This paper contributes a new fixed point theorem for multivalued mappings satisfying ^-contractive conditions in b-Menger spaces, leveraging
the concept of bounded orbits. As an application, these results are ex- S tended to establish a corresponding fixed point theorem in fuzzy b-metric m spaces. These authors' findings not only improve upon the work of Hadzic (Hadzic, 1989) and Fang (Fang, 1992) but also generalize their results. ^ The structure of this article unfolds as follows: Section 2 provides es- S sential concepts and lemmas in b-Menger spaces. Section 3 establishes the existence of fixed points for multivalued ^-contractions in b-Menger spaces, employing two distinct approaches and offering illustrative examples. Finally, Section 4 identifies a coincidence fixed point for multivalued ^-contraction mappings in fuzzy b-metric spaces.
Preliminaries
( ) _ , 0 if a € (—a] , £a(a)_i 1 if a € (a, +<).
s
To start with, here are some basic definitions and facts from b-Menger ion
spaces. act
tr n
Definition 1. Let A+ be the class of all distance distribution mappings y : 8
[0, ^ [0,1] such that: t
1. y is left continuous on [0, +rc>],
2. y is non-decreasing,
3. y(0) = 0 and y(+^) = 1. The subset D+ c A+ is the set D+ = <j y g A+ : lim y (a) = 1
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Definition 2. (Schweizer & Sklar, 1983) A triangular norm (briefly t-norm) is a binary operation 1 on [0,1] such that for all u,v,w € [0,1] the following conditions are verified:
1. 1(a,ß)_ 1(ß,a), <
2. 1(a, 1(ß,Y))_ 1(1(a,ß ),y),
3. 1(a,ß) < 1(a,Y) for ß < y,
4. 1(a, 1) _ 1(1, a) _ a.
Example 1. Here the most basic t-norms are cited:
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O >
(M o (M
1. The minimum t-norm 1M(a,/) = min(a,/).
2. The Lukasiewicz t-norm 1L (a, /3 ) = max (a + /3 - 1,0).
3. The product t-norm (a, /3) = a • /
Definition 3. (Pap et al., 1996) A t-norm 1 is said of H-type if the family On(a))neN is equi-continuous at the point a = 1, it means that :
ct
yy for all e G (0,1), there exists A G (0,1) : t > 1—A implies 1n(t) > 1—e for all n > 1,
g where for all a e [0,1] and n e N there exists: -j
O -,n/ X [1 if n = 0,
i 1 (a) \ 1(1n-1(a),a) otherwise.
>_ A simple example of H-type t-norm is 1M, unlike 1L is not of H-type. <c The readers are referred to (Pap et al., 1996) for more details.
Definition 4. (Mbarki & Oubrahim, 2017) A quadruple (r, F, 1, s) where r is a nonempty set, F is a function from r x r into A+, 1 is a t-norm and s > 1 So is a real number, is called a b-Menger space if the following requirements ^ are verified for all p,a,§ g r and t,v > 0:
g 1- Fp,p = eo,
2. Fp,a — eo if p — a, Fp,a — Fa,p,
4. Fp,a (s(t + v)) > 1 (FpAt), Fv,a (v)).
Note that a Menger space is a b-Menger space with s — 1. In the topology created by the family of (e, A)-neighborhoods:
N — {Np(e, A): p G r, e> 0 and A> 0} ,
where:
Np(e, A) — {q G r: FP,q(e) > 1 — A} .
The space (r, F, 1, s) is a Hausdorff topological space if the t-norm 1 is continuous, as demonstrated by Mbarki and Oubrahim (Mbarki & Oubrahim, 2017)
Definition 5. A sequence {wn} in a b-Menger space (r, F, 1, s) is said to be:
1. Convergent to w e r if for any given e > 0 and X > 0 there exist S N e N satisfying FUn, u (X) > 1 - e whenever n > N. d
2. Strong Cauchy sequence if for any e> 0 and X> 0 there exist N e N d.
CP
co <u
satisfying FUn,Um (X) > 1 - e whenever n, m > N. A b-Menger space (r, F, s) is complete if each Cauchy sequence in r is | convergent to some point in r.
In the following, it is assumed for the b-Menger space (r, F, s) that 1 is a continuous t-norm, and the class of all nonempty closed subsets of r is denoted by C(r), where for all U,V e C(r) and w e r the functions Fu,u(.) and FUvV(.) are defined as follows:
Fu, u (t) = sup Fu, v (t) for all t e R,
veu
Fu,v(t) = inf sup Fu,v (t) for all t G R. The first result is the following:
Lemma 1. Let (r, F, s) be a b-Menger space, then for all U g C(r) and g r there is
Fu,u(t) = 1 for all t > 0 if and only if w G U.
FM,u (t) = sup Fu,v (t)
veu
> Fu,u (t) = 1 for all t> 0. Hence Fu, u(t) = 1 for all t> 0. □
Definition 6. Let (r, F, 1, s) be a b-Menger space and U a nonempty set of r. The function Du defined on [0, by
( lim 6u(t) if 0 < w < Du (w) = < t yu-
1 if w =
<u
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Proof. If Fu, u(t) = 1 for all t > 0, then for any e > 0 and X e (0,1) there exists w0 e U such that Fu,U0 (e) > 1 - X. As U e C(r) then w e U. 5
On the other hand, if w e U, then
X LU
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o <
CM <u
CM
o >
where
Qu (t)—inf {Fa ,b(t) | a,b G U } is called the probabilistic diameter of U.
^ It is clear that DU g A+ for any U c r, and for all p,q g U. Also U is
° bounded if DU is into D+.
0£
yy Lemma 2. Let (r, F, 1, s) be a b-Menger space and U a nonempty set of
g r, the probabilistic diameter has the following proprieties: o 1. For all a,b G U, there exists Fabb > DU.
0 2. DU — e0 if and only if U is a singleton.
1 3. If U c V, then DU > DV.
o U ~ V
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Main result
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Throughout this section, a point z g r is said to be a fixed point of f : r ^ C(r) if z g fz. If for u0 g r, there exists a sequence {u,,} c r such that ui g fui-l, then pf (u0) — {u0,ul,u2,...} is called an orbit of f starting at u0.
« Let % denote the family of all function $ : [0, +rc>) ^ [0, +rc>) satisyfing
0 < $(t) < t and lim $n(t) — 0 for all t > 0.
X n—»+oo
LU I—
§ The definition of multivalued probabilistic ^-contraction is first intro-o duced in a b-Menger space.
Definition 7. Let (r, F, 1,s) be a b-Menger space and $ : [0, +rc>) [0, +ro).A a mapping f : r ^ C(r) is called a multi-valued probabilistic $-contraction if for every u,u g r and every p g fu there exists a g fu such that
Fp,a($(t)) > Fuu(st) for all t > 0. (1)
Remark 1. Note that if f is a multi-valued $-contraction, there exists
Ffu>,fv($(t)) > Fu,v(st) for all u,u G r, and t > 0. (2)
Before stating the main result, one will use later the following lemma.
<M
e
(r, F, 1, s) where RanF c D+. If there exists a function 0 e x such that
Lemma 3. Let {wn} be a bounded sequence in a b-Menger space S
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FUn, Um (0(t)) > FUn-1 ,Um-1 (st) for alln,m > 0 such that m > nand for all t
(3)
Then {wn} is a Cauchy sequence. ®
Proof. Let {un} be a bounded sequence in r that satisfying the condition (3). Then one obtains
> Fu- ,U,m-1 (r- (t))
> Fun-2,(Sr^(t))
CO
cu
Un, V^ 77 - 1 Un-1 , ! (0
— r 1 .Urn- 1
Wn-1 (t)) |
— (r-(t))
o
o CO
o
0
1
o
> FUo , um-n (st)
> FUo ,Um-n (t) >DP(u)(t)-
On the other hand, let e > 0 and 5 e (0,1) be given, since Vp(u)(t) ^ 1 as t ^ then there exist t0 > 0 such that
DpH(to) > 1 - 5.
From that 0n(t0) ^ 0 as n ^ then there exist n0 e N satisfying H
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0n(t0) < e whenever n > n0. By using the monotonicity of F, one obtains
Fun, Um (e) > Fun , Um ^ fofi *
> Dp(u) (t0)
> 1 - 5.
Therefore, {wn} is a Cauchy sequence. □
Theorem 1. Let (r, F, 1, s) be a complete b-Menger space where RanF c D+ and f : r ^ C(r) is a multivalued probabilistic 0-contraction mapping with 0 e x■ If all the orbits pf (u) for some w e r are bounded, then there exists z e r satisfying z e fz.
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Proof. Let w0 e r and u1 e fu0 such that the orbit pf starting at u0 is bounded. Then there exists u2 e fu1 thus by (1) one obtains
t)) > Fuo,»!(st) for all t> 0.
Inductively, one constructs a sequence {un} satisfying the following conditions:
w„+i e fun and Fun,un+1 t)) > (st) for all n e N, and t > 0.
o
o It will shown first that for each t> 0,
<
Fun-i,un+m-i(st) for all m> 0. (4)
x
o
w It is obvious that (4) is true for m = 1. > It is claimed that (4) holds for m> 0.
^ Since un e fun-i and wn+m+i e fwn+m, using Remark 1, one gets
(№) >
>Ff №))
o So, by induction, it is proved that (7) holds for all m> 0.
Therefore, from Lemma 3, it follows that {un} is a Cauchy sequence. As (r, F, s) is complete, then {wn} converge to some z e r. It will be demonstrated that z is a fixed point within f. For that, let t > 0, ^ then, from (1), one has
F^Jzm)) > Fu,n-1 ,Z(st)
> Fun-1 ,Z (t).
Since that ^ e x, it follows
F^nJz(t) > FUn-i ,z(t),
by letting n ^ one gets
Fz,fz(t) > 1 for each t> 0, which implies by Lemma 1 that z e fz. Hence, z is a fixed point of f.
□
Example 2. Let r = [0, +œ). Define r : r x r ^ A+ by S
en CD LT) £± Œ
CO cu
(i) = ÊQ(i -|w - u|2).
w u 2 - 2
|p - a| <
Case 2 If p > U, since 0 < p < |, one gets
n ^ u w u 0 < p — ~ < ~ — ~
w u
2 - 2
Therefore, from case 1 and case 2, one obtains that
"4
Fp,a m)) = eo(7 i -IP - ^|2)
,3
M
. ,3 W u - x
> eQ(7i - ô -ô ) m
2 2
> eo(3i - ^ - u|2)
> Fu,v (2i).
Since p/(0) is bounded, then all the conditions of this Theorem are satisfied. Hence f have a fixed point which is 0.
Remark 2. One should mark down that the condition propriety about the boundedness of the orbits p/(w) is an obligatory condition to prove the existence of a fixed point as the next example shows.
It is claimed that (r, F, 2) is a complete b-Menger space. Let us consider the mapping f : r ^ C(r) given by f (w) = [0, f ]. |
It will be proven that for all w,u e r and p e fw there exists a e fu such that p
FP,q(0(t)) > Ff,v(2t).
With 0(t) = 31. Let w,u e r and p e fw one has Case 1 If p < | then p e fu. so there exists a e [0, f] such that
CO
'.a
o
CO c o
o CO
o
0
1
2 _ 2 2 «
Then if one takse a = U, one obtains 1
2 ' çp
^ - aK
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>
3 . i2\ OT
O
2 &
CO
3 1
= eo(4i - - ^ - u|2) J
2\
o <
CM <D
Example 3. (Sherwood, 1971) Define the distribution function as:
r(t)=> 0 if t < 4,
S K(t) = { 1 - i si 2« <t< 2«+l, a > 1.
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yy 0£
Consider r = {1,2,3,..., a,...}. Define F :r x r —>V+ as follows :
CM
° i 0 if t = 0,
J 0 if t = 0,
' a'a+b(t) \ lbL K(2at), K(2a+lt),.., K(2a+bt)) if t> 0.
g Then, one obtains that (r,F, 1L, 1) is a complete b-Menger space, and ° since every single-valued mapping is a multi-valued mapping, one puts o f (a) = a + 1, which is ^-contractive with ^(t) = 11. However, f have no
2
fixed point, since there exists any n e r such that pf (u) is bounded.
Lemma 4. Every Cauchy sequence in a b-Menger space (r, F, s) such
oc that RanF c D+ is bounded.
<t
Proof. Let {un} be a Cauchy sequence. Taking e > 0, then for t > 0 there exists a positive integer N e N such that
<j (t) > 1 — e whenever n,m > N. (5)
CD
2 Since RanF c D+ there exists t0 >t such that
>o z
X LU I—
o Then, from (5) and (6), one obtains
,um(to) > 1 — e whenever n,m < N. (6)
id (6), one obtains
(t0) > 1 — e whenever n,m € N.
Hence,
^p(u)(t0) > 1 — e,
which implies also that for all tl > t0 one gets
0p(^)(ti) > ep{M)(ti) > 1 — e. Since that e > 0 is arbitrary, there is tl > 0 such that
Dp{u)(ti) > 1 — e.
Thus,
Dp(u) (ti) ^ 1 as ti ^ The proof is completed. □
Lemma 5. Let (r, F, s) be a complete b-Menger space where RanF c D+ and f : r ^ C(r) is a multivalued 0-probabilistic contraction mapping on r with 0 e x■ If the t-norm 1 is of H-type, then for all u e r, the orbit pf (u) is bounded.
Proof. Let u e r and {un} be a sequence of the orbit pf (u) starting at u. From Lemma 4, it suffices to show that {un} is a Cauchy sequence. Since f is 0-probabilistic, then there exists 0 e x such that
F„n,un+1 (0(t)) > F„n-1,„n (st) for all n e N, and t > 0.
Then, by induction, it is shown, as in the proof of the Theorem, that
F„n,un+k(0(t)) > Fun_i,Un+k-i(st) for al1 k>
Next, for k = 1, one obtains that
F„n,,n+i (0n(t)) > F^0,„1 (st)
Since lim F„0,„ (t) = 1.
t—
> F„0, U1 (t) for all n e N andt > 0.
Then, for any e e (0,1], there exists t0 > 0 such that
F „o ,„i (to) > 1 - e. As 0 e x, then there exists ti > t0 such that
lim 0n(t1) = 0.
n—
So for any t> 0, there exists n0 e N such that
0n(t1) < t for all n > n0. By the monotonicity of F, one has for all n > n0,
F„n„n+1 (t) > F„n,„n+1 (0n(ti))
> F„0,„1 (ti)
(7)
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cn CD LT) £± CP
CO
cu o CO CP CO
CD CT
CO
o
CO c o
o CO
o
0
1
o
.Q CO .Q O
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CU
CO
>
E
CT C
X LU
lo cu
o <
> Fu
> 1 e.
(t0)
573
CM <u
CM
o >
Hence
lim (t) = 1 for all t> 0. (8)
n—
Now, one proves by induction that for any k > 2,
S (st) > 1k-1 [F^n+1 (t — m)) . (9)
o CM
of Inequality (9) is satisfied for k = 3. ^ Now, suppose that (9) holds for k> 2.
o Using (7), the monotonicity of 1 and the induction hypothesis, one obtains
o
< F^n ,^n + k + 1 (st) = F^n, Un+k + 1 (s(t — ^(t)) + s^(t))
x > 1(F ^n (t—m),Fu
° > 1(Fun,Un+1 (t — m),F^n+k (st))
UJ
DC > ^(F(t "h(t) ^ l
<
> i(F^.n+1 (t—m), ik-1(F^+1 (t—m))
= ik (f^ (t—m))-
Hence, (9) is proved for all k > 2. co Now, let e e (0,1) be given. From that 1 is of H-type, there exists 5 > 0 ^ such that
1n(t) > 1 — e for all t e (1 — 5,1] and n e N. (10)
>o
X LU
As —^ > 0, then by (8), one has that
I Um (—^) = 1-
O ra^+TO s
>
So, there exists N € N such that for all n > N one obtains
F t—m) > i Ä
+ l (-S-) > 1 —
Finally, from (9) and (10), one gets
F^n+k(t) > lk~l(FUn,Un+i(t — S!(t))) > 1 — e for all n > N, and k> 1.
Therefore, {un} is a Cauchy sequence, which implies from Lemma 5 that pf (u) is bounded. □
As a direct consequence of Theorem 1, Lemma 4, and Lemma 5, one obtains the following result
574
Corollary 1. Let (r, F, 1, s) be a complete b-Menger space with 1 is of H-type and f : r ^ C (r) is a probabilistic 0-contraction mapping where 0 e x■ Then there exists z e r satisfying z e fz.
Next, here is an example to illustrate corollary 1. Example 4. Let r = [0, +rc>). Define r : r x r ^ A+ as follows
Fu,u(t) = eo(t -\w - v\2).
It is easy to check that (r, F, 1M, 2) is a complete b-Menger space with 1M is of H-type. And one considers the function f : r ^ C(r) given by
f (w) = {l U, 3).
Then, for any w,y e r and p e fw, there are the following cases: Case 1 If p = l e fw, then one chooses p = a. Case 2 If p = u e fw, then one chooses a = u.
Case 3 If p = 3 e fw, then one chooses a = 3.
<M oo LT) I
CO CD LT) £± cp
CO
cu o CO
cp
CO
CD CT
CO
o
CO c o
o CO
o
0
1
o
Now, it is necessary to show that the 0-contraction is satisfied with ^(t) 11.
For case 1, there is
Fp,a (21) = eo(11) = eo(2t),
hence
l 2 FP,a(-t) > eo(2t - \w - u\2)
2
= Fu, u (2t).
Case 2 gives
ll
Fp,a(2t) = e0(2t -
w V 2 - 2
Finally, for case 3, one obtains
) = eo(2t \w - u\2) = Fu,u(2t).
l
l
Fp,a(-t) = eo(-t -
2
2
w V
3 - 3
2 9 2
)= eo(21 - \w - u\2),
hence
Fp,a(^t) > eo(2t -\w - v\2)
.Q CO .Q O
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CU
CO
>
E
CT C
X LU
lo cu
o <
575
(M <D
(M
O >
(M o (M
= Fu,v (2t).
Thus, all the conditions of the above Corollary are satisfied, which implies that f admits a fixed point.
Coincidence point theorems in a fuzzy b-metric space
Prior to announcing the coincidence, recall first the Definition of a fuzzy ^ b-metric space.
o
" Definition 8. A quadruple (r, Z, 1, s) where r is an arbitrary nonempty set,
0 r is a continuous t-norm, Z is a fuzzy set on r x r x (0, +x>) and s > 1 is a
1 real number, is called a fuzzy b-metric space if the following conditions are w verified:
£ 1. Z(a,(, 0) = 0,
£ 2. Z(a, ¡3,t) = 1 for all t> 0 if and only if a = (3,
3. Z(a, (, t) = Z((, a, t),
4. Z(a, y, s(t + v)) > 1 (Z(a, (3, t), Z((, y, v)),
5. Z(a, (3,.) : [0, +rc>) —> [0,1] is left-continuous and nondecreasing cd for all a,(,Y e r and t,v > 0. ^ When s = 1 then (r, Z, 1, s) is a fuzzy metric space in the form ofKramosil x and Michalek (Kramosil & Michalek, 1975).
LU H
§ The definition of a multivalued ^-contraction in a fuzzy b-metric version o is given as the following.
Definition 9. Let (r, Z, 1, s) be a fuzzy b-metric space and fi : [0, +rc>) [0, +rc>). A mapping f : r ^ C(r) is called a multivalued fuzzy fi-contraction if for every u,u e r and every p e fu there exists a e fu such that
Z(p,a,fi(t)) >Z(u,u,st) for all t> 0.
Theorem 2. Let (r, Z, 1,s) be a complete fuzzy b-metric space where
lim Z(u,u,t) = 1 for all u,u e r and f : r ^ C(r) is a multivalued
t— + X
fuzzy fi-contraction mapping with fi e x. If all the orbits pf for some u e r are bounded, then there exists z e r satisfying z e fz.
Proof. From that Z(w,v,.) is left-continuous and nondecreasing mapping S
for all u,v e r, then by taking F^,v(t) = F(u,v,t) for all t > 0 and since m
the condition of Fp,a(+œ) = 1 has not been used in the proof of theorem ir>
1, it implies that this result holds. □ Œ
eu
Similarly, from Corollary 1, one obtains g,
CO
Corollary 2. Let (r, Z, s) be a complete fuzzy b-metric space with 1 is of S H-type and f : r ^ C(r) is a fuzzy 0-contraction mapping where 0 e Then there exists z e r satisfying z e fz.
Conclusion
References
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o
In summary, the novel approach applied in this study has led to significant advancements, generalizing and enhancing the results originally pro- o posed by Fang (Fang, 1992) and Hadzic (Hadzic, 1989). These achieve- § ments mark a notable contribution to the fixed point theory literature, partic- § ularly in the context of multivalued maps within probabilistic metric spaces. 4 Additionally, the authors introduced and defined the concept of multivalued 0-contraction in a b-Menger space, extending it to encompass fuzzy
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b-metric spaces. Moreover, this exploration uncovered a meaningful con- -§
nection between the boundedness of orbits and the H-type t-norms, providing valuable insights into the interplay between these concepts. As a consequential outcome, coincidence point theorems applicable to fuzzy b-metric spaces are derived, adding a new dimension to the understanding of e these spaces and their applications in the context of multivalued mappings. This comprehensive study not only broadens the theoretical foundations but also opens avenues for further research and exploration in the rich and w diverse field of the fixed point theory. ®
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^ Kramosil, I. & Michálek, J. 1975. Fuzzy metrics and statistical metric spaces.
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Mbarki, A. & Oubrahim, R. 2017. Probabilistic b-metric spaces and nonlinear ° contractions. Fixed Point Theory and Applications, 2017, art.number:29. Available ¡3 at: https://doi.org/10.1186/s13663-017-0624-x.
Menger, K. 2003. Statistical Metrics. In: Schweizer, B. et al. (Eds.) Selecta Mathematica. II, pp.433-435. Vienna: Springer. Available at: par https://doi.org/10.1007/978-3-7091-6045-9_35.
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Nadler Jr, S.B. 1969. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2), pp. 475-488. Available at: https://doi.org/10.2140/pjm.1969.30.475.
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Contractions and Applications in Symmetric and Probabilistic Spaces. Mathemat-
Explorando la probabilidad multivaluada ^-contracciones con órbitas en espacios b-Menger
CAMPO: matemáticas
TIPO DE ARTÍCULO: artículo científico original Resumen:
Introducción/objetivo: El artículo presenta un enfoque novedoso para ciertos teoremas bien establecidos de punto fijo para contracciones probabilísticas multivaluadas en espacios de b-
Patle, P., Patel, D., Aydi, H. & Radenovic, S. 2019. ON H+Type Multivalued ^
ics, 7(2), art.number:144. Available at: https://doi.org/10.3390/math7020144. |
<u
Schweizer, B. & Sklar, A. 1983. Probabilistic Metric Spaces. Mineola, New York: Dover Publications, Inc. ISBN: 0-486-44514-3.
Sherwood, H. 1971. Complete probabilistic metric spaces. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 20, pp. 117-128. Available at: https://doi.org/10.1007/BF00536289.
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h 2
Youssef Achtouna, autor de correspondencia, Stojan Radenovicb, 'tj
Ismail Tahiria, Mohammed Lamarti Sefiana ^
Universidad Abdelmalek Essaadi, Escuela Superior Normal, 8 Departamento de Matemáticas e Informática, Tetuán, Reino de Marruecos
b Universidad de Belgrado, Facultad de Ingeniería Mecánica, ^
Belgrado, República de Serbia .2
o
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<u
ro
>
E
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lo
Menger, aprovechando la acotación de las órbitas. El objetivo w
era generalizar y mejorar los resultados obtenidos anteriormen- w
et
te por Fang y Hadzic.
Métodos: La delimitación de las órbitas en los espacios b-Mengerse utiliza para establecer su enfoque para contracciones probabilísticas multivaluadas.
Resultados: Los hallazgos del estudio no sólo generalizaron los teoremas del punto fijo existentes, sino que también los mejoraron significativamente. Se demostró la eficacia del enfoque para ampliar los resultados propuestos originalmente por Fang y Adzic. Además, se demostró la aplicabilidad del teorema del punto fijo de coincidencia en espacios b-métricos difusos.
ф Conclusión: El estudio presentó una perspectiva novedosa so-
bre los teoremas del punto fijo en contracciones probabilísticas multivaluadas dentro de espacios de b-Menger. Al aprovechar la acotación e introducir un teorema de punto fijo de coincidencia o para espacios b-métricos difusos, el trabajo contribuyó al avance
en este campo.
(M
° Palabras claves: punto fijo, espacios b-Menger, contracción ф-
multivaluada, espacio b-métrico difuso.
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o
° Исследование многозначной вероятности ф-сжатия с
< орбитами в b-пространстве Менгера
х Юсеф Ахтуна, корреспондент, Стоян Раденович6
^ Исмаил Тахириа, Мохаммед Ламарти Сефиан
Университет Абдельмалека Эссаади, Высшая нормальная
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а
>
школа, факультет математики и компьютерных наук, г. Тетуан, Королевство Марокко б Белградский университет, факультет машиностроения, г Белград, Республика Сербия
РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций, <5 27.39.15 Линейные пространства,
О снабженные топологией,
порядком и другими структурами ВИД СТАТЬИ: оригинальная научная статья
х ш
Резюме:
Введение/цель: В данной статье представлен новый подход к некоторым общепризнанным теоремам о неподвижной точке для многозначных вероятностных сокращений в Ь-менгеровских пространствах с использованием ограниченности орбит. Целью данной статьи было обобщить и улучшить предыдущие результаты, полученные Фангом и Хаджичем.
Методы: Ограниченность орбит в Ь-пространствах Мен-гера используется для определения их подхода к многозначным вероятностным сжатиям.
Результаты: Результаты исследования не только обобщили существующие теоремы о неподвижной точке, но и существенно их усовершенствовали. Была продемонстрирована эффективность подхода в расширении результатов, первоначально предложенного Фангом и
Хаджичем. Кроме того, была продемонстрирована приме- га нимость теоремы о неподвижных точках и точках совпа- „ дения в нечетких Ь-метрических пространствах. ю
Выводы: В исследовании представлен новый взгляд на ^ теоремы о неподвижных точках в многозначных вероятностных сжатиях в Ь-пространствах Менгера. Используя ^ ограниченность и вводя теорему о совпадении неподвижных точек для нечетких Ь-метрических пространств, дан- ^ ная статья вносит большой вклад в изучение данной области.
Ключевые слова: неподвижная точка, Ь-пространства Менгера, многозначное ф-сжатие, нечеткое Ь-метрическое пространство.
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Истражива^е вишевредносне вероватнойе ^-контракци]е о
са орбитама у Ь-Менгеровим просторима
Jусуф Актуна, аутор за преписку, Сто]ан Раденовийб,
Езмеил Тахириа, Мухамад Ламарти Сефиана о
а Универзитет „Абделмалек Есади", Висока школа, Одсек за математику и рачунарство, Тетуан, Кра^евина Мароко го
б Универзитет у Београду, Машински факултет, 2
Београд, Република Срби]а тз
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ОБЛАСТ: математика го
КАТЕГОРИJА (ТИП) ЧЛАНКА: оригинални научни рад ~
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Сажетак: ^
Увод/цил: Рад представла нови приступ одре^еним до- .2 бро утвр^еним теоремама о фиксно] тачки за вишезнач- ^ не вероватносне контракци]е у Ь-Менгеровим просторима, я користеПи ограниченост орбите. Цил }е био да се гене-рализу/у и поболша}у претходни резултати ко}е су извели У Фанг и ХауиП.
Методе: КоришПене су ограничености орбита у Ь-Менгеровим просторима ко}е успоставла}у сво} приступ < за вишезначне вероватносне контракци]е.
Резултати: Налази студи}е нису само генерализовали по-сто}еЬе теореме о фиксно] тачки веЬ су их и знача]но по-болшали. Представлена }е и ефективност приступа у проширеъу резултата ко\и су првобитно предложили Фанг
<и
о
о
ф и ХауиЪ. Тако^е, демонстрирана jе применъивост те-
ореме коинциденц^е о фиксноj тачки у расплинутим Ь-метричким просторима.
Закъучак: Студ^а jе представила нову перспективу те> ореме фиксне тачке у вишезначним вероватносним кон-тракц^ама унутар Ь-Менгерових простора. КоришЯеше ограничености и уво^еше фиксне случаjности теорема ¡г тачке за расплинуте Ь-метричке просторе представка ^ допринос унапре^ешу ове области.
о Кьучне речи: фиксна тачка, Ь-Менгерови простори, ви-
шезначно пресликаваше ф-контракц^а, расплинута Ь-о метрика.
2
Paper received on: 03.02.2024. >r Manuscript corrections submitted on: 06.06.2024. ¡ft Paper accepted for publishing on: 07.06.2024.
© 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, http://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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