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VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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On some fixed point results for expansive mappings in S'-metric spaces
Nora Fetoucia, Stojan Radenovicb
aJijel University, LMPA Laboratory, Department of Mathematics,
Jijel, People's Democratic Republic of Algeria, e-mail: [email protected], corresponding author,
ORCID iD: ©https://orcid.org/0000-0002-1474-6554 bUniversity 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://doi.org/10.5937/vojtehg72-50064
FIELD: mathematics
ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: The aim of this paper is to establish some existence results of a fixed point for a class of expansive mappings defined on a complete S-metric space.
Methods: An iteration scheme was used.
Results: Some existing results of mappings satisfying contractive conditions are expanded to expansive ones, providing a new condition expressed in one variable under which the existence of a fixed point holds. Conclusions: This work provides new tools to fixed point theory together with their applications.
Key words: fixed point, S-metric space, expansive mapping.
Introduction and preliminaries
One of the most important instruments to treat nonlinear problems with the aid of functional analytic methods is the fixed point approach. Metric fixed point theory provides essential tools for solving problems emerging from various areas of mathematical analysis, such as variational and linear inequalities, equilibrium problems, complementarity problems, optimization and approximation theory, as well as problems of proving the existence of a solution of integral and differential equations. The first important result of fixed points in metric spaces was the well known contraction mapping theorem, established by S. Banach in his dissertation and published for the
first time in 1922. Later, some generalized metric spaces have been studied to obtain new fixed point theorems.
In 2006, Z. Mustafa and B. Sims (Mustafa & Sims, 2006) introduced the notion of G-metric spaces as a generalization of ordinary metric spaces, and analysed the topological structure of G-metric spaces. In this first paper, the authors developed some fixed point results for various classes of mappings in the setting of a G-metric space. For this and more details, the reader can see (Abbas et al., 2016; An et al., 2015; Dosenovic et al., 2018; Vujakovic et al., 2023). In 2012, S. Sedghi, N. Shobe and A. Aliouche (Sedghi et al., 2012) introduced S-metric spaces as a generalization of G-metric spaces and metric spaces, and proved several fixed point results in the setting of S-metric spaces. For other results, see (Mojaradiafra, 2016; Mojaradiafra & Sabbaghan, 2021; Sedghi & Dung, 2014). In 1984, (Wang et al., 1984), introduced the concept of expanding mappings and proved some fixed point theorems in metric spaces. For more details on expanding mappings and related results, we refer the reader to (Mohanta, 2012; Mojaradiafra et al., 2020; Mustafa et al., 2010). This paper establishes some existence results of a fixed point for a class of expansive mappings in S-metric spaces. Some existing results from G-metric and S-metric spaces are extended. The contribution of this paper is in providing new tools to fixed point theory together with their applications. Let us recall some basic definitions and properties of S-metric spaces.
Definition 1. (Sedghi etal., 2012) Let X be a non-empty set. An S-metric on X is a function
S : X x X x X ^ [0, +rc>[ that satisfies the following conditions for all
x, y,z,a £ X:
(51) S (x, y,z) = 0 if and only if x = y = z,
(52) S(x, y, z) < S(x, x, a) + S(y, y, a) + S(z, z, a).
The pair (X, S) is called an S-metric space.
Example 1. 1. Let X = Rn and || • || a norm on X, then
S(x, y, z) = lly + z - 2x|| + ||y - z|| is an S-metric on X.
2. Let X = Rn and ||-1| a norm on X, then S(x,y,z) = ||x - z|| + ||y - z|| is an S-metric on X.
3. Let X be a nonempty set, d is ordinary metric on X, then
S(x, y, z) = d(x, z) + d(y, z) is an S-metric on X.
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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Definition 2. (Bakhtin, 1989) Let X be a nonempty set. A b-metric on X is a function d : X x X ^ [0, +ж[ if there exists a real number b > 1 such that the following conditions hold for all x,y,z e X.
B1 d(x, y) = 0 if and only if x = y,
B2 d(x,y) = d(y, x),
B3 d(x, z) < b[d(x, y) + d(y, z)].
The pair (X, d) is called a b-metric space.
We will prove that every 5-metric space (X, S) will define a b-metric space (X, d).
Proposition 1. (Sedghi & Dung, 2014) Let (X, S) an S-metric space and let
d(x, y) = S(x, x, y)
for all x,y e X. Then
1. d is a b-metric on X,
2. xn ^ x in (X, S) if and only if xn ^ x in (X, d),
3. (xn) is a Cauchy sequence in (X, S) if and only if (xn) is a Cauchy sequence in (X, d).
Definition 3. (Sedghi et al., 2012) Let (X, S) be an S-metric space. For r > 0 and x e X, we define the open ball BS (x, r) and the closed ball BS [x, r] with the center x and the radius r as follows
Bs(x,r) = {y e X : S(y,y,x) < r} (1)
Bs[x,r] = {y e X : S(y,y,x) < r}. (2)
The topology induced by the S-metric is the topology generated by the base of all open balls in X.
Definition 4. (Sedghi et al., 2012) Let (X, S) be an S-metric space.
1. A sequence (xn) converges to x e X if S(xn, xn, x) ^ 0 as n ^ +ж. That is, for each e > 0, there exists n0 e N such that for all n > n0 one gets S(xn, xn, x) < e. We write xn ^ x for brevity.
2. A sequence (xn) is a Cauchy sequence if S(xn,xn,xm) ^ 0 as n,m ^ +ж. That is, for each e > 0, there exists n0 e N such that for all n,m > n0 one gets S(xn, xn, xm) < e.
3. The S-metric space (X, S) is complete if every Cauchy sequence is a convergent sequence.
Lemma 1. (Sedghi et al., 2012) Let (X,S) be an S-metric space. If the sequence (xn) in x converges to x, then x is unique.
The next three lemmas are well known, see for example (Sedghi et al.,
2012).
Lemma 2. In an S-metric space, there exists
S (x,x,y) = S (y,y,x),
for all x,y e X.
Lemma 3. Let (X, S) be an S-metric space. If xn ^ x and yn ^ y then
S (xn,xn,yn) ^ S(x,x,y').
Lemma 4. (Mojaradiafra et al., 2020) Any S-metric space is Hausdorff.
Definition 5. Let (X, Si) and (Y, S2) be S-metric spaces. A map f : X ^ Y is called continuous at x e X if for every e > 0 there exists 5 > 0 such that
Si(x, x,y) <5 implies that S2(f (x), f (x), f (y)) < e, or
f (Bsг (x, 5)) c Bs2 (f (x),e).
The next result is also known, see (Sedghi et al., 2012).
Lemma 5. Let (X, S) be an S-metric space. The map f : X ^ X is continuous at x e X if and only if f (xn) f (x) whenever xn x.
Definition 6. Let (X, S) be an S-metric space. A map T : X ^ X is said to be a contraction if there exists a constant 0 < к < 1 such that
S(Tx,Tx,Ty) < kS(x, x, y), for all x,y e X.
Theorem 1. Let (X, S) be a complete S-metric space and T : X ^ X be a contraction. Then T has a unique fixed point.
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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Definition 7. Let (X, S) be an S-metric space and T be a self-map on X. Then T is called an expansive map if there exists a constant a > 1 such that for all x,y e X, one gets
S(Tx,Tx,Ty) > aS(x,x,y).
The constant a is called the expansion coefficient.
Remark 1. Expansive map on an S-metric space does not need to be continuous.
Theorem 2. (Mojaradiafra et al., 2020) Let (X, S) be a complete S-metric space, and let T : X ^ X bean onto continuous mapping satisfying
S(Tx,Tx,T2x) > aS(x,x,Tx) (3)
for all x e X, where a > 1. Then T has a fixed point in X.
Example 2. (Mojaradiafra et al., 2020) Let
T : (R, S) ^ (R, S) be defined by
T (x) = I
4x for x < 2,
4x + 3 for x > 2,
where S(x,y, z) = max{|x - z\, \y - z\}. Then (R, S) is a complete S-metric space and T is an expansive map with the coefficient a = 2.
Main results
Our first main result is as follows:
Theorem 3. Let (X, S) be a complete S-metric space. Let T : X ^ X be an onto mapping such that,
S(Tx,Ty,Tz) > aN(x, y, z)
(4)
for all x, y, z, where a > 1 and
N(x, y, z) = min
S(x, y, Tx), S(Tx, Ty, y), S(Tx, Ty, x), S(Tx,Tx,y),
S(z, z, x),S(Tz, Tz, T2x),S(T2y, T2x, Tz),S(Tx, Ty, z), S(x, y, z),S(Tx, Tx, x),S(Ty, Ty, y),S(T2x, T2y, Tz), S(Tx, Tx,z), S(Ty, Ty, x), S(Tz, Tz, T2y)
Then T admits a fixed point.
Proof. Let x0 e X, since T is onto then there exists an element x\ satisfying x1 e T-1 (x0). Continuing in this way, one gets a sequence (xn), where xn e T-1(xn-1). If xn = xn-1 for some n, then xn is a fixed point of T. Assume xn = xn-1 for every n e N, then from (5) one obtains
S(xn—1j xn-1j xn-2) = S(Txn,Txn,Txn-1) ^ aN(xn,xn7 xn— 1)7
where
N (xn,xn,xn—l) = (6)
S (xni xnj Txn) 7 S (Txn 7 Txn7 xn) 7 S (Txn7 Txn7 xn) 7 S (Txni Txni xn) 7 S(xn— 1 7 xn— 17 xn) 7 S (Txn— 1 7 Txn—17 T xn) 7 min S (T xn7T xn 7Txn—1)7S (Txn 7Txn7xn—1)7S (xni xni xn—1)7 7
S (Txn7T xn7 xn) 7 S(T xn7 Txn7xn )7^S (T2xn7 T 2xn7 T xn—1) 7
S (Txn7 Txn7 xn—1)1 S(Txn7 Txn7 xn) 7 S(Txn—17 Txn—17 T xn)
so
N (xnixn7xn—1) --
S(xn7xn7xn—1)7S(xn—17xn—17xn)7S(xn—17xn—17xn)7 S(xn— 17 xn— 17 xn) 7 S(xn— 17 xn— 17 xn) 7 S(xn—27 xn—2i xn—2~) 7 — min S(xn—2,xn—27 xn—2^7 S(xn— 17 xn—17 xn—1) 7 S(xn7 xn7 xn—1)7
S(xn— 17 xn— 17 xn) 7 S(xn— 17 xn— 17 xn) 7 S(xn—27 xn—2i xn—2~) 7 S(xn— 17 xn— 17 xn— 1) 7 S(xn— 17 xn— 17 xn) 7 S(xn—27 xn—27 xn—2~)
(7)
then
this implies that
S(xn—17xn—17xn—2') ^ aS (xnixn7 xn—1) 7
S(xn7 xn7 xn—1) ^ aS(xn— 17 xn— 17 xn—2) ■
Set k = a, then k < 1. By induction one obtains
S(xn7 xn7 xn—1) ^ kS(xn—17xn—17xn—2')
< k2S(xn—27 xn—27 xn—з)
(8)
(9)
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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< ...
< kn-1 S (x1,x1,x0).
Then, by (S2) and Lemma 2, we get for all n,m e N; n <m,
S(xm,xm,xn) < 2S(xmi xmi xm-1) + S(xn,xn4 xm-1 ) (10)
= 2S(xm:> xm:> xm—l) + S(xm-1, xm—1j xn)
< 2S(xmj xmj xm-1) + 2S(xm-1i xm—1i xm-2)
+ S(xn xn xm-2)
— 2S(xmj xmj xm-1) + 2S(xm-1,xm—1,xm—2)
+ S (xm— 2, xm-2i xn)
< 2 ^ ^ S(xi1 xi1 xi-1) + S(xn+1i xn+1j xn),
i=n+2
by (10), we obtain
m
S(xm,xm,xn) < 2 ^ ^ S(xi,xijxi—1) + S(xn+1, xn+1, xn) (11)
i=n+2
m
< 2 ^ k 1S(x1 ,x1,x0) + knS(x1,x1,x0)
i=n+2
i _ km—n—1
< 2kn+1-----;---;--S(x1, x1,x0) + knS(x1,x1,x0)
1-k
k'n+1
< 2j fcS(x1 ,x1,xo) + knS (x1,x1,xo),
so, S(xm,xm,xn) ^ 0 as n,m ^ and (xn) is a Cauchy sequence. Since (X, S) is complete, then there exists u e X such that (xn) is convergent to u.
We need to show that Tu — u, let y e T-1(u), for n such that xn — u, we get
S (xn,xn,u) S(Txn+1,Txn+1,Ty) ^ (xn+1l xn+1l y) 1 where
N (xn+1,xn+1,y) —
mm
SiXn+x, Xn+1, Xn), S(Xn, Xn, Xn+\), S(Xn, Xn, Xn+\), S(Xn, Xnj Xn+1) , S (.УтУ] Xn+1) , S (Ty , Ту, Xn— 1) ,
S(Xn— 1, Xn—1] Ty') , S (Xn] Xn] y) , S (xn+1, Xn+1 ,y')] S(Xn, Xn] Xn+1)] S(Xn] Xn] Xn+1)] S(Xn— 1, Xn—1] Ty) , „ S(Xn, Xn, y) , S (Xn, Xn, Xn+1) , S(Tyi Tyj Xn—1)
taking the limit as n we obtain S(u,u,y) < 0, which implies that
Ty = u = y; hence u is a fixed point of T.
□
Our second result is given by
Theorem 4. Let (X, S) be a complete S-metric space and let T : X ^ X be an onto S-continuous mapping. Assume that there exist nonnegative reals a, b, c, d, e, f, g with b < 1, and a + b + c + d + e + f + g > 1 such that
S4(Tx,Ty,Tz) > (13)
> aS4(X, y, z) + bS4(T2x, T2y, z) + cS4(TX,Ty, x) +
+ dS2(Tz, T2y,Tx)S2(y, x,Tx) + eS2(z,Tx, y)S2(z,Tx, y) +
+ fS2(z, Tx, T2y)S2(Tx, Ty, y) + gS2(T2x, T2x, y)S2(y, x, Tx),
for all x e X, then T has a fixed point in X.
Proof. Let x0 e X, since T is onto there exists x1 e T—1(x0). Continuing in this way, we get a sequence (xn), where xn e T—1 (xn—1).
If, xn = xn—1 for some n, then we obtain xn as a fixed point of T. Hence, without loss of generality, we may assume that xn = xn—1 for every n e N. By (13), we get
S (Xn— 1 , Xn— 1 , Xn—2 )
= S4(TXn,TXn,TXn—1)
> aS4(Xn, Xn, Xn—1) + bS4(Xn—2,Xn—2,Xn—1) + +cS 4(Xn—1 ,Xn—1,Xn)
+ dS (xn—2,Xn—2,Xn—1)S (xn,Xn,Xn—1)
+ eS (xn—1,Xn—1,Xn)S (xn—1,Xn—1,Xn)
+ fS (xn—1,Xn—1,Xn—2')S (xn—1,Xn—1,Xn) + gS (xn—2,Xn—2 ,Xn)S (xni Xni Xn—1) i
hence, by Lemma 2 we obtain
(a + c)S4(xn,Xn,Xn—1)+ (14)
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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+ (d + e + f + g)S2(xn, xn, Xn-i)S2(xn-i,Xn-i,Xn-2) - (1 - b)S4(Xn-l,Xn-l,Xn-2) < 0, which is equivalent to
(a + c)r4 + (d + e + f + g)r2 — (1 — b) < 0,
where
S (XniXniXn—l )
S(xn—l, Xn— l, Xn— 2)
Let f : [0, R be the function given by
r
f (r) = (a + c)r4 + (d + e + f + g)r2 — (1 — b) = 0,
then, using assumptions, we get f (0) = b — 1 < 0 and f (1) = a + b + c +
d + e + f + g — 1 > 0.
One can deduce that, there exists k e]0,1[ for which inequality (15) holds whenever r < k, and hence
S(Xn,Xn,Xn—1) < kS(Xn—l,Xn—l,Xn—2')
< k2S(Xn—2,Xn—2,Xn—3)
< ...
< kn—l S (Xl,Xl,X0),
then, by (S2) and Lemma 2, we get for all n,m e N; n <m,
S(Xm,Xm,Xn) <
<
+
+
2S(Xm,Xm,Xm—l) + S (Xn,Xn,Xm—l) 2S(Xm,Xm,Xm—l) + S (Xm—l,Xm—l,Xn) 2S(Xmj Xmj Xm—l) + 2S(Xm—li Xm—lj Xm—2) S(Xnj Xn■> Xm—2)
2S(Xmj Xmj Xm—l) + 2S(Xm—li Xm—lj Xm—2) S(Xm—2, Xm—2j Xn)
(16)
m
< 2 ^ ' S (Xii Xii Xi—l) + S (Xn+li Xn+li Xn) i
i=n+2
by (16), we obtain
m
S(Xm,Xm,Xn) < 2 ^ ' S(Xi,Xi,Xi—l) + S(Xn+l} Xn+li Xn) (17)
i=n+2
< 2 ^ кг ,x1,x0) + knS(x1,x1,x0)
i=n+2
1 _ km—n—1
< 2kn+1-----;---;--S(x1, x1,x0) + knS(x1,x1,x0)
2
kn+1
T к
1 - к
S(x1 ,x1, x0) + knS(x1,x1,x0),
so, S(xm, xm, xn) ^ 0 as n,m ^ and (xn) is a Cauchy sequence, and by the completeness of (X, S), there exists u e X such that (xn) converges to u. By continuity of T, we get
T(xn) — xn— 1 ^ Tu,
thus u — Tu. We conclude that u is a fixed point of T. We state our third result in the sequel.
□
Theorem 5. Let (X, S) be a complete S-metric space and let T : X ^ X be an onto continuous mapping satisfying
S(Tx,Ty,Tz) > C1S(Tx,Ty,x) + c2S(Tz,T2y,Tx) + c3S(y,y,Ty)+ (18)
+ cSTx.Tx, Tz) + C5 'S(Ty, z,T2x) + S(z, Tx y> + 3 * SГ* T2y,z)
+
+ C6 + C7
S(z,Tx,T2y) + S (T2y,Tz,Tx) + S (x,y,z) 3
S(Tx, Ty, y) + S(Tz, T2x, Ty) + S(y, y, z)
3
+
where c1, c2, c3, c4, c5, c6, c7 are non negative reals that verify
C1 + c2 + c3 + c4 + c5 + c6 + c7 > 1, and 3 - 3c2 - 3c4 - 2c5 - 2вб - c7 > 0, then, T has a fixed point in X.
Proof. Replacing y by x and z by Tx in (18), we obtain
S (Tx,Tx,T2x) > c1S (Tx,Tx,x) + c2S (T2x,T2x,Tx) (19)
+ c3S (x,x,Tx) + c4S (Tx,Tx,T 2x)
+ c5
S(Tx, Tx, T2x) + S(Tx, Tx, x) + S(T2x, T2x, Tx)
3
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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+ c6 + C7
S(Tx, Tx, T2x) + S(T2x, T2x, Tx) + S(x, x, Tx)
3
S(Tx, Tx, x) + S(T2x, T2x, Tx) + S(x, x,Tx)
3
without loss of generality, we may assume that T(x) = T2(x), then (19) entails
S(Tx, Tx, T2x) > (ci + )S(Tx, Tx, x)
3
+ (c2 + C5 + C6 + C7 )S (T 2x,T 2x,Tx)
3
+ (C3 + C6 + C7 )S (x,x,Tx)
+ (C4 + C5+C6 )S (Tx,Tx,T2x). (20)
3
Using Lemma 2, we obtain
(1 - (C2 + C4 + ^+6 + c5+c36+c7 ))S(Tx, Tx, T2x) >
(Ci + C3 + £5++Cr + £6++Cr)S(x, x, Tx), (21)
which implies that
(3 — 3c2 — 3c4 — 2c5 — 2c6 — C7)S(Tx, Tx, T2x) > (3c1 + 3c3 +
+C5 + C6 + 2c7)S(x,x,Tx), (22)
hence
S(Tx, Tx, T2x) >
2 л ^ 3ci + 3C3 + C5 + C6 + 2C7
3 — 3c2 — 3c4 — 2c5 — 2c6 — c7
S(x, x,Tx).
(23)
Setting, a = ci + c2 + c3 + c4 + c5 + c6 + c7, then by assumption we obtain a > 1. So, (23) becomes condition (3); therefore, the result follows from Theorem 2. □
Now, we give an example illustrating our result in Theorem 5.
Example 3. Let X = R be the set of real numbers. Define
S(x,y,z) =
0 if x = y = z, max{|x|, \y\, \z\) otherwise.
Then (X, S) is an S-metric space. Assume
2x if x < 0, 2x if x > 0.
We get
T 2x = I
2x if x < 0, 25 if x > 0.
5(x, x, Tx) = \Tx\ = j ^2x if x < 0
2x if x > 0.
S(Tx, Tx, T2x) = \T2x\ = j
—2x if x < 0, 25x if x > 0.
Set: ci = 20 ,c2 = 8, сз = 5, c4 =
7
20 and c7
— so
40 ’ SO
20 8’ ^ 5 4 ’ C5 40 ’ C6 20
conditions of Theorem 5 are verified; hence T admits a fixed point in R. One can see that the fixed point here is zero.
References
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Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
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Sobre algunos resultados de punto fijo para mapeos expan-sivos en espacios S-metricos
Nora Fetoucia, Stojan Radenovicb
a Universidad de Jijel, Laboratorio LMPA, Departamento de Matematicas, Jijel, RepOblica Argelina Democratica y Popular, autor de correspondencia
b Universidad de Belgrado, Facultad de Ingenieria Mecanica, Belgrado, RepOblica de Serbia
CAMPO: matematicas
TIPO DE ARTiCULO: articulo cientifico original
Resumen:
Introduccion/objetivo: El objetivo de este artfculo es establecer algunos resultados de existencia de un punto fijo para una clase de mapeos expansivos definidos en un espacio S-metrico com-pleto.
Metodos: Se utilize un esquema de iteracion.
Resultados: Algunos resultados existentes de mapeos que sa-tisfacen condiciones contractivas se expanden a resultados expansivos, proporcionando una nueva condicion expresada en una variable bajo la cual se cumple la existencia de un punto fijo.
Conclusion: Este trabajo proporciona nuevas herramientas a la teorfa del punto fijo junto con sus aplicaciones.
Palabras claves: punto fijo, espacio S-metrico, mapeo expansi-vo.
О некоторых результатах с неподвижной точкой в расширенных отображениях
Нора Фетучи3 * *, Стоян Раденовичб
a Университет Джиджеля, лаборатория LMPA, математический факультет, г Джиджель, Алжирская Народная Демократическая Республика, корреспондент
б Белградский университет, машиностроительный факультет, г Белград, Республика Сербия
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: Цель данной статьи - выявить результаты о наличии неподвижной точки в классе расширяющихся отображений, определенных в полном S-метрическом пространстве.
Методы: В исследовании использована итерационная схема.
Результаты: В данной статье развернуты некоторые из существующих результатов расширяющихся отображений, которые были дополнены новыми сжимающими усло-
Fetouci, N. et al., On some fixed point results for expansive mappings in 5-metric spaces, pp.1004-1018
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
виями, выраженными одной переменной, при наличии неподвижной точки.
Выводы: В данной статье представлены способы применения новых инструментов в теории неподвижных точек.
Ключевые слова: неподвижная точка, S-метрическое пространство, расширенное отображение.
O нeким резултатима фиксне тачке за експанзивна пресликава^а у S-метричким просторима
Нора Фетучиа, Сто]ан РаденовиЬб
а Универзитету Цицелу, Лаборатори]а LMPA, Департман математике, Цицел, Народна Демократска Република Алжир, аутор за преписку
5 Универзитету Београду, Машински факултет, Београд, Р Срби]а ОБЛАСТ: математика
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:
Увод/ция>: Циъ овог рада }есте да установи неке резулта-те посто]ак>а фиксне тачке за класу експанзивних пресли-каваъа дефинисаних на комплетном S-метричком простору.
Методе: У истраживаъу]е применена шема итераци]е. Резултати: Проширени су неки посто]еЯи резултати за експанзивна пресликаваъа са новим контрактивним усло-вима израженим са }едном променъивом под ко}ом посто}и фиксна тачка.
Закъучак: Ауторски допринос пружа нове алате за теорбу фиксне тачке и ъихову примену.
Къучне речи: фиксна тачка, S-метрички простор, експан-зивно пресликаваъе.
Paper received on: 28.03.2024.
Manuscript corrections submitted on: 23.09.2024.
Paper accepted for publishing on: 24.09.2024.
© 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, httpV/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/).
