CC BY
20
INTERPOLATIVE GENERALISED MEIR-KEELER CONTRACTION
Shobha Jaina, Vuk N. Stojiljkovicb, Stojan N. Radenovicc
aShri Vaishnav Vidyapeeth Vishwavidyalaya, Department of Mathematics, Indore, Madhya Pradesh, Republic of India, e-mail: shobajain1@yahoo.com, ORCID iD: ©https://orcid.org/0000-0002-9253-8689
bUniversity of Novi Sad, Faculty of Science, Novi Sad, Republic of Serbia, e-mail: vuk.stojiljkovic999@gmail.com, corresponding author, ORCID iD: ©https://orcid.org/0000-0002-4244-4342 cUniversity of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia, e-mail: radens@beotel.net,
ORCID iD: ©https://orcid.org/0000-0001-8254-6688
DOI: 10.5937/vojtehg70-39820;https://doi.org/10.5937/vojtehg70-39820 FIELD: Mathematics
ARTICLE TYPE: Original scientific paper Abstract:
Introduction/purpose: The aim of this paper is to introduce the notion of an interpolative generalised Meir-Keeler contractive condition for a pair of self maps in a fuzzy metric space, which enlarges, unifies and generalizes the Meir-Keeler contraction which is for only one self map. Using this, we establish a unique common fixed point theorem for two self maps through weak compatibility. The article includes an example, which shows the validity of our results.
Methods: Functional analysis methods with a Meir-Keeler contraction.
Results: A unique fixed point for self maps in a fuzzy metric space is obtained.
Conclusions: A fixed point of the self maps is obtained.
Key words: Fuzzy metric space, common fixed points, weak compatibility, Interpolative generalised Meir-Keelar contraction.
Introduction
In 1965 L. Zadeh (Zadeh, 1965) introduced the theory of fuzzy sets. Later on, in 1978, the concept of a fuzzy metric space was introduced by Kramosil and Michalek in (Kramosil & Michalek, 1975), which was modified by George and Veeramani (George & Veeramani, 1994) in order
to obtain a Hausdorff topology for this class of fuzzy metric spaces. Then in year 1988, Grabiec (Grabiec, 1988) gave a fuzzy version of the Banach (Banach, 1922) contraction principle in the setting of a fuzzy metric space. Over the past years, various authors have tried to generalize the fixed point theorem by modifying and varying the contractive condition, see, e.g., (Gregori & Sapena, 2002), (Jain & Jain, 2021), (Mihet, 2008), (Saha et al, 2016), (Tirado, 2012) and (Wardowski, 2013) in the sense of George and Veeramani. In 2019, Zheng and Wang (Zheng & Wang, 2019) introduced a Meir-Keeler contraction in the setting of a fuzzy metric (Schweizer & Sklar, 1983) space and proved some fixed point results for a self map.
Inspired with the interpolative theory, Karapinar and Agrawal (Karapinar & Agarwal, 2019) introduced the notion of an interpolative Rus-Reich-Ciric type contraction via the simulation function in a metric space. Motivated by this paper, we introduce an interpolative generalised Meir-Keeler contraction (Gregori & Minana, 2014) for two self maps (Rhoades, 2001) in the setting of a fuzzy metric space, which enlarges, unifies and generalizes the existing Meir-Keelar contraction in a fuzzy metric (Mihet, 2010) space through weak compatibility (Banach, 1922).
The structure of the paper is as follows:
After the preliminaries, we introduce a interpolative generalised Meir-Keeler contraction in the setting of a fuzzy metric space. Then we study the Meir-Keeler contractive mapping due to Zheng and Wang (Zheng & Wang, 2019). In section 4, the existence of a unique common fixed point of an interpolative generalised Meir-Keeler contractive mapping has been established through weak compatibility followed by an example.
Preliminaries
Definition 1. (George & Veeramani, 1994) A mapping * : [0,1] x [0,1] ^ [0,1] is called a continuous triangular norm (t-norm for short) if * is continuous and satisfies the following conditions:
(i) * is commutative and associative, i.e. a * b = b * a and
a * (b * c) = (a * b) * c, for all a,b,c £ [0,1];
(ii) 1 * a = a, for all a £ [0,1];
(iii) a * c < b * d, for a < b,c < d for a,b,c,d £ [0,1].
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Definition 2. (George & Veeramani, 1994) A fuzzy metric space is an or> dered triple (X, M, *) such that X is a (nonempty) set, * is a continuous t-norm and M is a fuzzy set on X x X x (0, +œ) satisfying the following conditions, for all x,y,z e X and t,s > 0; (GV1) M(x,y,t) > 0;
=5 (GV2) M (x, y, t) = 1 if and only if x = y;
(GV3) M(x,y,t) = M(y,x,t);
o (GV4) M(x, z,t + s) > M(x, y, t) * M(y, z, s);
g (GV5) M(x, y,.) : (0, ^ (0,1] is continuous. EC
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¡^ x e X and t> 0 and M(x, y, t) < 1, for all x = y and t> 0.
The following notion was introduced by George and Veeramani in (George & Veeramani, 1994).
^ Definition 3. (George & Veeramani, 1994) A sequence {xn} in a fuzzy cd metric space (X, M, *) is said to be M-Cauchy, or simply Cauchy, if for g each e e (0,1) and each t > 0 there exists an n0 e N, such that M(xn,xm,t) > 1 - e, for all n,m > n0. Equivalently, {xn} is Cauchy if
M(xn,xm,t) = 1, for all t> 0.
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o Lemma 1. (Grabiec, 1988) Let (X, M, *) be a fuzzy metric space. Then M (x,y,.) is non-decreasing for all x,y e X.
Theorem 1. (George & Veeramani, 1994) Let (X,M, *) be a fuzzy metric space. A sequence {xn}neN in X converges to x e X if and only if
limn^^ M(xn,x,t) = 1.
Definition 4. (George & Veeramani, 1994) (X, M, *) (or simplyX) is called M-complete if every M-Cauchy sequence inXis convergent.
Lemma 2. (Saha et al, 2016) If * is a continuous t-norm {an}, {/n}, {jn} are sequences such that an ^ a,/n ^ / and jn ^ y as n ^ then
limfc^+^(afc * /k * Yk) = a * lim^+^/k * y, 820
and
!imfc^+œ(afc * A * Yk) = a * Mk^+^A * Y-
Denote A = {5|5 : (0,1] ^ (0,1]} where 5 is right continuous.
Definition 5. (Zheng & Wang, 2019) Let (X, M, *) be a fuzzy metric space. A mapping f : X ^ X is said to be a fuzzy Meir-Keeler contractive mapping with respect to 5 e A if the following condition holds:
for all e e (0,1), e - 5(e) <M (x,y,t) < e implies M (fx,fy,t) > e, (1)
for all x,y e X,t > 0.
Interpolative generalised Meir-Keeler contraction
Definition 7. Let (X, M, *) be a fuzzy metric space. A pair (f, g) of self maps in X is said to be an interpolative generalised Meir-Keeler contractive if there exists a, /3 £ [0,1) with a + / < 1 and for all x,y £ X,t> 0
for all e £ (0,1), e - 5(e) <M(x,y) < e implies M(fx,fy,t) > e, (2)
where
M (x,y) = (M (fx,gx,t))a(M (fy,gy,t)f (M (gx, gy,t))(l-a-l3).
Remark 1. From equation (2) for all x = y £ X,t > 0 the pair (f,g) is a strict contraction i.e.
M(fx,fy,t) > M(x,y).
Thus for x = y.
M (fx, fy, t) > (M (fx, gx, t))a(M (fy, gy, t)f (M (gx, gy, t))(l-a(3)
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Lemma3. (Saha etal, 2016) Let {f(k,.) : (0, ^ (0,1];k = 0,1,2,...,} be a sequence of functions such that f (k,.) is continuous and monotone increasing for each k > 0. Then limfc^+^f (k, t) is a left continuous function | in t and limfc^+^f (k, t) is a right continuous function in it. o
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Definition 6. (Jain et al, 2009) Two self maps f and g in a fuzzy metric space (X, M, *) are said to be weakly compatible if they commute at their io" coincidence points i.e. forx £ X, fx = gx = y implies gy = fy.
Remark 2. Taking g = I, the identity map in equation (2)we obtain
for all e e (0,1),e - 5(e) < M(x, y) < e implies M(fx, fy, t) > e, (4) where
M (x,y) = (M (fx,x,t))a(M (fy,y,t)f (M (x,y,t))(1-a-ii).
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which is an interpolative generalised Meir-Keeler contraction, for a self map f.
§ Remark 3. Taking a = 0,3 = 0 in equation (4) then M(x,y) = M(x,y,t) and we have
for all e e (0,1), e - 5(e) < M(x, y,t) < e implies M(fx, fy, t) > e, (5)
which is precisely the Meir-Keeler contraction, for a self map given by
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Lemma 4. (Zheng & Wang, 2019) If 5 e then for t e (0,1), there exists k = k(t) e N such that
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OT 0 1 k < 1 """ 5 1 k k > 0
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t + ^ < 1 and 5 (t + - ^ > 0.
Before we prove the main result, we prove the following lemma:
Lemma 5. Let (f,g) be a pair of an interpolative Meir-Keeler contractive mapping with respect to 5 e △ and f (X) c g(X). Construct a sequence {yn}, by fxn = gxn+1 = yn, for n = 0,1,2,.... Then
O M(yn, yn+i,t) = 1.
Proof. Suppose if possible on the contrary that limn—M(yn,yn+1,t) = p(< 1). For a, / e [0,1) we have
((M (fxn,gxn,t))a(M (fxn+1 ,gxn+i,t)Y
lim M(,n,Xn+i) = lim { ((MV*-'**-»™fX-Sr-^ I n—+^ ^ n—+^ I (M(gxn,gxn+1,t))(1 a p> J
(M(yn,yn-1,t))a(M(yn+1,yn,t))fi }
(M (yn-1,yn+,t))(1-—
= lim (pap^p(1-a-^A n—V J
= p.
By using lemma (4), for p < 1 and 5 e △ we can find k = k(p) e N such that
, + m < 1 and 5(pp + M) - M > 0. Since limn^+^ M(xn, xn+l) = p, we can find n0 such that when n > n0, M(xn, xn+l) >p + ® - 5 (p + ^ . (6)
M(xn,xn+i) <p + ®. (7)
Let n > max{no,nl}. Then both equations (6), and ( 7) hold for such n.
k
i. e. M(yn,yn+l,t) > p + , which contradicts the fact that
M(yn,yn+i,t) = p. Therefore, !imn^+M M(yn,yn+i,t) = 1. □
Main results
Our first new result is the next one:
Theorem 2. : Let f and g be self maps in a fuzzy metric space (X, M, *) satisfying the following conditions: (4.11) f (X) C g(X);
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Taking e = p + ^, in equation (2) we get 2
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(4.12) The pair (f, g) is an interpolative generalised Meir-Keeler contraction;
(4.13) f (X) is complete;
(4.14) The pair (f, g) is weakly compatible.
Then f and g have a unique common fixed point in X if and only if there exists x0 e X such that /\t>0 M(x0, f (x0),t) > 0.
Proof. : Suppose the pair (f,g) has a unique common fixed point u then u = fu = gu. Therefore, M(u,fu,t) = 1, yt > 0. Hence
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At>0 M(u, fu, t) > 0.
o Conversely, suppose that there exists x0 e X such that
f\t>0 M(x0,f (x0),t) > 0. Construct a sequence {yn}, by defining fxn = gxn+l = yn, for n = 0,1,2,.... First we show that if the two maps f and g have a common fixed point then it is unique. Let u and v be two common fixed points of f and g. Then u = fu = gu and v = fv = gv. We
^ show that u = v.
a:
O
° Suppose, on the contrary that u = v, then fu = fv. Now
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M (u, v) = (M (fu, gu, t))a(M (fv, gv, t))3 (M (gu, gv, t))(1-a-3\ = (M(u, u, t))a(M(v, v, t))3(M(u, v, t))(l-a-3\ = (M (u,v,t))(l-a-3\
Now
M (u, v, t) = M (fu,fv,t),
^ > M(u,v), using (3)
« = (M (u,v,t))(l-a-3)
>Q
i. e. M (u, v, t) > (M (u,v,t))(l-a-3\
w implies
(M(u,v,t))(a+3) > 1,
which is not true as the left hand quantity is less than 1.So u = v. Thus, if the pair (f,g) has a common fixed point then it is unique. Step 1 To see the existence of a common fixed point of the self maps f and g, we consider the following cases.
CASE I Suppose any two terms of the sequence {yn} are equal i. e. for some n € N, yn = yn+i- As yn = fxn = gxn+i = fxn+i = gxn+2 = yn+i we have fxn+i = gxn+i. Let fxn+i = gxn+i = So xn+i is a point of coincidence of the pair (f,g). As the pair (f,g) is weakly compatible we have fz = gz. Now we show that fz = Suppose, if possible on the contrary, that fz = z so fz = fxn+i. By using equation (3) we have
M (z,fz,t) = M (fxn+i ,fz,t),
> (M (fxn+i, gxn+i, t))a(M (fz, gz, t))3 (M (gxn+i, gz, t))(i-a-3),
= (M(z, z, t))a(M(fz, fz, t))13(M(z, fz, t))(1-a-3), = (M (z,fz,t))(1-a-3).
i.e.
M(z, fz, t) > (M(z,fz,t))(1-a-3), for all t > 0 implies (M(z, fz,t))(a+3) > 1, which is not possible. Hence fz = z. Therefore, z is a common fixed point of the pair (f,g) in this case.
So we can assume the consecutive terms of the sequence {yn} are distinct.
Again, to see the existence of a common fixed point in other cases, we first show that all the terms of the sequence {yn} are distinct. CASE II Suppose yn = ym, for some m > (n + 1), then as all the consecutive terms of the sequence {yn} are distinct, we claim that yn+1 = ym+1. Suppose if possible on the contrary yn+1 = ym+1 then yn = yn+1 = ym+1 = ym implies yn = ym which contradicts our assumption. So we have yn+1 = ym+1. Also and
M(yn+1,yn+2,t) = M(fxn+1, fxn+2,t)
>
(M (fxn+1, gxn+1,t))a(M (fxn+2,gxn+2,t))3
(M (gxn+1,gxn+2,t))(1-a-3)
(M(yn+1,yn, t))a(M(yn+2,yn+1,t))3 } (M (yn,yn+1,t))(1-a-3) J
= (M (yn+1 ,yn,t))1-3 (M (yn+2, yn+1 ,t))3
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i. e.
M (yn+1, yn+2, t) > (M (yn+1,yn,t))1 3 (M (yn+2,yn+1t))3
Thus
So
M(yn,yn+1,t) < M(yn+1, yn+2, t).
(8)
M(yn,yn+1,t) < M(yn+1, yn+2, t) < M(yn+2,yn+3,t) < ... < M(ym,ym+1,t).
i. e. M(yn,yn+1,t) < M(yn,yn+1 ,t), which is not possible. So this case does not arise.
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Thus, we conclude that for distinct n,m e N,yn = ym. Therefore, the
elements of the sequence {yn} are distinct. From equation (8) we have
^ M(yn,yn+1,t) <M(yn+1,yn+2,t),
for all t > 0. Thus, {M(yn,yn+1,t)}, for each t > 0, is a strictly increasing o sequence, which is bounded above by 1. Therefore, by lemma 5, for t> 0,
lim M(yn,yn+i,t) = 1, (9)
^ n—
o Now we prove that the sequence {yn} is M-Cauchy. Suppose if possible
< on the contrary that it is not true; then there exist n e (0,1), to > 0 and the ^ sequences {p(n)}, {q(n)} (p(n) being the smallest ones of the index)
X
ft n<p(n) <q(n),M(Vp(n),yq(n),t0) < 1 - n,M(Vp(n) — 1 ,yq(n),t0) > 1 - V-
>- (10) STEP 2: In this step, we show that limn—^> M(yp(n)—1,yq(n)—1,to) = 1 - n-Now for all n > 1,0 < A < t0/2, we obtain,
1 - n > M(yp{n), yq(n) ,to), using (10)
> M (yp{ n), yp(n)-1, A) * M(yp(n)-1,yq(n)-1,to, A - 2)* (11) ^ *M (yq(n)-1,yq(n),A)-
Let
h1(t) = J—taM (yp{
n) — 1, yq(n) — 1, t),t> 0.
o Taking the limit supremum on both sides of equation (11), and using the
properties of M and *, and by lemma 3, we obtain
1 - n > 1 * limn—+^M(yp(n)—1,yq(n)—1,to - 2A) * 1 using (10) W = h1(to - 2A).
Since M is bounded with the range in (0,1], continuous and non-decreasing in the third variable t, it follows from lemma 3, that h1 is continuous from the left. Therefore, for A ^ 0 , we obtain
h1(to) = nl™,M(yp(n)—1,yq(n)—1,to) < 1 - V- (12)
Let
h2(t) = ljm.n—+^M(yp(n)—1,yq(n)—1,t),t> 0.
Again, for all n > 1,A> 0 M (yp( n) — 1, yq(n) — 1, to + A) > M (yp(
n) — 1, yq(n), to) * M(yq(n),yq(n) — 1, A) 826
> (1 - n) * M(yq{n),yq(n)-1,to) using (10) 8
i
Taking the limit infimum as n ^ in the above inequality, we obtain
h2 (A + t0) = ljMn^+^M(yq( A + to),
> (1 - n) * ljmn^+^M(yp(„),yq(n)-1, A), using (9) = (1 - n) * 1
= 1 - n.
Since M is bounded with the range in (0,1] , continuous and non-decreasing in the third variable t, it follows from lemma 3, that h2 is continuous from the right. So letting A ^ 0 , we obtain
ljmn^+^M(yp(n)-1,yq(n)-1,to) > (1 - n). (13)
Combining the inequalities (12) and (13), we get
nlim^M(yp(n)-1 ,yq(n)-1,to) = (1 - n). (14)
STEP 3 In this step, we show that M(yp(n),yq(n),t0) = 1 - n.
From equation (10) we have
limn^+^ M (yp(
n), yq(n), fa) < 1 - n. (15)
Also for all n > 1 and A > 0 we have
M (yp(n), yq(n) ,to + 2 A) > M (yp(
n), yp(n)-1, A) * M(yp(n)-1,yq(n)-1,to) * * M (yq(n)-1 ,yq(n),A)
Taking the limit infimum as n ^ in the above inequality, using (9), (14) and the properties of M and * and by lemma 2, we obtain
liMn^+^M (yp(
n), yq(n), to +2A) > 1 * Vmn^+^M(yp(n)-1,yq(n)-1,to) * 1 = 1 - n, using (14)
(16)
Since M is bounded with the range in (0,1], continuous and non-decreasing in the third variable t, it follows from lemma 3 that limn^+^M(yp(n),yq(n),t0) is a continuous function of t from the right.
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Therefore, for A ^ 0 , we obtain
0 Vyp(n)j yq(n
ljm.n^+<x>M(yp(n),yq(n),to) > (1 - n). (17)
Combining inequalities (15) and (17), we get
o
>
CM CM
° Jm> M(yp(n), yq(n), to) = (1 - n). (18)
yy
a: STEP 4 In this step, we show that the sequence {yn} is an M-Cauchy
0 sequence.
-j
<
u Using equations (9) and (14) at t = t0, we have
1
w M (x x ) = j (M (fxp(n),gxp(n),t0))a(M (fxq(n),gxq(n),t0))3 \
H M (xp(n),xq{n)) = { (m (gxp{n),gxq(n),to))(1-a-3) }
<t f (M (yp(n),yp(n)-1,to))a(M (yq(n),yq(n)-1,to))3 1
I (M (yp(n)-1,yq(n)-1,to))(1-a-3) j
(19)
Therefore
g lim + M(xp{n),xq{n)) = (1 - n)(1-a-3). (20)
w And
T
—>
o
M (x x ) = f (M (fxp(n),gxp(n),t0))a (M (fxq(n),gxq(n),t0))3 \
M (xp(n) ,xqin)) = j (M(gxpn ,gxq(n),to)(1-a-3) )
^ > M (fxp(n),fxq(n),to)
= M (yp(n),yq(n),t0).
M (yp(n),yq(n),t0) > M (xp(n), xq(n)) (21)
For n ^ and using equations (20) and (21), we have
(1 - n) > (1 - n)(1-a-3),
implies that (1 - n)(a+3) > 1., which is not possible as(1 - n) < 1. So,{yn} is an M-Cauchy sequence in g(X) which is M-complete. Therefore, there
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{yn} ^ z- (22)
exists z e g(X) such that m
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i. e. £
STEP 5 Now we show that gv = fv. Suppose, on the contrary, that fv = gv(= w). Then exists a positive integer no such that (gv = gxn, for all n > no.
M(fxn,fv,t) > M(xn,v)
= (M(yn, yn—1,t))a(M(fv, z, t))13(M(gxn, z, t))(1—a—3).
For n ^ and using equations (9), (23) and (24) we get
M(z,fv,t) > [M(fv,z,t)]3 i. e. M(fv,z,t)]1—3 > 1, which is not possible if fv = z. Hence fv = u and we have
fv = gv = z. (25)
As the pair of self maps (f, g) is weakly compatible, we have
fz = gz. (26)
STEP 6 Now we show that fz = z. Suppose, on the contrary that fz = z. Then gz = z.
M (z,v) = (M (fz, gz, t))a(M (fv, gv,t))3 (M (gz,gv,t)(1—a—3) = (M(fz,z,t))(1—a—3) using (25, 26)
and
M (fz,z,t) = M (fz,fv,t)
{fxn} ^ z and {gxn+1} ^ z. (23)
As z e g(X) there exists v e X such that
z = gv. (24)
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M (xn ,v) = (M (fxn,gxn,t))a(M (fv,gv,t))3 (M (gxn,gv,t)(1—a—3) |
= (M(yn, yn—1,t))a(M(fv, z, t))3(M(gxn, z, t)(1—a—3).
Now
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> M(z,v), using (25)
= [M (fz,z,t](l-a-l).
i. e. M(fz, z, t)(a+l3) > 1 which is not possible as the left hand side is less than I.Thus, fu = gu = u. □
Taking g = I in Theorem 2, then the sequence {xn} = {x0,fx0, • • •f nx0,, • • •} becomes a Picard sequence for the self map f and we have
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o Corollary 1. Let f be an interpolate fuzzy Meir-Keeler contractive map
° on a M-complete fuzzy metric space (X, M, *).Then the map f has a unique
o fixed point in X.
ft Remark 4. If we take a = 0 and f = 0 in the above corollary, we obtain Theorem 3.1 of Zheng and Wang (Zheng & Wang, 2019).
Example 1. (of Theorem 4.1) Let X = [0,1] . Define a self map f : X ^ X by f (x) = ^, and g(x) = x, the identity map on X. Taking M(x,y) = l+d(x y), then (X, M,.) is a complete stationary fuzzy metric space with the
product t-norm. Define S as follows:
2 S(t) = f if
_ _ if 0 <t< 3,
° S(t) _ \ if n-T < t < n+T,for n > 4.
n(n+2y n — — n+T ' —
k Then S zA.
Taking a = 0 = f. observe that for all values of x,y e X, f (x), f (y) e [0, 3). We show that the quadruple (X,M,5,f) is an interpolative Meir-
Keeler contractive. For this we prove the following condition:
3
forall e e (4,, 1),e — 5(e) < M (x, y) < e =^ M (fx, fy) > e.
|f e e (4,1)=^ ^ < e < n+T, for n > 4, so 5(t) = n^•
Therefore, the inequality e— < M(x,y) < e gives (nr) — nnw) < e — 5(e) < T+dxy < e< n+T. Therefore n < d(x,y) < n
which implies that x,y e [0,1]. Hence
M (fxfy)=T+dkw)=i+k<) > ih=^>e-
tractive and x = 0 is the unique fixed point of the map f.
References
LT)
cn oo I
oo
CP CP
Thus, the quadruple (X, M, 5, f) is an interpolative Meir-Keeler con- ^
o o
JÜ <D
<u
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| ИНТЕРПОЛЯЦИОННОЕ ОБОБЩЕННОЕ СЖАТИЕ
МЕИРА-КЕЛЛЕРА
Собха Джейна, Вук Н. Стоилькович6, Сто]ан Н. Раденовичв
а Университет Шри Вайшнав Видьяпит Вишвавидьялайя в Индауре, математический факультет, кафедра математики, г Индаур, Мадхья-Прадеш, Республика Индия
6 Нови-Садский университет, факультет естественных наук, г Нови-Сад, Республика Сербия, корреспондент
в Белградский университет, Машиностроительный факультет, г Белград, Республика Сербия
РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций, 27.33.00 Интегральные уравнения, 27.39.29 Приближенные методы
функционального анализа
ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Методы: Методы функционального анализа с сокращением Меира-Келлера.
Результаты: Получена уникальная неподвижная точка для отображений в нечетком метрическом пространстве.
Введение/цель: Цель данной статьи заключается в вве- & дении понятия интерполяционного обобщенного условия сжатия Меира-Келлера для отображений в нечетком метрическом пространстве, которое расширяет, объединяет и обобщает многообразие Меира-Келлера, предназначенное только для одного отображения. При примене- ®
го
ф
нии устанавливается единая теорема о совместной неподвижной точке для двух отображений через слабую совместимость. В статье приведен пример, доказывающий достоверность результатов исследования.
"О ф
(Л
"го ф
с ф
ст ф
>
о
СР
Выводы: Получена неподвижная точка собственных отображений.
То
Ключевые слова: нечеткое метрическое пространство, ф общие фиксированные точки, слабая совместимость, ин- ю терполяционное обобщенное сокращение Меира-Келлера.
ИНТЕРПОЛАТИВНА УОПШТЕНА МЕИР-КЕЛЕРОВА КОНТРАКЦША
Собха иаин3, Вук Н. Сто^^кови^6, Сто]ан Н. Раденови^6
а Shri Vaishnav Vidyapeeth Vishwavidyalaya, Катедра математике,
Индоре, Мад]а Прадеш, Република Инди]а 6 Универзитету Новом Саду, Природно-математички факултет,
Нови Сад, Република Ср6и]а, ауторза преписку в Универзитету Београду, Машински факултет, Београд, Република Ср6и]а
ОБЛАСТ: математика
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад
<u
Сажетак:
Увод/цил: Цил овог рада je да се уведе nojaM интерпо-лативног генерализованог Меир-Келеровог контрактив-ног услова за пресликаваъа у фузиметричком простору. Он увепава, обjeди^>уje и гeнeрализуje Меир-Келерову кон-тракц^у и служи за само jeдно пресликаваъе. Користепи га, успоставламо jeдинствeну заjeдничку теорему фиксне тачке за два пресликаваъа кроз слабу компатибилност. Е Рад садржи пример ко\и показуje валидност наших резул-
о тата.
о
^ Методе: Методе функционалне анализе са Меир-
У Келеровом контракц^ом.
о Резултати: Jeдинствeна фиксна тачка за пресликаваъа у
ш фузипростору je доб^ена.
>
Заклучак: Фиксна тачка пресликаваъа самог у себе je до-б^ена.
Клучне речи: фузиметрички простор, заjeдничкe фиксне тачке, слаба компатибилност, интерполативна генера-^ лизована Меир-Келерова контракц^а.
о
S¿ EDITORIAL NOTE: The third author of this article, Stojan N. Radenovic, is a current member of the Editorial Board of the Military Technical Courier. Therefore, the Editorial Team x has ensured that the double blind reviewing process was even more transparent and more h rigorous. The Team made additional effort to maintain the integrity of the review and to ^ minimize any bias by having another associate editor handle the review procedure inde-g pendently of the editor - author in a completely transparent process. The Editorial Team has taken special care that the referee did not recognize the author's identity, thus avoiding the conflict of interest.
КОММЕНТАРИЙ РЕДКОЛЛЕГИИ: Третий автор данной статьи Стоян Н. Раденович является действующим членом редколлегии журнала «Военно-технический вестник». Поэтому редколлегия провела более открытое и более строгое двойное слепое рецензирование. Редколлегия приложила дополнительные усилия для того чтобы сохранить целостность рецензирования и свести к минимуму предвзятость, вследствие чего второй редактор-сотрудник управлял процессом рецензирования независимо от редактора-автора, таким образом процесс рецензирования был абсолютно прозрачным. Редколлегия во избежание конфликта интересов позаботилась о том, чтобы рецензент не узнал кто является автором статьи. РЕДАКЦШСКИ КОМЕНТАР: Трепи ауторовогчланка Стс^ан Н. Раденови^е актуелни члан Уре^ивачког одбора Во]нотехничког гласника. Због тога ]е уредништво спровело транспарентни|и и ригорозни|и двоструко слепи процес рецензи]е. Уложило ]е додатни напор да одржи интегритет рецензи]е и необ]ективност сведе на на]ма^у могупу мерутако штсфдругиуредниксарадникводио процедуру рецензи]е независно
од уредника аутора, при чему je Taj процесбио апсолутнотранспарентан. Уредништво je посебно водило рачуна да рецензент не препозна ко ]е написао рад и да не до^е до конфликта интереса.
а.
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© 2022 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier <u
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