CC BY
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EDN: TOMGFM УДК 517.10
Common Coupled Fixed Point Theorems for a Pair of S^-metric Spaces
Thounaojam Indubala*
Department of Mathematics D. M. college of Arts, Dhanamanjuri University
Imphal, India
Yumnam Rohen*
Department of Mathematics National Institute of Technology Manipur
Imphal, India
Mohammad Saeed Khan*
Department of Mathematics and Applied Mathematics Sefako Makgatho Health Sciences University Ga-Rankuwa, South Africa
Nicola Fabiano§
"Vinca" Institute of Nuclear Sciences National Institute of the Republic of Serbia University of Belgrade Belgrade, Serbia
Received 10.07.2022, received in revised form 15.09.2022, accepted 20.10.2022
Abstract. In this work, we investigate the existence of common coupled fixed point and coupled coincidence points in a setting of two SVmetric spaces. Here we use a pair of w-compatible mappings. Various results are also given in the form of corollaries.
Keywords: common coupled fixed point, coupled coincidence point, £f,-metric spaces, w-compatible mappings.
Citation: T. Indubala, Yu. Rohen, M.S. Khan, N. Fabiano, Common Coupled Fixed Point Theorems for a Pair of -metric Spaces , J. Sib. Fed. Univ. Math. Phys., 2023, 16(1), 121-134. EDN: TOMGFM.
1. Introduction and preliminaries
S. Sedghi, N. Shobe and A. Aliouche [1] introduced S-metric space as a generalisation of metric space. They also claimed that S-metric space is a generalisation of G-metric space. But some researchers commented that the claim is not true. Further it is claimed that the class of S-metric and the class of G-metric are all distinct. For detail results in this claim and more about S-metric space one can see research papers in [2-4] and references there in.
Bakhtin [5] introduced the concept of 5-metric space. The concept of Bakhtin is extensively used by S.Czerwick [6,7]. Nizar and Nabil [8] introduced the concept of S^-metric space by using the concept of both S-metric and 5-metric. Y. Rohen, T.Dosenovic and S.Radenovic [9]
* indubalathou1@gmail.com
tymnehor2008@yahoo.com https://orcid.org/0000-0002-1859-4332
tdrsaeed9@gmail.com https://orcid.org/0000-0003-0216-241X
§ nicola.fabiano@gmail.com https://orcid.org/0000-0003-1645-2071 © Siberian Federal University. All rights reserved
also gave a more general definition of Sb-metric space. For more results on Sb-metric space one can see research papers in [10-13].
In this paper we prove a common coupled fixed point and coupled coincidence point theorem for a pair of w-compatible mappings in the setting of two Sb-metric spaces in the line of the results obtained by Feng Gu [16].
Following definitions and properties will be needed in order to start the main result.
Definition 1 ([8]). Let W be a nonempty set and let b ^ 1 be a given number. A function S : W3 ^ [0, x) is said to be Sb-metric if and only if for all 9,$,X,^ G X, the following conditions hold:
(i) S(9, $,X)=0 if and only if 9 = $ = X,
(ii) S(9, $, X) < b[S(9,9, n) + S($, $, n) + S(X, X, n)],
(iii) S(9,9, $) = S($, $, 9) for all 9,$ G W.
The pair (W, S) is called a symmetric Sb-metric space. If the pair (W, S) does not fulfil (iii) then it is called an Sb-metric space.
Example 1 ([8]). Let W be a nonempty set and card (W) > 5. Suppose W = Wi U W2 a partition of W such that card (Wi) ^ 4. Let b ^ 1. Then
{0 if 9 = $ = X = 0 3b if (9, $, X) G W3 . 1 if (9,$,X) G W3
Definition 2 ([14,15]). Let W be a non-empty set, P : W x W ^ W and q : W ^ W are two mappings then
(i) an element (9, $) G W x W satisfying P(9, $) = 9 and P($, 9) = $ is called a coupled fixed
point of P,
(ii) an element (9, $) G W x W satisfying P(9, $) = q9, P($, 9) = q$ is called a couple coinci-
dence point of P and q. The point (q9, q$) is called a coupled point of coincidence,
(iii) an element (9, $) G W x W satisfying P(9, $) = q(9) = 9, P($, 9) = q$ = $ is called a common coupled fixed point of P and q,
(iv) the pair of mappings P and q is said to be w-compatible if P(9,$) = q9 and P($,9) = q$ implies qP(9, $) = P(q9, q$).
2. Main results
We prove the following theorems.
Theorem 2.1. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(9, 9, $) ^ Si(9, 9, $) for all 9,$ G W and b > 1 is a real number. Let P : W x W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si(P(9,$),P(9, $), P(Z,r,) + S2(P($, 9),P($, 9),P(v,0) <
< ki [S2(q9, q9, qO + S2(q$, q$, qV)]] + k2 [S2(q9, q9, P(9, $)) + S2(q$, q$, P($, 9))] +
+k3[S2(q9, q9, P(£, V)) + S2(q$, q$, P(V, £))],
where (9, $), (£,r/) G W x W and ki, k2,k3 in [0,1] such that 0 < ki + k2 + k3 < —-
b2
(ii) P(W X W) C q(W)
(iii) q(W) is Si complete
then P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Proof. Let (6o,fao) £ W x W. By (ii) there exists (6i,fai) £ W such that q6i = P(6o,fao), qfai = P(fao, 0o). Similarly, (62, fa) £ W such that q02 = P(61, fai), qfo = P(fai, Si). Continuing in this way sequences {6n} and {fan} can be constructed as
q6n+i = P(6n, fan), qfan+i = P(fan, 6n) for all n > 0.
From (i) we have,
Si(q6n+i, qSn+i, q6n+2) + S2(qfan+i, qfan+i, qfan+2) = = Si(P ((6n, fan), P (6n, fan),P (6n+i,fan+i)) + S2 (P ((fan, 6n), P (fan, 6n), P (fan+i,6n+i)) <
< ki[S2(q6n,q6n, q6n+i) + S2(qfan, qfan, qfan+i)] +
+ k2 [S2(q6n, q6n, P(6n, fan)) + S2(qfan, qfan, P(fan, 6n))] + +k3[S2(q6n+i, q6n+i ,P (6n+i, fan+i)) + S2 (qfan+i, qfan+i, P (fan+i, 6n+i))] = = ki[S2(q6n,q6n, q6n+i) + S2(qfan, qfan, qfan+i)] + + k2[S2(q6n,q6n, q6n+i) + S2(qfan, qfan, qfan+i)] + + k3[S2(q6n+i, q6n+i, q6n+2) + S2(qfan+i, qfan+i, qfan+2)] <
< ki[Si(q6n,qSn, q6n+i) + Si(qfan, qfan, qfan+i)] + + k2[Si(q6n,q6n, q6n+i) + Si(qfan, qfan, qfan+i)] +
+ k3[Si(q6n+i, q6n+i, q6n+2) + Si(qfan+i, qfan+i, qfan+2)] . (1)
It follows from (1) that
Si(q6n+i, q6n+i, q6n+2) + Si (qfan+i, qfan+i, qfan+2) <
< k + k [Si(q6n, q6n, qSn+i) + Si(qfan, qfan, qfan+i)} = 1 - k3
= k[Si(q6n,q6n, q6n+i) + S\(qfan, qfan, qfan+i)], (2)
where k = k + k , by the condition 0 < ki + k2 + k3 < 1, then we have 0 < k < -1. By taking
1 — k3 b2 b2
Sn = S1(qdn,qdn ,qdn+i) + Si(qfan,qfan,qfan+i),
thus,
Sn+i < kSn < k2Sn-i < ... < kn+1So. (3)
Next, we show that {q6n} and {qfan} are Cauchy sequences in q(W). For this, we consider Si(6n, 6n, 6n+p) into two cases. Firstly, considering p = 2l +1
Si(q6n,q6n,q6n+p) = Si(q6n,q6n,q6n+2l+i) <
< 2bSi(q6n, q6n, q6n+i) + b2Si(q6n+i, q6n+i, q6n+2l+i) <
< 2bS\(q6n ,qen,qOn+1) + 2b3S1(qen+ i, q6n+i,q0n+2) + +b4Si(q0n+2,qOn+2,qdn+2l+l) <
< ...
< 2bSi(q6n, qdn, qOn+l) + 2b3Si(qOn+i, qOn+i, qOn+2) + + 2b5Si(qdn+2, qOn+2, qOn+3) + • • • +
+ 2b2(2l-i')+iSi(q0n+2i-i, q0n+2i-i,q0n+2i) + +b2{2l-i)+2Si(q9n+2l, qdn+2l, qdn+2l+i) <
< 2{b(Si(q9n, q9n, q6n+i)) + b3S\(qOn+i, qOn+i, qOn+2) + + bZSi(qdn+2,qdn+2, qdn+3) + ••• +
+ b2(2l-i)+iSi(qBn+2l-i ,qe n+2l-i, q0n+2l) + + b2(2l-i)+3S\(qOn+2l,qOn+2l,qOn+2l+i)} . (4)
We can similarly prove the following result
Si{q&n, q&n, q&n+p) = Si(q$n,q&n,q&n+2l+i) <
< 2{bSi(q^n, q^n, q^n+i) + b3Si(q^n+i, q^n+i, q^n+2) + + b5Si(q^n+2, q^n+2, q^n+3) +-----+
+ b2(2l-i)+iSi(q^n+2l-i, qj>n+2l-i, q^n+2l) + + b2(2l-i)+3Si(q^n+2l ,qtn+2l, q^n+2l+i)} . (5)
Adding (4) and (5), we have
Si(qdn, qdn,qdn+p) + Si(q^n, q^n,q^n+p) = = Si(q0n,qdn,qdn+2l+i) + Si(q^n, q^n, q^n+2l+i) <
< 2[b{Si(qdn,q0n ,qOn+i) + Si(q^n ,q^n,q^n+i)} +
+ b3{Si(qdn+ i,qOn + i, qSn+2) + Si(q^n+i, q$n+i,q$n+2)} +
+ b5{Si(q0n+2 ,q0n+2, qdn+3) + Si (qftn+2, q^n+2,q^n+3)} + + ... +
+b2(2l-i)+3{Si(qdn+2l,q0n+2l, qdn+2l+i) + Si{q^n+2l,q^n+2l, q^n+2l+i)}] = = 2[bSn + b3Sn+i + b5Sn+2 + ••• + b2(2l-i)+3Sn+2l] =
= 2[bknSo + b3kn+iSo + b5kn+2So + ••• + b2(2l-i)+3kn+2lSo] =
= 2bkn50{\ + b2k + b4k2 + ••• + b4lk2} = = 2bkn. (6)
Secondly, considering p = 2l
Si(q9n,qdn,qdn+p) = Si(qOn,qOn,qOn+2l) <
< 2bSi(q6n, qdn, qdn+i)) + b2Si(q0n+i,qdn+i, qOn+2l) <
< 2b(Si(q0n, qdn, qdn+i)) + 2b3Si(qOn+i,q0n+i, qOn+2) + + b4Si (qdn+2, qdn+2, qdn+2l) <
< ...
< 2bSi(q0n, qdn, qdn+i) + 2b3Si(qOn+i, qOn+i, qOn+2) + + 2b5Si(q0n+2, qOn+2, qOn+3) + ••• +
+2 b2(2l-2)+1 S1(qen+2i-2,qen+2l-2,qen+2l-1) + + b2(2l-2)+2Si(qdn+2l-U q6n+2l-1,qen+2l) <
< 2{bSi(q0n, qdn, qdn+l) + b3Si(qOn+i, qOn+i, qdn+2) + + bZSi(qdn+2, q0n+2,q0n+3) + ••• +
n+2l-2, q0n+2l-i) +
n+2l-i,qOn+2l)} . (7)
By similar arguments as above,
Si(qfan,qfan,qfan+p) = Si(qfan,qfan,qfan+2l) <
< 2{bSi(qfan,qfan,qfan+i) + b3Si(qfan+i, qfan+i, qfan+2) + + b5Si( qfan+2, qfan+2, qfan+3) +-----+
+ b2(2l-2) + iSi(qfan+2l-2, qfan+2l-2, qfan+2l-i) + + b2(2l-2)+3Si(qfan+2l-i,qfan+2l-i,qfan+2l)} . (8)
Adding (7) and (8) we get
Si(qdn, qdn, qOn+p) + Si(qfan, qfan, qfan+p) = = Si(q0n,qdn,qdn+2l) + Si(qfan, qfan, qfan+2l) < < 2[b{Si(q0n, q9n, q9n+i) + Si(qfan, qfan, qfan+i)} +
+ b3{Si (qOn+i, qOn+i, qdn+2) + Si (qfan+i, qfan+i, qfan+2)} + + b5{Si (qOn+2, qOn+2, qOn+3) + Si (qfan+2, qfan+2, qfan+3)} + +... +
+ b2(2l-2)+3{Si(qOn+2l-i,qOn+2l-i, qdn+2l) + Si (qfan+2—i, qfan+2—i, qfan+2l)}] = = 2[bSn + b3Sn+i + b5Sn+2 + ••• + b2(2l-2)+3Sn+2l-i] =
n+
= 2[bknSo + b3kn+iSo + b5kn+2So + ••• + b2(2l-2)+3kn+2l-iSo] = = 2bknS0{l + b2k + b4k2 + ••• + b4l-2k2l-i} =
= 2bknT-~b?kS0 ' (9)
Since k e [0, i), so kn ^ 0 when n tends to infinity. From (6) and (9), we have lim [Si(qOn, qdn, qOn+p) + S\(qfan, qfan, qfan+p)] =0
which implies that {qOn} and {qfan} are Cauchy sequences in q(W). Since q(W) is complete, then there exists O,fa e W such that
limn^TO qdn = qO and limn^TO qfan = qfa.
It follows from (i) and (3) that
Si(qOn+i, qOn+i, P (O, fa)) + Si (qfan+i, qfan+i,P (fa, O)) = = Si(P (On, fan),P (On, fan),P (O,fa))+ Si (P (fan ,On),P (fan ,On),P (fa,O)) < < ki[S2(qOn,qOn,qO) + S2 (qfan, qfan, qfa)] +
+ k2 [S2(qOn, qOn, P(On, 4>n)) + S2^n, q$n, P($n, On))] + + k3[S2(qO, qO, P(O, $)) + S2(qq$, P($, d))] = = ki[S2(qOn, qOn, qO) + S2(q$n, q$n, q$)] +
+ k2[S2(qOn,qOn,qOn+i) + S2 (q$n,q$n,q$n+i)] + + k3[S2(qO, qO, P(O, $)) + S2(q& q$, P($, O))] <
< ki[Si(qOn, qOn, qO) + Si(q$n, q$n, q$)] +
+ k2[Si(qOn,qOn,qOn+i) + Siq$n,q$n,q$n+i)] + + k3[Si(qO, qO, P(O, $)) + Si(q$, q$, P($, O))] = = ki[Si(qOn, qOn, qO) + Si(q$n, q$n, q$)] +
+ k2Sn + k3[Si(qO, qO, P(O, $)) + Si(q$, q$, P($, O))] <
< ki[Si(qOn, qOn, qO) + Si(q$n, q$n, q$)] +
+ k2knSo + k3[Si(qO, qO, P(O, $)) + S\(q$, q$, P($, O))] . (10)
From (10) and (3), we have
Si (qO, qO, P(O, $)) + Si (q& q$, P($, O)) <
< 2b{Si(qO, qO, qOn+i) + b2 Si(qOn+i, qOn+i,P(O, $))} +
+ 2b{Si(q$, q$, q$n+i) + b2Si (q$n+i, q$n+i, P($, O))} = = 2b{Si(qO,qO, qOn+i) + Si(q$, q$, q$n+i)} +
+ b2{Si(P(On, $n), P(On, $n), P(O, $)) + Si(P($n, On), P($n, On), P($, O))} = = 2b{Si(qO,qO, qOn+i) + Si(q$, q$, q$n+i)} + + b2[ki{S2(qOn,qOn,qO) + S2(q$n, q$n, q$)} + + k2{S2(qOn, qOn, P (On, )) + S2(q$n, q$n, P ($n, On))} + + k3{S2(qO, qO, P(O, $)) + S2(q$, q$, P($, O))}] <
< 2b{Si(qO, qO, qOn+i) + Si(q$, q$, q$n+i)} + + b2ki{Si(qOn, qOn, qO) + Si(q$n, q$n, q$)} +
+ b2k2{Si(qOn, qOn, P(On, 4>n)) + Si(q$n, q$n, P($n, On))} + + b2k3{Si(qO, qO, P(O, $)) + Si(q$, q$, P($, O))} = = 2b{Si(qO,qO, qOn+i) + Si(q$, q$, q$n+i)} + + b2ki{Si(qOn, qOn, qO) + Si(q$n,q$n,q$)} + + b2k2{Si(qOn, qOn, qOn+i) + Si(q$n, q$n, q$n+i)} +
+ b2k3{Si(qO, qO, P(O, $)) + Si(q$, q$, P($, O))} . (11)
Taking limit as n ^ to we have
(1 - b2k3){Si(qO,qO,P(O,$)) + Si(q$,q$,P($,O))} < 2b X 0 + b2ki X 0 + b2k2 X 0 ^ Si(qO, qO, P(O, $)) + Si(q$, q$, P($, O)) = 0
[because 1 - b2k3 > 0, b2(ki + k2) > 0] . (12)
So, qO = P(O, $) and = P($, O) which shows that (O, $) is the coupled coincidence point of P and q.
In order to prove the uniqueness of coupled coincidence point, let (O* ,$*) be the second coupled coincidence point of P and q.
From (i),
Si(qO,qO,qO*) + Si(qfa, qfa, qfa*) = = Si(P(O, fa), P(O, fa), P(O*, fa*)) + Si(P(fa, O), P(fa, O), P(fa*,O*)) <
< ki{S2(qO,qO,qO*) + S^qfa, qfa, qfa*)} +
+ k2{S2(qO, qO, P(O, fa)) + S^qfa, qfa, P(fa, O))} + + k3{S2(qO*,qO*,P(O*,fa*)) + S2(qfa*, qfa*, P(fa*, O*))} = = ki{S2(qO,qO,qO*) + S2 (qfa, qfa, qfa*)} <
< ki{Si(qO,qO,qO*) + Si(qfa, qfa, qfa*)} . (13)
But 0 < ki + k2 + k3 < -1 ^ ki < 1. We have
b2
Si (qO, qO, qO*) + Si(qfa, qfa, qfa*) =0.
Thus, qO = qO* and qfa = qfa*, which shows that coupled point of coincidence of P and q is unique.
Now, we need to show qO = qfa. By (1), we have
Si(qO, qO, qfa) + Si(qfa, qfa, qO) = = Si(P (O,fa),P (O,fa),P (fa,O)) + Si (P (fa, O), P (fa, O), P (O, fa)) <
< ki{S2(qO,qO,qfa) + S2(qfa, qfa, qO)} +
+ k2{S2(qO, qO, P(O, fa)) + S^qfa, qfa, P(fa, O))} + + k3{S2(qfa, qfa, P(fa, O)) + S2(qO, qO, P(O, fa))} = = ki{S2(qO,qO,qfa) + S^qfa, qfa, qO)} <
< ki{Si(qO,qO,qfa) + Si(qfa, qfa, qO)} .
As 0 < ki < ki + k2 + k3 < -1 < 1 and (10), we deduce
b2
Si(qO, qO, qfa) + Si(qfa, qfa, qO) = 0.
Hence, qO = qfa.
By w-compatibility of P and q, we get q(P(O,fa)) = P(qO,qfa). Taking a = qO, we get
a = qO = P(O, fa) = qfa = P(fa, O),
therefore qa = qqO = q(P(O, fa)) = P(qO, qfa) = P(a, a).
Hence, (qa, qa) is a coupled point of coincidence of q and P. Due to uniqueness, qa = qO, therefore P(a, a) = qa = a,
therefore (a, a) is the unique common coupled fixed point of q and P. □
Corollary 2.2. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(O, O, fa) ^ Si(O, O, fa) for all O,fa e W and b > 1 is a real number. Let P : W x W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si(P(O, fa), P(O, fa), P(£,n)) <
< ki[S2(qO, qO, qO + S^qfa, qfa, qn)] + k2^(qO, qO, P (O, fa)) + S2 (qfa, qfa,P (fa, O))] +
+ k3 [S2(qi, qi, P (i, n)) + S2 (qn, qn, P (n, 0)],
where (O, fa), (£,n) e W x W and ki,k2, k3 in [0,1) such that 0 < 2(ki + k2 + k3) < 1
b2
(ii) P(W X W) C q(W)
(iii) q(W) is Si complete.
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Proof. It follows from (11) that
Si(P(O,$),P(O,$),P(Z,n)) <
< ki [S2(qO, qO, qZ) + S2 (q$, q$, qrj)] + k2 S2 (qO, qO, P(O, $)) + S2(q$, q$, P($, O))] +
+ k3 [S2(q£, qZ, P(Z, n)) + S2 (qn, qn, P(n, 0)] (14)
and
Si(P($,O),P($,O),P(n,0) <
< ki[S2(q$, q$, qn) + S2(qO, qO, q£)] + k2S (q$, q$, P ($, O)) + S2 (qO, qO, P (O, $))] +
+ k3[S2(qn,qn,P(n,Z)) + S2(qS,qS,P(Z,n))] ■ (15)
Adding (14) and (15) we have
Si(P(O,$),p(O,$),p(z,n)) + Si(P($,O),p($,O),p(n,Z)) < < 2ki[S2(qO, qO, qZ) + S2(q$, q$, qn)] + 2k2S(qO, qO, P(O, $)) + S2(q$, q$, P($, O))] +
+ 2k3 [S2(qZ, qZ, P(Z, n)) + S2 (qn, qn, P(n, Z))] -
By the Theorem 2.1, we get the conclusion. □
Taking Si(O, O, $) = S2(O, O, $) = S(O, O, $) for all O,$ e W, in Theorem 2.1, we have
Corollary 2.3. Let (W, S) be a complete Sb metric space and b ^ 1 is a real number. Let P : W X W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) S(P(O, $), p(O, $), p(Z, n)) + S(P($, O), p($, O), p(n, Z)) <
< ki [S(qO, qO, qZ) + S(q& q$, qn)] + k2 [S(qO, qO, P(O, $)) + S(q$, q$, P($, O))] +
+ k3[S(qZ, qZ, P(Z, n)) + S(qn, qn, P(n, Z))],
where (O, $), (Z,n) e W X W and ki,k2,k3 in [0,1) such that 0 < 2(ki + k2 + k3) <
b2
(ii) P(W X W) C q(W).
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Corollary 2.4. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(O, O, $) ^ Si(O, O, $) for all O,$ e W and b > 1 is a real number. Let P : W X W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si(P(O,$),P(O,$),P(Z,n)) <
< aiS2(qO, qO, qZ) + a2S2(q$, q$, qn) + a3S2(qO, qO, P(O, $)) + aiS2(q$, q$, P($, O)) + + a5S2(qZ, qZ, p (Z, n)) + a6S2(qn, qn, P (n, Z))
where (O, $), (Z,n) e W xW and aAi = 1, 2,..., 6) in [0,1) such that 0 ^ ai+a2+a3+• • < 72
b2
and 0 ^ b(a5 + a6) < 1
(ii) P(W x W) C q(W)
(iii) q(W) is Si complete.
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Proof. Since (O, fa), (i, n) e W x W, we have from (15) that
Si(P (O, fa), P (O, fa), P (i,n)) <
< aiS2(qO,qO,qi) + a2S2(qfa,qfa,qn) + a3S2(qO,qO,P(O,fa)) + a4S2(qfa, qfa, P(fa, O)) +
+ aS (qi, qi, P(£, n)) + a6S2 (qn, qn, P(n, i)) (16)
and
Si(P(fa, O), P(fa, O), P(n,i)) <
< aiS2(qfa, qfa, qn) + a2S2(qO, qO, qi) + a3S2(qfa, qfa, P (fa, O)) + a4 S2 (qO, qO, P (O, fa)) +
+ a5S2(qn,qn,P(n,i)) + aeS^qiP(i,n)) ■ (17)
Adding (16) and (17), we have
Si(P(O, fa), P(O, fa), P(i, n)) + Si(P(fa, O), P(fa, O), P(n, i)) < < (ai + a2){S2(qO, qO, qi) + S2(qfa, qfa, qn)} +
+ (a3 + a4){S2(qO, qO, P(O, fa)) + S2(qfa, qfa, P(fa, O))} + + (as + ae){S2(qi, qi, P(i, n)) + S2(qn, qn, P(n, i))} . By Theorem 2.1, required result follows. □
Remark 1. Taking Si(O, O, fa) = S2(O, O, fa) = S(O, O, fa) for all O,fa e W, where S is an Sb-metric on W, in Corollary 2.3 we can have another result.
We have following corollary from Theorem 2.1.
Corollary 2.5. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(O, O, fa) ^ Si(O, O, fa) for all O,fa e W and b > 1 is a real number. Let P : W x W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si(P (O, fa), P (O, fa), P (i,n)) + Si (P (fa, O), P (fa, O), P (n,i)) <
< k{S2(qO,qO, qi) + S2(qfa, qfa, qn)}
where (O, fa), (i,n) e W x W and k e [0, i)
(ii) P(W x W) c q(W)
(iii) q(W) is Si complete.
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Corollary 2.6. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(O, O, fa) ^ Si(O, O, fa) for all O,fa e W and b > 1 is a real number. Let P : W x W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si(P (O, fa), P (O, fa), P (i,n)) + Si (P (fa, O), P (fa, O), P (n,i)) <
< k{S2(qi, qi,P(i, n)) + S2(qn, qn,P(n,i))},
where (O, fa), (i,n) e W x W and k e [0, i)
(ii) P(W x W) C q(W)
(iii) q(W) is Si complete.
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Corollary 2.7. Let Si, S2 be two Sb metric spaces in a non-empty set W satisfying S2(O, O, fa) ^ Si(O, O, fa) for all O,fa e W and b > 1 is a real number. Let P : W x W ^ W and q : W ^ W be two mappings satisfying the following conditions
(i) Si (P (O, fa), P (O, fa), P (i,n)) + Si(P (fa, O),P (fa, O),P (n,i)) < < k{S2 (qi, qi, P(i, n)) + S2 (qn, qn, P(n, i))}
where (O, fa), (i,n) e W x W and k e [0, i)
(ii) P(W x W) C q(W)
(iii) q(W) is Si complete.
Then, P and q have a coupled coincidence point. Further, if P and q are w-compatible then P and q have a unique common coupled fixed point.
Remark 2. (i) Replacing q by identity mapping in the above results, we have corresponding coupled fixed point results.
(ii) Taking b = 1 in the above results, we can have corresponding results in S metric space.
3. Applications
Example 2. Let W = R, define S : W x W x W ^ R+ as
S(O,fa,X) = \O + fa - 2X\k, O,fa,X e W,
where k ^ 1. Here, (W,S) is an Sb-metric space.
Example 3. Let W = R and Si, S2 are two Sb-metrics in W such that
Let us define P : W x W ^ W and q : W ^ W by
Ofa
P(O, fa) = , qO = 2O for all O, fa e W.
We have P(W x W) C q(W), q(W) is Si-complete. Also, P and q are w-compatible. Now,
Si(P(O,fa),P(O, fa), P(i,n)) = {P(O, fa) + P(O, fa) - 2P(i, n)}2 =
= 4{P(O, fa) - P(i,n)}2 =
2
A\(29 - 2£)2 + (2^ - 2n)2 )
9 { 2 2
4 f(qe - qO2 + (q4 - qn)2 )
9 { 2 2
1 f (2qe - 2qj)2 (2q4 - 2qn)2 )1 9 f 2 + 2 )J 1 f (qe + qe - 2qQ2 (q4 + q4 -9 I 2 ' 2
1 9
9 { S2(qe,qe,qO + S2 (q4, q& qn^ •
Similarly,
s1(p(4,6),p(4,0),p(n,0) < 9{S2(q4,q4,qn) + S2(q°,^
Further, we have
Si(P(0,4),P(0,4),P(€,n)) + Si(P(4,0),P(4,0),P(n,0) < 2{S2(q0,q0,+ S2(q4,q4,qn)}■
Then, by corollary 2.6, (0,0) is the unique common coupled fixed point of P and q.
Now, let W = C[c, d] is the set of all continuous functions. Let S1(0,4,X) = max \0(p) + 4(p) - 2X(^)\k
S2(0, 4, X) = m*x*M\0(») + 4(- 2X(^)k for all 04 G W, (k > 1). Also, let b = 3k-1. Consider
0(r) = K(r)+ f G(r,ri){f(ri,0(ri)) + q(ri,4(ri))}dri
J c
4(r) = K(r)+ f G(r,ri){f(ri,4(ri)) + q(ri,0(ri))}dri. (18)
c
Next, we will analyse (18) under the following condition
(i) f,q : [c,d] x R ^ R be two continuous functions.
(ii) K : [c,d] ^ R is a continuous function.
(iii) G : [c,d] x R ^ [0, x) is a continuous function.
(iv) There exists u,v > 0 such that for all 0,4 G R,
I
\f(v,e(rf) - f (v,4(v))\ < u\e - 4\, \q(^e(v)) - q(^,4(p))\ < v\e - 4\.
( d V 1
(v) maxreM] ( J \G(r,v)\dpj < 2k+iLk, with L = xmAx[u,v}.
Theorem 1. Under the condition (i)-(v), the integral equation (18) has a unique common solution on [c,d].
Proof. Let P : W x W ^ W and q : W ^ W
P(0, 4>)(r) = K(r) + fd G(r, p)\f (p, 0(p)) + q(p, 4>(p))\dp,
J c
q0 = 20 V0 € W, p € [c, d], € W. Si(P(0, t),P(Ç,V),P(X,w)) =
= max \P(0,fa)(r) + P(Ç,v)(r) - 2P(X,w)(r)\k for all 0,j>,Ç,r),w,X € W,
rE[c,d]
S2(P (0,t),P (t,n),P (X,w)) =
maxre[c,d] \P(0, #)(r) + P(Ç, n)(r) - 2P(X, w)(r)\k
2
for all 0, € W.
Also, P (W x W ) c q(W ), q(W ) is Si-complete, and P and q are w-compatible. Now,
\P(0, fa)(r) + P(0, fa)(r) - 2P(Ç, n)(r)\k =
= 2k\P(0,fa)(r) - P(t,v)(r)\k =
! d
G(r,p){f (p,0(p)) - f (p,C(p))}dp + / G(r,p){q(p,fa(p)) - q(p,n(p))}dp
c
<■d k
<
< 2k-i x 2k
+ 2k x 2k-i d
G(r,p){f(p,0(p)) - f (p,£(p))}dp + / G(r,p){q(p,fa(p)) - q(p,n(p))}dp ^
< 2
+
G(r, p){f (P, 0(p)) - f (P, î(p))}dp
G(r, p){q(p, fa(p)) - q(p, n(p))}dp
+
x 2k-i <
< 2k-i x 2k
pk( max \0(p) - C(p)\)k + qk( max \4>(p) - n(ß)\Y
fj.£[c,d\ fiE[c,d]
G(r, p) dp
<
< 2k-i x 2k x L
max \0(p) - £(p)\k + max \fa(p) - n(p)\k
1
2k+iLk
<
<
2k
2k+2 1
2k+2 1
2k+i 1
2k+i
max \20(p) - 2£(p)\k + max \2fap) - 2V(p)\h
fj.£[c,d] fiE[c,d]
max \20(p) + 20(p) - 2(2£(p))\k + max \2fa(p) + 2fa(p) - 2(2n(p))\k
max^çd] \20(p) + 20(p) - 2(2Ç(p))\k + max^d] W(p) + 2fa(p) - 2(2V(p))\
k
[S2(q0, q0, qO + S2(qfa, qn)}.
Thus,
Si(P (0, 4>),p (0, 4>),p (£,n))
<
max \P(0,fa)(r) + P(0,4>)(r) - 2P(£,n)(r)\k <
re[c,d] 1
[S2(q0, q0, qO + S2(qfa, qn)] .
2k+i
(19)
k
k
d
k
k
Similarly,
Si(P ($,O),P ($,O),P (n,Z)) < T^Tl [S2(q$,q$,qn) + S2(qO, qO, qZ)] (20)
It follows from (29) and (30) that
Si(P (O, $), P (O, $), P (Z, n)) + Si(P ($, O),P ($, O),P (n, Z)) < 2k [S2 (qO, qO, qZ) + S2(q$, q$, qn)].
Consequently, all the conditions of Corollary 2.6 are satisfied. It follows from the result of Corollary 2.6 that P and q have a unique common coupled fixed point and hence integral equation in equation (18) has a unique solution. □
The first author (T.I.) would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
References
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Общие связанные теоремы о неподвижной точке для пары Sb-метрических пространств
Тунаоджам Индубала
Университет Дханаманджури Импхал, Индия
Юмнам Роэни
Национальный технологический институт Манипура
Импхал, Индия
Мохаммад Саид Ханз
Университет медицинских наук Сефако Макгато Га-Ранкува, Южная Африка
Никола Фабиано
Национальный институт Республики Сербия Белградский университет Белград, Сербия
Аннотация. В данной работе мы исследуем существование общих связанных неподвижных точек и связанных точек совпадения в сеттинге двух £{,-метрических пространств. Здесь мы используем пару ад-совместимых отображений. Различные результаты приводятся также в виде следствий.
Ключевые слова: общая связанная неподвижная точка, связанная точка совпадения, Яъ-метрические пространства, ад-совместимые отображения.
