ĆIRIĆ TYPE RESULTS IN QUASI-METRIC SPACES AND 𝐺-METRIC SPACES USING SIMULATION FUNCTION Текст научной статьи по специальности «Математика»

72

Probl. Anal. Issues Anal. Vol. 11 (29), No 2, 2022, pp. 72-90

DOI: 10.15393/j3.art.2022.11230

UDC 517.521.2, 517.988

Sejal V. Puvar, R. G. Vyas

CIRIC TYPE RESULTS IN QUASI-METRIC SPACES AND G-METRIC SPACES USING SIMULATION FUNCTION

Abstract. In this paper, we establish existence of some common fixed-point theorems for admissible mappings via a simulation function along with C-class functions in quasi-metric spaces. As a consequence, these results are extended to G-metric spaces and metric spaces.

Key words: quasi-metric space, G-metric space, simulation function, common fixed point, admissible mappings

2020 Mathematical Subject Classification: 47H10, 54H25

1. Introduction. Jleli and Samet [4], Samet et al. [13] have shown that a G-metric space has a quasi-metric type structure. Then many results for such spaces follow from results for quasi-metric spaces.

Khojasteh [7] introduced the simulation function and proved fixed-point theorems in metric spaces. Later, Roldan et al. [10] modified the definition of the simulation function by removing the symmetry condition, and introduced a (Z, ^-contraction. Roldan et al. [12] investigated the existence and uniqueness of coincidence points via simulation functions in the setting of quasi-metric spaces and deduced corresponding results in the framework of G-metric spaces.

Radenovic and Chandok [9] proved common fixed-point theorems for a (Zq , g)-contraction and a generalized (Zg, g)-contraction. They also introduced a (Zq , g)-quasi-contraction of Ciric-Das-Naik type and posed an open problem regarding common fixed point theorems for a (Zq , g)-quasi-contraction of Ciric-Das-Naik type in metric spaces.

In this paper, we use the a-admissible mapping, introduce a (Z(a,Q), ^)-quasi-contraction of Ciric type, prove common fixed-point theorems in quasi-metric spaces, and observe its consequences to G-metric

© Petrozavodsk State University, 2022

spaces.

2. Preliminaries.

Definition 1. [4] Let X be a non-empty set and let d : X x X ^ [0, to) be a function, such that the following conditions hold:

(i) d(x, y) = 0 if and only if x = y;

(ii) d(x, y) ^ d(x, z) + d(z, y), for any points x,y,z E X.

Then, d is called a quasi-metric on X and the pair (X, d) is called a quasi-metric space.

Definition 2. Let T,g : X ^ X be self maps on X. A point x E X is called a:

• fixed point of the operator T, if Tx = x; we denote x E Fix(T);

• coincidence point of T and g, if Tx = gx; we denote x E C(T,g);

• common fixed point of T and g, if Tx = gx = x.

Definition 3. Let (X, d) be a quasi-metric space, [xn] be a sequence in X and x E X. The sequence {xn} converges to x if and only if

lim d(xn,x) = lim d(x,xn) = 0. (1)

The limit of a sequence in a quasi-metric space is unique.

Definition 4. Let (X, d) be a quasi-metric space and {xn} be a sequence in X. We say that {xn} is

• left-Cauchy if and only if for every £ > 0, there exists a positive integer N = N(e) such that d(xn, xm) < £ for all n ^ m > N.

• right-Cauchy if and only if for every £ > 0, there exists a positive integer N = N(e) such that d(xn, xm) < £ for all m ^ n > N.

• Cauchy if and only if for every £ > 0, there exists a positive integer N = N (e) such that d(xn, xm) < £ for all m,n > N.

A sequence {xn} in a quasi-metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy.

Definition 5. Let (X, d) be a quasi-metric space. We say that (X, d) is complete if and only if each Cauchy sequence in X is convergent.

Lemma 1. [5] Let {xn} be a sequence in a quasi-metric space (X,d), such that

(i) d(xn+i,xn+2) ^ \d(xn,xn+i),n ^ 0,

(ii) d(xn+2,xn+i) ^ Xd(xn+i,xn),n ^ 0,

for some X E (0,1). Then {xn} is a Cauchy sequence in X.

Definition 6. [11] A subset E of a metric space (X,d) is said to be precomplete if every Cauchy sequence {un} in E converges to a point of X.

Similarly, precompleteness is defined for quasi-metric space.

Lemma 2. [12] Let (X, d) be a quasi-metric space and T: X ^ X be a given mapping. Suppose that T is continuous at u E X. Then, for each sequence {xn} in X, such that xn ^ u, we have Txn ^ Tu; that is,

lim d(Txn,Tu) = lim d(Tu,Txn) = 0.

Every quasi-metric induces a metric, that is, if (X, d) is a quasi-metric space, then the function 5: X x X ^ [0, ro), defined by

5(x, y) = max{d(x, y), d(y, x)}

is a metric on X (see [4]).

The following result is an immediate consequence of the above definition:

Theorem 1. [4] Let (X, d) be a quasi-metric space, 5: X x X ^ [0, ro) be the function defined by 5(x, y) = max{d(x, y), d(y, x)}. Then

(1) (X, #) is a metric space;

(2) {xn} C X is convergent to x in (X, d) if and only if {xn} is convergent to x in (X, 5);

(33) {xn} C X is Cauchy in (X, d) if and only if {xn} is Cauchy in (X, 5);

(4) (X, d) is complete if and only if (X, #) is complete.

Definition 7. [14] Let T,g: X ^ X and a: X x X ^ [0, ro) be mappings. We say that T is a-admissible for g if

a(gx,gy) ^ 1 =^ a(Tx,Ty) ^ 1 for all x,y E X.

For g = ix (identity mapping on X), T is an a-admissible mapping.

Definition 8. Let T,g: X ^ X and a: X x X ^ [0, to) be mappings. We say that T is triangular a-admissible for g if T is a-admissible for g and

a(gx,gy) ^ 1 and a(gy,gz) ^ 1 a(gx,gz) ^ 1 for all x,y,z E X.

Definition 9. [10] Let T,g: X ^ X be self-mappings on X. A sequence {xn} in X is said to be a Picard-Jungck sequence of the pair (T, g) (based on xo) if gxn+\ = Txn, for all n ^ 0.

If T(X) C g(X), then there exists a Picard-Jungck sequence of (T,g) based on any point x0 E X.

Definition 10. [12] Let T,g: X ^ X be mappings on a quasi-metric space (X, d). We say that T and g are compatible if and only if

lim d(Tgxn, gTxn) = 0 or lim d(gTxn,Tgxn) = 0

n^-TO n^-TO

for all sequences {xn} C X such that the sequences {gxn} and {Txn} are convergent and have the same limit.

Ciric [2] introduced the quasi-contraction and proved fixed point theorems for metric spaces.

Definition 11. [2] Let (X,d) be a metric space and T: X ^ X be a self-mapping on X. A mapping T is said to be a quasi-contraction if and only if there exists a number A, 0 ^ A < 1, such that

d(Tx,Ty) ^ \max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}

for all x,y E X.

Later, Das and Naik generalized the quasi-contraction of Ciric for two mappings and established the following result:

Theorem 2. [3] Let (X, d) be a complete metric space. Let T be a continuous self-mapping on X and g be any self-mapping on X that commutes with T. Further, g(X) C T(X) and there exists a constant X E (0,1), such that, for every x,y E X,

d(gx,gy) ^ Amax{d(Tx,Ty),d(Tx,gx),d(Ty,gy),d(Tx,gy),d(Ty,gx)}. Then T and g have a unique fixed point.

Roldan et al. [10] modified the definition of simulation function by Khojasteh et al. [7] as follows:

Definition 12. A simulation function is a function (: [0, to)x[0, to) ^ R satisfying the following conditions:

(<i) ((0, 0) = 0;

((2) ((t, s) < s — t for all t,s> 0;

((3) if {tn} and {sra} are sequences in (0, to) such that lim tn = lim sn > 0

n^TO n^TO

and tn < sn, then lim sup ((tn,sn) < 0.

n^-TO

Set of all simulation functions is denoted by Z. It is clear that a simulation function must satisfy ((s, s) < 0 for all s > 0.

Definition 13. [10] Let (X, d) be a metric space, T,g : X ^ X be self mappings. Then T is called a (Z, g)-contraction if there exists ( E Z, such that

((d(Tu, Tv), d(gu, gv)) ^ 0 for all u,v E X and gu = gv. If g is the identity mapping on X, we say that T is a Z-contraction

for c.

Example 4. Let (\ : [0, to) x [0, to) ^ R be the function defined by (x(t, s) = Xs - t, where A e (0,1). Then, E Z.

Ansari [1] introduced C-class functions as follows:

Definition 14. A function Q : [0, to)2 ^ R is called a C-class function if it is continuous and satisfies the following conditions:

(i) Q(s,t) ^ s;

(ii) Q(s,t) = s implies that either s = 0 or t = 0 for all s,t ^ 0.

Definition 15. [8] A function Q : [0,to)2 ^ R has the property Cg, if there exists Cg ^ 0, such that

(Qi) Q(s,t) > Cg implies s > t; (Q2) Q(t,t) ^ Cg for all t ^ 0.

Example 5. Q(s,t) = s — t, Cg = r, r ^ 0 is a C-class function that has property Cg.

Definition 16. [8] Cg simulation function is a function (: [0,to)2 ^ R satisfying the following conditions:

(Co) C(t,s) < Q(s,t) for all t,s > 0, where Q: [0,ro)2 ^ R is a C-class

function with the property Cg; ((b) if {tn} and K} are sequences in (0, ro), such that lim tn = lim sn > 0 and tn < sn, then limsup((tn,sn) < Cg.

Example 6. Let k E R be such that k ^ 1 and ( : [0, ro)2 ^ R be the function defined by ((t,s) = kQ(s,t). Here take Cg = 1. Then, ( is a Cg simulation function.

The family of all Cg simulation functions is denoted by Zg. Radenovic et al. [8] generalized the simulation function using C-class function for two operators, as follows:

Definition 17. [8] Let (X,d) be a metric space and T,g: X ^ X be self-mappings. A mapping T is called a (Zg, g)-contraction if there exists ( E Zg such that

((d(Tx,Ty),d(gx,gy)) ^ Cg for all x,y E X and gx = gy.

3. Main result. In this section, we use admissible mappings, simulation functions, and C-class functions to consider quasi-contraction of Ciric-type on quasi-metric spaces, and we establish related results on existence and uniqueness of the coincidence point.

Lemma 3. Let (X,d) be a quasi-metric space and T,g: X ^ X be mappings. Let {xn} be a Picard-Jungck sequence of (T,g). If T is triangular a-admissible for g with a(gx0,Tx0) ^ 1 and a(Tx0, gx0) ^ 1, then a(gxn,gxm) ^ 1 for n = m.

Proof. Let {xn} be a Picard sequence of (T,g) based at x0, that is,

Txn gxn+l, for all n ^ 0. Since T is a-admissible for g, we have

a(gxo,Txo) = a(gx0, gxl) ^ 1 a(Tx0,Txl) = a(gxl,gx2) ^ 1. By induction, we get

a(gxn,gxn+l) ^ 1 for all n ^ 0. Since T is triangular a-admissible for g, we have

a(gx0, gxl) ^ 1 and a(gxl,gx2) ^ 1 a(gx0, gx2) ^ 1.

Continuing this way, we get

a(gxn, gxm) ^ 1 for all m> n. Similarly, for a(Tx0, gx0) ^ 1, we get

a(gxn, gxm) ^ 1 for all m < n.

□

Definition 18. Let (X,d) be a metric space, a : X x X ^ [0, to) and T,g : X ^ X be given mappings. A mapping T is called a (Z(a,g),g)-quasi-contraction of Ciric type if there exist ( E Zg and X E (0,1) such that

((а(gx, gy)d(Tx, Ty),XM(gx, gy)) ^ Cg for all x,y E X, (2)

where

M(gx, gy) = mаx{d(gx, gy), d(gx, Tx), d(gy,Ty), d(gx,Ty), d(gy, Tx)}. Remark 1.

(i) If we take a(x,y) = 1, inequality (2) becomes a (Zg,g)-quasi-contraction of Ciric-Das-Naik type contraction.

(ii) For a(x,y) = 1, g = ix and Cg = 0, we get a Z-quasi-contraction of Ciric type.

(iii) For a(x,y) = 1 and ((t, s) < Q(s,t) = s — t, inequality (2) becomes a Das-Naik type quasi-contraction.

Theorem 3. Let (X, d) be a quasi-metric space, T,g : X ^ X be mappings with T(X) C g(X). If T is a (Z(a,g), g)-quasi-contraction of CiriC type satisfying the following conditions:

(i) T is triangular a-admissible for g;

(ii) there exists x0 E X such that a(gx0,Tx0) ^ 1 and a(Tx0, gx0) ^ 1;

(iii) at least one of the following conditions holds:

(a) T (X) is precomplete in g(X).

(b) (X,d) is a complete quasi-metric space and T and g are continuous and compatible.

Then, T and g have a point of coincidence.

Proof. Start with x0 E X; since T(X) C g(X), we get a sequence {xn} in X with Txn = gxn+l for all n ^ 0. If gxn = gxn+l for some n, then Txn = gxn; that is, xn is a coincidence point of T and g. Thus, we assume that d(gxn+l, gxn) > 0 and d(gxn, gxn+l) > 0 for all n ^ 0. In view of condition (i), by Lemma 3, we get

a(gxn, gxm) ^ 1 for all n = m. (3)

Now,

d(gxn,gxn+l) = d(Txn-l,Txn) ^ a(gxn-l, gxn)d(Txn-l,Txn). (4)

Since T is a (Z(a,g), g)-quasi-contraction of Ciric type,

Cg ^ ((a(gxn-l,gxn)d(Txn-l,Txn),XM(gxn-l,gxn)) < < Q(XM(gxn-1,gxn),a(gxn-i, gxn)d(Txn-1, Txn)).

Using (Ql), we get

a(gxn-l, gxn)d(Txn-l,Txn) ^ XM(gxn-l,gxn). (5)

From (4) and (5), we have

d(gxn,gxn+l) ^ XM(gxn-l,gxn), (6)

where

M (gxn-l,gxn) = max{d(gxn-l,gxn ),d(gxn-l,Txn-l),

d(gxn,Txn),d(gxn-l,Txn),d(gxn, Txn-l)} = = max{d(gxn-l,gxn),d(gxn, gxn+l),d(gxn-l,gxn+l))} ^

^ d(gxn-l,gxn) + d(gxn,gxn+l).

Hence,

d(gxn,gxn+l) ^ X(d(gxn-l, gxn) + d(gxn, gxn+l)), d(gxn,gxn+l) ^ j^jd(gxn-l, gxn),

d(gxn,gxn+l) ^ kd(gxn-l,gxn), (7)

where k = j—^ < 1. Similarly, we get

d(gxn+l,gxn) ^ kd(gxn, gxn-l) for k < 1.

By Lemma 1, the sequence [gxn] is a Cauchy sequence. Now, let us consider independently cases (a)-(b). Let us prove that T and g have a coincidence point.

(a): Assume T(X) is precomplete in g(X). The precompleteness of T(X) in g(X) ensures the existence of some v E X with

lim gxn = gv = lim Txn-l. (8)

We claim that v is a coincidence point of T and g. On contrary, assume that d(gv,Tv) > 0 and d(Tv,gv) > 0. We have

lim M(gxn,gv) = lim max{d(gxn,gv),d(gxn,Txn),d(gv,Tv),

d(gxn,Tv),d(gv,Txn)} = d(gv,Tv) > 0. (9)

Using (2), we get

Cg ^ ((a(gxn,gv)d(Txn,Tv),XM(gxn,gv)) < Q(XM(gxn, gv),a(gxn,gv)d(Txn, Tv)).

By (Q\) , we have

a(gxn, gv)d(Txn,Tv) < XM(gxn,gv) for all n E N

Letting n ^ to in above inequality and using (9), we get

lim a(gxn, gv)d(gxn,Tv) < Xd(gv,Tv).

Hence, d(gv,Tv) < Xd(gv,Tv) and so is a contradiction. Therefore, d(gv,Tv) = 0. So, v is a coincidence point of T and g.

(b): Assume that (X,d) is complete and T and g are continuous and compatible. In this case, the sequence {gxn} is a Cauchy sequence in the complete quasi-metric space (X, d), hence, there exists u E X such that lim gxn = u. That is,

lim d(gxn,u) = lim d(u,gxn) = 0.

Since Txn = gxn+l for all n ^ 0, we have

lim d(Txn,u) = lim d(u,Txn) = 0.

The continuity of T yields that

lim d(Tgxn,Tu) = lim d(Tu,Tgxn) = 0.

The continuity of g yields that

lim d(gTxn,gu) = lim d(gu, gTxn) = 0.

Moreover, as T and g are compatible and the sequences [Txn] and [gxn] have the same limit, we deduce that

lim d(Tgxn, gTxn) = 0 or lim d(gTxn,Tgxn) = 0.

Now,

d(Tu, gu) ^ d(Tu, Tgxn) + d(Tgxn, gTxn) + d(gTxn, gu).

By taking limit n ^ in above inequality, we get d(Tu,gu) = 0. Similarly, we can show that d(gu,Tu) = 0. In any case, Tu = gu and we conclude that u is a coincidence point of T and g.

□

If T(X) C g(X), then there exists a Picard-Jungck sequence of (T,g) based on any point x0 E X.

Corollary 1. Let (X,d) be a quasi-metric space, T,g: X ^ X be mappings and let {xn} be a Picard-Jungck sequence of (T,g). Let T be a (Z(a,g), g)-quasi-contraction of Ciric-type and satisfy the following conditions:

(i) T is triangular a-admissible for g;

(ii) there exists x0 E X such that a(gx0,Tx0) ^ 1 and a(Tx0,gx0) ^ 1;

(iii) at least one of the following conditions holds:

(a) (g(X),d) is complete.

(b) (X,d) is a complete quasi-metric space and T and g are continuous and compatible.

Then, T and g have a point of coincidence.

For the uniqueness of a coincidence point and existence and uniqueness of a fixed point of a (Z(a,g), g)-quasi-contraction of Ciric type, we propose the following conjecture.

Theorem 4. In addition to the assumptions of Theorem 3, suppose that for all u,v E C(T,g), there exists w E X, such that a(gu,gw) ^ 1, a(gw,gu) ^ 1, a(gw,gv) ^ 1, and a(gv,gw) ^ 1. Also, T,g commute at their coincidence points. Then, T and g have a unique common fixed point.

Proof. We claim that if u,v E C(T, g), then gu = gv. By the assumption, there exists w E X, such that

a(gw,gu) ^ 1 and a(gw,gv) ^ 1.

Let us define the Picard sequence {wn} in X by gwn+l = Twn for all n ^ 0 and w0 = w. Reasoning as in the proof of Theorem 3, we obtain that the sequence {gwn} converges to gz.

By condition (i) in Theorem 3, we have

a(gwn,gu) ^ 1 and a(gwn,gv) ^ 1 for all n ^ 1. (10)

Using (2), we have

Cq ^ ((a(gwn,gu)d(Twn,Tu),\M(gwn,gu)) <

< Q(XM(gwn, gu), a(gwn, gu)d(Twn,Tu)) =

= Q (XM (gwn,gu),a(gwn,gu)d(gwn+i ,gu)). (11)

By (Qi) and (10), we have

d(gwn+i,gu) ^ a(gwn, gu)d(gwn+i, gu) < XM (gwn,gu) for all n ^ 1,

(12)

where M (gwn,gu) =

= max{d(gwn, gu), d(gwn, Twn),d(gu, Tu),d(gwn, Tu), d(gu, Twn)} = = mox{d(gwn, gu),d(gwn, gwn+i),d(gu, gwn+i)}.

Passing to the limit n ^ to, we get

lim M(gwn,gu) = max{d(gz,gu),d(gu,gz)}.

Similarly, we get

d(gu, gwn+l) < XM(gu,gwn) for all n ^ 1, (13)

where

M(gu, gwn) = max{d(gu, gwn),d(gwn, gwn+i),d(gwn, gu)}. Letting n —y to, we obtain

lim M(gwn,gu) = max{d(gu,gz),d(gz,gu)}.

If gu = gz and we take the limit n — <x> in (12) and (13), we get d(gz,gu) < Xmax{d(gz,gu),d(gu,gz)},

d(gu,gz) < Xmax{d(gz,gu),d(gu,gz)}.

If d(gz,gu) < Xd(gz,gu) or d(gz,gu) < Xd(gu,gz) < X2d(gz,gu), we get a contradiction. Thus, d(gz,gu) = 0. Therefore, gu = gz. Similarly, gv = gz implies gu = gv. Hence, u is a unique coincidence point of T and g.

Existence of a common fixed point: Let u E C(T,g), that is, Tu = gu. Due to commutativity of T and g at their coincidence points, we get

ggu = gTu = Tgu.

Denote gu = z*. Then gz* = Tz*, and, thus, z* is a coincidence point of T and g. By uniqueness of the coincidence point, we have z* = gu = gz * = Tz*. Then, z* is a common fixed point of T and g.

Uniqueness: Assume that w* is another common fixed point of T and g. Then w* E C(T,g). Thus, we have w* = gw* = gz* = z*. This completes the proof. □

From Theorem 1 we see that the result above is valid also for metric spaces.

Corollary 1. Let (X,d) be a metric space, T,g: X — X be mappings, and let {xn} be a Picard-Jungck sequence of (T,g). Assume that T is a (Z(a,g), g)-quasi-contraction of Ciric type satisfying the following conditions:

(i) T is triangular a-admissible for g;

(ii) there exists x0 E X, such that a(gx0,Tx0) ^ 1;

(iii) for all u,v E C(T,g), there exists w E X, such that a(gu,gw) ^ 1, a(gv,gw) ^ 1 and T,g commute at their coincidence points.

(iv) at least one of the following conditions holds:

(a) T (X) is precomplete in g(X).

(b) (X, d) is a complete metric space and T and g are continuous and compatible.

Then, T and g have a unique common fixed point.

The following result is a solution to an open problem posed by Rade-novic and Chandok [9].

Corollary 2. [9] Let (X,d) be a metric space, T,g: X ^ X be mappings, and let {xn} be a Picard-Jungck sequence of (T,g). Let T be a (Zq, g)-quasi-contraction of Ciric-Das-Naik type. Assume that at least one of the following conditions holds:

(a) (g(X),d) is complete.

(b) (X,d) is a complete metric space and T and g are continuous and compatible.

Then, T and g have a unique point of coincidence. Moreover, if T and g commute at their coincidence point, then they have a unique common fixed point in X.

Proof. The result follows from Corollary 1 and Theorem 4, if we consider a(x,y) = 1 and (X,d) is a metric space with metric d as defined in Theorem 1. □

Now, if we take Q(s,t) = s — t, Cg = 0, we get the following result:

Corollary 3. Let (X,d) be a metric space, a: X x X ^ [0, to), and T,g: X ^ X be mappings. Let {xn} be a Picard-Jungck sequence of (T,g) and X E (0,1), such that

a(gx, gy)d(Tx,Ty) ^ XM(gx,gy) for all x,y E X,

M(gx, gy) = max{d(gx, gy),d(gx, Tx),d(gy, Ty),d(gx, Ty),d(gy,Tx)}. Assume that

(i) T is triangular a-admissible for g;

(ii) there exists x0 E X such that a(gx0,Tx0) ^ 1;

(iii) for all u,v E C(T,g), there exists w E X, such that a(gu,gw) ^ 1, a(gv,gw) ^ 1 and T,g commute at their coincidence points;

(iv) at least one of the following conditions holds:

(a) (g(X),d) is complete.

(b) (X, d) is a complete metric space and T and g are continuous and compatible.

Then T and g have a unique common fixed point.

4. Consequences: Common fixed point results in the context of G-metric spaces. In this section, we give some consequences of our main results. For this purpose, we first recollect the basic concepts on G-metric spaces.

Definition 19. [15] Let X be a nonempty set. Let G : X x X x X — R+ be a function satisfying the following properties:

(Gl) G(x, y, z) = 0, if x = y = z,

(G2) G(x, x,y) > 0 for all x,y E X with x = y,

(G3) G(x, x, y) ^ G(x, y, z) for all x,y,z E X with z = y,

(G4) G(x,y,z) = G(x,z,y) = G(y,z,x) = ... (symmetry in all three variables),

(G5) G(x,y,z) ^ G(x,a,a) + G(a,y,z) for all x,y,z,a E X(rectangle inequality).

The function G is called G-metric on X and the pair (X, G) is called a G-metric space.

Definition 20. A G-metric space (X, G) is said to be symmetric if G(x, y, y) = G(y, x, x) for all x,y E X.

The function defined by dc> (x,y) = G(x, y, y) + G(y, x, x) for all x,y E X, is a metric on X. Furthermore, (X, G) is G— complete if and only if (X,dc<) is complete.

Recently, Jleli and Samet [4] obtained the following results.

Theorem 5. Let (X, G) be a G—metric space. Let do: X x X — [0, to) be the function defined by dc(x, y) = G(x, y, y). Then,

(1) (X,dc) is a quasi-metric space;

(2) {xn} C X is G-convergent to x E X if and only if {xn} is convergent to x in (X, do);

(33) {xn} C X is G-Cauchy if and only if {xn} is Cauchy in (X, dG); (4) (X, G) is G-complete if and only if (X, do) is complete.

Definition 21. A subset E of a G-metric space (X,G) is said to be precomplete if every Cauchy sequence {un} in E converges to a point of X.

Furthermore, a subset E of X is precomplete in (X, G) if and only if it is precomplete in (X,dG).

Definition 22. For a nonempty set X, let T,g: X ^ X and aG: X3 ^ [0, to) be mappings. We say that T is aG-admissible for g, if for all x,y E X we have

®G(gx,gy,gy) ^ 1 aG(Tx,Ty,Ty) ^ L

Definition 23. For a nonempty set X, let aG: X3 ^ [0, to) and T: X ^ X be mappings. We say that T is triangular aG-admissible for g, if T is aG-admissible for g and for all x,y E X, we have

aG(gx,gy,gy) ^ 1 and aG(gy,gz,gz) ^ 1 aG(gx,gz,gz) ^ 1.

By using the above definition, we get the following corollary:

Corollary 1. Let X be a non-empty set. The mapping T: X ^ X is triangular aG-admissible for g if and only if T is triangular a-admissible for g.

Proof. It is obvious by taking a(x,y) = aG(x,y,y). □

Now, we present Theorem 3 and Theorem 4 in the context of G-metric spaces, using the quasi-metric dG as defined in Theorem 5.

Corollary 2. Let (X, G) be a G-metric space, aG: X xX x X ^ [0, to), and T,g: X ^ X be mappings with T(X) C g(X). Let ( E Zg, and X E (0,1), such that

C(aG(gx,gy,gy)G(Tx,Ty,Ty),XM(gx,gy,gy)) ^ Cg (14)

for all x,y E X, where

M(gx,gy,gy) =

= max{G(gx, gy., gy) ,G(gx,Tx, Tx), G(gy, Ty, Ty), G(gx, Ty, Ty),

G(gy,Tx,Tx)}.

Suppose that

(i) T is triangular aG-admissible for g;

(ii) there exists x0 E X, such that aG(gx0,Tx0,Tx0) ^ 1 and aG(Txo,gxo,gxo) ^ 1;

(iii) for all u, v E C(T, g), there exists w EX such that aG(gu, gw, gw) ^ 1, aG(gw,gu,gu) ^ 1, aG(gv,gw,gw) ^ 1 aG(gw,gv,gv) ^ 1 and T,g commute at their coincidence points;

(iv) at least one of the following conditions holds:

(a) T(X) is precomplete in g(X).

(b) (X, G) is a complete G-metric space and T and g are continuous and compatible.

Then, T and g have a unique common fixed point.

Proof. It suffices to take dG(x,y) = G(x,y,y) and a(x,y) = aG(x,y,y). From (14), we get (2). Since (X,G) is complete, (X,dG) is a complete quasi-metric space due to Theorem 5. Hence, the result follows from Lemma 1, Theorem 3, and Theorem 4. □

Corollary 3. Let (X, G) be a G-metric space, aG : X xX x X — [0, to), and T,g: X — X be mappings. Let {xn} be a Picard-Jungck sequence of (T,g), ( E Zq and X E (0,1), such that (1) is satisfied. Suppose that

(i) T is triangular aG-admissible for g;

(ii) there exists x0 E X such that ac(gx0,Tx0,Tx0) ^ 1 and aG(Txo,gxo,gxo) ^ 1;

(iii) for all u, v E C(T, g), there exists w EX such that aG(gu, gw, gw) ^ 1, ac(gw,gu,gu) ^ 1, aG(gv,gw,gw) ^ 1, aG(gw,gv,gv) ^ 1 and T, g commute at their coincidence points;

(iv) at least one of the following conditions holds:

(a) T(X) is precomplete in g(X).

(b) (X, G) is a complete G-metric space and T and g are continuous and compatible.

Then T and g have a unique common fixed point.

Corollary 4. Let (X, G) be a G-metric space and T,g: X — X be mappings. Let {xn} be a Picard-Jungck sequence of (T,g), ( E Zq, and X E (0,1) such that

C(G(Tx,Ty,Ty),XM(gx,gy,gy)) > Cg (15)

for all x,y E X, where

M (gx, gy,, gy) = max{G(gx, gy, gy) ,G(gx,Tx, Tx), G(gy, Ty, Ty),

G(gx, Ty,, Ty),G(gy, Tx, Tx)}.

Also assume that at least one of the following conditions holds:

(a) T (X) is precomplete in g(X).

(b) (X, G) is a complete G-metric space and T and g are continuous and compatible.

Then T and g have unique point of coincidence. Moreover, if T, g commute at their coincidence points, then T and g have a unique common fixed point in X.

Proof. In (1), if we take ac(x,y,y) = 1, we get (2). □

Corollary 5. Let (X, G) be a G-metric space, ac: X x X x X ^ [0, to), and T,g: X ^ X be mappings. Let {xn} be a Picard-Jungck sequence of (T,g) and X E (0,1), such that

®G(gx,gy,gy)G(Tx,Ty,Ty) ^ XM(gx,gy,gy)

for all x,y E X, where

M(gx, gy,, gy) = max{G(gx, gy, gy), G(gx, Tx, Tx),G(gy, Ty, Ty),

G(gx, Ty,, Ty),G(gy, Tx, Tx)}.

Suppose that

(i) T is triangular ac-admissible for g;

(ii) there exists x0 E X such that ac(gxo,Tx0,Tx0) ^ 1 and aG(Txo,gxo,gxo) ^ 1;

(iii) for all u, v E C(T, g), there exists w EX such that ac(gu, gw, gw) ^ 1, ac(gw,gu,gu) ^ 1, acj(gv, gw, gw) ^ 1, acj(gw, gv, gv) ^ 1 and T,g commute at their coincidence points;

(iv) at least one of the following conditions holds:

(a) T (X) is precomplete in g(X).

(b) (X, G) is a complete G-metric space and T and g are continuous and compatible.

Then T and g have a unique common fixed point.

Acknowledgment. The first author acknowledges the financial support by SHODH-Scheme (Gujarat Government) with student reference number 202001720096.

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Received December 15, 2021. In revised form, March 02, 2022. Accepted March 05, 2022. Published online March 28, 2022.

Department of Mathematics, Faculty of Science The Maharaja Sayajirao University of Baroda Vadodara, Gujarat 390002, India

Sejal Puvar

E-mail: puvarsejal@gmail.com R. G. Vyas

E-mail: drrgvyas@yahoo.com