RATIONAL TYPE CYCLIC CONTRACTION IN 𝐺-METRIC SPACES Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2023.13110 UDC 517.988

S. V. PuvAR, R. G. Vyas

RATIONAL TYPE CYCLIC CONTRACTION IN G-METRIC SPACES

Abstract. Rational type cyclic contraction via C-class function is established in G-metric spaces, which can not be reduced to the contractive condition in standard metric spaces. A common fixed-point result is obtained for the pair of (A, B)-weakly increasing mappings in G-metric spaces.

Key words: G-metric spaces, Cyclic maps, C-class function, Common fixed point

2020 Mathematical Subject Classification: 47H10,

54H25

1. Introduction. In 2012, Jleli and Samet [3] observed that some of the fixed-point theorems in G-metric spaces can be deduced from standard metric spaces or quasi-metric spaces (for details see [7], [8]). Shatanawi and Abodayeh [9] introduced a new contractive condition and proved fixed-point and common fixed-point results in G-metric spaces, for which the techniques of Jleli and Samet [3], Samet et al. [6] are inapplicable.

In this paper, we introduce rational type cyclic contraction via C-class function in G-metric space that generalizes the contractive condition of Shatanawi and Abodayeh [9] for larger class of auxiliary functions and deduced common fixed-point result in G-metric spaces. Some examples are also presented to show that our results are effective.

2. Preliminaries.

Definition 1. An altering distance function is a continuous, non-decreasing mapping : [0, to) ^ [0, to), such that 0-1(O) = 0.

Notation:

(i) $ is the family of all altering distance functions. © Petrozavodsk State University, 2023

(ii) ^ is the family of all mappings ^: [0, ro) ^ [0, ro) with the property: if |iTO}meN C [0, ro) and ^(tm) ^ 0, then tm ^ 0.

Note that $ C

Definition 2. [5] Let X be a nonempty set. Let G: X x X x X ^ [0, ro) be a function satisfying the following properties:

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

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

(G3) G(x, x, y) ^ G(x, y, z), Vx, 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), Wx,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 3. [5] A G-metric space (X, G) is said to be symmetric if

G(x,y,y) = G(y,x,x), Wx,y E X. Lemma 1. [5] If (X, G) is a G-metric space, then

G(x,y,y) ^ 2G(y,x,x),Wx,y E X.

Definition 4. [5] Let (X, G) be a G-metric space, x E X be a point, and {xn} C X be a sequence. We say that:

(1) a sequence {xn} G-converges to x, if lim G(xn,xm,x) =

0; that is, for every £ > 0 there exists n0 E N satisfying G(xn,xm,x) < e, Wn,m ^ n0.

(2) a sequence {xn} is G-Cauchy if lim G(xn,xm,xk) = 0; that

is, for every £ > 0 there exists n0 E N satisfying G(xn,xm, Xk) < e, Wn, m,k ^ n0.

(33) (X,G) is complete if every G-Cauchy sequence in X is G-convergent in X.

Proposition 1. [5] Let (X,G) be a G-metric space, {xn} C X be a sequence, and x E X. Then the following are equivalent:

(a) {xn} G-converges to x,

(b) lim G(xn,xn,x) = 0,

(c) lim G(xn,x,x) = 0.

n^œ

Proposition 2. [5] A sequence [xn] in a G-metric space (X, G) is G-Cauchy if and only if lim G(xn, xm,xm) = 0.

Definition 5. [1] A sequence {xn} in a G-metric space (X,G) is asymptotically regular if lim G(xn,xn+i,xn+i) = 0.

Lemma 2. [1] Let {xn} be an asymptotically regular sequence in a G-metric space (X, G) and suppose that {xn} is not Cauchy. Then there exist a positive real number £ > 0 and two subsequences {xnk} and {xmk} of {xn}, such that Wk E N:

k ^ nk <mk < nk+i,

, + Xmk — i) ^ £ < G(Xnk ,Xnk + i,Xmk ) and, also, for all given p1,p2,p3 E Z:

lim G(xnk+P! , Xmk +P2 ,Xmk +p3 ) £.

Definition 6. [5] Let (X, G) be a G-metric space. We say that a mapping T: X ^ X is G-continuous at x E X if {Txm} ^ Tx for all sequences {xm} C X, such that {xm} ^ x.

In 2013, Shatanawi and Postolache [10] introduced (A,B)-weakly increasing functions for a pair of mappings:

Definition 7. Let (X, be a partially ordered set and A, B be two closed subsets of X with X = A U B. Let f,g: X ^ X be two mappings. Then the pair (f, g) is said to be (A, B)-weakly increasing if fx ^ gfx, Wx E A and gx ^ fgx, Wx E B.

Kirk et al. [4] introduced cyclic mappings and proved fixed point results for cyclic mappings:

Definition 8. A self-map f: X ^ X is cyclic if there exist nonempty subsets A0, Ai,..., AP-1 C X, such that

p

X = y Ai and f (Ai) C Ai+1 for 0 ^ i ^ p - 1 (where Ap = A0).

i=1

1 + f s

Ansari [2] introduced C-class functions as follows:

Definition 9. A mapping F: [0, ro)2 ^ R is called a C-class function if it is continuous and satisfies the following conditions:

(Fi) F(s,t) ^ s, Ws,t ^ 0;

(F2) F(s, t) = s implies that either s = 0 or t = 0, W s,t ^ 0. Example 7. Let s,t E [0, ro); then we have:

(1) F(s,t) = s - t,

s — t

(2) F(s,t) = S '

(3) F M=i + f

(2) F(s,t) = ks,k E (0,1).

3. Main Results. Here we consider functions ^ E ^ and generalize the contractivity condition of Shatanawi and Abodayeh ( [9], Theorem 2.1) by using C-class function, and prove common fixed point theorems in G-metric spaces.

Theorem 1. Let ^ be an ordered relation in a set X. Let (X, G) be a complete G-metric space and X = Ay]B, where A and B are nonempty closed subsets of X. Let f,g be self mappings on X that satisfy the following conditions:

(i) The pair (f, g) is (A, B)-weakly increasing.

(ii) f(A) C B and g(B) C A.

(iii) There exist two functions E E such that

t(G(fx,gfx,gy)) ^ F(<(M(x, y)),^(M(x, y))) (1)

holds for all comparative elements x,y E X with x E A and y E B and

<(G(gx,fgx,fy)) ^ F(<(M'(x, y)),^(M'(x, y))) (2)

holds for all comparative elements x,y E X with x E B and y E A, where F is a C-class function,

M(x, y) = max |G(x, fx, y),

G{fx,fx,y)[l + G(x,x,gy)] 1 + G(x,fx,y) , G(gy, gy, y)[1 + G(fx, fx, x)] 1 + G(x,fx,y)

and

G(gx,gx,y)[1 + G(x,x,fy)] 1 + G(x, gx, y) ' G(fy, fy, y)[1 + G(gx, gx., x)]

M'(x, y) = max < G(x, gx, y),

1 + ^(x, gx, y) ........... "}•

1 + G(x,gx,y) (iv) f or g is continuous.

Then, f and g have a common fixed point in A[~\B.

Proof. Start with x0 E A. Since f (A) C B, there exists x1 E B, such that fx0 = x1 and, since g(B) C A, there exists x2 E A, such that gx1 = x2. Continuing this way, we construct a sequence {xn} in X, such that

fx2n = x2n+1, for x2n E A; and gx2n+1 = x2n+2, for x2n+1 E B,n ^ 0.

Using condition (i), we have xn — xn+1, Wn ^ 0.

If x2n = x2n+1, for some n E N, then x2n is a fixed point of f in

Af)B. Since x2n — x2n+1, from (1) we have:

4>(G(x2n+1,X2n+2 ,%2n+2)) = &(G(fX2n, gf%2n, g%2n+1)) ^

^ F((j)(M(x2n,X2n+1)),^(M(x2n,X2n+1))), (3)

where

M (x2n,X2n+1) =

max |G(x2n, fx2n,x2n+i),

G(fx2n, fX2n,X2n+l)[1 + G(X2n, X2n, 9%2n+l)]

1 + G(X2n, fX2n,X2n+l) G(gX2n+l, gX2n+l,X2n+l)[1+G(fX2n, fX2n, ^2ra)]l =

1 + G(X2 ra i fX2n,X2n+l) J

G(X2n+l,X2n+l,X2n+l )[1 + G(X2n, X2n, ^2ra+2)]

max < G(X2n, X2n+l, ^2n+l),

1 + G(X2 ni

G(X2n+2,X2n+2,X2n+i)\1 + G(X2n+1, X2n+1, X2n)]^ =

1 + G(X2n,X2n+1,X2n+i) J

= max{G(X2n,X2n+1,X2n+l),G(X2n+1,X2n+2,X2n+2)} =

= G(X2n+1,X2n+2,^2n+2).

From (3) and ( F1) we have:

4>(G(X2n+1,X2n+2,%2n+2)) ^

^ F (<(G(x2n+1,X2n+2,X2n+2 )), lKG(X2n+1, %2n+2, %2n+2))) ^

^ 4>(G(X2n+1,X2n+2,X2n+2)),

which implies

F(4>(G(X2n+1, X2n+2, %2n+2)) , '^(G(X2n+1, X2n+2, %2n+2))) =

= <(G(X2n+1,X2n+2,%2n+2)).

From ( F2) we have:

4>(G(X2n+1,X2n+2,X2n+2) = 0 or ^(G(X2n+1, X2n+2, %2n+2) = 0.

Since < E $ and ^ E we have G(x2n+1,x2n+2,x2n+2) = 0. That is, x2n = x2n+1 = x2n+2. Hence, x2n is a common fixed point of f and g in Af|B. Now, assume that xn = xn+1, Wn ^ 0. Since x2n ^ x2n+1, from (1) we have:

4>(G(x2n+1 X2n+2)) = <(G(fX2n,gfX2n,gX2n+1)) ^

^ F(<(M(x2n,X2n+1 )),^(M(x2n,X2n+1))),

(4)

where

M (X2n,X2n+1) = max{G(X2n ,X2n+1,X2n+1),G(X2n+1,X2n+2,X2n+2)}.

If M(x2n,X2n+1) = G(x2n+1,X2n+2,X2n+2), Wn ^ 0, then from (4) we have

<(G(x2n+1,X2n+2,X2n+2)) ^

^ F (4>(G(x2n+1, X2n+2, X2n+2 )),^(G (x2n+1 ,X2n+2, X2n+2 ))).

Since F is C-class function, we have:

F (<(G(x2n+l,x2n+2,x2n+2)),'(G(x2n+l,x2n+2,x2n+2))) =

= <(G(x2n+l,x2n+2,x2n+2)) <(G(x2n+l, x2n+2, x2n+2)) = 0

or

'(G(x2n+l , x2n+2, x2n+2))=0, Vn > 0.

Since < E $, we have G(x2n+l,x2n+2,x2n+2) = 0,Vn ^ 0; this implies x2n+l = x2n+2, Vn ^ 0: a contradiction. Therefore, M(x2n,x2n+l) = = G(x2n,x2n+l,x2n+l), Vn ^ 0. Now, from (4) and ( Fl), we get

<(G(x2n+l,x2n+2,x2n+2)) ^

^ F(0(G(x2 n 1 x2n+l,x2n+l)),'(G(x2n, x2n+l,x2n+l))) ^

^ 0(G(x2n,x2n+l,x2n+l)), Vn ^ 0. (5) Since x2n+l — x2n+2, from (2) we can prove:

0(G(x2n+2,x2n+3,x2n+3)) ^

^ F(0(G(x2n+l,x2n+2,x2n+2)),'(G(x2n+l,x2n+2,x2n+2))) ^

^ 4>(G(x2n+l , x2n+2i x2n+2)), Vn > 0. (6)

From (5) and (6), we conclude that

0(G(xn+l,xn+2,xn+2)) ^ F(0(G(xn,xn+l,xn+l))'(G(xn,xn+l,xn+l))) ^

^ 0(G(xn,xn+l,xn+l)),Vn ^ 0. (7)

Since < E $, we get G(xn+l,xn+2,xn+2) ^ G(xn,xn+l,xn+l),Vn ^ 0, which implies that the sequence {G(xn,xn+l,xn+l)} is a non-negative monotonically decreasing sequence. So, there exists r ^ 0, such that

lim G(xn,xn+l,xn+l) = r. (8)

By taking the limit as n ^ <x> in (7), we get

<(r) ^ F(<(r), lim '(G(xn,xn+l,xn+l))) ^ <(r),

which implies that F(<(r), lim '(G(xn,xn+l,xn+l))) = <(r). From ( F2), we get <(r) = 0 or lim '(G(xn,xn+l,xn+-])) = 0. Since < E $ and ' E we get

r = lim G(xn,xn+l,xn+l) = 0. (9)

From the definition of G-metric space, we have

lim G(xn,xn,xn+l) = 0. (10)

Now, we prove that {xn} is G-Cauchy. It is sufficient to show that {x2n} is a G-Cauchy sequence.

Suppose that {xn} is not Cauchy. Then, by (9), (10), and Lemma 2, there exist e > 0 and two subsequences {x2nk} and {x2mk} of {x2n}, such that Wk E N, k ^ nk < < nk+l and for all givenpl,p2,p3 E Z,

lim G(X2nk +pi ,X2mk +P2 ,x2mk +P3 ) = (11)

Since x2mk — x2nk+l, from (1) we have:

4>{G{x2mk+l,x2mk+2,x2nk+2)) = $(G(fX2mk ,gfx2mk , 3x2nk+l)) ^

^ F (0(M (X2mk ,X2nk + l))MM (X2mk ,X2nk + l))), (12)

where M (X2m

k , X2nk + l) — max G(X2mk , X2mk + l, x2nk+l) ,

G(X2mk+l,X2mk + l,X2nk+l)[l + G(X2mk , x2mk , x2nk +2)]

1 + G(x2mk ,X2mk+\,X2nk+i) G{x2nk +2,X2nk +2,X2nk+1)[1 + G(X2rnk+1,X2mk+1,X2mk )]

1 + G(x2mk ,X2mk+1,X2nk+1) Using (9), (10) and (11), we get lim M(X2mk ,X2nk+1 ) = £■ Taking

k^-œ

limit as k ^ <x> in (12), we get

0(e) ^ F(0(e), lim ^(M(X2mk,X2Uk+1))). Since F is a C-class function, we get

0(e) ^ F(0(e), lim ^(M(X2mk,X2Uk+1))) ^ 0(e);

k^-œ

this implies that

0(e) = 0 or lim ^(M(x2mk,x2nk+1)) = 0;

k^-œ

so we get e = lim M(x2mk, x2nk+1) = 0: a contradiction. Thus,

k^-œ

[x2n] is a G-Cauchy sequence in (X, G). So, the sequence [xn] is a G-Cauchy sequence in (X, G). Since (X, G) is complete, there exists u E

X, such that {xn} is G-convergent to u. Therefore, the subsequences {x2n} and {x2n+l} are G-convergent to u. Since {x2n} C A and A are closed, u E A. Also, {x2n+l} C B and B are closed, so u E B. Now, we may assume that f is continuous. So, we have fu = lim fx2n = lim x2n+l = u. By uniqueness of the limit, we have fu = u. Since u — u, from (1) we have:

<(G(u,gu,gu)) = <(G(fu,gfu,gu)) ^ F(<(M(u,u)),'(M(u,u))),

(13)

where

G(fu, fu, u)[1 + G(u, u, gu)}

M(u, u) = max < G(u, fu, u),

[1 + G(u, fu, u)] G(gu, gu, u)[1 + G(fu, fu, u)]

| = G(u, gu, gu).

[1 + G(u, fu, u)]

Using (13), we obtain

<(G(u,gu,gu)) ^ F(<p(G(u,gu,gu)),'(G(u,gu,gu))).

Since F is a C-class function, we have

<(G(u, gu, gu)) = 0 or '(G(u, gu, gu)) = 0.

This implies G(u,gu,gu) = 0. Hence, gu = u. Thus, u is a common fixed point of f and g in A P| B. □

The following example shows that the condition (iii) defined in Theorem 1 is more general than the condition (iii) of Theorem 2.1 in [9].

Example 8. Let X = {0,1} and define G: X x X x X ^ [0, <x>) as

G(0,0,0) = G(1,1,1) = 0, G(0,0,1) = 1 and G(0,1,1) = 2. Then the function G is a G-metric on X.

Take A = B = {0,1}, and x — y if and only if x ^ y. Define the mappings f, g: X ^ X as follows:

f(0) = 1, f(1) = 0 and 0(0) = 0, g(1) = 1.

Let 4>, ^: [0, to) ^ [0, to) and F: [0, to) x [0, to) ^ R be defined by 0(i) = t/2, ^(t) = t and F(s,t) = s/(1 +t), for all s,t E [0, to). For x = 0, y =1,

M (0,1)=max{ G(0,f 0,1)

G(f 0,f0,1)[1 + G(0, 0,g1)]

1 + G(0,f 0,1) G(91,91,1)[1 + G(f °J 0 0)], = = 2

1 + G(0,f 0,1) j = max{2,0} = 2.

Now,

F(0(M(0,1)),^(M(0,1))) = F(0(2),^(2)) = F(1, 2) = 1 ^

^ 0(G(/0,gf0,01)) = 0(0).

For x =1, y = 0:

G(f 1,f 1,0)[1 + G(1,1,g0)]

M (1, 0)= max{ G(1,f 1, 0),

1 + G(1,f 1, 0)

G(g0, g0,0)[1 + G(/1J1,1)] | = max{1,0} = 1. 1 + G(1,f1,0) J ^ }

Now,

F(0(M(1,0)),^(M(1, 0))) = F(0(1), ^(1)) = F(2, 1) = 1 ^

^ 0(G(/ 1,gf 1,g0)) = 0(0).

Hence, the condition (iii) of Theorem 1 is satisfied.

But 0(G(0, f 0,1)) - 4>(G(0, f 0,1)) = 0(2) - -0(2) = 1 - 2 = -1 ^ 0.

This shows that the condition (iii) of Theorem 2.1 in [9] does not

hold.

In Theorem 1, if we replace ^ E ^ with ^ E $ and take M(x,y) = G(x, fx,y), M'(x,y) = G(x,gx,y) and F(s,t) = s - t, then we get Theorem 2.1 of [9], as a particular case. Now, the following example validates Theorem 1.

Example 9. Let X = [0,1/2] and let f,g: X ^ X be given as 2

f (x) = and g(x) = -. Take A = [0,1/2] and B = [0,1/2].

1 + x 2

Define the function G : X x X x X ^ [0, œ) as

j0, ifx = y = z,

G(x, y, z) = <

\max{x,y, z}, otherwise.

Clearly, G is a complete G-metric on X. We introduce a relation on X by x ^ y if and only if y « x. Also, define the functions F : [0, œ)2 ^R by F (s, t) = s - t and : [0, œ) ^ [0,œ) by <(t) = 2t and

é(t) = . 1 + 2t

Note that f(A) = [0,1/4] ÇB and g(B) = [0,1/2] Ç A. To prove (i), given x E X,

gfx

x2

2(1+ x)'

x2 x2

Since x E [0,1/2], —-- < ---. Thus, a fx ^ fx and, hence,

L 1 1 2(1+x) (1+x)

x - x for all x E X. To prove (iii), given x E A and y E B with y^x. Then

x22 x2

G(fx, qfx, qy) = ma^^--, —--, — \ = -

(1 + x) 2(1 + x) 2 2

and

M( \ / y(1 + x) y(1 + 2) \ M(x, y) = max{y, jr^r, orwJ

Since

2

T «2!/ - (TT20!

we have

<(G(fx,gfx,fy)) « F(<(M(x, y)),^(M(x, y))).

Hence, all the conditions of Theorem 1 are satisfied. Notice that 0 is the unique common fixed point of f and g.

Corollary 1. Let ^ be an ordered relation in a set X. Let (X,G) be a complete G-metric space and X = A U B, where A and B are nonempty closed subsets of X. Let f be a continuous self map on X that satisfies the following conditions:

(1) fx X f2x, Vx E X.

(2) f (A) C B and f (B) C A.

(3) There exist two functions E E such that

0(G(fx,f2x,fy)) ^ F($(M(x,y)),^(M(x,y))) (14)

holds for all comparative elements x,y E X, where F is a C-class function,

Then f has a fixed point in A fi B.

Proof. The proof follows from Theorem 1 by taking g = f. □

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

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M (x, y) = max

G{fx,fx,y)[l + G{x,x,fy)] 1 + G(x,fx,y)

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Received January 9, 2023. In revised form, June 4, 2023. Accepted June 6, 2023. Published online July 2, 2023.

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

Sejal V. Puvar

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

E-mail: drrgvyas@yahoo.com