COMMON FIXED POINT IN 𝐺-METRIC SPACES VIA GENERALIZED Γ-𝐶𝐹-SIMULATION FUNCTION Текст научной статьи по специальности «Математика»

64

Probl. Anal. Issues Anal. Vol. 13 (31), No3, 2024, pp. 64-78

DOI: 10.15393/j3.art.2024.15970

UDC 517.521.2, 517.988

S. V. PuvAR, R. G. Vyas

COMMON FIXED POINT IN G-METRIC SPACES VIA GENERALIZED r-CF-SIMULATION FUNCTION

Abstract. We present the generalized r-Cf-simulation function and establish the common fixed point result for weak pqp, g)-contrac-tion in complete G-metric space. The exploration extends to its ramifications on both quasi-metric spaces and metric spaces. The study explores the existence of a solution for a non-linear integral equation as an application of these results.

Key words: r-Cp-simulation functions, G-metric spaces, quasi-

metric spaces, weak contraction, common fixed point

2020 Mathematical Subject Classification: 47H10, 54H25

1. Introduction. Expanding the Banach fixed point theorem to G-metric spaces marks a significant advancement in mathematical analysis. These extensions often entail adjusting the contraction condition to suit the properties of G-metrics. Since Samet et al.'s work [14], it has been recognized that G-metric spaces possess a quasi-metric structure. As a result, many fixed-point theorems established within the domain of G-metric spaces can be inferred from existing results in (quasi-)metric spaces. Specifically, when the contraction condition in a fixed-point theorem for a G-metric space can be simplified to involve only two variables instead of three, it becomes feasible to establish analogous fixed-point results in a metric space.

Further, Khojasteh et al. [9] introduced the notion of simulation functions in order to express different contractivity conditions in a simple, unified, coherent manner. By employing a unified language through simulation functions, researchers can convey and analyze a wide range of contractive mappings using a common set of principles. Later, this principle has been extended in various directions (see [1], [6], [8], [10], [11], [12], [13]).

In this context, we introduce the generalized r — Cp-simulation function by employing the r-C-class functions [11]. Additionally, we define

© Petrozavodsk State University, 2024

weak contraction and establish a common fixed point result applicable to G-metric spaces, along with its implications for quasi-metric spaces and metric spaces. This flexibility is crucial in addressing diverse mathematical problems and adapting to various settings, allowing researchers to tailor contractivity conditions to specific needs. Finally, we apply the derived fixed-point result to solve a specific type of integral equation.

2. Preliminaries. Let us recollect some basic definitions and results for G-metric space.

Definition 1. [15] Let X be a nonempty set, G: X x X x X ^ [0, +8) be a function satisfying the following properties:

(Gi) 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 2. [15] 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.

Lemma 1. [15] If (X,G) is a G-metric space, then

G(x, y, y) ^ 2G(y, x, x), for all x,y e X.

Proposition 1. [15] Let (X,G) be a G-metric space, {xn} Q X be a sequence, and x e X. Then

(i) {xn} G-converges to x ^^ lim G(xn,xn,x) = 0 ^^ lim G(xn,x,x)= 0.

(ii) {xn} is G-Cauchy ^^ lim G(xn,xm,xm) = 0.

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

in X.

Definition 3. [15] 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 sequence {xm} c X such that {xm} ^ x.

Definition 4. [2] A sequence {xn} in a G-metric space (X, G) is asymptotically regular if lim G(xn,xn+i,xn+i) = 0.

Barinde [5] introduced asymptotic regularity for two operators in metric spaces, which can be extended to G-metric spaces as follows:

Definition 5. Let (X,G) be a G-metric space and f,g: X ^ X be two operators. Then the operator g is called f -asymptotically regular in (X, G) if

G(gn(x),f (gn(x)),f (gn(x))) ^ 0 as n for all x e X.

Let r([0, +8)) be the set of all non-decreasing functions 7: [0, +<x>) ^ [0, +8), such that 7(t) = 0 if and only if t = 0.

Definition 6. [11] A function F: [0, +8)2 ^ R is called T-G-class function if it is continuous and there exists 7 e r([0, +8)), such that:

(i) F(s,t) ^ 7(s);

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

The collection of all T-G-class functions is denoted by Cr. For 7(t) = t, the T-G-class function reduces to C-class function of [3].

Definition 7. [11] A function F: [0, +8)2 ^ R has the property r-G^, if there exists 7 e r([0, +8)) and CF ^ 0, such that:

(Fx) F(s,t) > CF implies 7(5) > 7(t), for all s,t ^ 0; (F2) F(t,t) ^ CF, for all t ^ 0.

Example 1. The following functions Fi: [0, +8)2 ^ R are elements of Cr with property r-G^:

(i) Fx(s,t) = r^),CV " 1,2.

(ii) F2(s,t) " (1 + ,r e(0, +8),Cf " 1.

3. Main results. In this section, we introduce generalized r-G^-simulation function using T-G-class functions. Subsequently, the conditions for the existence and uniqueness of a common fixed point result

for weak contractions by using the generalized T-Cp-simulation function are established.

Definition 8. A function rq: [0, +œ) x [0, +œ) ^ R is a (generalized) T-Cf-simulation function of type II, if

(qi) There exists Cf ^ 0, such that

if y(t,s) ^ CF then y(t,s) ^ F(s,t), for all s,t ^ 0,

where F e Cr with property r — Cf ■ (rq2) If {tn} and {sra} are non increasing sequences in (0, +œ>) and y{tn,sn) ^ Cf, then

lim rq(tn, sn) ^ Cf implies sn ^ 0.

We say that rq is a (generalized) T-Cf-simulation function of type I, if it satisfies (,qi) and the following (y2)* condition:

(rq2)* If {tn} and {sra} are non increasing sequences in (0, +8), such that lim tn = lim sn > 0, then limsuprq(tn,sn) < Cf.

n^+8 n^+8 ra—>+oo

Remark 1.

(i) Every simulation function is a (generalized) T-Cf-simulation function of type I.

It follows from definition 8 for Cp = 0 and F(s, t) = j(s) — 7(t) and 7(t) = t.

(ii) Also, condition (rq2)* is different from (rq2).

Now, we introduce weak (rqF, ^-contraction for G-metric spaces.

Definition 9. Let (X, G) be a G-metric space and f, g be self mappings on X. For a function rq: [0, +8) x [0, +8) ^ R, f is called

(i) an (qp, g)-contraction if

V(G(fx,gy,gy),G(x,y,y)) ^ CF, for ah x,y e X, (1) y(G(gx,fy,fy),G(x,y,y)) ^ CF, for ah x,y e X, (2)

(ii) a weak (rqp ,g)-contraction if

y(G(fx,gfx,gfx),G(x,fx,fx)) ^ Cf, for all x e X, (3) rq(G(gx, fgx, fgx),G(x,gx,gx)) ^ Cp, for all x e X, (4)

(iii) a generalize weak non-expansive map if

G(fx,gfx,gfx) ^ G(x,fx,fx), for all x e X, (5)

G(gx, fgx, fgx) ^ G(x, gx, gx), for all x e X. (6)

For g = f in (1)—(6), we get the following contractions: A mapping f is called

(a) an yF-contraction if

V(G(fx,fy,fy),G(x,y,y)) ^ CF, for aU x,y p X, (7)

(b) a weak yF-contraction if

y(G(fx, f2x, f2x),G(x, fx, fx)) ^ CF, for all x e X, (8)

(c) a weak non-expansive map if

G(fx, f2x, f2x) ^ G(x, fx, fx), for all x e X. (9)

Remark 2. In definition 9, for dG(x, y) = G(x, y, y), an (yF, g)-contraction for G-metric spaces reduced to (yF, g)-contraction for quasi-metric spaces (X,dG).

Now, we prove common fixed point result for the pair of mappings in G-metric spaces.

Theorem 1. Let (X, G) be a complete G-metric space, f and g be self mappings on X and y: [0, +8) x [0, +8) ^ R be a function.

(i) Let f be an (yF, g)-contraction. If y satisfies (yx), then f and g have at most one common fixed point. Also, if 7 e r([0, +8)), then

G(fx,gy,gy) < G(x,y,y), for afi x ^ y.

(ii) Let y be a r-CF-simulation function of type II; if f(gf)n° and (gf )n°, n0 e N is a weak (yF, g)-contraction, then f is g-asymptotically regular. The same result holds if y is a r-CF-simulation function of type I and f is a generalized weak non-expansive map.

(iii) Let f be an (yF, g)-contraction with f or g continuous, and y be a r-CF-simulation function of type II (or type I with f being a generalized weak non-expansive map), then f and g have a unique common fixed point.

Proof.

(i) Suppose that gx = fx = x, gy = fy = y and x ^ y; then G(x,y,y) = G(fx,gy,gy) = t(say) > 0. From (yi) and (F2), we get

which is a contradiction. Hence, common fixed point of f and g is unique if exists.

Suppose that 0 < s = G(x,y,y) ^ t = G(fx,gy,gy), where x ^ y. From (1) and (yi), we have

CF ^ y(G(fx,gy,gy),G(x,y,y)) < F(G(x,y,y),G(fx,gy,gy)).

From (F1), we get

Since 7 is non-decreasing, G(fx,gy,gy) < G(x,y,y), which is a contradiction. Hence, G(fx,gy,gy) < G(x,y,y).

(ii) For any fixed x0 in X, construct a sequence {xn} with %2n = (gf )n(Xo), %2n+1 = f (X2n), for all U ^ 0.

Let ti = G(xi,xi+]_,xi+1) for all i ^ 0. Suppose tk = 0, for some k e N. If %2k = x2k+l, then x2k is a fixed point of f. If x2k+l = %2k+2, then x2k+l is a fixed point of g. Thus, at least one mapping of f or g has a fixed point. Now, assume that tk ^ 0, for all k ^ 0. Put x = x2no+2k = (gf )n°+k(x0), k = 0,1,... in (3) to get

y(t,t) ^ CF y(t,t) < F(t,t) ^ CF,

1 (G(fx,gy,gy)) < 7(G(x,y,y)).

— y(t2n0+2k+l,t2n0 + 2k) < < F (t2n0 + 2k ,t2n0+2k+\).

(10)

Put x = X2no+2k+i = f (gf )n0+k(xo),k = 0,1,... in (4) to get

Cp ^ У(G(gX2no+2k+l, fgX2no+2k+l, fgX2no+2k+l),

^ {x2no+2k+l, 9X2nQ + 2k+l, 9x2no+2k+l)) "

= Tj(G{x2n0 + 2k+2,X2n0+2k+3, x2n0+2k+3) , GpX2no+2k+l, %2n0 + 2k+2, %2n0+2k+2)) "

= V{t2n0+2k+2,t2n0+2k+l) <

< F {t2n0 + 2k+1 ,t2n0 + 2k+2). (11)

From (10) and (11), we get

CF ^ 'q{ti+ i,ti) < F(U,ti+1), for all i ^ no. (12)

From (Fl), we get j{ti+^ < 7{ti). Since, 7 is non-decreasing U+1 < ti, that is,

G{xi+i,xi+2,xi+2) < G{xi,xi+i,xi+1), for all i ^ no.

Hence {G{xi,xi+]_,Xi+is monotonically decreasing sequence of nonnegative real numbers. Thus, there exists r ^ 0, such that

lim G{xi,xi+l,Xi+^ = r.

i—> + 8

Let us prove that r = 0. Suppose, on the contrary, that r > 0. Taking limit as i ^ +8 in (12) and using {F2), we get

CF ^ lim 'q{ti+l,ti) ^ F{ lim ti, lim ti+1) = F{r,r) ^ CF.

i— + 8 i— + 8 i— + 8

Hence,

lim ri{ti+1 ,ti) = CF. (13)

i— + 8

Type II: From {rq2), we get r = lim ti = 0: a contradiction.

i— + 8

Type I: From (5) and (6), we have U+1 ^ ti, for all i ^ 0. Using {rq2)*,

we get lim sup rq{ti+l,ti) < CF: a contradiction to (13). Hence, r = 0.

i— + 8 Therefore,

lim G{xi,xi+l,xi+1) = 0. (14)

i— + 8

Since G{xi,Xi,Xi+^ ^ 2G{xi,Xi+]_,Xi+1), we get

lim G{xi,xi,xi+1) = 0. (15)

i— + 8

(iii) We now show that {xn} is a Cauchy sequence. It is sufficient to show that {x2n} is Cauchy in X. On the contrary, assume that {x2n} is not Cauchy. Then, from Lemma 4.1.5 in [2], there exists e > 0 and

two subsequences {x2n(k)} and {x2mpk)} of {x2n}, such that, for all k p N, k ^ n(k) < m(k) < n(k + 1) and for all given pi,p2,p3 p Z,

lim G(X2n(k)+P1 ,X2m(k)+P2 ,x2m(k)+P3)= £. (16)

k^+8

Considering two non-increasing subequences

at = G(X2n{ k)pi ),x2m{k){l),x2m{ k)(l))

and

a'l = G(X2n( k)[l) + 2 ,x2m( k)(l )+2,x2m{k){l )+2) of G(x2n{k),X2m{k) ,X2m{ k)) and G(x2n( k)+2,x2m( k)+2,x2m{k) + 2), such that

lim al = lim al = e. (17)

From (1) and (yi), we have

CF ^ v(al, al) < F(ahal).

Letting I ^ +8, we get

CF ^ lim y(al,al) ^ F( lim al, lim al) = F(e,e) ^ CF.

This implies

lim rq(dl,al) = Cf. (18)

l ^ + 8

Type II: From (rq2), lim al = 0: a contradiction to (17).

l^+8

Type I: From (rq2)*, we get limsup,q(al,al) < Cp: a contradiction to (18).

l ^+ 8

Thus {x2n} is a Cauchy sequence in (X,G). Hence, {xn} is Cauchy in (X,G). Since (X,G) is complete, X, implies that

lim X2n = lim X2n+1 = u.

n^ + 8 n^ + 8

Assume f is continuous; then lim fx2n = lim x2n+\ = fu. This implies

n^+rn n^+rn

that fu = u. From (1), we have

CF ^ y(G(fu,gfu,gfu),G(u,fu,fu)) = = rq(G(u, gu, gu),G(u,u,u)) < < F(G(u, u, u), G(u, gu, gu)).

This implies that 0 ^ G{u, gu, gu)) < -/{G{u,u,u)) = j{0) = 0. Hence, G{u, gu, gu) = 0, and so gu = u. The uniqueness follows from part (i). □

The following example validates our result.

Example 2. Let X = [0,1]. Define G: X3 ^ [0, +8) as

.10, if x = y = z, . .

G{x,y, z) = \ ' u (19)

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

Then {X, G) is a complete G-metric space. Define f,g: X ^ X as f{x) = and g{x) = \, @x p X. Also define 7: [0, +8) ^ [0, +8) by #

it, if 0 ^t < 1, { ) [2t, if 1 ^ t,

and r]: [0, +8)2 ^ R by

r]{t, s) = l{SI . - -At), for all t, s p [0, +8). 1 + { )

Take F{s, t) = j{s) - j{t) with CF = 0, for all s, t p [0, +8). Then r] is a (generalized) r-Cp-simulation function of type I and all the conditions of Theorem 1 are satisfied, and x = 0 is the unique common fixed point of and .

4. From G-metric space to quasi-metric space and metric space. We recollect some basic definitions and results for quasi-metric spaces.

Definition 10. [7] Let X be a non-empty set and let d: X x X ^ [0, +8) be a function, such that

(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 p X.

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

Definition 11. [7] 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) < e 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, that is, a sequence {xn} in a quasi-metric space is Cauchy if and only if it is both left-Cauchy and right-Cauchy.

Jleli and Samet [7] gave the following results:

Theorem 2. Let (X, G) be a G-metric space. Let do: X xX ^ [0, +8) be the function defined by do(x,y) = G(x,y,y). Then

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

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

(3) {xn} c X is G-Cauchy if and only if {xn} is Cauchy in (X, do);

(4) (X, G) is G-complete if and only if (X, do) is complete.

Definition 12. [7] 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.

Asymptotic regularity for two operators for quasi-metric spaces is defined as follows:

Definition 13. Let (X,d) be a quasi-metric space and f,g: X ^ X be two operators. The operator g is called f -asymptotically regular on X if

lim d(gn(x),f (gn(x))) = 0 = lim d(f (gn(x)),gn(x)), for all x p X.

n—> + 8 n—> + 8

Theorem 1 in context of quasi-metric spaces is stated as follows. For proving the following result in quasi-metric spaces, we need contractive conditions:

V(d(fx,gfx),d(x,fx)) ^ Cf,

y(d(gx, fgx), d(x, gx)) ^ CF, for all x p X,

and two more contractive conditions got by changing the order of d(x,y). But here, we can directly derive the result from G-metric space without changing the order of d(x, y) in the contractivity conditions.

Theorem 3. Let (X, d) be a complete metric space, f and g be self mappings on X, and y: [0, +8) x [0, +8) ^ R be a function.

(i) Let f be an (yF, g)-contraction. If y satisfies (y\), then f and g have at most one common fixed point (if any).

Also, if 7 e r([0, +8)), then

d(fx, gy) < d(x, y), for all x ^ y.

(ii) Let y be a r — CF-simulation function of type II. If f (gf )n0 and (gf )n0, n0 e N is a weak (yF, g)-contraction, then f is g-asymptotically regular. The same result holds if y is a r-CF-simulation function of type I and f is a generalized weak non-expansive map.

(iii) Let f be an (yF, g)-contraction with f or g continuous, and y be a r-CF-simulation function of type II (or type I, then f is generalized weak non-expansive map). Then f and g have a unique common fixed point.

Proof. In Theorem 1, take dc(x,y) = G(x,y,y); then result follows from Theorem 2. □

Theorem 3 is also valid in the context of metric spaces.

Corollary 1. Let (X, d) be a complete metric space, f be a self mapping on X, and (: [0, +8) x [0, +8) ^ R be a function.

(i) Let f be an (-contraction. If ( satisfies ((\), then f has at most one common fixed point (if any).

Also, if 7 e r([0, + 8)), then

d(fxjy) < d(x,y), for all x " y.

(ii) Let ( be a simulation function of type II; if fn°, n0 e N is a weak (-contraction, then f is asymptotically regular. The same result holds if ( is a simulation function of type I and f is a weak nonexpansive map.

(iii) Let f be an (-contraction with f continuous, and ( be a simulation function of type II (or type I, then f is weak non-expansive map). Then f has a unique fixed point.

Proof. In Theorem 3, take g = f, F(s,t) = j(s) — 7(t),^f(t) = t, and CF = 0; then (yF, g)-contraction reduces to (-contraction [9]. □

5. An application. In this section, we present an application of Theorem 1: we guarantee the existence of a solution to an integral equation.

Let X = C[0,1] be the set of all continuous functions defined on [0,1] and let G: X x X x X ^ R be defined by

G{x,y, z) = sup | x{t)-y{t) | + sup | y{t)-z{t) | + sup | z{t)-x{t) | .

ie[0,1] ie[0,1] ie[0,1]

Then {X, G) is a complete G-metric space. Consider the integral equation:

i

x{t) = I H{t, s)K(s,T{x{s)))ds, (20)

where H: [0,1] x [0,1] ^ R+ and K: [0,1] x R+ ^ R+ are continuous functions and T: X ^ X is a self mapping on X. Now we present the following theorem:

Theorem 4. Suppose the following assumptions hold:

(1) for all sp[0, 1] and x,y p X, we have

^(s,x) — K{s,y)) -y[,

(2) for all t,s p [0,1], we have

1

1

sup H{t, s)ds = -. ie[0,1] J 4

0

Then the integral equation (20) has a solution. Proof. Let T: X ^ X be a mapping defined by

1

nxWi-jMM.))*, l],xpx

0

From condition (1) and (2), we have G(fx,gfx,gfx) = 2 sup | f (x(t)) - gf (x(t)) | =

ie[0,1]

1 1

= 2 sup | iff(t,s)K(s,x(s))ds - \H(t,s)K(s,f (x(s)))ds

ie[0,1] J J

00

2 sup [h(t,s)\ K(s,x{s)) - K(s,f (x(s))) \ ds < 0

1

2 sup H(t, s) \ x(s) - f (x(s)) \ ds ^

iero,1l J 0

1

G(x,fx,fx) sup \H(t,s)ds. ielOJl J

0

So, we get

G{fx, gfx, g fx) ^ 1 G{x, fx, fx). (21)

Let g(t,s) = (s) - lit), F(s,t) = 7(s) - 7(t) for all s,t e [0, +œ),

CF = 0, 7(t) = 2t for all t e [0, +œ).

Now,

g(G(fx,gfx,gfx),G(x,fx,fx)) =

= 1 /l{G(x,fx,fx)) - >y(G(fx,gfx,gfx)) =

= 1 (2G(x,fx,fx)) - 2G(fx,gfx,gfx).

Then, from (21), we have

V[G(fx,gfx,gfx),G(x,fxJx)J ^

Thus all the conditions of Theorem 3 are satisfied and hence f and g have a unique common fixed point x e X. Thus x is a solution of the integral equation (20). □

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Received April 09, 2024. In revised form, August 28 , 2024. Accepted September 03, 2024. Published online September 18, 2024.

Department of Mathematics, Faculty of Science The Maharaja Sayajirao University of Baroda Vadodara, Gujarat 390002, India Sejal V. Puvar

E-mail: [email protected] R. G. Vyas

E-mail: [email protected]