Common fixed point theorem for two selfmaps of a G-metric space

J. Math. Comput. Sci. 10 (2020), No. 2, 412-417 https://doi.org/10.28919/jmcs/4338

ISSN: 1927-5307

COMMON FIXED POINT THEOREM FOR TWO SELFMAPS OF A G-METRIC SPACE

K. RAJANI DEVI1, V. KIRAN2,∗, J. NIRANJAN GOUD2

1_{Department of Mathematics, K.V.R (W) Government Degree College, Kurnool, India} 2_{Department of Mathematics, Osmania University, Hyderabad, 500007, India}

Copyright c2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper, we prove a common fixed point theorem for two compatible self maps of a G-metric space. Keywords:G-metric space; compatible mappings; fixed point; associated sequence of a point relative to two self maps; contractive modulus.

2010 AMS Subject Classification:47H10, 54H25.

1.

commuting mappings were generalizied as weakly commuting maps by Sessa[8]. Later

G.Jungck[4, 5] introduced compatibility as a further generalization of weakly commuting maps.

Among all generalizations[1,2,3,9] of metric spaces,G- metric spaces initiated by Zead Mustafa

and Brailey Sims[6, 7] evinced interest in many researchers.

in the present paper we prove a common fixed point theorem for two compatible self maps of a

G-metric space.

∗_{Corresponding author}

E-mail address: [email protected] Received October 17, 2019

2.

Before proving the main result we begin with,

Definition 2.1:LetX be a non empty set and

G:X3→[0,∞)be a function satisfying

(G1)G(x,y,z) =0 ifx=y=z

(G2)0<G(x,x,y)for allx,y∈X withx6=y

(G3)G(x,x,y)≤G(x,y,z)for allx,y,z∈X withz6=y

(G4) G(x,y,z) = G(σ(x,y,z)) for all x,y,z∈ X where σ(x,y,z) is a permutation of the set

{x,y,z}and

(G5)G(x,y,z)≤G(x,w,w) +G(w,y,z)for allx,y,z,w∈X

ThenGis called aG-metric onX and the pair(X,G)is called aG- metric space.

Definition 2.2:Let(X,G)be aG-metric Space. A sequence{xn}inX is said to be

G-convergent if there is ax₀∈X such that to eachε>0 there is a natural numberN for which G(x_n,x_n,x₀)<ε for alln≥N.

Definition 2.3: Let(X,G)be aG-metric Space. A sequence{x_n}inX is said to beG-Cauchy

if for eachε>0 there exists is a natural numberNsuch thatG(xn,xm,xl)<εfor alln,m,l≥N.

Note that everyG-convergent sequence in aG-metric space(X,G)isG-Cauchy.

Definition 2.4:Let f andgbe two self maps of aG-metric space(X,G)such that

lim

n→∞

G(f gx_n,g f x_n,g f x_n) =0 for every sequence{x_n}in X with lim

n→∞

f x_n= lim

n→∞

gx_n=tfor some

t∈X, then the functions f andgare said to be compatible.

Definition 2.5: A functionψ :[0,∞)→[0,∞)is said to be a contractive modulus ifψ(0) =0

andψ(t)<t fort>0

Definition 2.6: Let f and g be self maps of a non-empty set X and let x0∈X,we can find

3.

Theorem 3.1: Suppose f is continuous selfmap of a G-metric space(X,G), then f has a

fixed point inX if and only if there is a contractive modulusψ and a continuous selfmapgofX

such that:

(i) f andgare compatible

(ii) G(gx,gy,gy)≤ψ(G(f x,f y,f y))for allx,y∈X

and

(iii) there is a pointx₀∈X and an associated sequence{x_n}ofx₀ relative to the selfmaps f andgsuch that the sequence{f xn}converges to some pointt ofX.

Furthergtis the unique common fixed point of f andg.

Proof: To prove the necessary part, suppose that f has a fixed point, say ’a’,a∈X, then f a=a.

Defineg:X→X bygx=afor allx∈X. Now for anyx∈X, we have(g f)x=g(f x) =aand

(f g)x= f gx= f a=afor anyx∈X, giving that f g=g f, so that f and g are compatible.

Now letψ be a contractive modulus, thenψ(0) =0 andψ(t)<t fort>0 and for anyx,y∈X

G(gx,gy,gy) =G(a,a,a) =0≤ψ(G(f x,f y,f y).

Further an associated sequence ofx0=arelative to the selfmaps f andgis given byxn=afor n=0,1,2,3· · · , and since the sequence{f x_n}is a constant sequence converging toa, which is

a point inX.

Thus the conditions (i) (ii) and (iii) of the theorem are satisfied.

Conversely, suppose that there is a contractive modulusψ and a selfmapg ofX satisfying (i)

(ii) and (iii) of the theorem hold.

From the condition (iii) of the theorem there is an associated sequence{xn}ofx0relative to the

selfmaps f andgsuch that f x_n=gx_n−1forn=1,2,3· · · and f x_n→t asn→∞for somet∈X.

Then sincegx_n= f x_n+1, it follows thatgx_n→t asn→∞.

Now we show thatgis continuous onX. To see this, suppose that{y_n}is a sequence inX with

y_n→yasn→∞,y∈X. Since f is continuous f yn→ f yasn→∞, this together with inequality

(ii) of the theorem, we getG(gy_n,gy,gy)≤ψ(G(f yn,f y,f y))→0 asn→∞, which implies that

gy_n→gyasn→∞, showing thatgis continuous.

as n→∞ and f and g are compatible, we have lim

n→∞

G(f gx_n,g f x_n,g f x_n) =0 which implies

that G(f t,gt,gt) = 0 gives f t =gt. To show that f gt =g f t, take z_n=t for n= 1,2,3· · ·

so that f zn → f t and gzn →gt as n→∞. Since f t =gt, f and g are compatible , we get

lim

n→∞

G(f gz_n,g f z_n,g f z_n) =0.

Using the continuity of G, f andg, we get g f zn→g f t and f gzn→ f gt asn→∞. It follows

thatG(f gt,g f t,g f t) =0 and hence f gt=g f t

Consequently

(1) f f t = f gt=g f t=ggt

If possible suppose thatgt6=ggt, thenG(gt,ggt,ggt)>0 and hence

(2) ψ(G(gt,ggt,ggt))<G(gt,ggt,ggt)

But from (ii)of the theorem and 1 we get

G(gt,ggt,ggt)≤ψ(G(f t,f gt,f gt)) =ψ(G(gt,ggt,ggt))

which is contradicts to 2, hencegt=ggt.

Using this in 1 we getggt =gt= f gt, showing thatgtis a common fixed point of f andg.

Uniqueness:Suppose thatu= f u=guandv= f v=gvfor someu,v∈X.

if possible suppose thatu6=v,thenG(u,v,v)6=0 so that

(3) ψ(G(u,v,v))<G(u,v,v)

from (ii) of the theorem we have

G(u,v,v) =G(gu,gv,gv)≤ψ(G(f u,f v,f v)) =ψ(G(u,v,v))

which is contradiction to 3, henceu=v, proving the theorem.

Corollary 3.2: Suppose f is continuous selfmap of aG-metric space(X,G),then f has a fixed

point inX if and only if there is a contractive modulusψ and a selfmapgofX such that

(i) f g=g f

(ii) G(gx,gy,gy)≤ψ(G(f x,f y,f y))for allx,y∈X

(iii) there is a pointx₀∈X and an associated sequence{x_n}ofx₀ relative to the selfmaps f and gsuch that the sequence {f x_n} converges to some pointt of X. Further gt is the

unique common fixed point of f andg.

Proof: From the fact that the commutativity implies the compatibility of a pair of selfmaps

proof of the corollary follows from the Theorem 3.

Corollary 3.3: Suppose f andgare selfmaps of aG-metric space(X,G).Let f is continuous

and if there is a contractive modulusψ and a positive integer k such that:

(i) f g=g f

(ii) G(gkx,gky,gky)≤ψ(G(f x,f y,f y))for allx,y∈X

and

(iii) there is a pointx₀∈X and an associated sequence{x_n}ofx₀ relative to the selfmaps f andgk such that the sequence{f x_n}converges to some pointt ofX. Furthergt is the

unique common fixed point of f andg.

Proof: From the condition (i) of the corollary 3 we get f gk =gkf. Thus f and gk are

commuting and hence satisfying the hypothesis of 3, and therefore f and gk have a unique

common fixed point sayb, thengkb=b= f b.

Nowgkgb=gk+1b=ggkb=gband f gb=g f b=gb.

This shows thatgbis a common fixed point of f andgk. The uniqueness ofbimplies thatgb=b

since f b=b,bis a common fixed point of f andg.

To prove that f andghave unique common fixed point, suppose thatu= f u=guand

v= f v=gvfor some u,v∈X, so thatgku=u andgkv=v, this shows that u,v are common

fixed points of f andgk. The uniqueness of common fixed point of f andgk impliesu=v.

Corollary 3.4: Let p be a positive integer. If g is continuous selfmap of a G-metric space

(X,G),such that:

(i) G(f x,f y,f y)≤ψ(G(gPx,gpy,gpy))for allx,y∈X

and

(ii) there is a pointx₀ ∈X and an associated sequence {x_n}of x₀ relative to the selfmaps gp andI( whereI is the identity map onX) such that the sequence{gpx_n}converges to

Proof: We know thatgpI=Igp. From (ii) of the corollary 3, we have

G(x,y,y) =G(Ix,Iy,Iy)≤ψG(gpx,gpy,gpy)for allx,y∈X.

Sincegis continuous, gp is continuous. Applying corollary 5.3.1 to the functiongp andI, we

have unique common fixed point, showing thatghas unique fixed point as every point ofX is a

fixed point ofI.

CONFLICT OF INTERESTS

The author(s) declare that there is no conflict of interests.

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