Fixed Point as a G-contractive fixed point

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 11 No.1 (2016) © Research India Publications; http/www.ripublication.com/ijaer.htm

Fixed Point as a G-contractive fixed point S. Saravanan

T. Phaneendra

Department of Mathematics, VIT University, Vellore - 632 014, Tamil Nadu State, India [email protected]

Department of Mathematics, VIT University, Vellore - 632 014, Tamil Nadu State, India [email protected]

ABSTRACT. A fixed point theorem of Mustafa and Obiedat (2010) is proved for a self-map f on a G-metric space (X,G), using the wellknown infimum property of real numbers without an appeal to the iterative procedure. An additional interesting consequence is that the obtained fixed point is shown as a G-contractive fixed point for f in

Lemma 1.1.The following statements are equivalent in a Gmetric space ( X , G) : (a)  xn   n 1 X is G-convergent with limit p  X , (b) lim G( xn , xn , p)  0 , n

the sense that the orbit x, fx,..., f n x,... at each x  X is Gconvergent with limit p.

(c) lim G( xn , p, p)  0 . n

It is also known that G( x, y, z) is jointly continuous in all the three variables x, y and z. Definition 1.3.A sequence  xn  n 1 in a G -metric space ( X , G) is said to be G-Cauchy if lim G( xn , xm , xm )  0 .

Key words: The Infimum Property, G-Metric Space, G-Cauchy Sequence, Fixed Point, G-contractive Fixed Point

I. INTRODUCTION Let X be a nonempty set and G : X  X  X   such that (G1) G( x, y, z )  0 for all x, y, z  X with G( x, y, z )  0 if x  y  z, (G2) G( x, x, y)  0 for all x, y  X with x  y , (G3) G( x, x, y)  G( x, y, z) for all x, y, z  X with z  y , (G4) G( x, y, z)  G( x, z, y)  G( y, x, z)  G( z, x, y)  G( y, z, x)  G( z, y, x) for all x, y, z  X (G5) G( x, y, z)  G( x, w, w)  G(w, y, z) for all x, y, z, w  X . Then G is called a G -metric on X and the pair ( X , G) , a G metric space. Axiom (G4) reveals that G is symmetric in the three variables x , y and z and Axiom (G5) is referred to as the rectangle inequality (of G). This notion was introduced by Mustafa and Sims [10] in 2006. From the definition of Gmetric space, it immediately follows that G( x, y, y)  0 whenever x  y and G( x, y, y)  2G( x, x, y) for all (1.1) x, y  X .… We use the following notions, developed in [10]: Definition 1.1. Let ( X , G) be a G -metric space. A G -ball

n, m

Definition 1.4.A G-metric space ( X , G) is said to be Gcomplete if every G-Cauchy sequence in X converges in it. Definition 1.5.A self-map f on a G -metric space ( X , G) is G -continuous at a point p  X if f 1  BG ( fx0 , r )   (G) for all r  0 , and f is G -continuous on X if it is G -continuous at every p  X . Since a G-metric topology is also a metric topology, we have Lemma 1.2. A self-map f on a G -metric space ( X , G) is G continuous at a point p  X if and only if the sequence   fpn   n 1  X G -converges to fp whenever  pn  n 1 is a sequence in X which G -converges to p. An extensive research has been done in recent years in G-metric spaces. For instance, one can refer to [1], [2], [3], [4], [5], [6], [9] and [12]. In this sequel, the following three results were proved in succession by Mustafa and Obiedat [7]: Theorem 1.1.Let ( X , G) be a complete G-metric space and

f : X  X satisfying the following condition G( fx, fy, fz)  k G( x, fx, fx)  G( y, fy, fy)  G( z, fz, fz )

in X is defined by BG ( x, r )   y  X : G( x, y, y)  r It is easy to see that the family of all G -balls forms a base topology, called the G -metric topology  (G) on X . Also G ( x, y)  G( x, y, y)  G( x, x, y) for all x, y  X .… (1.2) Induces a metric on X , and the G -metric topology coincides with the metric topology induced by the metric G . This allows us to readily transform many concepts from metric space into the setting of G -metric space. Definition1.2. A sequence  xn  n 1 in a G-metric space ( X , G) is said to be G -convergent with limit p  X if it converges to p in the G -metric topology  (G) .

for all x, y, z  X ,

…

(1.3)

where k is a nonnegative number such that 0  k  13 . Then

f has a unique fixed point p and f is G -continuous at p. Theorem1.2.Let ( X , G) be a complete G-metric space and f : X  X satisfying the following condition G( fx, fy, fz)  aG( fx, fy, fz)  b[G( x, fx, fx)

G( y, fy, fy)  G( z, fz, fz)] for all x, y, z  X , …

316

(1.4)

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 11 No.1 (2016) © Research India Publications; http/www.ripublication.com/ijaer.htm

2 where a and b are nonnegative real numbers with 0  a  b  1 . Then f has a unique fixed point p and f will be G continuous at p .

Definition 2.1.A fixed point p of f on a G -metric space ( X , G) is a G-contractive fixed point of it if the orbital sequence x0 , fx0 ,..., f n x0 ,... at each x0  X is G-convergent with limit p .With these notions, we restate Theorem 1.3 in a modified form as follows: Theorem 2.1.Let ( X , G) be a complete G -metric space and f, a self-map on G satisfying (1.5), where  ,   0 , not both zero with 0      1 . Then f has a unique fixed point p and

Theorem 1.3. Let ( X , G) be a complete G-metric space and f : X  X satisfying the following condition

G( fx, fy, fz)   G( fx, fy, fz )   max G( x, fx, fx), G( y, fy, fy), G( z, fz, fz), G( y, fy, fy), G( z, fz, fz) for all x, y, z  X ,… (1.5) where  and   0 such that 0  a  b  1 . Then f has a unique fixed point p and f will be G -continuous at p . Writing a  0 and b 

k 3

f will be G-continuous at p . Further, if 0    3  1 , then p will be a G -contractive fixed point of f. Analytical Proof of Theorem 2.1: Step 1 – Existence of the infimum Define S  {G( x, fx, fx) : x  X } . Each S is a nonempty set of nonnegative numbers which is bounded below. Hence by the Lemma 2.1,inf S  a exists.

in (1.4) then 0  k  13 , we get (1.1).

Since a  3b  1 , (1.4) is a generalization of (1.1). Also, since the arithmetic average of any three non-negative numbers cannot exceed their maximum, we see from (1.4), G( fx, fy, fz)  aG( fx, fy, fz)

Step 2 – Vanishing infimum We establish that a  0 . If it is possible, Let a  0 . Now G( fx, f 2 x, f 2 x)  S . Writing y  z  fx in (1.5), we have G( fx, f 2 x, f 2 x)   G( x, fx, fx) 

G( x, fx, fx)  G( y, fy, fy)  G( z, fz, fz ) 3   G( fx, fy, fz)   max{G( x, fx, fx), G( y, fy, fy), 3b 

G( z, fz, fz)} for all x, y, z  X , Where   a and   3b .

…(1.6)

 max{G( x, fx, fx), G( fx, f 2 x, f 2 x), G( fx, f 2 x, f 2 x)} … (2.1)   G( x, fx, fx)   M

Thus (1.4) implies (1.5). Therefore, Theorem 1.2 is a special case of Theorem 1.3. In other words, the proof of Theorem 1.1 and Theorem 1.2 are redundant, since they follow from that of Theorem 1.3. Remark 1.1.If     0 , then with z  y  fx , (1.5) gives G( fx, f 2 x, f 2 x)  0 so that f 2 x  fx for each x  X . That is every fx is a fixed point of f or the fixed point is not unique. Therefore, Theorem 1.3 needs a minor modification. Several researchers including Mustafa and Obiedat [7] employed the usual iteration technique. The iterative procedure will be cumbersome if the inequality involves more number of terms. The present paper aims at an elegant proof of Theorem 1.3, using the well-known infimum property of non-negative real numbers wherein no iterative procedure is needed. This analytical technique was first employed by Phaneendra and Kumarsamy in [11] to prove the Banach contraction principle in a G-metric space.

2. Main Result We begin with the infimum property of real numbers, as stated below: Lemma 2.1. Let S   be nonempty and bounded below. Then   inf S exists. An immediate consequence of Lemma 2.1 is: Lemma 2.2. Let  be the infimum of S   .Then there exists a sequence  pn  n 1 in S with lim pn   .We utilize the n

where M  max{G( x, fx, fx), G( fx, f 2 x, f 2 x)}  Case (a): If M  G( fx, f 2 x, f 2 x) , from Lemma 2.1, we have G( fx, f 2 x, f 2 x)   G( x, fx, fx)   G( fx, f 2 x, f 2 x) or

G( fx, f 2 x, f 2 x) 

 1 

 G( x, fx, fx)  a ,

which is a contradiction. Case (b): If M  G( x, fx, fx) , from again Lemma 2.1, G( fx, f 2 x, f 2 x)   G( x, fx, fx)   G( x, fx, fx)

.  (   )G( x, fx, fx)  a This shows that a is not a lower bound of S , which is again a contradiction. Hence a  0 . Step 3 – Existence of a sequence By Lemma 2.2, there exists a sequence  xn  n 1 in X such that G( xn , fxn , fxn )  S for all n  1, 2,3,... and .… (2.2) lim G( xn , fxn , fxn )  0 n

Step 4 –  xn  n 1 is Cauchy In fact, by the rectangle inequality of G and (1.1), we have G( xn , xm , xm )  G( xn , fxn , fxn )  G( fxn , xm , xm )  G( xn , fxn , fxn )  [G( fxn , fxm , fxm ) 

G( fxm , xm , xm )]

following notion of G -contractive fixed point [11]:

317

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 11 No.1 (2016) © Research India Publications; http/www.ripublication.com/ijaer.htm

3  G( xn , fxn , fxn )  G( fxn , fxm , fxm )  .… (2.3) 2G( xm , fxm , fxm ) Now, with x  xn and y  z  xm , (1.5) gives, G( fxn , fxm , fxm )   G( xn , xm , xm )   max{G( xn , fxn , fxn ),

G( p, xn , xn )   G( xn , p, p) or In the limiting case as n   , this in view of (2.2),(2.4) and Lemma 1.1 implies G( p, fp, fp)  0 or fp  p . Thus p is a fixed point.

G( xm , fxm , fxm ), G( xm , fxm , fxm )}   G( xn , xm , xm ) 

Step 7 – Uniqueness of the fixed point Let q be another fixed point of f writing x  p and y  z  q in (1.5),

{

 max G( xn , fxn , fxn ), G( xm , fxm , fxm )}

G( p, q, q)  G( fp, fq, fq)   G( p, q, q)   max{G( p. fp, fp), G(q. fq, fq), G(q. fq, fq)}   G( p, q, q)   max{G ( p , fp, fp), G (q, fq, fq)}   G( p, q, q)   max{G ( p , p , p ), G (q , q , q )} or (1   )G( p, q, q)  0 so that p  q . That is p is the unique fixed point of f . Step 8 – Fixed point as a G -contractive fixed point Writing y  z  p in (1.5) and using (G5), we get

  G( xn , xm , xm )   [G( xn , fxn , fxn )  G( xm , fxm , fxm )] Substituting this in (2.3), we get G( xn , xm , xm )  G( xn , fxn , fxn )  G( xn , xm , xm )  [G( xn , fxn , fxn )  G( xm , fxm , fxm )] 

2G( xm , fxm , fxm ) (1   )G( xn , xm , xm )  G( xn , fxn , fxn )  [G( xn , fxn , fxn )  G( xm , fxm , fxm )]  2G( xm , fxm , fxm ) or  1   G( xn , xm , xm )    G( xn , fxn , fxn )   1   2   1 

so that fq  q . Then

G( f n x, p, p)  G( f n x, fp, fp)   G( f n1 x, p, p)   max{G( f n1x, f n x, f n x), G( p, fp, fp), G( p, fp, fp)}

  G( xm , fxm , fxm ) 

  G( f n1 x, p, p) 

or Applying the limit as m, n   in this and using (2.2) we

 max{G( f n1 x, f n x, f n x), G( p, fp, fp)}   G( f n1 x, p, p)   G( f n1x, f n x, f n x)

obtain that  xn  n 1 is a Cauchy sequence in X .

  G( f n1 x, p, p) 

Step 5 – G -convergence Since, X is G -complete, we find the point p in X such that … (2.4) lim xn  p .

 [G( f n1 x, p, p)  G( p, f n x, f n x)]  (   )G( f n1 x, p, p)   G( p, f n x, f n x)  (   )G( f n1 x, p, p)  2 G( f n x, p, p)

n 

Step 6 – G -convergent limit as a fixed point Again repeatedly using (G5), G( p. fp, fp)  G( p, fxn , fxn )  G( fxn , fp, fp)

or

(1  2 )G( f n x, p, p)  (   )G( f n1 x, p, p) so that

G( f n x, p, p)  cG( f n1 x, p, p) ,

 [G( p, xn , xn )  G( xn , fxn , fxn )]  G( fxn , fp, fp) … (2.5) Now, from (1.5) with x  xn and y  z  p , it follows that G( fxn , fp, fp)   G( xn , p, p)   max{G( xn , fxn , fxn ), G( p, fp, fp), G( p, fp, fp)}   G( xn , p, p)   max{G( xn , fxn , fxn ), G( p, fp, fp)}   G( xn , p, p)   [G( xn , fxn , fxn )  G( p, fp, fp)] . … (2.6) 1. Substituting (2.6) in (2.5) and then using (G5), G( p, fp, fp)  [G( p, xn , xn )  G( xn , fxn , fxn )]   G( xn , p, p) 

 [G( xn , fxn , fxn )  G( p, fp, fp)] (1   )G( p. fp, fp)  (1   )G( xn , fxn , fxn ) 

Where c 

   . By induction on n , 1  2

G( f n x, p, p)  cG( x, p, p) for n  1, 2,3,... (2.7) Since,   3  1 , we find that c  1 so that c n  0 as n   . Proceeding the limit as n   in (2.7),

we get G( f n x, p, p)  0 as n   for each x  X . Thus p is a G -Contractive fixed point of f . Example and Discussion Example 3.1.Let ( X , d ) be a metric space where X   and d ( x, y)  x  y .Define G( x, y, z )  max  x  y , y  z , z  x 

for all x, y, z  X .Then ( X , G) is a G -metric space. Consider

f : X  X by fx 

318

x for all x  X . Then 0 is the only fixed 2

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 11 No.1 (2016) © Research India Publications; http/www.ripublication.com/ijaer.htm

4 x X ,

have

[7]

x  x x  O f ( x)   x, , 2 ,... and for any x  X ,  0 as 2 2 2n   n   .Therefore, O f ( x)  0 for all x . In other words,

[8]

point

of

f .

Also

for

each

we

every f -orbit converges to the fixed point 0 . Therefore, 0 is the only fixed point.

[9]

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