zamfirescu maps and it's stability on generalized space

Abha Singh et al. / International Journal of Engineering Science and Technology (IJEST)

ZAMFIRESCU MAPS AND IT’S STABILITY ON GENERALIZED SPACE Abha Singh Department of Mathematics, Prep Year College, Hail University, Hail, Kingdom of Saudi Arabia Postal Addresses: Department of Mathematics, SRM University, Modinagar 201204, Ghaziabad, U.P., India Email : [email protected];

Aftab Alam Department of Mathematics, SRM University, Modinagar 201204, Ghaziabad, U.P., India Email: [email protected]

Abstract In this paper we discus about the fixed point iteration function and the stability of Zamfirescu contraction maps on the setting of generalized metric spaces. All process has seen from the point of view of a feedback system. For this class of feedback system the contraction mapping principle give this remarkable result of stability in view of the attractor, the invariance and the estimate.

Key words: fixed point; function iteration; Zamfirescu contraction; The Attractor; The Invariance; The Estimate; stability.

1. Introduction The Banach’s fixed point theorem (or the contractions mapping principle) is the most important metrical fixed point theorem (see [2]). In the field of computer and mathematical sciences, the most used iteration procedure to approximate fixed points is the method of successive approximation (function or Picard iteration), given by (2.1). And also it is used in fractal and superior fractal studies. In a recent paper [2] give its full statement and proof. Concept of stability of fixed point iteration procedures firstly introduced and established by Harder and Hicks [6], they give some stability results for The Mann and Kirk function iterations under various contractive conditions. Extension of the results of Harder and Hicks to other classes of contractive mappings may refer to Birende [2], Rhoades ([8] – [11]), Singh et al. ([12] – [13], Zamfirescu [14]. Main aim of this paper is to see the existence of Zamfirescu contraction mapping and obtain it stability results of fixed point iteration procedures with changes or new conditions for b-metric space setting. 2. Preliminaries Let x0 , x1 , x2 , ... be a sequence of elements defined by

xn1  T xn , n  0,1,2,...

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(2.1)

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but due to computational errors (rounding errors, numerical approximations of function, derivatives or integrals) we do not get the exact value of x1 and instead of the x1 , we takes y1 which is very closed to y1 ≈ x1 , similarly y2 ≈ x2 , and so on yn ≈ xn . Definition 2.1. ([6]) Let

( X , d ) be a metric space T : X  X a self mapping, x0  X and the iteration

procedure defined by (2.1). Such that the generated sequence {xn } converges to a fixed point u of

T . Let

{ yn } be an arbitrary sequence in X and set  n  d ( yn 1 , f (T , yn )), n  0,1, 2,. .. We say that the iteration (2.1) is

(2.2)

T -stable or stable with respect to T

if and only if

li m n  n  0  li m n yn  u . Lemma 2.1. (cf.[2]) If  is a real number such that such

that

0    1 and { n } is a sequence of positive numbers

li m n  n  0 , then for any sequence of positive numbers

{un }

satisfying

un1   un   n , n  0,1, 2,. . . we have li m n un  0 . Definition 2.2. (cf.[2],[14]) A self mapping T : X  X is said to be Zamfirescu contraction if there exist real numbers

,  and  satisfying 0    1, 0  ,   0.5 such that for x, y  X at least one of the

following is true: (Z1)

d (T x, Ty )   d ( x, y ) ;

(Z2)

d (T x, T y )  [d ( x,T x)  d ( y, T y )] ;

(Z3)

d (T x, Ty )  [d ( x, Ty )  d ( y, Tx)] .

So all idea, we can be seen from the point of view of a feedback system. For this class of feedback system the contraction mapping principle gives the following remarkable result: The Attractor. For any initial object x0 the feedback systems xn 1  T xn , n  0,1,2,... will always have a predictable long-term behavior. There is an object x (the limit of the feedback system). The Invariance. The feedback system leaves x invariant. In other words, if we start with x then x is returned. x is a fixed point of

T , i.e. T x  x (like u is a fixed point).

The Estimate. We can predict how fast the feedback system will arrive close to x when it is started at x0 . We have to test the feedback loop once on the initial object. That means, if we measure the distance between x0 and x1  T x0 , we can already safely predict how often we have to run the system to arrive near x within a prescribed accuracy. Moreover, we can estimate the distance between x0 and x (see[7]). Definition 2.3. (cf.[3]) Let X be a set and s  1 a given real number. A function

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d : X  X  R  is called

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a b -metric provided that for all x, y , z  X ,

d ( x, y )  0, i ff x  y ,

(bm-1)

(bm-2) d ( x, y )  d ( y , x ) ,

d ( x, z )  s[d ( x, y )  d ( y, z )] .

(bm-3)

The pair ( X , d ) is called a b -metric space or a generalized metric space. In all that follows, ( X , d ) is a b metric space. We remarked that the class of b -metric spaces is effectively larger than that of metric spaces. Lemma 2.2. [13] Let

( X , d ) be a b -metric space and { yn } a sequence in X such that

d ( yn1 , yn 2 )  qd ( yn , yn 1 ), n  0,1,2,... Then the sequence { yn } is Cauchy sequence in X , provided that s q  1 where q  (0,1) and s  1 . The following examples show that b -metric on X need not be a metric on X . Example 2.1. (cf. [4]) Let

d ( x1 , x3 )  d ( x2 , x3 )  1 ,

X  x1 , x2 , x3 and d : X  X  R  such that d ( x1 , x2 )  x  2 , d ( xn , xn )  0 ,

2

d ( xn , xk )  d ( xk , xn )

n, k  1, 2,3 .

for

Then

x d ( xn , xk )  [ d ( xn , xi )  d ( xi , xk )] , n, k , i  1, 2,3 . And if x  2 , the ordinary triangle inequality does 2 not hold. Example 2.2. Let x

X  x1 , x2 , x3 and d : X  X  R  such that d ( x1 , x2 )  3 , d ( x1 , x3 ) 

d ( x2 , x3 )  1 , d ( xn , xn )  0 , d ( xn , xk )  d ( xk , xn ) . Then d ( xn , xk ) 

x [d ( xn , xi )  d ( xi , xk )], 2

n, k , i  1, 2,3 . Then ( X , d ) is a b -metric space. Example x

2.3.

Let

X  x1 , x2 , x3

d ( x1 , x3 )  d ( x2 , x3 )  1 ,

and

d ( xn , xn )  0 ,

d : X  X  R

such

d ( xn , xk )  d ( xk , xn ) .

that

d ( x1 , x2 )  x  3 , d ( xn , xk ) 

Then

x [d ( xn , xi )  d ( xi , xk )] , n, k , i  1, 2,3 . Then ( X , d ) is a b -metric space. 3 3. Main Results We shall start with following theorem. Theorem 3.1. Let ( X , d ) be a complete b -metric space. A mapping T : X  X is a Zamfirescu contraction

to

the

complete

b -metric

space,

provided

that

there

is

a

constant

0    1, 0  ,   0.5,  s  0.5 such that for all x, y in X , that is (Z1), (Z2) or (Z3). The constant

, ,  is called the Zamfirescu contraction factor for T . The following holds true:

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(3-i)

x is invariant, T ( x )  x .

(3-ii)

There is a unique attractor x  li m n xn 1 .

i.e., the function iteration converges to unique attractor for any initial approximation x0  X . Proof. If s  1 then the conclusion follows from metric space setting, so we need to take s  1 . We have the contraction mapping

d ( x, y )  d (Tx, Ty )   d ( x, y )  d ( x, y ) , to prove the existence of the attractor x (fixed point x ), we will show that for any given x0  X the function iteration {xn } is a Cauchy sequence, If (Z1) holds by (2.1) we have,

d ( x2 , x1 )  d (T x1 , T x0 )   d ( x1 , x0 ) , d ( x3 , x2 )  d (T x2 , T x1 )

  2 d ( x0 , x1 ) , similarly we get

d ( xn , xn 1 )   n d ( x0 , x1 ), n  0,1,2,...

(3.1)

If (Z2) holds, prove the existence of the attractor {x } , we will show that the function iteration {xn } is a Cauchy sequence, so

d ( x2 , x1 )  d (T x1 , T x0 )   [d ( x0 , T x0 )  d ( x1 , T x1 )]      d ( x1 , x0 ) , 1    d ( x3 , x2 )  d (T x2 , T x1 )   [d ( x1 , T x1 )  d ( x2 , T x2 )] 2

     d ( x0 , x1 ) , 1    similarly we get n

   d ( xn , xn 1 )    d ( x0 , x1 ), n  0,1,2,... 1   

(3.2)

If (Z3) holds, by Lemma 2.2

d ( x2 , x1 )  d (T x1 , T x0 )   [d ( x0 , T x1 )  d ( x1 , T x0 )]

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  [ s d ( x0 , x1 )  s d ( x1 , x2 )]  s    d ( x1 , x0 ) , 1  s   d ( x3 , x2 )  d (T x2 , T x1 )   [d ( x1 , T x2 )  d ( x2 , T x1 )]   [ sd ( x1 , x2 )  sd ( x2 , x3 )] 2

 s    d ( x0 , x1 ) , 1  s   then similarly, n

 s  d ( xn , xn 1 )    d ( x0 , x1 ), n  0,1, 2,. . . 1  s   Let

(3.3)

 n    n  s   n    max  ,   ,  . 1    1  s    

Thus, for any positive numbers m, n, m  n , we have

d ( xn , xm )  sd ( xn , xn 1 )  s 2 d ( xn 1 , xn 2 )  ...  s mn d ( xm1 , xm )

 s ( n  s n 1  ... s mn1  m1 ) d ( x0 , x1 )  sn (1  s  s 2  2  ...  s mn1 mn 1 ) d ( x0 , x1 ) s n d ( x0 , x1 ) .  1  s Take n sufficiently large, we obtain li m n   0 . Since s  1,   n

that d ( xn , xm )   . Thus {xn } is a Cauchy sequence. But

1 and s  (0, 0.5) , it follows 1 s

( X , d ) is a complete b -metric space, therefore

{xn } converges to attractor {x } in X . Since any Lipschitzian mapping is continuous, so li m n xn  x . We find x  li m n xn 1  li m n (T xn )  T (li m n xn )  T x , which gives x  T ( x ) , i.e. x is a attractor of T . This shows that for any x0  X the function iteration convergence in X and x is a attractor of T . Since T has at most one attractor we deduced that for every choice of x0  X the function iteration convergence to the same attractor, that is the unique attractor of Theorem 3.2. If { yn } is a sequence in

T:X X

on a complete

T . So we proved (3-i) and (3-ii).

X and { n } the →∞ sequence defined by (2.2) to a Zamfirescu contraction

b -metric space, provided that there is 0  s , s, sL "  1 , then

li m n  n  0  li m n yn  x . Proof. Firstly, we prove that T is a Zamfirescu contraction to a complete b -metric space (in similar manner to

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paper [2], Theorem 2). Let {x } be the iteration attractor and { yn } be a sequence in X and { n } the sequence given by (2.2), for each x, y  X . At least one of (Z1), (Z2) or (Z3) is true. If (Z1) holds, then

d (Tx, T y )   d ( x, y )

(3.4)

If (Z2) holds, we get

d (T x, Ty )   [d ( x, Tx)  d ( y, Ty )]   [d ( x, T x)  sd ( y, x)  s 2 d ( x, T x)  s 2 d (T x, T y )] , d (T x, T y ) 

s (1  s 2 ) d ( x, T x )  d ( x, y ) 2 1 s  1  s 2

(3.5)

If (Z3) holds, then similarly we get,

d (T x, Ty )   [d ( x, T y )  d ( y, T x)] . d (T x, T y ) 

2s s d ( x, T x )  d ( y, x) . 1  s 1  s

(3.6)

Therefore from (3.4)-(3.6), we have,

 (1  s 2 ) 2 s    s s  d (T x, T y )  max 0, , ,  d ( x, T x)  max ,  d ( x, y ) 2 2  1  s  1  s   1  s  1  s 



Let L  max 0,



 s s  (1  s 2 ) 2 s   , ,  , implies that 0  L, L "  1 .  and L "  max , 2 2 1  s  1  s   1  s  1  s 

Then

d (T x, T y )  Ld ( x, T x)  L " d ( x, y )

(3.7)

by conditions (3.7), prove that T is a Zamfirescu contraction, i.e. T holds a conditions (3.7). Now by triangle inequality and condition (3.7) we get,

d ( yn1 , x )  s d ( yn 1 ,T yn )  s d (T yn , x )

 s n  s[ L d ( x , T ( x ))  L " d ( x , yn )] , since x  T ( x ) , it results that

d ( yn1 , x )  s [ n  L " d ( x , yn )], n  0,1,2,... Suppose li m n  n  0 , then by (3.8) and Lemma 2.1 we obtain li m n yn  0 , since

(3.8)

0  sL "  1 .

The reverse implication is estimated,

 n  d ( yn 1 , T yn )  s d ( yn1 , x )  s d ( x , T yn )  s d ( yn 1 , x )  s[ L d ( x , T ( x ))  L " d ( x , yn )] , hence 0   n  s d ( yn 1 , x )  s L "∞ d ( x , yn ) ,

∞

n

which shows that li m n yn  x implies li m n  n  0 . Remark. Theorem 3.2 shows the Picard iteration or the function iteration corresponding to a strict contraction is

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stable for all such mappings in b -metric space (see [2]) and also we found some changes in conditions. Furthers it may be give some remarkable results. Acknowledgement. The Author first wishes to thanks Professor S. L. Singh for his precious suggestions for this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

V. Berinde, Contractii generalizate si aplicatii, Cub Press 22, Baia Mare, 1997. V. Berinde, On the stability of some fixed point procedures, Bul. Stiint. Univ. Baia Mare, Ser. B, Mathematica-Informatica, Vol.XVIII (2002), Nr.1, 7-14. Stefan Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5–11. MR1250922. Stefan Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena 46(2) (1998), 263– 276. MR1665883 (99j:54043). Stefan Czerwik, Krzysztof Dlutek, and S. L. Singh, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci. 11 (1997), 87–94. MR1659318. Alberta M. Harder, Troy L. Hicks, Stability results for fixed point iteration procedures, Math. Japon. 33(5) (1988), 693–706. MR0972379 (90a:54109a) Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe, Chaos and Fractals, new frontiers of science, second edition, Springer. B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226(1977), 257–290. MR0433430. B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1) (1990), 1–9. MR1048010 (91e:54101). B. E. Rhoades, Some fixed point iteration procedures, Internat. J. Math. Math. Sci. 14(1) (1991), 1–16. MR1087396 (91k:47148) B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, II. Indian J. Pure Appl. Math. 24 (11) (1993), 691–703. MR1251180 (95a:47064) S. L. Singh, Charu Bhatnagar, S. N. Mishra, Stability of iterative procedures for multivalued maps in metric spaces, Demonstratio Math. 38(4) (2005), 905–916. MR2199144 (2006j:49034) S. L. Singh, S. Czerwik, Krzysztof Krol and Abha Singh, Coincidences and Fixed points of hybrid contractions, Tamsui Oxford Journal of Mathematical Sciences 24 (4) (2008) 401-416. Tudor Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (Basel) 23 (1972), 292–298. MR0310859 (46 #9957).

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