10-IJAEST-Volume-No-2-Issue-No-2-Some-Common-Fixed-Point-Results-In-Relative-Superior-Julia-Sets-Wit by ISERP ISERP

Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 175 - 180

Some Common Fixed Point Results In Relative Superior Julia Sets With Ishikawa Iteration And S-Convexity (SCFSC) *

Devdutt Sharma

Manish Kumar Mishra , Deo Brat Ojha

mkm2781@rediffmail.com, deobratojha@rediffmail.com

Abstract— In this paper, we established some common fixed point results in relative superior Julia sets with Ishikawa iteration with s- convexity. Many authors have presented the

Keywords- Complex dynamics, Ishikawa Iteration, s- Convexity.

relative

Takahashi [8] first introduced a notion of convex metric space, which is more general space, and each linear normed space is a special example of the space. Very recently Chauhan and rana[9] discussed the dynamics and the method of generating fractal image for Ishikawa iteration procedure. Recently Ojha [10] discussed an application of fixed point theorem for s-convex function. In this work, we want to investigate relative superior using Ishikawa iteration for sconvexity as an application.

papers on several “orbit traps” rendering methods to create the artistic fractal images. An orbit trap is a bounded area in complex plane into which an orbiting point may fall. Motivated by this idea of “orbit traps”, In this paper, we want to investigate relative superior using Ishikawa iteration for sconvexity as an application.

Research Scholar Mewar University Department of Computer Science & Engineering

Department of Mathematics R.K.G.Institute of Technology Ghaziabad,U.P.,INDIA

superior

Julia

Set,

II.

INTRODUCTION

IJ A

As we feel, with growing need of human beings and high expectation for mutual fulfillment in our day to day computational life through the help of computer, robotics control and optimization techniques. It is a well known fact that convexity and its generalization plays important role in different part of mathematics, mainly in optimization theory. In our paper we deal with a common generalization of sconvexity, approximate convexity, and results of Bernstein and Doetsch [5]. The concept of s-convexity and rational sconvexity was introduced by Breckner [6]. In 1978 Breckner[6] and H. Hudzik and L. Maligranda [7] it was proved that s-convex functions are nonnegative, when 0 < s <1, moreover the set of s-convex functions increases as s decreases. In the paper 1994 H. Hudzik and L. Maligranda[7] discussed a few results connecting with s-convex functions in second sense and some new results about Hadamard‟s inequality for s-convex functions are discussed in (M. Alomari and M. Darus [1-2], U. S. Kirmaci [4]), In 1999, S. S. Dragomir[3] et al. proved a variant of Hermite-Hadamard‟s inequality fors¡convex functions in second sense.

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Definition 1.1

[8].

PRELIMINARIES

Let

 X , d  be

a metric space, and

I  [0,1] . A mapping  : X 2  I  X is said to be x, y,  X and convex structure on X, if for any u  X , the following inequality holds:

d x, y, , u    s d x, u   1    d  y, u  If  X , d  is a metric space with a convex structure  , then  X , d  is called a convex metric space. Moreover, a nonempty subset E of X is said to be convex if x, y,  X for all x, y, E 2  I . s

 X , d  be a metric space, and an ,bn , cn  real sequences in

Definition 1.2 [36]. Let

I  [0,1]

and

[0,1] with an  bn  cn  1 . A mapping  : X 3  I 3  X is said to be convex structure on X, if for any

( x, y, z, an , bn , cn )  X 3  I 3 and u  X , the following

inequality holds:

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d   x, y, z, an , bn , cn  , u   (an ) s d  x, u   (bn ) s d  y, u  (cn ) s d  z, u  is a metric space with a convex structure

 , then

called a generalized convex metric space.

Moreover, a nonempty subset

E of X is said to be convex if

 ( x, y, z, an , bn , cn )  E 3

for

all

( x, y, z, an , bn , cn )  E  I . Let

zn : n  1, 2,.......

, denoted by

of complex numbers. Then , we say

 zn 

be a sequence

lim zn   if for given n 

M  0 , there exits N  0 , such that for all n  N , we must have zn  M . Thus all the values of z n , lies outside a circle of radius M, for sufficiently large values of n. Let

are allowed to be complex numbers. In other words, it follows

Qc  z   z 2  c . Definition: Let X be a nonempty set and f : X  X ,for any that

point x0  X , the Picard‟s orbit is defined as the set of of

point

that

is;

O  f , x0   xn ; xn  f  xn1  , n  1, 2,3, 4.......

IJ A

In functional dynamics, we have existence of two different types of points. Points that leave the interval after a finite number are in stable set of infinity. Points that never leave the interval after any number of iterations have bounded orbits. So, an orbit is bounded if there exists a positive real number, such that the modulus of every point in the orbit is less than this number. The collection of points that are bounded, i.e. there exists M, such that Qn  z   M for

all n, is called as a prisoner set while the collection of points that are in the stable set of infinity is called the escape set. Hence, the boundary of the prisoner set is simultaneously the boundary of escape set and that is Julia set for Q. Definition 2.2: Definition 2.2: The set of points K whose orbits are bounded under the iteration function of

Qc  z  is called the Julia set. We choose the initial point

0, as 0 is the only critical point of III.

Qc  z  .

MAIN RESULTS

Ishikawa Iteration For Relative Superior Julia Sets

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where 0  pn  1 and pn is convergent to non zero number and

x1   p0  f  y0   1  p0  x0 s

x2   p1  f  y1   1  p1  x1....... s

xn   pn 1  f  yn 1   1  pn 1  xn 1 s

where 0  pn  1 and number[21].

is convergent to non zero

Definition 3.1: The sequences xn and yn constructed above is called Ishikawa sequences of iteration or relative superior sequences of iterates. We denote it by RSO  x0 , pn , pn , t  .

Notice that RSO  x0 , pn , pn , t  with pn is RSO  x0 , pn , t  i.e

Q  z   a0 z n  a1 z n1  a2 z n2 .....  an1 z1  an z 0 ; a0  0 be a polynomial of degree n, where n  2 . The coefficients

iterates

 xn  and  yn  in X, in the following manner: s s y0   p0  f  x0   1  p0  x0 s s y1   p1  f  x1   1  p1  x1....... s s yn   pn  f  xn   1  pn  xn

the sequences

X , d   X , d  is

Let X be a subset of real or complex numbers and f : X  X , For x0  X , we construct

Mann‟s

orbit

and

pn  pn  1

place

then

RSO  x0 , pn , pn , t  reduces to O  x0 , t  .We remark that

Ishikawa orbit RSO  x0 , pn , pn , t  with pn  1/ 2 is Relative superior orbit. Now we define Julia set for function with respect to Ishikawa iterates. We call them as Relative Superior Julia sets. Definition 3.2: The set of points SK whose orbits are bounded under Relative superior iteration of function Q(z) is called Relative Superior Julia sets. Relative Superior Julia set of Q is boundary of Julia set RSK. We now define escape criterions for these sets. Relative Superior Escape Criterions for Quadratics The following theorem gives us an escape Criterions for

function Qc  z   z  c in respect to Ishikawa iteration 2

procedure. Theorem 3.1: Let‟s assume that z  c  2 / p; z  c  2 / p ' , where 0  p  1, 0  p '  1 and c is a complex number. Define z1  1  p  z  p s Qc  z  . . s

zn  1  p  zn 1  p s Qc  zn 1  s

Where Qc  z  can be a quadratic, cubic or biquadratic polynomial in terms of as

n  .

p ' and n  2,3, 4..... then zn   ,

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Proof: Let‟s take n, Qc  z  

exceeds this bound, then by definition of the Relative Superior Julia sets, denoted by RSKc . We can make extensive use of this algorithm in the next section.

z 2  c and for s  1

 p ' z 2  1  p ' z  p ' c

Relative Superior Escape Criterion for Cubic polynomials:

 p ' z 2  1  p '  z  p ' c

 p ' z  1  p '  p ' z  

 p ' z  1  p '  p ' z  z  p ' z  1

First, we prove the following theorem for the function Qa,b  z   z 3  az  b with respect to the Ishikawa iteration

z  c

 z

procedure. .... 1

Theorem

Now since, zn  1  p  zn 1  pQc  z 

z  b   a  2 / p

1/ 2

, z  b   a  2 / p '

1/ 2

that ,where

z1  1  p  z  p wQa,b  z  . . w

 p ' z  1

 z  pz  p z . p ' z  p z

zn  1  p  zn 1  p s Qa,b  zn 1  , n  2,3,..... s

  z  pz    p z . p ' z  p z 

Where Qa,b  z  is the function of p ' , then zn   , as

 z  pz  p z . p ' z  p z

n  .

 z 1  pp ' z  pp ' z  1  1   consequently

 0

, such that

Proof: Let‟s take , ,

Since s z  2, so, ss ' z  2 , there exits

Qa,b  z   1  p ' z   p ' Qa ,b  z  for s  1 s



z1  1    z

 1  p '  z  p ' z 3  az  b

. .



 z  p ' z  p ' z 3  p ' az  bp '  z  p ' z  p ' z 3  p ' az  bp '

zn  1    z n

Thus, the Ishikawa orbit of z, under the quadratic function tends to infinity. This completes the proof.

IJ A

Corollary 3.1: Suppose that c  2 / p; c  2 / p ' . Then, the

relative superior orbit of Ishikawa RSO  Qc ,0, p, p ' . In the proof of the theorem, we used z  c and z  2 / p as well as z  2 / p '

the facts . Hence,

that the

following corollary is the refinement of the escape criterion discussed in the above theorem. Corollary

assume

0  p  1,0  p '  1 and a and b are in complex plane. Define

So, z1  1  p  z  pQc  z  on substituting 1  1  p  z  p z

Let‟s

3.2:

 z

3.2(Escape

z  max  c , 2 / p, 2 / p '

Criterion):

then

Suppose

zn  1    z n

that and

zn   as n  .

 p'z  z  p'z  z  p'z  z  p' z  z p ' z  z p ' z  z p ' z  z

2 2

2 2 2 2 2

  ap '  1  p '   p ' z  ap '  1  p ' p '   a  1  a  1/ p '   a  1/ p '    a  1/ p '    p 'a 1 p '  p ' z

z b

Now since z1  1  p  z  p s Qa,b  z  for s = 1, s

(1.1)

Corollary 3.3: Suppose that zk  max  c , 2 / p, 2 / p ' , for some k  0 . Then zk 1  1    zk and zn   as n  . n

This corollary gives us an algorithm for computing the Relative Superior Julia sets of Qc , for any c. Given any point z  c , we have computed the superior orbit of z. If for some

n, zn lies outside the circle of radius max  c , 2 / p, 2 / p ' , we guarantee that the orbit escapes. Hence, z is not in the Relative Superior Julia sets. On the other hand, if

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zn never

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Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 175 - 180

 

 

Thus

 1  p  z  p z . p ' z 2  a  1











 a









For n = 2, we get Gc  z   z  c . So, the escape criterion is z  max  c ,  2 / p  ,  2 / p '  (See Theorem 3.1)

For n = 3, we get Gc  z   z 3  c . So, the result follows from

Theorem 3.2 with a = 0 and b = c, such that the escape criterion



and z   a  2 / p '

1/ 2



exits and so

 1/ pp ' such that



pp ' z   a  1/ pp '

1/ 2

 1/ pp '  1 . Hence, there exists

we get zn  

z . Therefore, the relative Superior orbit of z,

under the cubic polynomial Qa,b  z  , tends to infinite. This completes the proof.

Corollary 3.4: Suppose that b   a  2 / p 

1/ 2

b   a  2 / p '

1/ 2

and

exits. Then, the relative superior orbit of

Ishikawa RSO  Qc ,0, p, p ' escapes to infinity.



IJ A

z  max b   a  2 / p 

1/ 2

, b   a  2 / p '

1/ 2

, then

zn   as n  . Corollary 3.5 gives an escape criterion for

cubic polynomials. Corollary 3.6: Suppose that



zk  max b   a  2 / p 

1/ 2

for , n  1, 2,3, 4...... Now, suppose that theorem is true for any n. Let‟s Gc  z   z

n 1

 c and z  c   2 / p 

z  c   2 / p '

1/ n 1

, b   a  2 / p '

1/ 2

,where

Gc  z   z n1  c and for s  1



 z  zp ' p ' z n 1  c



 z  zp ' p ' z n 1  p ' c

 p ' z  p ' 1   p ' z  z  p' z  p'  1  p' z  z  p ' z  p ' 1  p '   z  p ' z 1   z



z  c

z1  1  p  z   p ' Gn  z  for s  1 s



 z  pz  p z . p ' z n  p z



Now ,  z  p z pp ' z n 1  p z





Theorem 3.1: The general function Gc  z   z n  c

 z pp ' z n  1

, n  1, 2,3, 4...... where 0  p  1,0  p '  1 and c is a complex number. Define z1  1  p  z  p Gc  z  . . s

zn  1  p  zn 1  p Gc  zn 1 



 1  p  z  p z p ' z n  1

 pp ' z n 1  z

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Then

3.3 A General Escape Criterion: We will obtain a general escape Criterion polynomials of the formn Gc  z   z n  c .

as well as

exists.

 , for some

3.6, we find an algorithm for computing the superior Julia sets of Qa,b  z  , for any a and b.

1/ n 1

Gn  z   1  p ' z   p ' Gc  z 

k  0 . Then zk 1   zk and zn   as n  . Corollary

 . Hence, the theorem is true

Corollary 3.5(Escape Criterion): Suppose that



1/ 2

,  2 / p '

  1 , such that z1   z . Repeating this argument n time, n

1/ 2

z  max c ,  2 / p 

follows. Therefore, z 2   a  1/ pp ' 2



   z p. p '1/ p. p ' z  a   z p. p ' z   a  1/ p. p '  Since z   a  2 / p 

.

is c ,which is obvious, i.e z  max c , 0, 0 .

 z p. p ' 1/ p. p ' z 2  a

1/ 2

,  2 / p '

Criterion

Proof: We shall prove this theorem by induction: For n = 1, we get Gc  z   z  c . So, the escape criterion

 z  p z  p. p ' z z 2  a  p z  z 1  p. p ' z 2

escape

1/ n 1



general 1/ n 1

max c ,  2 / p 

 z  pz  p z . p ' z  a  p z  z  p z  p z .p ' z2  a  p z

the



 pp ' z n 1  z



 1/ n

z  2 / p

Since 1/ n

z   2 / pp '

1/ n

; z  2 / p ' p

1/ n

; z   2 / p '



, therefore pp ' z n  1  1 .





Hence, for some   0 , we have pp ' z n  1  1   .



and

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Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 175 - 180

z1  1    z

[11]. R.Abraham and C.Shaw, Dynamics: “The Geometry of Behaviour, Part One: Periodic Behavior, Part Two: Chaotic Behavior”, Aerial Press,

. Thus, .

Santa Cruz, Calif, (1982). [12]. Alan F.Beardon, “Iteration of Rational functions”, Springer Verlag,

zn  1    z n

N.York, Inc.(1991).

Therefore, the Ishikawa orbit of z under the iteration of z 1/ n

1/ n

,  2 / p '

c

 is the

escape criterion. This proves the theorem.

Corollary 1/ n 1

c  2 / p

1/ n 1

and c   2 / p '

that

exits. Then, the relative

superior orbit of Ishikawa RSO  Gc ,0, p, p ' escapes to infinity. Corollary



1/ k 1

zk  max c   2 / p 

k  0 . Then

Assume

3.8:

, b   a  2 / p '

zk 1   zk

1/ k 1

and

“Iterative

approximation of

fixed points”, Editura

Efermeide, Baia Mare, (2002). [14]. S,Beddings and K.Briggs, “Itreation of quaterinian maps”, Int J. Bifur Chaos. Appl. Sci. and Engg.5 (1995), 877-881.

Suppose

3.7:

[13]. V.Berinde,

that

 , for some

zn   as n  . This

[15]. B.Branner and J.Hubbard, “The Iteration of Cubic Polynomials”. Part I,Acta. Math. 66(1998), 143-206. [16]. G.V.R.Babu and K.N.V.Prasad, “Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators”, Fixed Point Theory and its Applications, Vol. 2006, Art. ID 49615, 1-6(2006). [17]. P.W.Carlson, “Pseudo 3D rendering methods for fractals

in the

complex plane, Computer and Graphics”, 20(5), (1996), 757-758.



tends to infinity. Hence z  max c ,  2 / p 

n 1

[18]. R.M.Crownover, “Introduction to Fractals and Chaos, Mandelbrot sets”, Jones and Barlett Publishers, (1995).

Corollary gives an algorithm for computing the Relative superior Julia sets for the functions of the form Gc  z   z  c ,

[19]. R.L.Daveney,

, n  1, 2,3, 4......

[20]. K.Heinz, Becker and Michael Dorfler, “Dynamical Systems and

CONCLUSSION

IV.

“An Introduction to Chaotic

Dynamical Systems”,

Springer–Verlag, N.York.Inc.1994.

Fractals”, Cambridge Univ. Press, 1989.

As we know various applications of fractal sets in cryptography, stegnography digital signature, commitment scheme, key agreement protocol, that means all communications between two communicators with security. The results of [9] are as special case of our results for s =1.

[21]. R.A.Hohngren, “A First Course in Discrete Dynamical Systems”,

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