7-IJAEST-Volume-No-2-Issue-No-2-Fixed-Point-Results-In-Tricorn-&-Multicorns-of-Ishikawa-Iteration-an by ISERP ISERP

Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 157 - 160

Fixed Point Results In Tricorn & Multicorns of Ishikawa Iteration and s-Convexity (FTMISC)

Manish Kumar Mishra , Deo Brat Ojha

mkm2781@rediffmail.com, deobratojha@rediffmail.com

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

Devdutt Sharma Research Scholar Mewar University Department of Computer Science & Engineering

Mandelbrot set, given by Mandelbrot in 1979 and its relative object Julia set due to their beauty and complexity of their nature have become elite area of research nowadays.

Keywords- Complex

Recently Shizuo [18], has presented the various properties of Multicorns and Tricorn along with beautiful figures. Shizuo has quoted the Multicorns as the generalized Tricorn or the Tricorn of

INTRODUCTION

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Human beings always intend to understand the events easily with the help of geometrical aspects. It is due to high expectation for mutual fulfillment in our day to day computational life, specially in the developments of robotics control and optimization techniques. It is a well known fact that s-convexity and Ishikawa iteration plays vital role in the development of geometrical picturesque of fractal sets[9]. Further, we know very well about the applications of fractal sets in cryptography and other useful areas in our modern era. In this paper, we dealt with generalization of s-convexity, approximate convexity, and results of Bernstein and Doetsch [5]. The concept of s-convexity and rational s-convexity 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 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 HermiteHadamard’s inequality fors¡convex functions in second sense. 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 Ishikawa iteration procedure. Recently Ojha [10] discussed an application of fixed point theorem for s-convex function. The word “Fractal” from Latin word “fractus” meaning “broken” was introduced in 1975 by mathematician Benoit B. Mandelbrot to describe irregular and intricate natural phenomenon as lunar landscapes, mountains, trees branching and coastlines etc. The object

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higher order. The dynamics of antipolynomial z  z d  c of

dynamics, Ishikawa Iteration, Relative Superior Tricorn and Relative Fuzzy metric space, Superior Multicorns, s-convex function.

Abstract— In this paper, we established fixed point results in Tricorn and Multicorns of Ishikawa iteration and s- convexity.

complex polynomial z d  c , where d  2 , leads to interesting tricorn and multicorns antifractals with respect to function iteration (see [12] and [17, 18]). Tricorn are being used for commercial purpose, e.g. Tricorn mugs and Tricorn T shirts. Multicorns are symmetrical objects. Their symmetry has been studied by Lau and Schleicher [15]. The study of connectedness locus for antiholomorphic polynomials

z 2  c defined as Tricorn, coined by Milnor, plays intermediate role

between quadratic and cubic polynomials. Crowe etal.[11] considered it as in formal analogy with Mandelbrot set and named it as Mandelbar set and also brought its features bifurcations along axes rather than at points. Milnor [16] found it as a real slice of cubic connected locus. Winters [26] showed it as boundary along the smooth arc. Superior Tricorn and Superior Multicorns using the Mann iterates rather than function iterates is studied and explored by A. Ishikawa iterates and also study their corresponding Relative Superior Julia sets. .In this work, we want to investigate in Tricorn and Multicorns of using Ishikawa iteration for s-convexity as an application.

II.

PRELIMINARIES

2.1 Tricorn and Multicorns: Following the Milnor’s study, Shizuo [18] has defined the Tricorn, as n

the connectedness locus for antiholomorphic polynomials, z  c , where n  2 .

Definition2.1: The Multicorns c A , for the quadratic Ac  z   z n  c is defined as the collection of all c  C for

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Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 157 - 160

which the orbit of the point 0 is bounded, that is,

Definition: Relative Superior Mandelbort set RSM for the function of the form where Qc  z   z n  c ,

n  1, 2,3, 4......... is defined as the collection of c  C for which the orbit of 0 is bounded that is



equivalent



formulation



Ac  c  C :  Ac  0  not tends to  as n  

The Tricorn are special Multicorns when n = 2. As quoted by Shizuo [18], the Tricorn plays an intermediate role between quadratic and cubic polynomials. As quoted by Devaney [12], iterations of the function Ac  z   z 2  c , using the Escape Time Algorithm, results in many strange and surprising structures. Devaney [12] has named it Tricorn and observed that f  z   , the conjugate function of f  z  , is antipolynomial. Further, its second iterates is a polynomial of degree 4. Taking the initial choice z0 , one can iterate Ac1  z  , resulting z1 equals z0 2  c , which can be written as

z

/ z0



 c , since z0  z0 is equivalent to

gives z1 equals

z

/ z02



 z  , which 0

 c . Using this value one can



state the conjugate of z1 as z1  z0 / z02





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

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

z02  c . Now the second iterate can be stated as Ac2  z 

I  [0,1]



 c

[28].

 c , further, z04  2 z02c  c2  c , which is a

polynomial of degree 4 in z. Further, Devaney [12], has observed that

the function z12  c is conjugate of z12  d , where d  e  , which shows that the Tricorn is symmetric under rotations through angle 2 / 3 . The critical point for Ac is 0, since c  Ac  0  has 2 i/3

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only one preimage whereas any other w  C , has two preimages.

and

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

 xn  and  yn  in X, in the following manner: s s yn   pn  f  xn   1  pn  xn s s xn   pn 1  f  yn 1   1  pn 1  xn 1

the sequences

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

pn &

pn are both

Definition: 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 orbit

and

place

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

pn  pn  1

then

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.

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metric space, and

real sequences in

[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:

d   x, y, z, an , bn , cn  , u   (an ) s d  x, u   (bn ) s d  y, u  (cn ) s d  z, u 

Ishikawa s-Iteration

Mann’s

 X , d  be a an ,bn , cn 

Let

Definition

which is equal to z12  c , on simplifying, one can get

Is bounded .

 X , d  be a metric space, and I  [0,1]

Definition [8]. Let

 c , resulting

z



RSM  c  C : Qck  0  : k  0,1, 2...........



Ac  c  C :  Ac  0  : n  0,1, 2...........is bounded .

III.

MAIN RESULTS

Here we present the study of Relative superior Julia sets of Relative Superior Tricron by using the Escape Time Algorithm with respect to Ishikawa Iterates. Now we define escape criterions for these sets. Escape Criterion: We obtain a general escape Criterions for polynomials of the form Gc  z   z  c . n

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

, n  1, 2,3, 4...... where 0  p  1,0  p '  1 and c is a complex s

plane. Define z1  1  p  z  p s Gc  z  . . s

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

Thus



the

general 1/ n 1

max c ,  2 / p 

escape

1/ n 1

,  2 / p '

.

Criterion

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Manish Kumar Mishra* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 2, 157 - 160

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





is c ,which is obvious, i.e z  max c , 0, 0 . For n = 2, we get Gc  z   z  c . So, the escape criterion is 2

z  max  c ,  2 / p  ,  2 / p '  (See Theorem 3.1)



z  max c ,  2 / p 

1/ 2

,  2 / p '

.

z n1  c and z  c   2 / p 

1/ n 1

z  c   2 / p '

1/ n 1

as well as

exists. s

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

Then,

,where



z  c

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



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 1  p  z  p z p ' z n  1



pz



 pp ' z n 1  z



1/ n

z  2 / p

z   2 / pp '

; z  2 / p ' p

1/ n

; z   2 / p '

1/ n



Thus,

. . n

zn  1    z



and

, therefore pp ' z n  1  1 .



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

z1  1    z

,

for some k  0 .

zk 1   zk and zn   as n  . This Corollary

gives an algorithm for computing the Relative superior Julia sets for the functions of the form Gc  z   z  c , , n  1, 2,3, 4...... CONCLUSSION

[7]. Hudzik H. and L. Maligranda , Some remarks on si-convex functions, Aequationes Math. 48 ,1994,100–111.



1/ n

Then

,  2 / p '

that

[6]. Breckner W. W. and G. Orb´an, Continuity properties of rationally s-convex mappings with values in ordered topological liner space, ”Babes-Bolyai” University, Kolozsv´ar,1978

 z pp ' z n  1

Since

zk  max c ,  2 / p 

Assume

1/ k 1

[5]. Bernstein F. and G. Doetsch, Zur Theorie der konvexen Funktionen, Math. Annalen 76 , 1915,514–526.

 pp ' z n 1  z





3.2: 1/ k 1

[4]. Kirmaci U. S. et al., Hadamard-type inequalities for s-convex functions, Appl. Math. Comp., 2007,193,26-35.

 z  pz  p z . p ' z  p z



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

[3]. Dragomir S. S. and S. Fitzpatrick, The Hadamard‟s inequality for s¡convex functions in the second sense,Demonstratio Math. 32 (4) (1999) 687-696.

 z  p z pp ' z

exits. Then, the relative

[2]. Alomari M. and M. Darus , Hadamard-type inequalities for s¡convex functions, Inter. Math. Forum, 3(40) ,2008, 1965-1970.

Now,

n 1

and c   2 / p '

that

[1]. Alomari M. and M. Darus , On Co-ordinated s¡convex functions, Inter. Math. Forum, 3(40) (2008) 1977-1989.



c  2 / p

Suppose 1/ n 1

REFERENCES

3.1:

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

 is the

Application of here established results can be viewed as generalization of plotting process, to the RSM set for the function Ac using Ishikawa iterates. In our results, If we take s = 1, it pr ovides pervious existed results in the relative literature.



 z  zp ' p ' z n 1  p ' c  z

1/ n

,  2 / p '



 z  zp ' p ' z n 1  c

Corollary

IV.

Gc  z   z n1  c and for s  1

1/ n

escape criterion. This proves the theorem.

Corollary

Assume that the theorem is true for , n  1, 2,3, 4...... So, Let’s Gc  z  



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

1/ n 1

For n = 3, we get Gc  z   z 3  c . So, the escape criterion is 1/ 2

Therefore, the Ishikawa orbit of z under the iteration of z n 1  c

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[18]. Shizuo Nakane, and Dierk Schleicher, “On multicorns and unicorns: I. Antiholomorphic dynamics. hyperbolic components and real cubic polynomials”, Internat. J. Bifur. Chaos Appl. Sci. Engrg, (13)(10)(2003), 2825-2844. MR2020986. [19]. Ashish Negi, “Generation of Fractals and Applications”, Thesis, Gurukul Kangri Vishwvidyalaya, (2005).

[20]. M.O.Osilike, “Stability results for Ishikawa fixed point iteration procedure”, Indian Journal of Pure and Appl. Math., 26(1995), 937-945.

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[22]. Pitgen, Jurgens and Saupe, “Chaos and Fractals, SpringerVerlag”, NewYork, Inc., 1992. [23]. Mamta Rani, and Vinod Kumar, “Superior Mandelbrot sets”, J. Korea Soc. Math. Educ. Ser. D; Res. Math. Educ. (8)(4)(2004), 279-291.

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[26]. R. Winters, “Bifurcations in families of Antiholomorphic and biquadratic maps”, Thesis, Boston Univ. (1990). [27]. Rana, Dimri and tomar, “Fixed point theorems in fuzzy metric spaces using implicit relation” , International journal of computer applications vol. 8- no.1, 2010, pp. 16-21.

[28].

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