Hindawi Publishing Corporation Journal of Function Spaces Volume 2015, Article ID 963016, 7 pages http://dx.doi.org/10.1155/2015/963016
Research Article New Fixed Point Results for Fractal Generation in Jungck Noor Orbit with đ -Convexity Shin Min Kang,1 Waqas Nazeer,2 Muhmmad Tanveer,3 and Abdul Aziz Shahid3 1
Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea Division of Science and Technology, University of Education, Lahore 54000, Pakistan 3 Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan 2
Correspondence should be addressed to Waqas Nazeer;
[email protected] Received 26 May 2015; Accepted 14 July 2015 Academic Editor: Pasquale Vetro Copyright Š 2015 Shin Min Kang et al. 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. We establish new fixed point results in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) by using Jungck Noor iteration with đ -convexity.
1. Introduction The fractal geometry in mathematics has presented some attractive complex graphs and objects to computer graphics. Fractal is a Latin word, derived from the word âFractusâ which means âBroken.â The term âfractalâ was first used by a young mathematician, Julia [1], when he was studying Cayleyâs problem related to the behavior of Newtonâs method in complex plane. Julia introduced the concept of iterative function system (IFS) and, by using it, he derived the Julia set in 1919. After that, in 1982, Mandelbrot [2] extended the work of Gaston Julia and introduced the Mandelbrot set, a set of all connected Julia sets. The fractal structures of Mandelbrot and Julia sets have been demonstrated for quadratic, cubic, and higher degree polynomials, by using Picard orbit which is an application of one-step feedback process [3]. Julia and Mandelbrot sets have been studied under the effect of noises [1â4] arising in the objects. In 1982, Mandelbrot [2] introduced superior iterates (a two-step feedback process) in the study of fractal theory and created superior Julia and Mandelbrot sets. Rani et al. [5â7] generated and analyzed superior Julia and superior Mandelbrot sets for quadratic, cubic, and đth degree complex polynomials. After creation of superior Mandelbrot sets, Negi and Rani [5] collected the properties of midgets of quadratic superior Mandelbrot sets. Negi and Rani [6] simulated the behavior of Julia sets using switching processes. Chauhan et al. [4]
obtained new Julia and Mandelbrot sets via Ishikawa iterates (an example of three-step feedback process). Kang et al. [8] introduced Julia and Mandelbrot sets in Jungck Mann and Jungck Ishikawa orbits. In 1994, Hudzik and Maligranda [9] discussed a few results connecting with đ -convex functions in second sense and some new results about Hadamardâs inequality for đ convex functions are discussed in [10, 11]. In 1915, Bernstein and Doetsch [12] proved a variant of Hermite-Hadamardâs inequality for đ -convex functions in second sense. Takahashi [13] 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. Recently, Ojha and Mishra [14] discussed an application of fixed point theorem for đ -convex function. It is a well known fact that đ -convexity and Ishikawa iteration play a vital role in the development of geometrical picturesque of fractal sets. The applications of fractal sets are in cryptography and other useful areas in our modern era. Our aim is to deal with generalization of đ -convexity, approximate convexity, and results of Bernstein and Doetsch [12]. The concept of đ -convexity and rational đ -convexity was introduced by Breckner and Orb´an [15] in 1978. In this paper, we establish some new fixed point results in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) by using Jungck Noor iteration with đ -convexity. We define the
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Journal of Function Spaces
Jungck Noor orbit and escape criterions for quadratic, cubic, and đth degree complex polynomials by using Jungck Noor iteration with đ -convexity.
Definition 7 (Jungck Noor orbit [8]). Let us consider the sequence {đĽđ } of iterates for any initial point đĽ0 â đ such that đđĽđ+1 = (1 â đź)đ đđĽđ + đźđ đđŚđ ;
2. Preliminaries
đ
Definition 1 (Mandelbrot set [2, 6]). The Mandelbrot set đ for the quadratic polynomial đđ (đ§) = đ§2 + đ is defined as the collection of all đ â đś for which the orbit of the point 0 is bounded; that is, đ = {đ â đś : {đđđ (0)} ; đ = 0, 1, 2, . . .}
(1)
is bounded. An equivalent formulation is
đđŚđ = (1 â đ˝) đđ§đ + đ˝đ đđ˘đ ,
(8)
đ
đđ˘đ = (1 â đž) đđ§đ + đžđ đđ§đ , where đź, đ˝, đž, đ â (0, 1) for đ = 0, 1, 2, . . .. The above sequence of iterates is called Jungck three-step orbit or Jungck Noor orbit with đ -convexity, denoted by JNO, which is a function of six tuples (đ, đĽ0 , đź, đ˝, đž, and đ ). Remark 8. The JNO reduces to the following:
đ = {đ â đś : đđđ (0) does not tend to â as đ ół¨â â} .
(2)
(2) The Jungck Mann orbit when đ˝ = đž = 0, đ = 1.
We choose the initial point 0, as 0 is the only critical point of đđ . Definition 2 (Julia set [1]). The attractor basin of infinity is never all of đś, since đđ has fixed points đ§đ = 1/2 Âą â1/4 + đ (and also points of period đ, which satisfy a polynomial equation of degree 2đ ; namely, đđ (đ§) = đ§). The nonempty compact boundary of the attractor basin of infinity is called the Julia set of đđ : đ˝đ = đđ´ â (đ) .
(3)
Definition 3 (filled Julia set [1, 3, 7]). The filled Julia set of the function đ is denoted by đž(đ) and is defined as đž (đ) = {đ§ â đś : đđ (đ§) ół¨â â} .
(4)
Definition 4 (see [1, 3, 7]). The Julia set of the function đ is defined to be the boundary of filled Julia set đž(đ). That is, đ˝ (đ) = đđž (đ) .
(5)
Definition 5 (see [3]). Let {đ§đ : đ = 1, 2, 3, 4, . . .} be a sequence of complex numbers. Then, one says limđ â â đ§đ = â if, for given đ > 0, there exists đ > 0, such that, for all đ > đ, one must have |đ§đ | > đ. Thus all the values of đ§đ lie outside a circle of radius đ, for sufficiently large values of đ. Let đ (đ§) = đ0 đ§đ + đ1 đ§đâ1 + đ2 đ§đâ2 + â
â
â
+ đđâ1 đ§1 + đđ đ§0 ; đ0 ≠ 0
(1) The Jungck Ishikawa orbit when đž = 0, đ = 1.
(6)
be a polynomial of degree đ, where đ ⼠2. The coefficients are allowed to be complex numbers. In other words, it follows that đđ (đ§) = đ§2 + đ.
(3) The Jungck orbit when đ˝ = đž = 0 and đ , đź = 1. In nonlinear dynamics, we have 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 đ, such that |đđ (đ§)| ⤠đ, for all đ) is called 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 Mandelbrot set for đ.
3. Escape Criterions for the Complex Polynomials in Jungck Noor Orbit Now we prove the escape criterions of Julia and Mandelbrot sets for quadratic, cubic, and the higher degree complex polynomials in Jungck Noor orbit with đ -convexity. 3.1. Escape Criterion for the Quadratic Complex Polynomials. For quadratic complex polynomial đđ đ§ = đ§2 â đđ§ + đ, we will choose đđ§ = đ§2 + đ and đđ§ = đđ§, where đ and đ are complex numbers. Theorem 9. Assume that |đ§| ⼠|đ| > 2(1 + |đ|)/đ đź, |đ§| ⼠|đ| > 2(1+|đ|)/đ đ˝, and |đ§| ⼠|đ| > 2(1+|đ|)/đ đž, where 0 < đź, đ˝, đž, đ < 1 and đ is a complex parameter. Define đđ§1 = (1 â đź)đ đđ§ + đźđ đđŚ, .. .
Definition 6 (Picard orbit [3]). Let đ be a nonempty set and đ : đ â đ. For any point đĽ0 â đ, Picardâs orbit is defined as the set of iterates of a point đĽ0 ; that is,
đđ§đ = (1 â đź)đ đđ§đâ1 + đźđ đđŚđâ1 ;
đ (đ, đĽ0 ) = {đĽđ ; đĽđ = đ (đĽđâ1 ) , đ = 1, 2, 3, . . .} .
.. .
(7)
đ
đđŚ = (1 â đ˝) đđ§ + đ˝đ đđ˘,
Journal of Function Spaces
3 đ
đđŚđâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ˘đâ1 ,
Also, we have óľ¨ đ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨óľ¨ đ óľ¨óľ¨đđŚóľ¨óľ¨ = óľ¨óľ¨(1 â đ˝) đđ§ + đ˝ đđ˘óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨óľ¨(1 â đ˝) đđ§ + đ˝đ (đ˘2 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘2 óľ¨óľ¨óľ¨óľ¨ â đ đ˝ |đ| â |đđ§| + óľ¨óľ¨óľ¨đ đ˝đđ§óľ¨óľ¨óľ¨ ,
đ
đđ˘ = (1 â đ˝) đđ§ + đ˝đ đđ§, .. . đ
đđ˘đâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ§đâ1 , (9)
because đ < 1 óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘2 óľ¨óľ¨óľ¨óľ¨ â đ đ˝ |đ§| â |đđ§| ,
where đđ§ is injective, đđ§ is a quadratic polynomial, and đ = 2, 3, 4, . . .. Then, |đ§đ | â â as đ â â.
because |đ| ⼠0, |đ§| ⼠|đ| ,
Proof. Let đđ§ = đ§2 + đ and for đ§0 = đ§, đŚ0 = đŚ, and đ˘0 = đ˘, we have considered that đ
đđ˘đâ1 = (1 â đž) đđ§đâ1 + đžđ đđ§đâ1 óľ¨ óľ¨ đ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ đđ§óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ (đ§2 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ đ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + (1 â (1 â đž)) (đ§2 + đ)óľ¨óľ¨óľ¨óľ¨ .
(11)
because đ đ˝ < 1. Since |đ§| > 2(1+|đ|)/đ đž, this implies |đ§|2 (đ đž|đ§|/(1+|đ|)â1)2 > |đ§|2 . Hence |đ˘|2 > |đ§|2 (đ đž|đ§|/(1 + |đ|) â 1)2 > |đ§|2 > đ đž|đ§|2 ; this implies óľ¨ óľ¨ (16) đđŚ ⼠đ 2 đ˝đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| . Thus
Using binomialâs series up to linear terms of đź and (1 â đź), we get that óľ¨ óľ¨ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đ đž) đđ§ + (1 â đ (1 â đž)) (đ§2 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ (1 â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§2 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ (đ â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§2 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨ , because đ < 1 óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨
óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| ,
(10)
implies
(12)
2
đ đ˝đž |đ§| óľ¨óľ¨ óľ¨óľ¨ â 1) . óľ¨óľ¨đŚóľ¨óľ¨ ⼠|đ§| ( 1 + |đ| For đ§0 = đ§ and đŚ0 = đŚ, consider óľ¨óľ¨óľ¨đđ§đ óľ¨óľ¨óľ¨ = óľ¨óľ¨óľ¨(1 â đź)đ đđ§đâ1 + đźđ đđŚđâ1 óľ¨óľ¨óľ¨ , óľ¨ óľ¨ óľ¨ óľ¨ which implies that óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨ đ đ óľ¨óľ¨đđ§1 óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§ + đź đđŚóľ¨óľ¨óľ¨
(17)
(18)
(19)
yields óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨ đ đ óľ¨óľ¨đđ§1 óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§ + đź đđŚóľ¨óľ¨óľ¨
óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨ ,
óľ¨ óľ¨ = óľ¨óľ¨óľ¨óľ¨(1 â đź)đ đđ§ + (1 â (1 â đź))đ (đŚ2 + đ)óľ¨óľ¨óľ¨óľ¨
because |đ§| ⼠|đ|
óľ¨ óľ¨ âĽ (1 â đ (1 â đź)) óľ¨óľ¨óľ¨óľ¨đŚ2 + đóľ¨óľ¨óľ¨óľ¨ â |(1 â đ đź) đđ§|
óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| ,
óľ¨ óľ¨ âĽ đ đź óľ¨óľ¨óľ¨óľ¨đŚ2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§|
because |đ| ⼠0
(20)
óľ¨ óľ¨ âĽ đ 3 đźđ˝đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§|
gives us óľ¨ óľ¨ |đđ˘| ⼠đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â |đ§| â |đđ§| ,
⼠|đ§| (đ 3 đźđ˝đž |đ§| â (1 + |đ|)) . Hence
because đ đž < 1 óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â |đ| |đ§| â |đ§|
(13)
óľ¨ óľ¨ = đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| = |đ§| (đ đž |đ§| â (1 + |đ|)) . Thus |đ˘| ⼠|đ§| (
(15)
đ đž |đ§| â 1) . 1 + |đ|
(14)
đ 3 đźđ˝đž |đ§| óľ¨óľ¨ óľ¨óľ¨ â 1) , óľ¨óľ¨đ§1 óľ¨óľ¨ ⼠|đ§| ( 1 + |đ|
(21)
since |đ§| ⼠|đ| > 2(1 + |đ|)/đ đź, |đ§| ⼠|đ| > 2(1 + |đ|)/đ đ˝, and |đ§| ⼠|đ| > 2(1 + |đ|)/đ đž, so that |đ§| > 2(1 + |đ|)/đ 3 đźđ˝đž. Therefore there exist đ > 0, such that đ 3 đźđ˝đž|đ§|/(1 + |đ|) â 1 > 1 + đ. Consequently |đ§1 | > (1 + đ)|đ§|. In particular, |đ§đ | > |đ§|. So we may apply the same argument repeatedly to find |đ§đ | > (1+đ)đ |đ§|. Thus, the orbit of đ§ tends to infinity. This completes the proof.
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Journal of Function Spaces đ
Corollary 10. Suppose that |đ| > 2(1 + |đ|)/đ đź, |đ| > 2(1 + |đ|)/đ đ˝, and |đ| > 2(1 + |đ|)/đ đž; then the orbit of Jungck Noor JNO(đđ , 0, đź, đ˝, đž, đ ) escapes to infinity.
đđŚđâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ˘đâ1 ,
In the proof of theorem we used the facts that |đ§| ⼠|đ| > 2(1 + |đ|)/đ đź, |đ§| ⼠|đ| > 2(1 + |đ|)/đ đ˝, and |đ§| ⼠|đ| > 2(1 + |đ|)/đ đž. Hence the following corollary is the refinement of the escape criterion discussed in the above theorem.
.. .
Corollary 11 (escape criterion). Let |đ§| > max {|đ| ,
2 (1 + |đ|) 2 (1 + |đ|) 2 (1 + |đ|) , , } ; (22) đ đź đ đ˝ đ đž
đ
đđ˘ = (1 â đ˝) đđ§ + đ˝đ đđ§,
đ
đđ˘đâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ§đâ1 , (25) where đđ§ is injective, đđ§ is a cubic polynomial, and đ = 2, 3, 4, . . .. Then, |đ§đ | â â as đ â â. Proof. Let đđ§ = đ§3 + đ and for đ§0 = đ§, đŚ0 = đŚ, and đ˘0 = đ˘, we have considered that đ
đđ˘đâ1 = (1 â đž) đđ§đâ1 + đžđ đđ§đâ1
then |đ§đ | > (1 + đ)đ |đ§| and |đ§đ | â â as đ â â. implies
Corollary 12. Suppose that
óľ¨ óľ¨ đ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ đđ§óľ¨óľ¨óľ¨óľ¨
2 (1 + |đ|) 2 (1 + |đ|) 2 (1 + |đ|) óľ¨óľ¨ óľ¨óľ¨ , , } , (23) óľ¨óľ¨đ§đ óľ¨óľ¨ > max {|đ| , đ đź đ đ˝ đ đž for some đ ⼠0. Then |đ§đ+1 | > (1 + đ)đ |đ§đ | and |đ§đ+1 | â â as đ â â. This corollary gives us an algorithm in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) for đđ đ§. Given any point |đ§| ⤠|đ|, we have computed the orbit âJNOâ of đ§. If, for some đ, |đ§đ | lies outside the circle of radius max {|đ| ,
(26)
2 (1 + |đ|) 2 (1 + |đ|) 2 (1 + |đ|) , , }, đ đź đ đ˝ đ đž
óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ (đ§3 + đ)óľ¨óľ¨óľ¨óľ¨
óľ¨ óľ¨ đ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + (1 â (1 â đž)) (đ§3 + đ)óľ¨óľ¨óľ¨óľ¨ .
Using binomialâs series up to linear terms of đź and (1 â đź), we get that óľ¨ óľ¨ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đ đž) đđ§ + (1 â đ (1 â đž)) (đ§3 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ (1 â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§3 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ (đ â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§3 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨ ,
(24)
because đ < 1 óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨
we guarantee that the orbit escapes. Hence, đ§ is not in the Julia sets and also is not in the Mandelbrot sets. On the other hand, if |đ§đ | never exceeds this bound, then by definition of the Julia sets and the Mandelbrot sets, we can make extensive use of this algorithm in the next section. 3.2. Escape Criterion for the Cubic Complex Polynomials. For cubic complex polynomial đđ đ§ = đ§3 â đđ§ + đ, we will choose đđ§ = đ§3 + đ and đđ§ = đđ§, where đ and đ are complex numbers. Theorem 13. Assume that |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/2 , (2(1 + |đ|)/đ đ˝)1/2 and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/2 , where 0 < đź, đ˝, đž, đ < 1 and đ is a complex parameter. Define đ
because |đ§| ⼠|đ| óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| , because |đ| ⼠0 gives us óľ¨ óľ¨ |đđ˘| ⼠đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â |đ§| â |đđ§| , because đ đž < 1 óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â |đ| |đ§| â |đ§|
đđ§1 = (1 â đź) đđ§ + đź đđŚ,
đ
.. .
(29)
óľ¨ óľ¨ = đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§|
.. .
đđŚ = (1 â đ˝) đđ§ + đ˝đ đđ˘,
(28)
óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨ ,
đ
đđ§đ = (1 â đź)đ đđ§đâ1 + đźđ đđŚđâ1 ;
(27)
óľ¨ óľ¨ = |đ§| (đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|)) . Thus |đ˘| ⼠|đ§| (
óľ¨ óľ¨ đ đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨
1 + |đ|
â 1) .
(30)
Journal of Function Spaces
5 Corollary 14. Suppose that |đ§| ⼠|đ| > (2(1+|đ|)/đ đź)1/2 , (2(1+ |đ|)/đ đ˝)1/2 and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/2 ; then the orbit of Jungck Noor JNO(đđ , 0, đź, đ˝, đž, đ ) escapes to infinity.
Also, we have óľ¨ đ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨óľ¨ đ óľ¨óľ¨đđŚóľ¨óľ¨ = óľ¨óľ¨(1 â đ˝) đđ§ + đ˝ đđ˘óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨óľ¨(1 â đ˝) đđ§ + đ˝đ (đ˘3 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘3 óľ¨óľ¨óľ¨óľ¨ â đ đ˝ |đ| â |đđ§| + óľ¨óľ¨óľ¨đ đ˝đđ§óľ¨óľ¨óľ¨ , because đ < 1 óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘3 óľ¨óľ¨óľ¨óľ¨ â đ đ˝ |đ§| â |đđ§| ,
In the proof of theorem we used the facts that |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/2 , (2(1 + |đ|)/đ đ˝)1/2 and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/2 . Hence the following corollary is the refinement of the escape criterion discussed in the above theorem. (31)
Corollary 15 (escape criterion). Let
because |đ| ⼠0, |đ§| ⼠|đ| , |đ§| > max {|đ| , (
óľ¨ óľ¨ âĽ đ đ˝ óľ¨óľ¨óľ¨óľ¨đ˘3 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| , because đ đ˝ < 1. Since |đ§| > (2(1+|đ|)/đ đž)1/2 , this implies |đ§|3 (đ đž|đ§|2 /(1+|đ|)â 1)3 > |đ§|3 . Hence |đ˘|3 > |đ§|3 (đ đž|đ§|2 /(1 + |đ|) â 1)3 > |đ§|3 > đ đž|đ§|3 ; this implies óľ¨ óľ¨ (32) đđŚ ⼠đ 2 đ˝đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| . Thus
óľ¨ óľ¨ đ 2 đ˝đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ â 1) . óľ¨óľ¨đŚóľ¨óľ¨ ⼠|đ§| ( 1 + |đ|
For đ§0 = đ§ and đŚ0 = đŚ, consider óľ¨óľ¨óľ¨đđ§đ óľ¨óľ¨óľ¨ = óľ¨óľ¨óľ¨(1 â đź)đ đđ§đâ1 + đźđ đđŚđâ1 óľ¨óľ¨óľ¨ , óľ¨ óľ¨ óľ¨ óľ¨ which implies that óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨ đ đ óľ¨óľ¨đđ§1 óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§ + đź đđŚóľ¨óľ¨óľ¨
(33)
(34)
(35)
yields
2 (1 + |đ|) 1/2 2 (1 + |đ|) 1/2 ) ,( ) , đ đź đ đ˝
2 (1 + |đ|) 1/2 ( ) }; đ đž
(38)
then |đ§đ | > (1 + đ)đ |đ§| and |đ§đ | â â as đ â â. Corollary 16. Suppose that 2 (1 + |đ|) óľ¨óľ¨ óľ¨óľ¨ ) óľ¨óľ¨đ§đ óľ¨óľ¨ > max {|đ| , ( đ đź
1/2
,(
2 (1 + |đ|) 1/2 ) , đ đ˝
2 (1 + |đ|) 1/2 ( ) }, đ đž
(39)
for some đ ⼠0. Then |đ§đ+1 | > (1 + đ)đ |đ§đ | and |đ§đ+1 | â â as đ â â. This corollary gives us an algorithm in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) for đđ đ§.
óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨ đ đ óľ¨óľ¨đđ§1 óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§ + đź đđŚóľ¨óľ¨óľ¨ óľ¨ óľ¨ = óľ¨óľ¨óľ¨óľ¨(1 â đź)đ đđ§ + (1 â (1 â đź))đ (đŚ3 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ âĽ (1 â đ (1 â đź)) óľ¨óľ¨óľ¨óľ¨đŚ3 + đóľ¨óľ¨óľ¨óľ¨ â |(1 â đ đź) đđ§| óľ¨ óľ¨ âĽ đ đź óľ¨óľ¨óľ¨óľ¨đŚ3 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| óľ¨ óľ¨ âĽ đ 3 đźđ˝đž óľ¨óľ¨óľ¨óľ¨đ§3 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| óľ¨ óľ¨ âĽ |đ§| (đ 3 đźđ˝đž óľ¨óľ¨óľ¨óľ¨đ§2 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|)) .
(36)
đ 3 đźđ˝đž |đ§|2 óľ¨óľ¨ óľ¨óľ¨ â 1) , óľ¨óľ¨đ§1 óľ¨óľ¨ ⼠|đ§| ( 1 + |đ|
(37)
3.3. Escape Criterion for Higher Degree Complex Polynomials. For higher degree complex polynomial đđ đ§ = đ§đ â đđ§ + đ, we will choose đđ§ = đ§đ + đ and đđ§ = đđ§, where đ = 2, 3, 4, . . . and đ and đ are complex numbers. Theorem 17. Assume that |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/(đâ1) , |đ§| ⼠|đ| > (2(1 + |đ|)/đ đ˝)1/(đâ1) , and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/(đâ1) , where 0 < đź, đ˝, đž, đ < 1 and đ is a complex parameter. Define
Hence
since |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/2 , (2(1 + |đ|)/đ đ˝)1/2 , and |đ§| ⼠|đ| > (2(1+|đ|)/đ đž)1/2 , so that |đ§| > (2(1+|đ|)/đ 3 đźđ˝đž)1/2 . Therefore there exist đ > 0, such that đ 3 đźđ˝đž|đ§2 |/(1 + |đ|) â 1 > 1 + đ. Consequently |đ§1 | > (1 + đ)|đ§|. In particular, |đ§đ | > |đ§|. So we may apply the same argument repeatedly to find |đ§đ | > (1+đ)đ |đ§|. Thus, the orbit of đ§ tends to infinity. This completes the proof.
đđ§1 = (1 â đź)đ đđ§ + đźđ đđŚ, .. . đđ§đ = (1 â đź)đ đđ§đâ1 + đźđ đđŚđâ1 ; đ
đđŚ = (1 â đ˝) đđ§ + đ˝đ đđ˘, .. .
6
Journal of Function Spaces đ
đđŚđâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ˘đâ1 ,
gives us óľ¨ óľ¨ |đđ˘| ⼠đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â |đ§| â |đđ§| ,
đ
đđ˘ = (1 â đ˝) đđ§ + đ˝đ đđ§,
because đ đž < 1
.. .
óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â |đ| |đ§| â |đ§|
đ
đđ˘đâ1 = (1 â đ˝) đđ§đâ1 + đ˝đ đđ§đâ1 ,
óľ¨ óľ¨ = đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§|
(40) where đđ§ is injective, đđ§ = đ§đ + đ, and đ = 2, 3, 4, . . .. Then, |đ§đ | â â as đ â â. Proof. To prove the theorem, we follow the mathematical induction. For đ = 2, đđ§ = đ§2 + đ, so the escape criterion is |đ§| > max{|đ|, 2(1 + |đ|)/đ đź, 2(1 + |đ|)/đ đ˝, 2(1 + |đ|)/đ đž}. For đ = 3, đđ§ = đ§3 + đ, so escape criterion is |đ§| > max {|đ| , (
2 (1 + |đ|) 1/2 2 (1 + |đ|) 1/2 ) ,( ) , đ đź đ đ˝
2 (1 + |đ|) 1/2 ( ) }. đ đž
(41)
Hence the theorem is true for đ = 2, 3, . . . . Now suppose that theorem is true for any đ. Let đđ§ = đ§đ+1 + đ, đ§0 = đ§, đŚ0 = đŚ, |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/đ , |đ§| ⼠|đ| > (2(1 + |đ|)/đ đ˝)1/đ , and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/đ exist; then we have considered that đ
đđ˘đâ1 = (1 â đž) đđ§đâ1 + đžđ đđ§đâ1
(42)
implies óľ¨ óľ¨ đ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ đđ§óľ¨óľ¨óľ¨óľ¨
óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + đžđ (đ§đ+1 + đ)óľ¨óľ¨óľ¨óľ¨
(43)
óľ¨ óľ¨ đ đ = óľ¨óľ¨óľ¨óľ¨(1 â đž) đđ§ + (1 â (1 â đž)) (đ§đ+1 + đ)óľ¨óľ¨óľ¨óľ¨ .
Using binomialâs series up to linear terms of đź and (1 â đź), we get that óľ¨ óľ¨ |đđ˘| = óľ¨óľ¨óľ¨óľ¨(1 â đ đž) đđ§ + (1 â đ (1 â đž)) (đ§đ+1 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ (1 â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§đ+1 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨
óľ¨ óľ¨ óľ¨ óľ¨ âĽ (đ â đ (1 â đž)) óľ¨óľ¨óľ¨óľ¨đ§đ+1 + đóľ¨óľ¨óľ¨óľ¨ â óľ¨óľ¨óľ¨(1 â đ đž) đđ§óľ¨óľ¨óľ¨ , because đ < 1 óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| + óľ¨óľ¨óľ¨đ đžđđ§óľ¨óľ¨óľ¨ , because |đ§| ⼠|đ| óľ¨ óľ¨ âĽ đ đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â đ đž |đ§| â |đđ§| , because |đ| ⼠0
(44)
(45)
óľ¨ óľ¨ = |đ§| (đ đž óľ¨óľ¨óľ¨đ§đ óľ¨óľ¨óľ¨ â (1 + |đ|)) . Thus
óľ¨ óľ¨ đ 2 đ˝đž óľ¨óľ¨óľ¨đ§đ óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ â 1) . óľ¨óľ¨đŚóľ¨óľ¨ ⼠|đ§| ( 1 + |đ|
(46)
For đ§0 = đ§ and đŚ0 = đŚ, consider óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ đ đ óľ¨óľ¨đđ§đ óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§đâ1 + đź đđŚđâ1 óľ¨óľ¨ , which implies that óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨ đ đ óľ¨óľ¨đđ§1 óľ¨óľ¨ = óľ¨óľ¨(1 â đź) đđ§ + đź đđŚóľ¨óľ¨óľ¨ yields óľ¨óľ¨óľ¨đđ§1 óľ¨óľ¨óľ¨ = óľ¨óľ¨óľ¨(1 â đź)đ đđ§ + đźđ đđŚóľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľ¨óľ¨ óľ¨ đ = óľ¨óľ¨óľ¨(1 â đź) đđ§ + (1 â (1 â đź))đ (đŚđ+1 + đ)óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ âĽ (1 â đ (1 â đź)) óľ¨óľ¨óľ¨óľ¨đŚđ+1 + đóľ¨óľ¨óľ¨óľ¨ â |(1 â đ đź) đđ§| óľ¨ óľ¨ âĽ đ đź óľ¨óľ¨óľ¨óľ¨đŚđ+1 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| óľ¨ óľ¨ âĽ đ 3 đźđ˝đž óľ¨óľ¨óľ¨óľ¨đ§đ+1 óľ¨óľ¨óľ¨óľ¨ â (1 + |đ|) |đ§| óľ¨ óľ¨ âĽ |đ§| (đ 3 đźđ˝đž óľ¨óľ¨óľ¨đ§đ óľ¨óľ¨óľ¨ â (1 + |đ|)) . Hence óľ¨ óľ¨ đ 3 đźđ˝đž óľ¨óľ¨óľ¨đ§đ óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ ⼠( đ§ â 1) , |đ§| óľ¨óľ¨ 1 óľ¨óľ¨ 1 + |đ|
(47) (48)
(49)
(50)
since |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/đ , |đ§| ⼠|đ| > (2(1 + |đ|)/đ đ˝)1/đ , and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/đ , so that |đ§| > (2(1 + |đ|)/đ 3 đźđ˝đž)1/đ . Therefore there exist đ > 0, such that đ 3 đźđ˝|đ§đ |/(1 + |đ|) â 1 > 1 + đ. Consequently |đ§1 | > (1 + đ)|đ§|. In particular, |đ§đ | > |đ§|. So we may apply the same argument repeatedly to find |đ§đ | > (1 + đ)đ |đ§|. Thus, the orbit of đ§ tends to infinity. This completes the proof. Corollary 18. Suppose that |đ§| ⼠|đ| > (2(1 + |đ|)/đ đź)1/(đâ1) , |đ§| ⼠|đ| > (2(1 + |đ|)/đ đ˝)1/(đâ1) , and |đ§| ⼠|đ| > (2(1 + |đ|)/đ đž)1/(đâ1) ; then the orbit JNO(đđ , 0, đź, đ˝, đž, đ ) escapes to infinity. Corollary 19 (escape criterion). Suppose that 2 (1 + |đ|) óľ¨óľ¨ óľ¨óľ¨ ) óľ¨óľ¨đ§đ óľ¨óľ¨ > max {|đ| , ( đ đź
1/(đâ1)
,
2 (1 + |đ|) 1/(đâ1) 2 (1 + |đ|) 1/(đâ1) ,( }, ( ) ) đ đ˝ đ đž
(51)
Journal of Function Spaces for some đ ⼠0. Then |đ§đ+1 | > (1 + đ)đ |đ§đ | and |đ§đ+1 | â â as đ â â. This corollary gives us an algorithm in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) for đđ đ§.
4. Conclusions In this paper, new fixed point results for Jungck Noor iteration with đ -convexity have been introduced in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics). The new escape criterions for complex quadratic, cubic, and đth degree polynomials have been established. If we take đ = 1, it provides previous existing results in the relative literature [8].
Conflict of Interests The authors declare that they have no competing interests.
Authorsâ Contribution All authors read and approved the final paper.
Acknowledgment This research is supported by Gyeongsang National University.
References [1] G. Julia, âSur lâiteration des functions rationnelles,â Journal de Math´ematiques Pures et Appliqu´ees, vol. 8, pp. 737â747, 1918. [2] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, NY, USA, 1982. [3] R. L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992. [4] Y. S. Chauhan, R. Rana, and A. Negi, âComplex dynamics of Ishikawa iterates for non integer values,â International Journal of Computer Applications, vol. 9, no. 2, pp. 9â16, 2010. [5] A. Negi and M. Rani, âMidgets of superior Mandelbrot set,â Chaos, Solitons and Fractals, vol. 36, no. 2, pp. 237â245, 2008. [6] A. Negi and M. Rani, âA new approach to dynamic noise on superior Mandelbrot set,â Chaos, Solitons & Fractals, vol. 36, no. 4, pp. 1089â1096, 2008. [7] M. Rani and R. Agarwal, âEffect of noise on Julia sets generated by logistic map,â in Proceedings of the 2nd International Conference on Computer and Automation Engineering (ICCAE â10), pp. 55â59, IEEE, Singapore, February 2010. [8] S. M. Kang, A. Rafiq, M. Tanveer, F. Ali, and Y. C. Kwun, âJulia and Mandelbrot sets in modified Jungck three-step orbit,â Wulfenia Journal (Klagentrut, Austria), vol. 22, no. 1, pp. 167â185, 2015. [9] H. Hudzik and L. Maligranda, âSome remarks on s-convex functions,â Aequationes Mathematicae, vol. 48, no. 1, pp. 100â 111, 1994. [10] M. Alomari and M. Darus, âHadamard-type inequalities for đ convex functions,â International Mathematical Forum, vol. 3, no. 40, pp. 1965â1975, 2008.
7 ¨ [11] U. S. Kirmaci, M. KlariËci´c Bakula, M. E. Ozdemir, and J. PeËcari´c, âHadamard-type inequalities for s-convex functions,â Applied Mathematics and Computation, vol. 193, no. 1, pp. 26â35, 2007. [12] F. Bernstein and G. Doetsch, âZur Theorie der konvexen Funktionen,â Mathematische Annalen, vol. 76, no. 4, pp. 514â 526, 1915. [13] W. Takahashi, âA convexity in metric space and nonexpansive mappings. I,â Kodai Mathematical Seminar Reports, vol. 22, no. 2, pp. 142â149, 1970. [14] D. B. Ojha and M. K. Mishra, âAn application of fixed point theorem for s-convex function,â International Journal of Engineering Science and Technology, vol. 2, no. 8, pp. 3371â3375, 2010. [15] W. W. Breckner and G. Orb´an, Continuity Properties of Rationally S-Convex Mappings with Values in Ordered Topological Liner Space, Babes, -Bolyai University, Kolozsv´ar, Romania, 1978.
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