R E S E A R C H
Open Access
Fixed point results in the generation of
Julia and Mandelbrot sets
Waqas Nazeer
1, Shin Min Kang
2*, Muhmmad Tanveer
3and Abdul Aziz Shahid
3*Correspondence:
2Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract
The aim of this paper is to establish some fixed point results in the generation of Julia and Mandelbrot sets by using Jungck Mann and Jungck Ishikawa iterations with s-convexity.
Keywords: Julia set; Mandelbrot set; Jungck Mann iteration; Jungck Ishikawa iteration
1 Introduction
In mathematics, the fractal geometry has presented some attractive complex graphs and objects to computer graphics. Fractal is a Latin word, derived from the word ‘fractus’ which means ‘broken’. First time the term fractal was used by a young mathematician Gaston Julia [], when he was studying Cayley ’s problem related to the behavior of Newton’s method in complex plane. Julia introduced the concept of iterative function system (IFS), and, by using it, Julia derived the Julia set in . After that, in , Benoit Mandelbrot extended the ideas of Julia. He introduced the Mandelbrot set by using the complex functionz+ cwith usingzas a complex function andcas a complex parameter [–]. The fractal structure of Mandelbrot and Julia sets has been demonstrated for quadratic, cubic and higher degree polynomials, by using the Picard orbit which is an application of one-step feedback process [].
Julia and Mandelbrot sets have been studied under the effect of noises [–] arising in the objects. In , Rani and Kumar [, ] introduced superior iterates (a two-step feedback process) in the study of fractal theory and created superior Julia and Mandelbrot sets. Later on, in a series of papers Raniet al.generated and analyzed superior Julia and superior Mandelbrot sets for quadratic [–, –], cubic [], andnth degree [, , , –] complex polynomials. After creation of superior Mandelbrot sets, Negi and Rani [] collected the properties of midgets of quadratic superior Mandelbrot sets. Negi and Rani [] simulated the behavior of Julia sets using switching processes. Superior Julia and superior Mandelbrot sets have also been studied under the effect of noises [–, , , – ]. Chauhanet al.[, ] obtained new Julia and Mandelbrot sets via Ishikawa iterates (an example of three-step feedback process). Kanget al.[] introduced Julia and Mandelbrot sets in Jungck Mann and Jungck Ishikawa orbits.
Mandelbrot set serves as a lexicon for the Julia set. The location of the parameter c within the Mandelbrot set furnishes information on properties of the corresponding Julia
set. There are similarities between magnified positions of the Mandelbrot set, and the corresponding filled Julia set holds only near certaincvalues such as the central junctions of the antenna. These arecvalues, for which is eventually periodic; suchcvalues are called Misiurewicz points [].
It is a well-known fact thats-convexity and Ishikawa iteration play a vital role in the de-velopment of geometrical pictures of fractal sets []. 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 deal with generalization ofs-convexity, approximate convexity and the results of Bernstein and Doetsch []. The concept ofs-convexity and rationals-convexity was introduced by Breckner and Orbán []. In Breckner and Orbán [] and Hudzik and Maligranda [] proved thats-convex functions are nonnegative when <s< ; more-over, the set ofs-convex functions increases as s decreases.
In , Hudzik and Maligranda [] discussed a few results related withs-convex func-tions in the second sense, and some new results about Hadamard’s inequality fors-convex functions were discussed by Alomari and Darus [, ] and Kirmaciet al.[]. In , Dragomir and Fitzpatrick [] proved a variant of Hermite-Hadamard’s inequality fors -convex functions in the second sense. Takahashi [] first introduced a notion of -convex metric space, which is a more general space, and each linear normed space is a special example of the space. Very recently Ranaet al.[] discussed the dynamics of Ishikawa iteration procedure. Recently Ojha and Mishra [] discussed an application of a fixed point theorem fors-convex function.
In this paper, we establish some new fixed point results in the generation of Julia and Mandelbrot sets by using Jungck Mann and Jungck Ishikawa iterations with s -convexity. We define the Jungck Mann and Jungck Ishikawa orbits and escape criterions for quadratic, cubic and nth degree complex polynomials by using Jungck Mann and Jungck Ishikawa iterations withs-convexity.
2 Preliminaries
Definition .(Mandelbrot set [–]) The Mandelbrot set M for the quadraticQc(z) =
z+cis defined as the collection of allc∈Cfor which the orbit of the point is bounded,
that is,
M=c∈C:Qnc();n= , , , . . .
is bounded. An equivalent formulation is
M=c∈C:Qnc() does not tend to∞asn→ ∞.
We choose the initial point , as is the only critical point ofQc.
Definition .(Julia set []) The attractor basin of infinity is never all ofCsincefchas
fixed pointszf = /±
√
/ +c(and also points of periodnthat satisfy a polynomial equa-tion of degree n, namelyfn(z) =z). The nonempty, compact boundary of the attractor
basin of infinity is called the Julia set offc,
Definition .(Filled Julia set [, , –]) The filled in Julia set of the functionf is defined as
K(f) =z∈C:fk(z)→ ∞.
Definition .([, , –]) The Julia set of the functionf is defined to be the boundary
ofK(f),i.e.,
J(f) =∂K(f).
Definition . [] Let{zn:n= , , , , . . .} denoted by{zn} be a sequence of complex
numbers. Then we sayLimn→∞zn=∞if for givenM> , there existsN> such that for
alln>N, we must have|zn|>M. Thus all the values ofznlie outside a circle of radiusM
for sufficiently large values ofn. Let
Q(z) =azn+azn–+azn–+· · ·+an–z+anz; a=
be a polynomial of degree n, wheren≥. The coefficients are allowed to be complex numbers. In other words, it follows thatQc(z) =z+c.
Definition .(Picard orbit []) LetXbe a nonempty set andf :X→X. For any point x∈X, the Picard orbit is defined as the set of iterates of a pointx, that is,
O(f,x) =
xn;xn=f(xn–),n= , , , . . .
.
Definition .(Jungck Mann orbit []) Let us consider the sequence{xn}of iterates for
any initial pointx∈Xsuch that
Sxn+:Sxn+= ( –α)sSxn+αsTxn
,
where α,s∈(, ) forn= , , , . . . . The above sequence of iterates with s-convexity is called Jungck Mann orbit, denoted byJMO, which is a function of four tuple (T,x,α,s).
Definition .(Jungck Ishikawa orbit []) Let us consider the sequence{xn}of iterates
for any initial pointx∈Xsuch that
Sxn+:Sxn+= ( –α)sSxn+αsTyn;
Syn= ( –β)sSxn+βsTxn
,
whereα,β,s∈(, ) forn= , , , . . . . The above sequence of iterates withs-convexity is called Jungck Ishikawa orbit, denoted byJIO, which is a function of six tuple (T,x,α,β,s).
Remark . TheJIOreduces to:
In nonlinear dynamics, we have two different types of points. Points that leave the inter-val after a finite number are in a stable set of infinity. Points that never leave the interinter-val 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 num-ber. The collection of points that are bounded,i.e., there existsMsuch that|Qn(z)| ≤M,
for alln, is called a prisoner set, while the collection of points that are in the stable set of infinity is called an escape set. Hence, the boundary of the prisoner set is simultaneously the boundary of the escape set and that is the Mandelbrot set forQ.
3 Escape criterions for the complex polynomials in Jungck Mann orbit
The escape criterion is the key to generate the Julia sets and Mandelbrot sets. In this paper, we prove the escape criterions of Julia and Mandelbrot sets for quadratics, cubics and the higher degree complex polynomials in Jungck Mann orbit.
3.1 Escape criterion for the quadratic complex polynomials
For quadratic complex polynomialp(z) =z–az+c, we will chooseTz=z+candSz=az,
whereaandcare complex numbers.
Theorem . Assume that|z| ≥ |c|>(+sα|a|),where <α,s< and c is a complex param-eter.Define
Sz= ( –α)sSz+αsTz,
.. .
Szn= ( –α)sSzn–+αsTzn–,
where Sz is injective,Tz is a quadratic polynomial and n= , , , . . . ,then|zn| → ∞as
n→ ∞.
Proof LetTz=z+candz
=z, then we have
Szn= ( –α)sSzn–+αsTzn–
implies
|Sz|=( –α)sSz+αsTz
=( –α)saz+αsz+c
=( –α)saz+ – ( –α)sz+c.
Using binomial series up to linear terms ofαand ( –α) , we get
|Sz|=( –sα)az+
≥sαz–sα|c|–|az|+|sαaz|
≥sαz–sα|z|–|az|+|sαaz|, |z| ≥ |c| ≥sαz–sα|z|–|az|, |a| ≥
gives us
|az| ≥sαz–|z|–|az|, sα<
≥sαz–|a||z|–|z| =sαz– +|a||z| =|z|sα|z|– +|a|. Thus
|z| ≥ |z|
sα|z| +|a|–
since|z| ≥ |c|>(+sα|a|), therefore there existsλ> such thatsα+||za||– > +λ. Consequently, |z|> ( +λ)|z|. In particular,|zn|>|z|. So we may apply the same argument repeatedly to
find|zn|> ( +λ)n|z|. Thus, the orbit ofztends to infinity. This completes the proof.
Corollary . Suppose that|c|> (+sα|a|),then the orbit of Jungck Mann JMO(Tc, ,α,s)
escapes to infinity.
In the proof of the theorem we used the fact that|z| ≥ |c|>(+sα|a|). Hence the following corollary is the refinement of the escape criterion discussed in the above theorem.
Corollary . (Escape criterion) Let |z| >max{|c|,(+sα|a|)}, then |zn| > ( +λ)n|z| and
|zn| −→ ∞as n→ ∞.
Corollary . Suppose that|zk|>max{|c|,(+sα|a|)}for some k≥.Then|zk+|> ( +λ)n|zk|
and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTc. Given any point|z| ≤ |c|, we have computed the orbit ‘JMO’ ofz. If for somen,|zn|
lies outside the circle of radiusmax{|c|,(+sα|a|)}, we guarantee that the orbit escapes. Hence, zis not in the Julia sets and also it is not in the Mandelbrot sets. On the other hand, if|zn|
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 polynomialp(z) =z–az+c, we will chooseTz=z+candSz=az, whereaandcare complex numbers.
Theorem . Assume that|z| ≥ |c|> ((+sα|a|)),where <α,s< and c is a complex pa-rameter.Define
.. .
Szn= ( –α)sSzn–+αsTzn–,
where Sz is injective,Tz is a cubic polynomial and n= , , , . . . ,then|zn| → ∞as n→ ∞.
Proof LetTz=z+candz
=z, then we have
Szn= ( –α)sSzn–+αsTzn–
implies
|Sz|=( –α)sSz+αsTz
=( –α)saz+αsz+c
=( –α)saz+ – ( –α)sz+c.
Using binomial series up to linear terms ofαand ( –α), we get
|Sz|=( –sα)az+
–s( –α)z+c ≥ –s( –α)z+c–( –sα)az ≥s–s( –α)z+c–( –sα)az, s< ≥sαz–sα|c|–|az|+|sαaz|
≥sαz–sα|z|–|az|+|sαaz|, |z| ≥ |c| ≥sαz–sα|z|–|az|, |a| ≥
gives us
|az| ≥sαz–|z|–|az|, sα<
≥sαz–|a||z|–|z| =sαz– +|a||z| =|z|sαz– +|a|. Thus
|z| ≥ |z|
sα|z|
+|a|–
since|z| ≥ |c|> ((+sα|a|)), therefore there existsλ> such that sα|z |
+|a| – > +λ.
Con-sequently,|z|> ( +λ)|z|. In particular,|zn|>|z|. So we may apply the same argument
repeatedly to find|zn|> ( +λ)n|z|. Thus, the orbit ofztends to infinity. This completes
the proof.
In the proof of the theorem we used the fact that|z| ≥ |c|> ((+sα|a|)). Hence the follow-ing corollary is the refinement of the escape criterion discussed in the above theorem.
Corollary . (Escape criterion) Let|z|>max{|c|, ((+sα|a|))},then |zn|> ( +λ)n|z| and
|zn| −→ ∞as n→ ∞.
Corollary . Suppose that|zk|>max{|c|, ((+sα|a|))
}for some k≥.Then|zk+|> ( + λ)n|z
k|and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTcz=z+c.
3.3 Escape criterion for higher degree complex polynomials
For higher degree complex polynomialp(z) =zn–az+c, we will chooseTz=zn+cand Sz=az, wheren= , , , . . . ,aandcare complex numbers.
Theorem . Assume that|z| ≥ |c|> ((+sα|a|))n–,where <α,s< and c is a complex parameter.Define
Sz= ( –α)sSz+αsTz,
.. .
Szn= ( –α)sSzn–+αsTzn–,
where Sz is injective,Tz=zn+c and n= , , , . . . ,then|z
n| → ∞as n→ ∞.
Proof To prove the theorem, we follow the mathematical induction. Forn= ,Tz=z+c,
so the escape criterion is|z|>max{|c|,(+sα|a|)}. Forn= ,Tz=z+c, so the escape criterion
is|z|>max{|c|, ((+sα|a|))/}. Hence the theorem is true forn= , , . . . . Now suppose that
the theorem is true for anyn. LetTz=zn++c,z
=zand|z| ≥ |c|> ((+sα|a|))
n exist, then
we have
Szn= ( –α)sSzn–+αsTzn–
implies
|Sz|=( –α)sSz+αsTz
=( –α)saz+αszn++c
=( –α)saz+ – ( –α)szn++c.
Using binomial series up to linear terms ofαand ( –α), we get
|Sz|=( –sα)az+
–s( –α)zn++c ≥ –s( –α)zn++c–( –sα)az
≥sαzn+–sα|c|–|az|+|sαaz|
≥sαzn+–sα|z|–|az|+|sαaz| because|z| ≥ |c| ≥sαzn+–sα|z|–|az| because|a| ≥
gives us
|az| ≥sαzn+–|z|–|az| becausesα<
≥sαzn+–|a||z|–|z| =sαzn+– +|a||z| =|z|sαzn– +|a|. Thus
|z| ≥ |z|
sα|zn|
+|a|–
since|z| ≥ |c|> ((+sα|a|))n, so that|z|> ((+|a|) sα )
n. Therefore there existsλ> such that sα|zn|
+|a| – > +λ. Consequently,|z|> ( +λ)|z|. In particular,|zn|>|z|. So we may apply the
same argument repeatedly to find|zn|> ( +λ)n|z|. Thus, the orbit ofztends to infinity.
This completes the proof.
Corollary . Suppose that|c|> ((+sα|a|))n–,then the orbit JMO(Tc, ,α,s)escapes to in-finity.
Corollary .(Escape criterion) Suppose that|zk|>max{|c|, ((+sα|a|))
n–}for some k≥. Then|zk+|> ( +λ)n|zk|and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTcz=zn+c.
4 Escape criterions for the complex polynomials in Jungck Ishikawa orbit Now we prove the escape criterions of Julia and Mandelbrot sets for quadratics, cubics and the higher degree complex polynomials in Jungck Ishikawa orbit.
4.1 Escape criterion for the quadratic complex polynomials
For quadratic complex polynomialp(z) =z–az+c, we will chooseTz=z+candSz=az,
whereaandcare complex numbers.
Theorem . Assume that|z| ≥ |c|>(+sα|a|),|z| ≥ |c|>(+sβ|a|),where <α,β,s< and c is a complex parameter.Define
Sz= ( –α)sSz+αsTy,
.. .
where
Sy= ( –β)sSz+βsTz,
.. .
Syn–= ( –β)sSzn–+βsTzn–,
Sz is injective,Tz is a quadratic polynomial and n= , , , . . . ,then|zn| → ∞as n→ ∞.
Proof LetTz=z+cand forz
=zandy=y, then we have
Syn–= ( –β)sSzn–+βsTzn–,
implies
|Sy|=( –β)sSz+βsTz =( –β)saz+βsz+c
=( –β)saz+ – ( –β)sz+c.
Using binomial series up to linear terms ofαand ( –α), we get
|Sy|=( –sβ)az+ –s( –β)z+c ≥ –s( –β)z+c–( –sβ)az
≥s–s( –β)z+c–( –sβ)az becauses< ≥sβz–sβ|c|–|az|+|sβaz|
≥sβz–sβ|z|–|az|+|sβaz| because|z| ≥ |c| ≥sβz–sβ|z|–|az| because|a| ≥
gives us
|ay| ≥sβz–|z|–|az| becausesβ< ≥sβz–|a||z|–|z|
=sβz– +|a||z| =|z|sβ|z|– +|a|.
Thus
|y| ≥ |z|
sβ|z| +|a|–
.
Since|z|>(+sβ|a|)implies|z|(sβ|z|
Forz=zandy=y, consider
|Szn|=( –α)sSzn–+αsTyn–
implies
|Sz|=( –α)sSz+αsTy,
which yields
|az|=( –α)sSz+αsTy
=( –α)saz+ – ( –α)sy+c ≥ –s( –α)y+c–( –sα)az ≥sαy– +|a||z|
≥sαβz– +|a||z| ≥ |z|sαβ|z|– +|a|.
Hence
|z| ≥ |z|
sαβ|z|
+|a| –
,
since|z| ≥ |c|> (+sα|a|) and|z| ≥ |c|> (+sβ|a|), so that|z|> (+sαβ|a|). Therefore there exists
λ> such thatss+αβ|a|z||– > +λ. Consequently,|z|> ( +λ)|z|. In particular,|zn|>|z|. So
we may apply the same argument repeatedly to find|zn|> ( +λ)n|z|. Thus, the orbit ofz
tends to infinity. This completes the proof.
Corollary . Suppose that|c|>(+sα|a|)and|c|>(+sβ|a|),then the orbit of Jungck Ishikawa JIO(Tc, ,α,β,s)escapes to infinity.
In the proof of the theorem we used the facts that|z| ≥ |c|>(+sα|a|)and|z| ≥ |c|>(+sβ|a|). Hence the following corollary is the refinement of the escape criterion discussed in the above theorem.
Corollary . (Escape criterion) Let|z|>max{|c|,(+sα|a|),(+sβ|a|)},then|zn|> ( +λ)n|z|
and|zn| −→ ∞as n→ ∞.
Corollary . Suppose that|zk|>max{|c|,(+sα|a|),(+sβ|a|)}for some k≥.Then|zk+|> ( +
λ)n|z
k|and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTc. Given any point|z| ≤ |c|, we have computed the orbit ‘JIO’ ofz. If for somen,|zn|
lies outside the circle of radiusmax{|c|,(+sα|a|),(+sβ|a|)}, we guarantee that the orbit escapes. Hence,zis not in the Julia sets and also it is not in the Mandelbrot sets. On the other hand, if|zn|never exceeds this bound, then by definition of the Julia sets and the Mandelbrot sets
4.2 Escape criterion for the cubic complex polynomials
For cubic complex polynomialp(z) =z–az+c, we will chooseTz=z+candSz=az,
whereaandcare complex numbers.
Theorem . Assume that|z| ≥ |c|> ((+sα|a|)) and|z| ≥ |c|> ((+|a|)
sβ )
,where <α,β,s< and c is a complex parameter.Define
Sz= ( –α)sSz+αsTy,
.. .
Szn= ( –α)sSzn–+αsTyn–,
where
Sy= ( –β)sSz+βsTz,
.. .
Syn–= ( –β)sSzn–+βsTzn–,
Sz is injective,Tz is a cubic polynomial and n= , , , . . . ,then|zn| → ∞as n→ ∞.
Proof LetTz=z+cand forz=zandy=y, then we have
Syn–= ( –β)sSzn–+βsTzn–
implies
|Sy|=( –β)sSz+βsTz =( –β)saz+βsz+c
=( –β)saz+ – ( –β)sz+c.
Using binomial series up to linear terms ofαand ( –α), we get
|Sy|=( –sβ)az+ –s( –β)z+c ≥ –s( –β)z+c–( –sβ)az
≥s–s( –β)z+c–( –sβ)az becauses< ≥sβz–sβ|c|–|az|+|sβaz|
≥sβz–sβ|z|–|az|+|sβaz| because|z| ≥ |c| ≥sβz–sβ|z|–|az| because|a| ≥
gives us
=sβz– +|a||z| =|z|sβz– +|a|. Thus
|y| ≥ |z|
sβ|z|
+|a|–
.
Since|z| ≥ |c|> ((+sβ|a|)) implies|z|(sβ|z|
+|a| – )>|z|. Hence|y|>|z|(sβ+||az||– )>|z|> sβ|z|.
Forz=zandy=y, consider
|Szn|=( –α)sSzn–+αsTyn–
implies
|Sz|=( –α)sSz+αsTy,
which yields
|az|=( –α)sSz+αsTy
=( –α)saz+ – ( –α)sy+c ≥ –s( –α)y+c–( –sα)az ≥sαy– +|a||z|
≥sαβz– +|a||z| ≥ |z|sαβz– +|a|.
Hence
|z| ≥ |z|
sαβ|z| +|a| –
,
since|z| ≥ |c|> ((+sα|a|)) and|z| ≥ |c|> ((+|a|)
sβ )
, so that|z|> ((+|a|)
sαβ )
. Therefore there
existsλ> such thats+αβ||az||– > +λ. Consequently,|z|> ( +λ)|z|. In particular,|zn|>
|z|. So we may apply the same argument repeatedly to find|zn|> ( +λ)n|z|. Thus, the orbit
ofztends to infinity. This completes the proof.
Corollary . Suppose that|c|> ((+sα|a|)) and|c|> ((+|a|)
sβ )
,then the orbit of Jungck Ishikawa JIO(Tc, ,α,β,s)escapes to infinity.
In the proof of the theorem we used the facts that|z| ≥ |c|> ((+sα|a|)) and|z| ≥ |c|> ((+sβ|a|)). Hence the following corollary is the refinement of the escape criterion discussed in the above theorem.
Corollary . (Escape criterion) Let|z|>max{|c|, ((+sα|a|)), ((+|a|)
sβ )
}, then|zn|> ( + λ)n|z|and|z
Corollary . Suppose that |zk| >max{|c|, ((+sα|a|))
, ((+|a|)
sβ )
} for some k≥. Then |zk+|> ( +λ)n|zk|and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTcz=z+c.
4.3 Escape criterion for higher degree complex polynomials
For higher degree complex polynomialp(z) =zn–az+c, we will chooseTz=zn+cand Sz=az, wheren= , , , . . . ,aandcare complex numbers.
Theorem . Assume that|z| ≥ |c|> ((+sα|a|))n– and|z| ≥ |c|> ((+|a|)
sβ )
n–, where <
α,β,s< and c is a complex parameter.Define
Sz= ( –α)sSz+αsTy,
.. .
Szn= ( –α)sSzn–+αsTyn–,
where
Sy= ( –β)sSz+βsTz,
.. .
Syn–= ( –β)sSzn–+βsTzn–,
Sz is injective,Tz=zn+c and n= , , , . . . ,then|z
n| → ∞as n→ ∞.
Proof To prove the theorem, we follow the mathematical induction. Forn= ,Tz=z+c, so the escape criterion is|z|>max{|c|,(+sα|a|),(+sβ|a|)}. Forn= ,Tz=z+c, so the escape criterion is|z|>max{|c|, ((+sα|a|))/, ((+|a|)
sβ )/}. Hence the theorem is true forn= , , . . . .
Now suppose that the theorem is true for anyn. LetTz=zn++c,z
=z,y=yand|z| ≥
|c|> ((+sα|a|))n,|z| ≥ |c|> ((+|a|) sβ )
n exist, then we have
Syn–= ( –β)sSzn–+βsTzn–
implies
|Sy|=( –β)sSz+βsTz =( –β)saz+βszn++c
=( –β)saz+ – ( –β)szn++c.
Using binomial series up to linear terms ofαand ( –α), we get
≥s–s( –β)zn++c–( –sβ)az becauses< ≥sβzn+–sβ|c|–|az|+|sβaz|
≥sβzn+–sβ|z|–|az|+|sβaz| because|z| ≥ |c| ≥sβzn+–sβ|z|–|az| because|a| ≥
gives us
|ay| ≥sβzn+–|z|–|az| becausesβ< ≥sβzn+–|a||z|–|z|
=sβzn+– +|a||z| =|z|sβzn– +|a|.
Thus
|y| ≥ |z|
sβ|zn| +|a|–
.
Since|z|> ((+sβ|a|))n, it implies|y|n+>|z|n+(sβ|z| n
+|a| – )n+>sβ|z|n+. Forz=zandy=y, consider
|Szn|=( –α)sSzn–+αsTyn–,
from which we obtain
|Sz|=( –α)sSz+αsTy
yields
|az|=( –α)sSz+αsTy
=( –α)saz+ – ( –α)syn++c ≥ –s( –α)yn++c–( –sα)az ≥sαyn+– +|a||z|
≥sαβzn+– +|a||z| ≥ |z|sαβzn– +|a|.
Hence
|z| ≥ |z|
sαβ|zn|
+|a| –
,
since|z| ≥ |c|> ((+sα|a|))n and|z| ≥ |c|> ((+|a|) sβ )
n, so that|z|> ((+|a|) sαβ )
n. Therefore there
|z|. So we may apply the same argument repeatedly to find|zn|> ( +λ)n|z|. Thus, the orbit
ofztends to infinity. This completes the proof.
Corollary . Suppose that|c|> ((+sα|a|))n– and|c|> ((+|a|)
sβ )
n–,then the orbit JIO(Tc, ,
α,β,s)escapes to infinity.
Corollary .(Escape criterion) Suppose that|zk|>max{|c|, ((+sα|a|))
n–, ((+|a|)
sβ )
n–}for some k≥.Then|zk+|> ( +λ)n|zk|and|zk+| −→ ∞as n→ ∞.
This corollary gives us an algorithm for the generation of Julia sets and Mandelbrot sets ofTcz=zn+c.
5 Conclusions
In this paper, some fixed point results for Jungck Mann and Jungck Ishikawa iterations withs-convexity have been introduced in the study of Julia and Mandelbrot sets. The new escape criterions for complex quadratic, cubic and nth degree polynomials have been es-tablished. If we takes= , it provides previous results existing in the relative literature.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Division of Science and Technology, University of Education, Lahore, Pakistan.2Department of Mathematics and RINS,
Gyeongsang National University, Jinju, 660-701, Korea.3Department of Mathematics, Lahore Leads University, Lahore, 54810, Pakistan.
Acknowledgements
This research is supported by Gyeongsang National University, Korea.
Received: 26 May 2015 Accepted: 10 September 2015
References
1. Julia, G: Sur l’itération des functions rationnelles. J. Math. Pures Appl.8, 737-747 (1918) 2. Mandelbrot, BB: The Fractal Geometry of Nature. Freeman, New York (1983)
3. Negi, A, Rani, M: A new approach to dynamic noise on superior Mandelbrot set. Chaos Solitons Fractals36(4), 1089-1096 (2008)
4. Negi, A, Rani, M: Midgets of superior Mandelbrot set. Chaos Solitons Fractals36(2), 237-245 (2008)
5. Rani, M, Kumar, V: Superior Mandelbrot set. J. Korea Soc. Math. Educ. Ser. D Res. Math. Educ.8(4), 279-291 (2004) 6. Rani, M, Kumar, M: Circular saw Mandelbrot sets. In: WSEAS Proc. 14th Int. Conf. on Applied Mathematics (Math ’09):
Recent, Advances in Applied Mathematics, pp. 131-136 (2009)
7. Devaney, RL: A First Course in Chaotic Dynamical Systems: Theory and Experiment. Addison-Wesley, Reading (1992) 8. Chauhan, YS, Rana, R, Negi, A: Complex dynamics of Ishikawa iterates for non integer values. Int. J. Comput. Appl.9(2),
9-16 (2010)
9. Chauhan, YS, Rana, R, Negi, A: New Julia sets of Ishikawa iterates. Int. J. Comput. Appl.7(13), 34-42 (2010) 10. Rani, M: Cubic superior Julia sets. In: Proc. European Computing Conference, pp. 80-84 (2011) 11. Rani, M: Superior antifractals. In: IEEE Proc. ICCAE 2010, vol. 1, pp. 798-802 (2010)
12. Rani, M: Superior tricorns and multicorns. In: WSEAS Proc. 9th Int. Conf. on Applications of Computer Engineering (ACE ’10): Recent Advances & Applications of Computer Engineering, pp. 58-61 (2010)
13. Rani, M, Agarwal, R: Effect of noise on Julia sets generated by logistic map. In: Proc. 2nd IEEE International Conference on Computer and Automation Engineering (ICCAE 2010), vol. 2, pp. 55-59 (2010)
14. Rani, M, Agarwal, R: Effect of stochastic noise on superior Julia sets. J. Math. Imaging Vis.36, 63-68 (2010) 15. Rani, M, Agarwal, R: Generation of fractals from complex logistic map. Chaos Solitons Fractals42(1), 447-452 (2009) 16. Rani, M, Kumar, V: Superior Julia set. J. Korea Soc. Math. Educ. Ser. D Res. Math. Educ.8(4), 261-277 (2004)
17. Rani, M, Negi, A: New Julia sets for complex Carotid-Kundalini function. Chaos Solitons Fractals36(2), 226-236 (2008) 18. Kang, SM, Rafiq, A, Tanveer, M, Ali, F, Kwun, YC: Julia and Mandelbrot sets in Jungck Mann and Jungck Ishikawa orbits.
Mitteilungen Klosterneuburg65, 1 (2015)
21. Breckner, WW, Orbán, G: Continuity Properties of Rationallys-Convex Mappings with Values in Ordered Topological Liner Space. Babes-Bolyai University, Kolozsvár (1978)
22. Hudzik, H, Maligranda, L: Some remarks onsi-convex functions. Aequ. Math.48, 100-111 (1994) 23. Alomari, M, Darus, M: On Co-ordinateds-convex functions. Int. Math. Forum3(40), 1977-1989 (2008)
24. Alomari, M, Darus, M: Hadamard-type inequalities fors-convex functions. Int. Math. Forum3(40), 1965-1970 (2008) 25. Kirmaci, US, et al.: Hadamard-type inequalities fors-convex functions. Appl. Math. Comput.193, 26-35 (2007) 26. Dragomir, SS, Fitzpatrick, S: The Hadamard’s inequality fors-convex functions in the second sense. Demonstr. Math.
32(4), 687-696 (1999)
27. Takahashi, W: A convexity in metric space and nonexpansive mappings, I. Kodai Math. Semin. Rep.22, 142-149 (1970) 28. Ojha, DB, Mishra, MK: An application of fixed point theorem fors-convex function. Int. J. Eng. Sci. Technol.2(8),
3371-3375 (2010)