Open J. Math. Sci., Vol.
2
(2018), No. 1, pp. 56  72 ISSN 25230212(online)
Website: http://www.openmathscience.com
FIXED POINT RESULTS FOR THE COMPLEX FRACTALGENERATION IN THE
S
ITERATION ORBIT WITH
s
CONVEXITY
KRZYSZTOF GDAWIEC
1
, ABDUL AZIZ SHAHID
Abstract.
Since the introduction of complex fractals by Mandelbrot theygained much attention by the researchers. One of the most studied complexfractals are Mandelbrot and Julia sets. In the literature one can ﬁnd manygeneralizations of those sets. One of such generalizations is the use of theresults from ﬁxed point theory. In this paper we introduce in the generationprocess of Mandelbrot and Julia sets a combination of the
S
iteration,known from the ﬁxed point theory, and the
s
convex combination. Wederive the escape criteria needed in the generation process of those fractalsand present some graphical examples.Mathematics Subject Classiﬁcation :37F45, 37F50, 47J25
Key words and phrases:
itration schemes; Julia set; Mandelbrot set; escapecriterion.
1. Introduction
Mandelbrot and Julia sets are some of the best known illustrations of a highlycomplicated chaotic systems generated by a very simple mathematical process.They were introduced by Benoit Mandelbrot in the late 1970’s [1], but Juliasets were studied much earlier, namely in the early 20th century by Frenchmathematicians Pierre Fatou and Gaston Julia. Mandelbrot working at IBMhas studied their works and plotted the Julia sets for
z
2
+
c
and correspondingto them the Mandelbrot set. He was surprised by the result that he obtained.Since then many mathematicians have studied diﬀerent properties of Mandelbrotand Julia sets and proposed various generalizations of those sets. The ﬁrst andthe most obvious generalization was the use of
z
p
+
c
function instead of the
Received 15 November 2017. Revised 29 March 2018.
1
Corresponding Authorc
2018 Krzysztof Gdawiec, Abdul Aziz Shahid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the srcinal work is properly cited.56
Fixed point results for the complex fractal generation in the
S
iteration orbit with
s
convexity57
quadratic one used by Mandelbrot [2, 3]. Then some other types of functionswere studied: rational [4], transcendental [5], elliptic [6], antipolynomials [7]etc. Another step in the studies on Mandelbrot and Julia sets was the extensionfrom complex numbers to other algebras, e.g., quaternions [8], octonions [9],bicomplex numbers [10] etc.Another interesting generalization of Mandelbrot and Julia sets is the use of the results from ﬁxed point theory. In the ﬁxed point theory there exist manyapproximate methods of ﬁnding ﬁxed points of a given mapping, that are basedon the use of diﬀerent feedback iteration processes. These methods can be usedin the generalization of Mandelbrot and Julia sets. In 2004, Rani and Kumar[11, 12] introduced superior Julia and Mandelbrot sets using Mann iterationscheme. Chauhan
et al.
in [13] introduced the relative superior Julia sets using Ishikawa iteration scheme. Also, relative superior Julia sets, Mandelbrotsets and tricorn, multicorns by using the
S
iteration scheme were presented in[14, 15]. Recently, Ashish
et al.
in [16] introduced Julia and Mandelbrot setsusing the Noor iteration scheme, which is a threestep iterative procedure. The junction of a
s
convex combination [17] and various iteration schemes was studied in many papers. Mishra
et al.
[18, 19] developed ﬁxed point results in relativesuperior Julia sets, tricorn and multicorns by using the Ishikawa iteration with
s
convexity. In [20] Kang
et al.
introduced new ﬁxed point results for fractal generation using the implicit JungckNoor orbit with
s
convexity, whereasNazeer
et al.
in [21] used the JungckMann and JungckIshikawa iterations with
s
convexity. The use of Noor iteration and
s
convexity was shown in [22].In this paper, we present some ﬁxed point results for Julia and Mandelbrot setsby using the
S
iteration scheme with
s
convexity. We derive the escape criteriafor quadratic, cubic and the (
k
+ 1)th degree complex polynomial.The remainder of this paper is outlined as follows. In Sec. 2 we present somebasic deﬁnitions used in the paper. Next, in Sec. 3, we present the main results of this paper, namely we prove the escape criteria for the quadratic, cubic and the(
k
+ 1)th degree complex polynomial in the
S
iteration with
s
convexity. Somegraphical examples of complex fractals generated using the escape criteria arepresented in Sec. 4. The paper ends with some concluding remarks and futurework (Sec. 5).
2. PreliminariesDeﬁnition 2.1
(see [23], Julia set)
.
Let
f
:
C
→
C
be a polynomial of degree
≥
2. Let
F
f
be the set of points in
C
whose orbits do not converge to the pointat inﬁnity, i.e.,
F
f
=
{
z
∈
C
:
{
f
n
(
z
)
}
∞
n
=0
is bounded
}
. F
f
is called as ﬁlledJulia set of the polynomial
f
. The boundary points of
F
f
are called the pointsof Julia set of the polynomial
f
or simply the Julia set.
Deﬁnition 2.2
(see [24], Mandelbrot set)
.
The Mandelbrot set
M
consists of all parameters
c
for which the ﬁlled Julia set of
Q
c
(
z
) =
z
2
+
c
is connected, i.e.,
M
=
{
c
∈
C
:
F
Q
c
is connected
}
.
(1)
58 K. Gdawiec, A. A. Shahid
In fact,
M
contains an enormous amount of information about the structure of Julia sets.The Mandelbrot set
M
for the quadratic
Q
c
(
z
) =
z
2
+
c
can be equivalentlydeﬁned in the following way:
M
=
{
c
∈
C
:
{
Q
nc
(0)
}
does not tend to
∞
as
n
→∞}
,
(2)We choose the initial point 0, because 0 is the only critical point of
Q
c
, i.e.,
Q
′
c
(0) = 0.
Deﬁnition 2.3.
Let
C
⊂
C
be a nonempty set and
f
:
C
→
C
. For any point
z
0
∈
C
the Picard orbit is deﬁned as the set of iterates of the point
z
0
, i.e.,
O
(
f,z
0
) =
{
z
n
:
z
n
=
f
(
z
n
−
1
)
,n
= 1
,
2
,
3
,...
}
,
(3)where the orbit
O
(
f,z
0
) of
f
at the initial point
z
0
is the sequence
{
f
n
(
z
0
)
}
∞
n
=1
.
Deﬁnition 2.4
(see [25],
S
iteration orbit)
.
Consider a sequence
{
z
n
}
of iteratesfor initial point
z
0
∈
C
such that
z
n
+1
= (1
−
γ
n
)
f
(
z
n
) +
γ
n
f
(
w
n
)
,w
n
= (1
−
δ
n
)
z
n
+
δ
n
f
(
z
n
)
,
(4)where
n
= 0
,
1
,
2
,...
and
γ
n
,δ
n
∈
(0
,
1]. This sequence of iterates is called the
S
orbit, which is a function of four arguments (
f,z
0
,γ
n
,δ
n
) and we will denoteit by
SO
(
f,z
0
,γ
n
,δ
n
).In [14] Kang
et al.
have proved the escape criterion for the Mandelbrot andJulia sets in
S
orbit.
Theorem 2.5
(Escape Criterion for
S
iteration)
.
Let
Q
c
(
z
) =
z
k
+1
+
c
, where
k
= 1
,
2
,
3
,...
and
c
∈
C
. Iterate
Q
c
using
(4)
with
γ
n
=
γ,δ
n
=
δ
, where
γ,δ
∈
(0
,
1]
. Suppose that

z

>
max

c

,
2
γ
1
k
,
2
δ
1
k
,
(5)
then there exist
λ >
0
such that

z
n

>
(1 +
λ
)
n

z

and

z
n
→∞
as
n
→∞
.
3. Main results
In each of the two steps of
S
iteration we use a convex combination of twoelements. In the literature we can ﬁnd some generalizations of the convex combination. One of such generalizations is the
s
convex combination.
Deﬁnition 3.1
(
s
convex combination [17])
.
Let
z
1
,z
2
,...,z
n
∈
C
and
s
∈
(0
,
1]. The
s
convex combination is deﬁned in the following way:
λ
s
1
z
1
+
λ
s
2
z
2
+
...
+
λ
sn
z
n
,
(6)where
λ
k
≥
0 for
k
∈{
1
,
2
,...,n
}
and
nk
=1
λ
k
= 1.
Fixed point results for the complex fractal generation in the
S
iteration orbit with
s
convexity59
Let us notice that the
s
convex combination for
s
= 1 reduces to the standardconvex combination. Now, we will replace the convex combination in the
S
iteration with the
s
convex one.Let
Q
c
be a polynomial and
z
0
∈
C
. We deﬁne the
S
iteration with
s
convexityas follows:
z
n
+1
= (1
−
γ
)
s
Q
c
(
z
n
) +
γ
s
Q
c
(
w
n
)
,w
n
= (1
−
δ
)
s
z
n
+
δ
s
Q
c
(
z
n
)
,
(7)where
n
= 0
,
1
,
2
,...
and
γ,δ,s
∈
(0
,
1]. We will denote the
S
iteration with
s
convexity by
SO
s
(
f,z
0
,γ,δ,s
).In the following subsections we prove escape criteria for some classes of polynomials using the
S
iteration with
s
convexity.
3.1. Escape criterion for quadratic function.Theorem 3.2.
Assume that

z
 ≥ 
c

>
2
sγ
and

z
 ≥ 
c

>
2
sδ
, where
γ,δ,s
∈
(0
,
1]
and
c
∈
C
. Let
z
0
=
z
. Then for
(7)
with
Q
c
(
z
) =
z
2
+
c
we have

z
n
→∞
as
n
→∞
.Proof.
Consider

w

=

(1
−
δ
)
s
z
+
δ
s
Q
c
(
z
)

.
For
Q
c
(
z
) =
z
2
+
c
,

w

=
(1
−
δ
)
s
z
+
δ
s
(
z
2
+
c
)
=
(1
−
δ
)
s
z
+ (1
−
(1
−
δ
))
s
(
z
2
+
c
)
.
By binomial expansion upto linear terms of
δ
and (1
−
δ
)
,
we obtain

w
 ≥
(1
−
sδ
)
z
+ (1
−
s
(1
−
δ
))(
z
2
+
c
)
=
(1
−
sδ
)
z
+ (1
−
s
+
sδ
)(
z
2
+
c
)
≥
(1
−
sδ
)
z
+
sδ
(
z
2
+
c
)
,
because 1
−
s
+
sδ
≥
sδ
≥
sδz
2
+ (1
−
sδ
)
z
−
sδc
≥
sδz
2
+ (1
−
sδ
)
z
−
sδz

,
because

z
≥
c
≥
sδz
2
−
(1
−
sδ
)
z
−
sδz
≥
sδz
2
−
z

+

sδz
−
sδz

=

z

(
sδ

z
−
1)
.
(8)In the second step of the
S
iteration with
s
convexity we have

z
1

=

(1
−
γ
)
s
Q
c
(
z
) +
γ
s
Q
c
(
w
)

=
(1
−
γ
)
s
(
z
2
+
c
) + (1
−
(1
−
γ
))
s
(
w
2
+
c
)
.
(9)By binomial expansion upto linear terms of
γ
and (1
−
γ
)
,
we obtain

z
1
 ≥
(1
−
sγ
)(
z
2
+
c
) + (1
−
s
(1
−
γ
))(
w
2
+
c
)
=
(1
−
sγ
)(
z
2
+
c
) + (1
−
s
+
sγ
)(
w
2
+
c
)
60 K. Gdawiec, A. A. Shahid
≥
(1
−
sγ
)(
z
2
+
c
) +
sγ
((

z

(
sδ

z
−
1))
2
+
c
)
,
because 1
−
s
+
sγ
≥
sγ.
(10)Since

z

>
2
/
(
sδ
), which implies
sδ

z

>
2 and (
sδ

z
−
1)
2
>
1. Thus

z

2
(
sδ

z
−
1)
2
>

z

2
> δ

z

2
(
∵
0
< δ <
1) (11)Using (11) in (10) we have

z
1
 ≥
(1
−
sγ
)(
z
2
+
c
) +
sγ
(
δ

z

2
+
c
)
=
sγδ

z

2
+ (1
−
sγ
)
z
2
+ (1
−
sγ
)
c
+
sγc
=
sγδ

z

2
+ (1
−
sγ
)
z
2
+
c
≥
sγδ

z

2
−
(
sγ
−
1)
z
2
−
c
≥
sγδ

z

2
−
(
sγ
−
1)
z
2
−
z

(
∵

z

>

c

)=

z

((
sγδ
−
sγ
+ 1)

z
−
1)
.
Because

z

>
2
/
(
sγ
) and

z

>
2
/
(
sδ
), so

z

>
2
sγδ >
2
sγδ
−
sγ
+ 1
,
which implies(
sγδ
−
sγ
+ 1)

z
−
1
>
1
.
Therefore, there exists
λ >
0, such that (
sγδ
−
γ
+ 1)

z
−
1
>
1 +
λ >
1
.
Consequently

z
1

>
(1 +
λ
)

z

.
We may apply the same argument repeatedly to obtain:

z
2

>
(1 +
λ
)
2

z

,
...

z
n

>
(1 +
λ
)
n

z

.
Hence,

z
n
→∞
as
n
→∞
. This completes the proof.
Corollary 3.3.
Suppose that

c

>
2
sγ
and

c

>
2
sδ ,
(12)
then the orbit
SO
s
(
Q
c
,
0
,γ,δ,s
)
escapes to inﬁnity.
The following corollary is the reﬁnement of the escape criterion.
Corollary 3.4
(Escape Criterion)
.
Let
γ,δ,s
∈
(0
,
1]
. Suppose that

z

>
max

c

,
2
sγ ,
2
sδ
,
(13)
then there exist
λ >
0
such that

z
n

>
(1 +
λ
)
n

z

and

z
n
→∞
as
n
→∞
.