Fixed point results for fractal generation in Noor orbit and s-convexity

In this note, we give fixed point results in fractal generation (Julia sets and Mandelbrot sets) by using Noor iteration scheme with s-convexity. Researchers have already presented fixed point results in Mann and Ishikawa orbits that are examples of one-step and two-step feedback processes respectively. In this paper we present fixed point results in Noor orbit, which is a three-step iterative procedure.

Julia and Mandelbrot sets using Mann iteration scheme. Rana et al. (2010a, b) introduced relative superior Julia and Mandelbrot sets using Ishikawa iteration scheme. Also, relative superior Julia sets, Mandelbrot sets and tricorn and multicorns by using S-iteration scheme are presented in Kang et al. (2015a). Recently, (Ashish and Chugh 2014) introduced Julia and Mandelbrot sets using Noor iteration scheme which is a three-step iterative procedure.
Fixed point results can be found in the generation of the different types of fractals: for example, Iterated Function Systems (Prasad and Katiyar 2011;Singh et al. 2011), V-variable fractals (Singh et al. 2011), Inversion fractals (Gdawiec 2015) and Biomorphs (Gdawiec et al. 2016). Some polynomiographs are types of fractals which can be obtained via different iterative schemes, for more detail (see Kang et al. 2015cKang et al. , 2016Rafiq et al. 2014;Kotarski et al. 2012;Nazeer et al. 2016) and references therein. Kang et al. (2015b) introduced new fixed point results for fractal generation in Jungck Noor orbit with s-convexity. Mishra et al. (2011a, b) develop fixed point results in relative superior Julia sets and tricorn and multicorns by using Ishikawa iteration with s-convexity. Nazeer et al. (2015) introduced fixed point results in the generation of Julia and Mandelbrot sets.
In this paper we present some fixed point results for Julia and Mandelbrot sets by using Noor iteration scheme with s-convexity. The results of Ashish and Chugh (2014) are a special case of the results of this paper for s = 1, so in this article we extend the results from Ashish and Chugh (2014). We define the Noor orbit and escape criterions for quadratic, cubic, and k + 1th degree polynomials by using Noor iteration with s-convexity.

Preliminaries
Definition 1 (see Barnsley 1993, Julia set) Let f : C −→ C symbolize a polynomial of degree ≥ 2. Let F f be the set of points in C whose orbits do not converge to the point at infinity. That is, F f = {x ∈ C : { f n (x) , n varies from 0 to ∞} is bounded}. F f is called as filled Julia set of the polynomial f. The boundary points of F f can be called as the points of Julia set of the polynomial f or simply the Julia set.
Definition 2 (see Devaney 1992, 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, that is 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 is defined as the collection of all c ∈ C for which the orbit of the point 0 is bounded, that is We choose the initial point 0, as 0 is the only critical point of Q c .
where the orbit O(f , z 0 )of f at the initial point z 0 is the sequence {f n z 0 }. Definition 4 (see Noor 2000, Noor orbit). Consider a sequence {z n } of iterates for initial point z 0 ∈ C such that where α n , β n , γ n ∈ [0, 1] and {α n }, {β n }, {γ n } are sequences of positive numbers. The above sequence of iterates is called Noor orbit, which is a function of five arguments (f , z 0 , α n , β n , γ n ) which can be written as NO(f , z 0 , α n , β n , γ n ).

Main results
The definition of the Mandelbrot set gives us an algorithm for computing it. We simply consider a square in the complex plane. We overlay a grid of equally spaced points in this square. Each of these points is to be considered a complex c-value. Then, for each such c, we ask the computer to check whether the corresponding orbit of 0 goes to infinity (escapes) or does not go to infinity (remains bounded). In the former case, we leave the corresponding c-value (pixel) white. In the latter case, we paint the c-value dark. Thus the dark points represent the Mandelbrot set. Indeed, it is not possible to determine whether certain c-values lie in the Mandelbrot set. We can only iterate a finite number of times to determine if a point lies in M . Certain c-values close to the boundary of M have orbits that escape only after a very large number of iterations.
Corollary 1 (The Escape Criterion) Suppose |c| is less than or equal to 2. If the orbit of 0 under z 2 + c ever lands outside of the circle of radius 2 centered at the origin, then this orbit definitely tends to infinity.
When calculating Julia sets, z is a variable representing a Cartesian coordinate within the image and c is a constant complex number, c does not change during the calculation of the entire image. However, when different values of c are used, different images representing different Julia sets will result.
The escape criterion plays a vital role in the generation and analysis of Julia sets and Mandelbrot sets. We now define escape criterions for Julia sets and Mandelbrot sets in Noor orbit with s-convexity.
We take z o = z ∈ C, α n = α, β n = β and γ n = γ then can write Noor iteration scheme with s-convexity in the following manner where Q c (z n ) be a quadratic, cubic or nth degree polynomial.
We used the notion NO s (Q c , 0, α, β, γ , s) for the Noor iteration with s-convexity.
Hence the following corollary is the refinement of the escape criterion.

Escape criterions for cubic polynomials
We prove the following result for the cubic polynomial Q a,b (z) = z 3 + az + b, where a and b are complex numbers, as it is conjugate to all other cubic polynomials.

A general escape criterion
We will obtain a general escape criterion for polynomials of the form G c (z) = z k+1 + c.

Generation of Julia sets and Mandelbrot sets
In this section we present some Mandelbrot sets for quadratic and cubic functions by using the computational work in Mathematica 9.0. and following code   In Figs. 7,8,9,10,11,12,13, and 14 cubic Mandelbrot sets are presented in Noor orbit with s-convexity by using maximum number of iterations 30 and grid [−3.5, 3.5] × [−6, 6].
Julia sets for the quadratic polynomial Q c (z) = z 2 + c Quadratic Julia sets are presented in Figs. 15, 16, 17, and 18 for Noor iteration scheme with s-convexity by using maximum number of iterations 20 and s = 1.

Conclusions
In this paper we presented new fixed point results for Noor iteration with s-convexity in the generation of fractals (Julia sets and Mandelbrot sets). The new escape criterions have been established for complex quadratic, cubic, and (k + 1)th degree polynomials.