1 Introduction

Let \({\mathscr {Z}}_1\) and \({\mathscr {Z}}_2\) denote Banach spaces. Consider a convex and open subset \({\mathscr {H}} (\not = \emptyset )\) of \({\mathscr {Z}}_1\). Suppose \(BL({\mathscr {Z}}_1, {\mathscr {Z}}_2)\) stands for the set \(\{L_B: {\mathscr {Z}}_1 \rightarrow {\mathscr {Z}}_2 \ \text {linear and bounded operators}\}\). Define the following equation for a Fréchet derivable operator \(F:{\mathscr {H}} \subseteq {\mathscr {Z}}_1 \rightarrow {\mathscr {Z}}_2\) by

$$\begin{aligned} F(b)=0. \end{aligned}$$
(1.1)

Nonlinear equations of the above form (1.1) are often used to tackle various challenging tasks that appear in many disciplines of science and other applied topics. The analytical solution denoted by \(b_*\) of equation (1.1) can be found only in rare instances. This is why nonlinear equations are typically solved iteratively to find an approximate solution. Generally, constructing highly effective iterative procedures to produce the solutions of these equations is a tough responsibility. The most often used method for this problem is the traditional Newton’s iterative approach. Additionally, a lot of researchers have been designing higher order modifications of conventional procedures like Newton’s, Chebyshev’s, Jarratt’s, etc. [1,2,3, 5,6,7, 9, 11,12,17, 21, 23, 25,26,27, 31]. Kou and Li [18] presented a sixth order variant of Jarratt’s algorithm to address nonlinear equations in \({\mathbb {R}}\). They added Newton’s iterate as the third step in Jarratt’s iterate and used linear interpolation formula to eliminate the additional evaluation of the first derivative. Another modification of Jarratt’s method is designed in [33]. This method needs two evaluations of the function and two of its first derivatives per iteration. Cordero et al. [12] proposed an efficient family of nonlinear system solvers using a reduced composition technique on Newton’s and Jarratt’s algorithm. This family is given for an initial choice \(b_0 \in {\mathscr {H}}\) by

$$\begin{aligned} a_n&=b_n-\frac{2}{3}F'(b_n)^{-1}F(b_n), \nonumber \\ z_n&=b_n-\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(a_n))F'(a_n)^{-1}F(a_n), \nonumber \\ b_{n+1}&=z_n-(\alpha _1F'(b_n)+\alpha _2F'(a_n))^{-1}F(z_n), \end{aligned}$$
(1.2)

where \(\alpha _1\) and \(\alpha _2\) are real parameters. Further, Soleymani composed Newton’s procedure in Jarratt’s one and used Taylor expansion technique to derive a sixth order method. Two new families of Jarratt-type algorithms with sixth order convergence are studied in [30]. Sharma and Arora [25] constructed iterative algorithms of fourth and sixth convergence order for solving nonlinear systems. Two Jarratt-like steps are used to obtain the fourth order method and the sixth order algorithm is the composition of three Jarratt-like steps. The sixth order iterative procedure for a starter \(b_0 \in {\mathscr {H}}\) is written as follows.

$$\begin{aligned} a_n&=b_n-\frac{2}{3}F'(b_n)^{-1}F(b_n), \nonumber \\ z_n&=b_n-\Bigg [\frac{23}{8}I-3F'(b_n)^{-1}F'(a_n) \Bigg (3I-\frac{9}{8}F'(b_n)^{-1}F'(a_n)\Bigg ) \Bigg ]F'(b_n)F(b_n), \nonumber \\ b_{n+1}&=z_n-\frac{1}{2}(5I-3F'(b_n)^{-1}F'(a_n))F'(b_n)^{-1}F(z_n). \end{aligned}$$
(1.3)

Additional studies on Jarratt-like and other iterative processes with their dynamics and convergence are available in [4, 8, 10, 19, 20, 22, 24, 28, 32].

In this work, we are particularly focused on two sixth convergence order Jarratt-like solvers (JLM1 and JLM2) discussed by Soleymani et al. [29]. For an initial point \(b_0 \in {\mathscr {H}}\), these solvers are expressed as follows.

JLM1:

$$\begin{aligned} a_n&=b_n-\frac{2}{3}F'(b_n)^{-1}F(b_n), \nonumber \\ z_n&=b_n-\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(b_n))F'(b_n)^{-1}F(b_n), \nonumber \\ b_{n+1}&=z_n-\frac{1}{2}(3F'(a_n)^{-1}-F'(b_n)^{-1})F(z_n). \end{aligned}$$
(1.4)

JLM2:

$$\begin{aligned} a_n&=b_n-\frac{2}{3}F'(b_n)^{-1}F(b_n), \nonumber \\ z_n&=b_n-\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(b_n))F'(b_n)^{-1}F(b_n), \nonumber \\ b_{n+1}&=z_n-[((3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(b_n))]^2F'(b_n)^{-1}F(z_n). \end{aligned}$$
(1.5)

Taylor’s expansion formula and conditions on derivatives of the seventh order have been used in [29] to prove their convergence. Such results make these solvers very hard to use, since their usefulness is minimized due to the assumptions on derivatives of higher order. To illustrate this, we have chosen

$$\begin{aligned} F(b)= \left\{ \begin{array}{ll} b^3 \ln (b^2) + b^5 -b^4, &{} \hbox {if } b \not = 0 \\ 0, &{} \hbox {if } b=0 \end{array} \right. , \end{aligned}$$
(1.6)

where \({\mathscr {Z}}_1={\mathscr {Z}}_2={\mathbb {R}}\) and F is defined on \({\mathscr {H}}=[-\frac{1}{2}, \frac{3}{2}]\). Then, it is worth mentioning that the previously established theorems for the convergence of JLM1 and JLM2 do not apply to this case, since the third derivative of F is not bounded. Besides this, these convergence findings yield little information about \(||b_n-b_*||\), the convergence ball and the exact position of the solution \(b_*\). The analysis of ball convergence of an iterative algorithm is essential for estimating the radii of convergence balls, establishing error limits, and evaluating the uniqueness area for \(b_*\). Notably, the consequences of the ball analysis are highly helpful, since they demonstrate the degree of difficulty in choosing starting choices. This inspires us to analyze and compare the convergence balls of JLM1 and JLM2 using conditions only on \(F'\). Our convergence findings enable the calculation of convergence radii and estimation of error \(||b_n-b_*||\), as well as provide a precise location of \(b_*\).

The arrangement of this paper can be described by summarizing the remainder of the text into the following statements. In Sect. 2, the analytical results on the ball convergence of JLM1 and JLM2 are described. Section 3 compares the attraction basins for these algorithms. Section 4 contains numerical applications. Concluding remarks are discussed at the end of this paper.

2 Ball convergence analysis

We first study the ball convergence of JLM1 using some real valued functions. Set \(Q=[0, \infty )\).

Suppose equation

$$\begin{aligned} \delta _0(t)-1=0 \end{aligned}$$
(2.1)

has a least solution \(\rho _0 \in Q{\setminus } \{0\}\) for some function \(\delta _0:Q \rightarrow Q\) continuous and non-decreasing. Set \(Q_0=[0, 2\rho _0)\).

Suppose equation

$$\begin{aligned} {\mathscr {M}}_1(t)-1=0 \end{aligned}$$
(2.2)

has a least solution \(r_1 \in (0, \rho _0)\), for some functions \(\delta : Q_0 \rightarrow Q\) and \(\delta _1: Q_0 \rightarrow Q\) continuous and non-decreasing, where

$$\begin{aligned} {\mathscr {M}}_1(t)=\frac{\int _{0}^{1}\delta ((1-\sigma )t)\ \mathrm{{d}}\sigma + \frac{1}{3}\int _{0}^{1}\delta _1(\sigma t)\ \mathrm{{d}}\sigma }{1-\delta _0(t)}. \end{aligned}$$

Suppose equation

$$\begin{aligned} p(t)-1=0 \end{aligned}$$
(2.3)

has a least solution \(\rho _p \in (0, \rho _0)\), where

$$\begin{aligned} p(t)=\frac{1}{2}(\delta _0(t)+3\delta _0({\mathscr {M}}_1(t)t)). \end{aligned}$$

Set \(\rho _1= \min \{\rho _0, \rho _p \}\) and \(Q_1=[0, \rho _1)\).

Suppose equation

$$\begin{aligned} {\mathscr {M}}_2(t)-1=0 \end{aligned}$$
(2.4)

has a least solution \(r_2 \in (0, \rho _1)\), where \({\mathscr {M}}_2:Q_1 \rightarrow Q\) is given by

$$\begin{aligned} {\mathscr {M}}_2(t)=\frac{\int _{0}^{1}\delta ((1-\sigma )t)\ \mathrm{{d}}\sigma }{1-\delta _0(t)}+\frac{3(\delta _0(t)+\delta _0({\mathscr {M}}_1(t)t))\int _{0}^{1}\delta _1(\sigma t)\ \mathrm{{d}}\sigma }{4(1-\delta _0(t)) (1-p(t))}. \end{aligned}$$

Suppose equation

$$\begin{aligned} \delta _0({\mathscr {M}}_1(t)t)-1=0 \end{aligned}$$
(2.5)

has a least solution \(\rho _2 \in (0, \rho _0)\). Set \(\rho = \min \{\rho _1, \rho _2 \}\) and \(Q_2=[0, \rho )\).

Suppose equation

$$\begin{aligned} {\mathscr {M}}_3(t)-1=0 \end{aligned}$$
(2.6)

has a least solution \(r_3 \in (0, \rho )\), where \({\mathscr {M}}_3:Q_2 \rightarrow Q\) is given by

$$\begin{aligned} {\mathscr {M}}_3(t)&=\bigg [\frac{\int _{0}^{1}\delta ((1-\sigma ){\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{1-\delta _0({\mathscr {M}}_2(t)t)} \nonumber \\&\quad +\frac{(\delta _0{\mathscr {M}}_2(t)t)+\delta _0({\mathscr {M}}_1(t)t))\int _{0}^{1}\delta _1(\sigma {\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{(1-\delta _0({\mathscr {M}}_1(t)t))(1-\delta _0({\mathscr {M}}_2(t)t))} \nonumber \\&\quad +\frac{(\delta _0(t)+\delta _0({\mathscr {M}}_1(t)t))\int _{0}^{1}\delta _1(\sigma {\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{2(1-\delta _0(t))(1-\delta _0({\mathscr {M}}_2(t)t))}\bigg ]{\mathscr {M}}_2(t). \end{aligned}$$

We shall show that

$$\begin{aligned} r=\min \{r_i\}, i=1, 2, 3 \end{aligned}$$
(2.7)

is a radius of convergence for JLM1.

Definition (2.7) implies that for all \(t \in [0, r)\)

$$\begin{aligned} 0\le \delta _0(t)<1, \end{aligned}$$
(2.8)
$$\begin{aligned} 0\le p(t)<1, \end{aligned}$$
(2.9)
$$\begin{aligned} 0\le \delta _0({\mathscr {M}}_1(t)t)<1 \end{aligned}$$
(2.10)

and

$$\begin{aligned} 0\le {\mathscr {M}}_i(t)<1, i=1, 2, 3. \end{aligned}$$
(2.11)

The notations \(B(b_*, \epsilon )\), \({\overline{B}}(b_*, \epsilon )\) stand for the open and closed balls in \({\mathscr {Z}}_1\) with center \(b_* \in {\mathscr {Z}}_1\) and of radius \(\epsilon >0\).

The conditions (C) are needed with functions \(``\delta "\) as developed previously. Assume:

\((C_1)\) \(F:{\mathscr {H}} \rightarrow {\mathscr {Z}}_2\) is continuously differentiable and \(b_*\) is a simple solution of equation \(F(b)=0\).

\((C_2)\) \(||F'(b_*)^{-1}(F'(b)-F'(b_*))||\le \delta _0(||b-b_*||)\)

for all \(b \in {\mathscr {H}}\).

Set \({\mathscr {H}}_0={\mathscr {H}} \cap B(b_*, \rho _0)\).

\((C_3)\) \(||F'(b_*)^{-1}(F'(a)-F'(b))||\le \delta (||a-b||)\)

and

\(||F'(b_*)^{-1}F'(b)||\le \delta _1(||b-b_*||)\)

for all \(b, a \in {\mathscr {H}}_0\).

\((C_4)\) \({\overline{B}}(b_*, R)\subset {\mathscr {H}}\) for some \(R>0\) to be determined.

\((C_5)\) There exists \(r_*\ge R\) such that

$$\begin{aligned} \int _{0}^{1}\delta _0(\sigma r_*)\ \mathrm{{d}}\sigma < 1. \end{aligned}$$

Set \({\mathscr {H}}_1={\mathscr {H}} \cap {\overline{B}}(b_*, r_*)\).

The convergence analysis of JLM1 follows based on conditions (C) with the developed notation.

Theorem 2.1

Under the conditions (C) with \(R=r\), further choose \(b_0 \in B(b_*, r){\setminus } \{b_*\}\). Then, sequence \(\{b_n\}\) exists in \(B(b_*, r)\), stays in \(B(b_*, r)\) for all \(n=0\), \({\lim _{n \rightarrow \infty } b_n=b_*}\),and

$$\begin{aligned}&||a_n-b_*||\le {\mathscr {M}}_1(||b_n-b_*||)||b_n-b_*||\le ||b_n-b_*||<r, \end{aligned}$$
(2.12)
$$\begin{aligned}&||z_n-b_*||\le {\mathscr {M}}_2(||b_n-b_*||)||b_n-b_*||\le ||b_n-b_*|| \end{aligned}$$
(2.13)

and

$$\begin{aligned} ||b_{n+1}-b_*||\le {\mathscr {M}}_3(||b_n-b_*||)||b_n-b_*||\le ||b_n-b_*||, \end{aligned}$$
(2.14)

where functions \(''{\mathscr {M}}_i''\) are as before and radius r is given in (2.7).

Proof

Inequalities (2.12)–(2.14) are proved using induction on k. Using (2.7), (2.8), \((C_1)\) and \((C_2)\), we get in turn that for \(v \in B(b_*, r){\setminus } \{b_*\}\)

$$\begin{aligned} ||F'(b_*)^{-1}(F'(v)-F'(b_*))||\le \delta _0(||v-b_*||)\le \delta _0(r)<1. \end{aligned}$$
(2.15)

Estimate (2.15) and the lemma by Banach on linear invertible operators [3, 7] imply \(F'(v)^{-1}\) exists with

$$\begin{aligned} ||F'(v)^{-1}F'(b_*)||\le \frac{1}{1-\delta _0(||v-b_*||)}. \end{aligned}$$
(2.16)

We also have \(a_0\) exists and is given by the first substep of JLM1 for \(n=0\). Moreover, we can have

$$\begin{aligned} a_0-b_*=b_0-b_*-F'(b_0)^{-1}F(b_0)+\frac{1}{3}F'(b_0)^{-1}F(b_0). \end{aligned}$$
(2.17)

By (2.7), (2.11) (for \(i=1\)), (2.16) (for \(v=b_0\)), \((C_3)\) and (2.17), we have in turn that

$$\begin{aligned} ||a_0-b_*||&\le ||a_0-b_*-F'(b_0)^{-1}F(b_0)||+\frac{1}{3}||F'(b_0)^{-1}F(b_0)|| \nonumber \\&\le ||(F'(b_0)^{-1}F'(b_*))\int _{0}^{1}F'(b_*)^{-1}(F'(b_*+\sigma (b_0-b_*))-F'(b_0))\ \mathrm{{d}}\sigma (b_0-b_*))|| \nonumber \\&\quad +\frac{1}{3}||(F'(b_0)^{-1}F'(b_*)) (F'(b_*)^{-1}F(b_0))|| \nonumber \\&\le \frac{\int _{0}^{1}\delta ((1-\sigma )||b_0-b_*||)\ \mathrm{{d}}\sigma +\frac{1}{3}\int _{0}^{1}\delta _1 (\sigma ||b_0-b_*||)\ \mathrm{{d}}\sigma }{1-\delta _0(||b_0-b_*||)} \nonumber \\&\le {\mathscr {M}}_1(||b_0-b_*||)||b_0-b_*||\le ||b_0-b_*||<r \end{aligned}$$
(2.18)

showing \(a_0 \in B(b_*, r)\) and (2.12) for \(n=0\).

Next, we show that linear operator \(3F'(a_0)-F'(b_0)\) is invertible. Indeed, using (2.7), (2.9), \((C_2)\) and (2.18), we obtain in turn that

$$\begin{aligned}&||(2F'(b_*))^{-1}(3F'(a_0)-F'(b_0)-2F'(b_*)|| \nonumber \\&\quad \le \frac{1}{2}(3||F'(b_*)^{-1}(F'(a_0)-F'(b_*))||+||F'(b_*)^{-1}(F'(b_0)-F'(b_*))||) \nonumber \\&\quad \le \frac{1}{3}(3\delta _0(||a_0-b_*||)+\delta _0(||b_0-b_*||))\le p(||b_0-b_*||) \nonumber \\&\quad \le p(r)<1, \end{aligned}$$
(2.19)

so

$$\begin{aligned} ||(3F'(a_0)-F'(b_0))^{-1}F'(b_*)||\le \frac{1}{2(1-p(||b_0-b_*||)}. \end{aligned}$$

We also have that \(z_0\) exists by the second substep of JLM1 for \(n=0\), from which we can write

$$\begin{aligned} z_0-b_*&=b_0-b_*-F'(b_0)^{-1}F(b_0) \nonumber \\&\quad +(I-\frac{1}{2}(3F'(a_0)-F'(b_0))^{-1}(3F'(a_0)+F'(b_0)))F'(b_0)^{-1}F(b_0) \nonumber \\&=b_0-b_*-F'(b_0)^{-1}F(b_0) \nonumber \\&\quad +(3F'(a_0)-F'(b_0))^{-1}(3F'(a_0)-F'(b_0)-\frac{1}{2}(3F'(a_0)+F'(b_0)))F'(b_0))^{-1}F(b_0) \nonumber \\&=b_0-b_*-F'(b_0)^{-1}F(b_0) \nonumber \\&\quad +\frac{3}{2}(3F'(a_0)-F'(b_0)^{-1}(F'(a_0)-F'(b_0))F'(b_0)^{-1}F(b_0). \end{aligned}$$
(2.20)

In view of (2.7), (2.11) (for \(i=2\)), (2.16) (for \(v=b_0\)) and (2.18–2.20), we get in turn that

$$\begin{aligned} ||z_0-b_*||&\le \bigg [\frac{\int _{0}^{1}\delta ((1-\sigma )||b_0-b_*||)\ \mathrm{{d}}\sigma }{1-\delta _0(||b_0-b_*||)}\nonumber \\&\quad +\frac{3(\delta _0(||b_0-b_*||)+\delta _0(||a_0-b_*||))\int _{0}^{1}\delta _1(\sigma ||b_0-b_*||)\ \mathrm{{d}}\sigma }{4(1-\delta _0(||b_0-b_*||)) (1-p(||b_0-b_*||))}\bigg ]||b_0-b_*|| \nonumber \\&\le {\mathscr {M}}_2(||b_0-b_*||)||b_0-b_*||\le ||b_0-b_*||, \end{aligned}$$
(2.21)

showing \(z_0 \in B(b_*, r)\) and (2.13) for \(n=0\).

Furthermore, notice that \(b_1\) is well defined by the third substep of JLM1, from which we can also write

$$\begin{aligned} b_1-b_*&=z_0-b_*-F'(z_0)^{-1}F(z_0) \nonumber \\&\quad +[F'(z_0)^{-1}-\frac{3}{2}F'(a_0)^{-1}+\frac{1}{2}F'(b_0)^{-1}]F(z_0) \nonumber \\&=z_0-b_*-F'(z_0)^{-1}F(z_0) \nonumber \\&\quad +F'(z_0)^{-1}(F'(a_0)-F'(z_0))F'(a_0)^{-1}F(z_0) \nonumber \\&\quad +\frac{1}{2}F'(b_0)^{-1}(F'(a_0)-F'(b_0))F'(a_0)^{-1}F(z_0). \end{aligned}$$
(2.22)

Then, by (2.7), (2.11) (for \(i=3\)), (2.16) (for \(u=b_0, z_0\)), (2.18), (2.21) and (2.22), we obtain in turn that

$$\begin{aligned} |b_1-b_*||&\le \bigg [\frac{\int _{0}^{1}\delta ((1-\sigma )||z_0-b_*||)\ \mathrm{{d}}\sigma }{1-\delta _0(||z_0-b_*||)}\nonumber \\&\quad +\frac{(\delta _0(||z_0-b_*||)+\delta _0(||a_0-b_*||))\int _{0}^{1}\delta _1(\sigma ||z_0-b_*||)\ \mathrm{{d}}\sigma }{(1-\delta _0(||z_0-b_*||))(1-\delta _0(||a_0-b_*||))} \nonumber \\&\quad +\frac{(\delta _0(||b_0-b_*||)+\delta _0(||a_0-b_*||))\int _{0}^{1}\delta _1(\sigma ||z_0-b_*||)\ \mathrm{{d}}\sigma }{2(1-\delta _0(||b_0-b_*||))(1-\delta _0(||a_0-b_*||))}\bigg ]||z_0-b_*|| \nonumber \\&\le {\mathscr {M}}_3(||b_0-b_*||)||b_0-b_*||\le ||b_0-b_*||, \end{aligned}$$
(2.23)

showing \(b_1 \in B(b_*, r)\) and (2.14) for \(n=0\). The induction for inequalities (2.12)–(2.14) is terminated, if we replace \(b_0, a_0, z_0, b_1\) by \(b_k, a_k, z_k, b_{k+1}\) in the previous calculations. It then follows from the estimate

$$\begin{aligned} ||b_{k+1}-b_*||\le \Delta ||b_k-b_*||<r, \end{aligned}$$
(2.24)

where \(\Delta ={\mathscr {M}}_3(||b_0-b_*||) \in [0, 1)\) that \(b_{k+1} \in B(b_*, r)\) and \({\lim _{k \rightarrow \infty } b_k=b_*}\). Next, set \(G=\int _{0}^{1}F'(b_*+\sigma (u-b_*))\ \mathrm{{d}}\sigma \) for \(u \in {\mathscr {H}}_1\) with \(F(u)=0\). By \((C_2)\) and \((C_5)\), we have in turn that

$$\begin{aligned} ||F'(b_*)^{-1}(G-F'(b_*))||&\le \int _{0}^{1}\delta _0(\sigma ||u-b_*||)\ \mathrm{{d}}\sigma \nonumber \\&\le \int _{0}^{1}\delta _0(\sigma r_*)\ \mathrm{{d}}\sigma <1, \end{aligned}$$
(2.25)

so \(u=b_*\) follows from the estimate \(0=F(u)-F(b_*)=G(u-b_*)\). \(\square \)

Next, we present the local convergence analysis of JLM2 along the same lines. Let us introduce function

$$\begin{aligned} \overline{{\mathscr {M}}_3}(t)&= \Bigg [\frac{\int _{0}^{1}\delta ((1-\sigma ){\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{1-\delta _0({\mathscr {M}}_2(t)t)} \\&\quad + \frac{(\delta _0(t)+\delta _0({\mathscr {M}}_2(t)t))\int _{0}^{1}\delta _1(\sigma {\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{(1-\delta _0(t))(1-\delta _0({\mathscr {M}}_2(t)t))} \\&\quad + \frac{3(\delta _0(t)+\delta _0({\mathscr {M}}_1(t)t))(\delta _0(t)+8+9\delta _0({\mathscr {M}}_1(t)t))\int _{0}^{1}\delta _1(\sigma {\mathscr {M}}_2(t)t)\ \mathrm{{d}}\sigma }{16(1-p(t))^2 (1-\delta _0(t))} \Bigg ] {\mathscr {M}}_2(t). \end{aligned}$$

Suppose equation

$$\begin{aligned} \overline{{\mathscr {M}}_3}(t)-1=0 \end{aligned}$$
(2.26)

has a least solution \(\overline{r_3} \in (0, r_1)\).

Set

$$\begin{aligned} {\overline{r}}=\min \{r_1, r_2, \overline{r_3}\}. \end{aligned}$$
(2.27)

We shall show \({\overline{r}}\) is a radius of convergence for JLM2.

The following estimations are needed

$$\begin{aligned} b_{n+1}-b_*&=z_n-b_*-F'(z_n)^{-1}F(z_n)+(F'(z_n)^{-1}-F'(b_n)^{-1})F(z_n) \\&\quad +\bigg [I-\bigg (\frac{1}{2}(3F'(a_n)-F'(b_n)\bigg )^{-1}(3F'(a_n)+F'(b_n))\bigg )^2\bigg ]F'(b_n)^{-1}F(z_n), \end{aligned}$$

so

$$\begin{aligned} ||b_{n+1}-b_*||&\le \bigg [\frac{\int _{0}^{1}\delta ((1-\sigma )||z_n-b_*||)\ \mathrm{{d}}\sigma }{1-\delta _0(||z_n-b_*||)} \nonumber \\&\quad +\frac{(\delta _0(||b_n-b_*||)+\delta _0(||z_n-b_*||))\int _{0}^{1}\delta _1(\sigma ||z_n-b_*||)\ \mathrm{{d}}\sigma }{(1-\delta _0(||b_n-b_*||))(1-\delta _0(||z_n-b_*||))} \nonumber \\&\quad +\frac{||A_n||||B_n||\int _{0}^{1}\delta _1(\sigma ||z_n-b_*||)\ \mathrm{{d}}\sigma }{1-\delta _0(||b_n-b_*||)} \bigg ]||z_n-b_*|| \nonumber \\&\le \overline{{\mathscr {M}}_3}(||b_n-b_*||)||b_n-b_*||\le ||b_n-b_*|| \end{aligned}$$
(2.28)

showing \(b_{n+1} \in B(b_*, {\overline{r}})\) and (2.14) for \(n=0\) (for \({\mathscr {M}}_3=\overline{{\mathscr {M}}_3}\) and \(r=\rho ={\overline{r}}\)), where we used

$$\begin{aligned} A_n&=I-\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(b_n)) \\&=(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)-F'(b_n)-\frac{1}{2}(3F'(a_n)+F'(b_n))) \\&=\frac{3}{2}(3F'(a_n)-F'(b_n))^{-1}(F'(a_n)-F'(b_n)), \end{aligned}$$

so

$$\begin{aligned} ||A_n||&\le \frac{3(\delta _0(||b_n-b_*||)+(\delta _0(||a_n-b_*||))}{3(1-p(||b_n-b_*||))}, \\ B_n&=1+\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(3F'(a_n)+F'(b_n)) \\&=\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}(9F'(a_n)-F'(b_n)) \\&=\frac{1}{2}(3F'(a_n)-F'(b_n))^{-1}[9(F'(a_n)-F'(b_n)) \\&\quad +8F'(b_*)+(F'(b_*)-F'(b_n))], \end{aligned}$$

so

$$\begin{aligned} ||B_n||\le \frac{9\delta _0(||a_n-b_*||)+8+\delta _0(||b_n-b_*||)}{4(1-p(||b_n-b_*||))}. \end{aligned}$$

Hence, we arrived at the following local convergence result for JLM2.

Theorem 2.2

Under the conditions (C) for \(R={\overline{r}}\), the conclusions of Theorem 1 ***hold for JLM2 with \(\overline{{\mathscr {M}}_3}, {\overline{r}}, \overline{r_*}\) replacing \({\mathscr {M}}_3, r, r_*\), respectively.

Remark 2.3

In existing research works on iterative algorithms, the assumption

$$\begin{aligned} ||F'(b_*)^{-1} (F'(b)-F'(a))|| \le {\overline{\delta }}(||b-a||), \ {\text{ f }or \ all} \ b, a \in {\mathscr {H}} \end{aligned}$$

is used instead of the condition \(||F'(b_*)^{-1}(F'(b)-F'(a))|| \le \delta (||b-a||), \ {\text{ f }or \ all} \ b, a \in {\mathscr {H}}_0\). But then, since \({\mathscr {H}}_0 \subseteq {\mathscr {H}}\), we have

$$\begin{aligned} \delta (t) \le {\overline{\delta }}(t), \ {\text{ f }or \ all} \ t \in [0, 2\rho _0). \end{aligned}$$

This is a significant achievement. All earlier works can be rewritten in terms of \(\delta \), since the iterates belong to \({\mathscr {H}}_0\) which is a more accurate location about the iterates \(b_n\). This enhances the radius of convergence ball; tightens the upper error distances \(||b_n-b_*||\) and offers a better knowledge about \(b_*\). If we look at the example \(F(b)=e^b-1\) for \({\mathscr {H}}=B(0,1)\), then we have

$$\begin{aligned} \delta _0(t)=(e-1)t< \delta (t)=e^{\frac{1}{e-1}}t < {\overline{\delta }}(t)=et, \end{aligned}$$

and using Rheinboldt or Traub [23, 31] (for \(\delta _0=\delta ={\overline{\delta }}\)) we get \(R_{TR}=0.242529\). Moreover, Argyros et al. [3, 6] (for \(\delta ={\overline{\delta }}\)), estimated \(R_E=0.324947\). But under the proposed analysis \(r_1=0.382692\), so

$$\begin{aligned} R_{TR}<R_E<r_1. \end{aligned}$$

3 Comparison of attraction basins

To compare the dynamical properties of JLM1 and JLM2, we apply the well-known attraction basin approach. For a second or higher degree complex polynomial \({\mathscr {A}}(z)\), the set of points \(\{z_0 \in {\mathbb {C}}: z_j \rightarrow z_*\ as\ j \rightarrow \infty \}\) forms the attraction basin corresponding to a zero \(z_*\) of \({\mathscr {A}}\), where \(\{z_j\}_{j=0}^{\infty }\) is constructed by an iterative formula starting from \(z_0 \in {\mathbb {C}}\). Let us consider an area \({\mathscr {E}} =[-4, 4]\) \(\times \) \([-4, 4]\) on \({\mathbb {C}}\) with a grid of \(400 \times 400\) points. To prepare attraction basins, we apply JLM1 and JLM2 on various polynomials by employing each point \(z_0 \in {\mathscr {E}}\) as an initial estimation. The point \(z_0\) belongs to the basin of a zero \(z_*\) of a considered polynomial if \(\displaystyle {\lim _{j \rightarrow \infty } z_j=z_*}\). We demonstrate the starter \(z_0\) using a typical color corresponding to \(z_*\). According to the number of iterations, we apply the light to dark colors to every \(z_0\). Black color is used to indicate the non-convergence regions. The condition to terminate the iteration process is \(||z_j-z_*|| < 10^{-6}\) with maximum 400 iterations. With the help of MATLAB 2019a, the fractal diagrams are produced.

Fig. 1
figure 1

Comparison of attraction basins associated to second degree polynomial \({\mathscr {A}}_{1}(z)=z^2-1\)

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Fig. 2
figure 2

Comparison of attraction basins corresponding to second degree polynomial \({\mathscr {A}}_{2}(z)=z^2-z-1\)

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Fig. 3
figure 3

Comparison of attraction basins associated to third degree polynomial \(\mathscr {A}_{3}(z)=z^3-1\)

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Fig. 4
figure 4

Comparison of attraction basins related to third degree polynomial \({\mathscr {A}}_{4}(z)=z^3-(-0.7250+1.6500i)z-0.275-1.65i\)

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Fig. 5
figure 5

Comparison of attraction basins associated to fourth degree polynomial \({\mathscr {A}}_{5}(z)=z^4-1\)

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Fig. 6
figure 6

Comparison of attraction basins related to fourth degree polynomial \({\mathscr {A}}_{6}(z)=z^4-10z^2+9\)

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Fig. 7
figure 7

Comparison of attraction basins associated to fifth degree polynomial \({\mathscr {A}}_{7}(z)=z^5-1\)

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Fig. 8
figure 8

Comparison of attraction basins corresponding to fifth degree polynomial \({\mathscr {A}}_{8}(z)=z^5+z\)

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Fig. 9
figure 9

Comparison of attraction basins associated to sixth degree polynomial \({\mathscr {A}}_{9}(z)=z^6-1\)

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First of all, polynomials \({\mathscr {A}}_{1}(z)=z^2-1\) and \({\mathscr {A}}_{2}(z)=z^2-z-1\) of degree two are used to compare the attraction basins for JLM1 and JLM2. The comparison results are shown in Figs. 1 and 2. In Fig. 1, green and pink areas indicate the attraction basins corresponding to the zeros \(-1\) and 1, respectively, of \({\mathscr {A}}_{1}(z)\). The basins related to the solutions \(\frac{1+\sqrt{5}}{2}\) and \(\frac{1-\sqrt{5}}{2}\) of \({\mathscr {A}}_{2}(z)=0\) are displayed in Fig. 2 using pink and green colors, respectively. Figs. 3 and 4 provide the attraction basins for JLM1 and JLM2 associated to the zeros of \({\mathscr {A}}_{3}(z)=z^3-1\) and \({\mathscr {A}}_{4}(z)=z^3-(-0.7250+1.6500i)z-0.275-1.65i\). In Fig. 3, the basins of the solutions \(-\frac{1}{2}-0.866025i\), 1 and \(-\frac{1}{2}+0.866025i\) of \({\mathscr {A}}_{3}(z)=0\) are shown in blue, green and pink, respectively. The basins for JLM1 and JLM2 related to the zeros 1, \(-1.401440+0.915201i\) and \(0.4014403-0.915201i\) of \({\mathscr {A}}_{4}(z)\) are displayed in Fig. 4 by means of green, pink and blue regions, respectively. Next, polynomials \({\mathscr {A}}_{5}(z)=z^4-1\) and \({\mathscr {A}}_{6}(z)=z^4-10z^2+9\) of degree four are used to compare the attraction basins for JLM1 and JLM2. Fig. 5 shows the comparison of basins for these schemes associated to the solutions 1, \(-i\), i and 1 of \({\mathscr {A}}_{5}(z)=0\) are denoted in yellow, green, pink and blue regions. The basins for for JLM1 and JLM2 corresponding to the zeros \(-1\), 3, \(-3\) and 1 of \({\mathscr {A}}_{6}(z)\) are presented in Fig. 6 using yellow, pink, green and blue colors, respectively. Furthermore, we take polynomials \({\mathscr {A}}_{7}(z)=z^5-1\) and \({\mathscr {A}}_{8}(z)=z^5+z\) of degree five to design the attraction basins for JLM1 and JLM2. Fig. 7 provides the basins of zeros of 1, \(0.309016+0.951056i\), \(0.309016-0.951056i\), \(-0.809016+0.587785i\) and \(-0.809016-0.587785i\) of \({\mathscr {A}}_{7}(z)\) in cyan, red, yellow, pink and green colors, respectively. In Fig. 8, green, cyan, red, pink and yellow regions indicate the attraction basins of the solutions \(-0.707106-0.707106i\), \(-0.707106+0.707106i\), \(0.707106+0.707106i\), \(0.707106-0.707106i\) and 0, respectively, of \({\mathscr {A}}_{8}(z)=0\). Finally, degree six complex polynomials \({\mathscr {A}}_{9}(z)=z^6-1\) and \({\mathscr {A}}_{10}(z)=z^6+z\) are taken. In Fig. 9, the attraction basins for JLM1 and JLM2 associated to the zeros \(-0.500000-0.866025i\), \(-1\), \(0.500000-0.866025i\), 1, \(0.500000+0.866025i\) and \(-0.500000+0.866025i\) of \({\mathscr {A}}_{9}(z)\) are displayed in pink, green, yellow, red, cyan and blue colors, respectively. In Fig. 10, green, pink, red, yellow, cyan and blue colors are applied to illustrate the basins associated to the solutions \(-1.134724\), \(0.629372-0.735755i\), 0.7780895, \(-0.451055-1.002364i\), \(0.629372+0.735755i\) and \(-0.451055+1.002364i\) of \({\mathscr {A}}_{10}(z)=0\), respectively.

It is noticed from Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 that JLM1 has the bigger attraction basins in comparison to JLM2. Also, JLM1 shows less chaotic behavior than JLM2 on the boundary points of these basins. Hence, the numerical stability of JLM1 is higher than JLM2.

Fig. 10
figure 10

Comparison of attraction basins related to sixth degree polynomial \({\mathscr {A}}_{10}(z)=z^6+z\)

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4 Numerical examples

We apply the proposed techniques to estimate the convergence radii of JLM1 and JLM2.

Example 4.1

Let us consider \({\mathscr {Z}}_1={\mathscr {Z}}_2=C[0, 1]\) and \({\mathscr {H}}={\overline{S}}(0, 1)\). Define F on \({\mathscr {H}}\) by

$$\begin{aligned} F(b)(a)=b(a)-5 \int _{0}^{1} a w \ b(w)^{3}\ \mathrm{{d}}w, \end{aligned}$$

where \(b(a) \in C[0, 1]\). We have \(b_*=0\). Also, \(\delta _0(t)=7.5t\), \(\delta (t)=15t\) and \(\delta _1(t)=2\). The values of r and \({\overline{r}}\) are produced by the application of proposed theorems and summarized in Table 1.

Example 4.2

Let \({\mathscr {Z}}_1={\mathscr {Z}}_2={\mathbb {R}}^3\) and \({\mathscr {H}}={\overline{S}}(0, 1)\). Consider F on \({\mathscr {H}}\) for \(b=(b_1, b_2, b_3)^{t}\) as

$$\begin{aligned} F(b)=\left( e^{b_1}-1, \frac{e-1}{2}b_2^2 + b_2, b_3\right) ^t \end{aligned}$$

We have \(b_*=(0, 0, 0)^t\). Also, \(\delta _0(t)=(e-1)t\), \(\delta (t)=e^{\frac{1}{e-1}}t\) and \(\delta _1(t)=2\). Using the newly proposed theorems the values of r and \({\overline{r}}\) are calculated and displayed in Table 2.

Example 4.3

Finally, the motivational problem described in the first section is addressed with \(b_*=0\), \(\delta _0(t)=\delta (t)=96.662907t\) and \(\delta _1(t)=2\). We apply the suggested theorems to compute r and \({\overline{r}}\). These values are shown in Table 3.

Table 1 Comparison of convergence radii for Example 1
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Table 2 Comparison of convergence radii for Example 2
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Table 3 Comparison of convergence radii for Example 3
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5 Conclusions

The convergence balls of two sixth order Jarratt-like equation solvers (JLM1 and JLM2) are compared. For the ball analysis results, the first derivative and generalized Lipschitz conditions are used. The complex dynamical characteristics of these algorithms are also compared via the attraction basin approach. It is seen that JLM1 has the bigger basins than JLM2. Finally, the new analytical outcomes are validated on numerical problems. We obtained that JLM1 has the larger convergence balls in comparison to JLM2.