1 Introduction

The study presented in this paper is based on the problem of finding a locally unique solution \(x^{*}\) of the equation

$$\begin{aligned} F(x)=0, \end{aligned}$$
(1)

where \(F:\varOmega \subseteq X \rightarrow Y\) is a Fréchet differentiable function and \(\varOmega\) is a convex subset of X. X and Y are Banach spaces. In the field of applied science and engineering, a large number of problems can be solved by transforming them into nonlinear equations of the form (1). For instance, the boundary value problems occur in Kinetic theory of gases, the integral equations related to radiative transfer theory, problems in optimization and many others can be reduced to the problem of solving nonlinear equations. Usually, the solutions of these nonlinear equations can be found in closed form. So, the most frequently used solution techniques are iterative in nature.

A commonly used iterative technique for solving (1) is Newton’s scheme, which can be expressed as:

$$\begin{aligned} x_{n+1}=x_{n}-[F'(x_{n})]^{-1}F(x_{n}),\ n \ge 0. \end{aligned}$$
(2)

Evaluation of second and more order derivatives is a major drawback of higher-order iterative schemes and are not appropriate for practical use. Due to the calculation of \(F''\) in each iteration, the cubically convergent classical schemes are not suitable in terms of computational cost. Some classical third-order algorithms include Chebyshev’s, the Halley’s and Super-Halley’s schemes are produced by putting (\(\alpha =0\)), (\(\alpha =\frac{1}{2}\)) and (\(\alpha =1\)) respectively in

$$\begin{aligned} x_{n+1}=x_{n}-\left( 1+\frac{1}{2}(1-\alpha H_{F}(x_{n}))^{-1} H_{F}(x_{n}) \right) [F'(x_{n})]^{-1}F(x_{n}), \end{aligned}$$
(3)

where \(H_{F}(x_{n})=F'(x_{n})^{-1}F''(x_{n})F'(x_{n})^{-1}F(x_{n})\).

The local convergence analysis of many varieties of the methods defined in (3) has been studied by numerous authors in Refs. [1,2,3,4,5,6]. Also, the local convergence analysis for various iterative algorithms is studied in Banach spaces in Refs. [7,8,9,10,11,12]. In this paper, we use the Lipschitz continuity condition on the first derivative only to enhance the applicability of modified Weerakoon’s method in Banach spaces.

In Ref. [13], the authors studied the modification of Weerakoon’s method [14] with fifth-order convergence to solve systems of nonlinear equations in \({\mathbb {R}}^{n}\). The method is given as:

$$\begin{aligned}&y_n =x_n-F'(x_n)^{-1}F(x_n) \nonumber \\&z_n =x_n-2[F'(x_n)+F'(y_n)]^{-1}F(x_n) \nonumber \\&x_{n+1} =z_n-F'(y_n)^{-1}F(z_n) \end{aligned}$$
(4)

In this method, only the first-order derivative occurs in the iteration function but the convergence is proved with the assumption on higher-order derivatives for which the applicability of the method is restricted. For instance, consider a function F defined on \(\varOmega =\left[ -\frac{1}{2}, \frac{5}{2}\right]\) by

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

Notice that \(F'''\) is unbounded on \(\varOmega\). Therefore, the previous studies [13,14,15] based on higher-order derivatives fail to solve this problem. Also, no information is mentioned regarding the radius of convergence ball in these studies. The local convergence analysis of iterative algorithms provides essential information about the radius of convergence ball. In this paper, we provide the local convergence analysis of the method (4) using the hypotheses only on \(F'\) to avoid the use of higher-order derivatives. Particularly, it is assumed that the first derivative is Lipschitz continuous. This study extends the applicability of the method (4) and helps in obtaining the solution of such problems for which previous studies fail.

The rest portion of this paper is arranged as follows: The local convergence analysis of the method (4) is placed in Sect. 2. Section 3 is devoted to demonstrating the applications of our theoretical outcomes on some numerical examples. Conclusions are discussed in the last section.

2 Local convergence analysis

The local convergence analysis of modified Weerakoon’s method (4) is studied in this section. Let the open and closed balls in X are denoted as \(B(c, \rho )\) and \({\bar{B}}(c, \rho )\) respectively with center c and radius \(\rho >0\). Suppose the parameters \(k_0>0\) and \(k>0\) be given with \(k_0 \le k\). To study the local convergence of the scheme (4), we introduce the function \(J_1\) on the interval \(\left[ 0, \frac{1}{k_0}\right)\) by

$$\begin{aligned} J_1(s)=\frac{ks}{2(1-k_0 s)} \end{aligned}$$
(5)

and the parameter

$$\begin{aligned} R_1=\frac{2}{2k_0 + k} < \frac{1}{k_0}. \end{aligned}$$

Observe that \(J_1(R_1)=1\). Again, we define functions \(J_2\) and \(K_2\) on \(\left[ 0, \frac{1}{k_0}\right)\) by

$$\begin{aligned} J_2(s)= \frac{k_0}{2}(1+J_1(s))s \end{aligned}$$
(6)

and

$$\begin{aligned} K_2(s)=J_2(s)-1. \end{aligned}$$

Now, \(K_2(0)=-1<0\) and \(\displaystyle {\lim _{s \rightarrow \left( \frac{1}{k_0}\right) ^{-}}K_2(s)=+\infty }\). According to the intermediate value theorem, the interval \((0,\frac{1}{k_0})\) contains the zeros of the function \(K_2(s)\). Let the smallest zero of \(K_2(s)\) in \(\left( 0, \frac{1}{k_0}\right)\) is \(R_2\). Also, we introduce functions \(J_3\) and \(K_3\) on \([0, R_2)\) by

$$\begin{aligned} J_3(s)= \frac{k[1+J_1(s)]s}{2(1-J_2(s))} \end{aligned}$$
(7)

and

$$\begin{aligned} K_3(s)=J_3(s)-1. \end{aligned}$$

Now, \(K_3(0)=-1<0\) and \(\displaystyle {\lim _{s \rightarrow R_2^{-}}K_3(s)=+\infty }\). The intermediate value theorem confirms that the interval \((0, R_2)\) contains the zeros of the function \(K_3(s)\). Let the smallest zero of \(K_3(s)\) in \((0, R_2)\) is \(R_3\). Again, we define \(J_4\) and \(K_4\) on \(\left[ 0, \frac{1}{k_0}\right)\) by

$$\begin{aligned} J_4(s)= k_0J_1(s)s \end{aligned}$$
(8)

and

$$\begin{aligned} K_4(s)=J_4(s)-1. \end{aligned}$$

Now, \(K_4(0)=-1<0\) and \(\displaystyle {\lim _{s \rightarrow \left( \frac{1}{k_0}\right) ^{-}}K_4(s)=+\infty }\). According to the intermediate value theorem, the interval \(\left( 0, \frac{1}{k_0}\right)\) contains the zeros of the function \(K_4(s)\). Let the smallest zero of \(K_4(s)\) in \(\left( 0, \frac{1}{k_0}\right)\) is \(R_4\). Finally, let us define \(J_5\) and \(K_5\) on \([0, R_4)\) by

$$\begin{aligned} J_5(s)= \left( 1+\frac{1+k_0J_3(s)s}{1-J_4(s)}\right) J_3(s) \end{aligned}$$
(9)

and

$$\begin{aligned} K_5(s)=J_5(s)-1. \end{aligned}$$

Now, \(K_5(0)=-1<0\) and \(\displaystyle {\lim _{s \rightarrow R_4^{-}}K_5(s)=+\infty }\). The intermediate value theorem confirms that the interval \((0, R_4)\) contains the zeros of the function \(K_5(s)\). Let the smallest zero of \(K_5(s)\) in \((0, R_4)\) is \(R_5\). Consider

$$\begin{aligned} R=min\{R_1, R_3, R_5\} \end{aligned}$$
(10)

Now, we have

$$\begin{aligned} 0\le & {} J_1(s) <1, \end{aligned}$$
(11)
$$\begin{aligned} 0\le & {} J_2(s) <1, \end{aligned}$$
(12)
$$\begin{aligned} 0\le & {} J_3(s) <1, \end{aligned}$$
(13)
$$\begin{aligned} 0\le & {} J_4(s) <1, \end{aligned}$$
(14)

and

$$\begin{aligned} 0\le J_5(s) <1 \end{aligned}$$
(15)

for each \(s \in [0, R)\). Furthermore, let us assume the followings hold for the Fréchet differentiable function \(F:\varOmega \subseteq X \rightarrow Y\).

$$\begin{aligned} F(x^{*})=0,\ F'(x^{*})^{-1} \in BL(Y, X), \end{aligned}$$
$$\begin{aligned} ||F'(x^{*})^{-1} (F'(x)-F'(x^{*}))|| \le k_0 ||x-x^{*}||,\ \forall x \in \varOmega \end{aligned}$$
(16)

and

$$\begin{aligned} ||F'(x^{*})^{-1} (F'(x)-F'(y))|| \le k ||x-y||,\ \forall x, y \in \varOmega , \end{aligned}$$
(17)

where BL(Y, X) is the set of all bounded linear operators from Y to X.

In several studies [1, 2, 9, 16, 17], a third condition assumed is

$$\begin{aligned} ||F'(x^{*})^{-1} F'(x)|| \le M,\ \forall x \in B\left( x^{*}, \frac{1}{k_0}\right) . \end{aligned}$$
(18)

This assumption is not taken in our study. We use the following results to avoid this extra condition.

Lemma 1

If F obeys (16) and \({\bar{B}}(x^{*}, R) \subseteq \varOmega\), then \(\forall x \in B(x^{*}, R)\), we get

$$\begin{aligned} ||F'(x^{*})^{-1} F'(x)|| \le 1 + k_0||x-x^{*}|| \end{aligned}$$
(19)

and

$$\begin{aligned} ||F'(x^{*})^{-1} F(x)|| \le (1 + k_0||x-x^{*}||) ||x-x^{*}|| \end{aligned}$$
(20)

Proof

Applying (16), we obtain

$$\begin{aligned} ||F'(x^{*})^{-1} F'(x)|| \le 1 + ||F'(x^{*})^{-1}(F'(x)-F'(x^{*}))|| \le 1 + k_0||x-x^{*}||. \end{aligned}$$

For \(\theta \in [0, 1]\),

$$\begin{aligned} ||F'(x^{*})^{-1}F'(x^{*}+\theta (x-x^{*}))|| \le 1 + k_0 \theta ||x-x^{*}|| \le 1 + k_0||x-x^{*}|| \end{aligned}$$

The mean value theorem is used to obtain

$$\begin{aligned} ||F'(x^{*})^{-1} F(x)||&= ||F'(x^{*})^{-1}(F(x)-F(x^{*}))|| \\&\le ||F'(x^{*})^{-1}F'(x^{*}+\theta (x-x^{*}))(x-x^{*})|| \\&\le (1 + k_0||x-x^{*}||) ||x-x^{*}||. \end{aligned}$$

\(\square\)

Next, the local convergence analysis of the method (4) is presented in Theorem 1.

Theorem 1

Let \(F:\varOmega \subseteq X \rightarrow Y\)be a Fréchet differentiable function. Suppose \(x^{*} \in \varOmega, F(x^{*})=0, F'(x^{*})^{-1} \in BL(Y, X)\), F obeys (16), (17) and

$$\begin{aligned} {\bar{B}}(x^{*}, R) \subseteq \varOmega , \end{aligned}$$
(21)

where R is defined in (10). Starting from \(x_0 \in B(x^{*}, R)\)the method (4) generates the sequence of iterates \(\{x_n\}\)which is well defined, \(\{x_n\}_{n \ge 0} \in B(x^{*}, R)\)and converges to the solution \(x^{*}\)of (1). Moreover, the following estimations hold \(\forall n \ge 0\)

$$\begin{aligned} ||y_n-x^{*}||\le & {} J_1(||x_n-x^{*}||)||x_n-x^{*}||<||x_n-x^{*}||<R, \end{aligned}$$
(22)
$$\begin{aligned} ||z_n-x^{*}||\le & {} J_3(||x_n-x^{*}||)||x_n-x^{*}||<||x_n-x^{*}||<R, \end{aligned}$$
(23)

and

$$\begin{aligned} ||x_{n+1}-x^{*}|| \le J_5(||x_n-x^{*}||)||x_n-x^{*}||<||x_n-x^{*}||<R, \end{aligned}$$
(24)

where the functions \(J_1\), \(J_3\)and \(J_5\)are given in (5), (7) and (9) respectively. Furthermore, the solution \(x^{*}\)of the equation \(F(x)=0\)is unique in \({\bar{B}}(x^{*}, \varDelta ) \cap \varOmega\), where \(\varDelta \in [R, \frac{2}{k_0})\).

Proof

Using the definition of R, the equation (16) and the assumption \(x_0 \in B(x^{*}, R)\), we find

$$\begin{aligned} ||F'(x^{*})^{-1} (F'(x_0)-F'(x^{*}))|| \le k_0 ||x_0-x^{*}||<k_0R<1. \end{aligned}$$
(25)

Now, Banach Lemma on invertible functions [18,19,20,21,22,23] confirms that \(F'(x_0)^{-1} \in BL(Y, X)\) and

$$\begin{aligned} ||F'(x_0)^{-1}F'(x^{*})|| \le \frac{1}{1-k_0 ||x_0-x^{*}||} < \frac{1}{1-k_0R}. \end{aligned}$$
(26)

Hence, it follows from the first step of the method (4) for \(n=0\) that \(y_0\) is well defined. Again,

$$\begin{aligned} y_0-x^{*}&=x_0-x^{*}-F'(x_0)^{-1}F(x_0) \nonumber \\&=-\left[ F'(x_0)^{-1}F'(x^{*})\right] \left[ \int _{0}^{1} F'(x^{*})^{-1} (F'(x^{*}+\theta (x_0-x^{*}))\nonumber \right. \\&\left. \quad -F'(x_0)) (x_0-x^{*}) d\theta \right] . \end{aligned}$$
(27)

Using (5), (10), (11) and (17), we find

$$\begin{aligned} ||y_0-x^{*}||&\le \left[ \left|\left|F'(x_0)^{-1}F'(x^{*})\right|\right| \right] \left[ \Bigg|\Bigg|\int _{0}^{1} F'(x^{*})^{-1} (F'(x^{*}+\theta (x_0-x^{*}))\nonumber \right. \\&\left. \quad -F'(x_0)) (x_0-x^{*})\ d\theta \Bigg|\Bigg| \right] \nonumber \\&\le \frac{k ||x_0-x^{*}||}{2(1-k_0||x_0-x^{*}||)}||x_0-x^{*}|| \nonumber \\&= J_1(||x_0-x^{*}||) ||x_0-x^{*}|| < ||x_0-x^{*}|| < R \end{aligned}$$
(28)

and this shows (22) for \(n=0\). Then we show \([F'(x_0)+F'(y_0)]^{-1} \in BL(Y, X)\). The equations (6), (10), (12), (16) and (28) are used to obtain

$$\begin{aligned}&||(2F'(x^{*}))^{-1} (F'(x_0)+F'(y_0)-2F'(x^{*}))|| \\&\quad \le \frac{1}{2}[||F'(x^{*})^{-1}(F'(x_0)-F'(x^{*}))||+ ||F'(x^{*})^{-1}(F'(y_0)-F'(x^{*}))||] \\&\quad \le \frac{k_0}{2}[||x_0-x^{*}||+||y_0-x^{*}||] \\&\quad \le \frac{k_0}{2}[||x_0-x^{*}||+J_1(||x_0-x^{*}||||x_0-x^{*}||] \\&\quad = \frac{k_0}{2}[1+J_1(||x_0-x^{*}||)] ||x_0-x^{*}||\\&\quad = J_2(||x_0-x^{*}||)<J_2(R)<1. \end{aligned}$$

Now, we obtain \([F'(x_0)+F'(y_0)]^{-1} \in BL(Y, X)\) using Banach Lemma on invertible functions. Also,

$$\begin{aligned} ||[F'(x_0)+F'(y_0)]^{-1}F'(x^{*})|| \le \frac{1}{2(1-J_2(||x_0-x^{*}||))}. \end{aligned}$$
(29)

Now, it follows from the second step of the method (4) for \(n=0\) that \(z_0\) is well defined. Using the definition of R, (13), (17), (28) and (29), we get

$$\begin{aligned} ||z_0-x^{*}||&\le \left( ||[F'(x_0)+F'(y_0)]^{-1}F'(x^{*})|| \right) \left( ||\int _{0}^{1} F'(x^{*})^{-1} \left( F'(x_0)\nonumber \right. \right. \\&\left. \quad -F'(x^{*}+\theta (x_0-x^{*}))\right) (x_0-x^{*}) d\theta || \nonumber \\&\left. \quad + ||\int _{0}^{1} F'(x^{*})^{-1} \left( F'(y_0)-F'(x^{*}+\theta (x_0-x^{*}))\right) (x_0-x^{*}) d\theta ||\right) \nonumber \\&\le \frac{\frac{k}{2} ||x_0-x^{*}||^{2} + k \int _{0}^{1}(||y_0-x^{*}-\theta (x_0-x^{*})||) d\theta ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&\le \frac{\frac{k}{2} ||x_0-x^{*}||^{2} + k (||y_0-x^{*}||+\frac{||x_0-x^{*}||}{2}) ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&\le \frac{\frac{k}{2} ||x_0-x^{*}||^{2} + k [J_1(||x_0-x^{*}||) ||x_0-x^{*}||+\frac{||x_0-x^{*}||}{2}] ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&\le \frac{(k ||x_0-x^{*}|| + kJ_1(||x_0-x^{*}||) ||x_0-x^{*}||) ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&= \frac{[k ||x_0-x^{*}|| + kJ_1(||x_0-x^{*}||)||x_0-x^{*}||] ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&= \frac{k[(1 + J_1(||x_0-x^{*}||))||x_0-x^{*}||] ||x_0-x^{*}||}{2(1-J_2(||x_0-x^{*}||))} \nonumber \\&= J_3(||x_0-x^{*}||)||x_0-x^{*}||< ||x_0-x^{*}||<R. \end{aligned}$$
(30)

Hence, we establish (23) for \(n=0\). Again,

$$\begin{aligned} ||F'(x^{*})^{-1} (F'(y_0)-F'(x^{*}))||\le & {} k_0 ||y_0-x^{*}||<k_0J_1(||x_0-x^{*}||) ||x_0-x^{*}||\nonumber \\= \, & {} J_4(||x_0-x^{*}||)<1. \end{aligned}$$
(31)

So, \(F'(y_0)^{-1} \in BL(Y, X)\) with

$$\begin{aligned} ||F'(y_0)^{-1}F'(x^{*})|| \le \frac{1}{1-J_4(||x_0-x^{*}||)}. \end{aligned}$$
(32)

Now, it follows from the last step of the method (4) for \(n=0\) that \(x_1\) is well define. Finally, we use (10), (15), (20), (30) and (32) to get

$$\begin{aligned} ||x_1-x^{*}||&\le ||z_0-x^{*}|| + ||F'(y_0)^{-1}F(z_0)|| \nonumber \\&\le ||z_0-x^{*}|| + ||F'(y_0)^{-1}F'(x^{*})||\ ||F'(x^{*})^{-1}F(z_0)|| \nonumber \\&\le ||z_0-x^{*}|| + \frac{(1+k_0||z_0-x^{*}||)||z_0-x^{*}||}{1-J_4(||x_0-x^{*}||)} \nonumber \\&\le \left( 1 + \frac{(1+k_0||z_0-x^{*}||)}{1-J_4(||x_0-x^{*}||)} \right) ||z_0-x^{*}|| \nonumber \\&\le \left( 1 + \frac{(1+k_0J_3(||x_0-x^{*}||)||x_0-x^{*}||)}{1-J_4(||x_0-x^{*}||)} \right) J_3(||x_0-x^{*}||)||x_0-x^{*}|| \nonumber \\&= J_5(||x_0-x^{*}||)||x_0-x^{*}||< ||x_0-x^{*}|| < R. \end{aligned}$$
(33)

Thus, we show the estimate (24) for \(n=0\). We get the estimates (22)-(24) by substituting \(x_n\), \(y_n\), \(z_n\) and \(x_{n+1}\) in place of \(x_0\), \(y_0\), \(z_0\) and \(x_1\) respectively in the previous estimations. Using the fact \(||x_{n+1}-x^{*}|| \le J_5(R)||x_n-x^{*}||<R\), we derive that \(x_{n+1} \in B(x^{*}, R)\) and \(\displaystyle {\lim _{n \rightarrow \infty } x_n=x^{*}}\). Now, we want to show the uniqueness of the solution \(x^{*}\). Suppose there exist another solution \(y^{*}\) of \(F(x)=0\) in \({B}(x^{*}, \varDelta )\). Consider \(T=\int _{0}^{1} F'(y^{*}+\theta (x^{*}-y^{*})) d\theta\). From equation (16), we get

$$\begin{aligned} ||F'(x^{*})^{-1} (T-F'(x^{*}))||&\le \int _{0}^{1} k_0 ||y^{*}+\theta (x^{*}-y^{*})-x^{*}||\ d\theta \\ &\le \frac{k_0}{2}||x^{*}-y^{*}|| \\ &\le \frac{k_0 \varDelta }{2} <1. \end{aligned}$$

Applying Banach Lemma, we find \(T^{-1} \in BL(Y, X)\). Now, Using the identity \(0=F(x^{*})-F(y^{*})=T(x^{*}-y^{*})\), it is concluded that \(x^{*}=y^{*}\). This ends the proof. \(\square\)

3 Numerical examples

Example 1

Define F on \(\varOmega =[-\frac{1}{2}, \frac{5}{2}]\) by

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

We have \(x^{*}=1\). Also, \(k_0=k=96.6628\). The value of R is determined using the definitions of \(``J''\) functions (Table 1).

Table 1 Parameters for example 1
Full size table

Example 2

Let us define F on \({\bar{B}}(0, 1)\) for \((x_1, x_2, x_3)^{t}\) by

$$\begin{aligned} F(x)= \left(e^{x_1}-1, \frac{e-1}{2}x_2^2 + x_2, x_3\right)^t \end{aligned}$$

We have \(x^{*}=(0, 0, 0)^t\). Also, we have \(k_0=e-1\) and \(k=e\). We determine the value of R using \(``J''\) functions (Table 2).

Table 2 Parameters for example 2
Full size table

Example 3

Let us define F on \(\varOmega =[-1, 1]\) by

$$\begin{aligned} F(x)=\sin (x) \end{aligned}$$

We have \(x^{*}=0\). Also, we have \(k_0=k=1\). R is determined using \(``J''\) functions (Table 3).

Table 3 Parameters for example 3
Full size table

Example 4

Consider the nonlinear Hammerstein type integral equation given by

$$\begin{aligned} F(x)(s)=x(s)-5 \int _{0}^{1} s t x(t)^{3}\ dt, \end{aligned}$$

where \(x(s) \in C[0, 1]\). We have \(x^{*}=0\). Also, \(k_0=7.5\) and \(k=15\). Using the definitions of \(``J''\) functions the value of R is determined (Table 4).

Table 4 Parameters for example 4
Full size table

4 Conclusions

We studied the local convergence analysis of the method (4) to find a locally unique solution of a nonlinear equation in Banach spaces. The Lipschitz continuity condition on the first derivative is used to enhance the applicability of these methods. This study helps in solving those problems for which higher-order derivative based previous studies fail. Lastly, the theoretical outcomes are applied on standard numerical examples like Hammerstein equation and system of nonlinear equations.