Extending the applicability of a third-order scheme with Lipschitz and Hölder continuous derivative in Banach spaces

Abstract

We extend the applicability of a cubically convergent nonlinear system solver using Lipschitz continuous first-order Fréchet derivative in Banach spaces. This analysis avoids the usual application of Taylor expansion in convergence analysis and extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, our study provides the radius of convergence ball and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is provided. Various numerical tests confirm that our analysis produces better results and it is useful in solving such problems where previous methods can not be implemented.

Introduction

The main aim of this paper is to extend the applicability of a third-order nonlinear system solver for estimating a locally unique solution s^∗ of

$$ G(s)=0, $$

(1)

where G:Ω⊆X→Y is a Fréchet differentiable operator with values in the Banach space Y and Ω is a convex subset of the Banach space X. In the domain of engineering and applied sciences, plenty of problems are solved by converting into nonlinear equations of the form (1). For example, the integral equations arise in radiative transfer theory, problems in optimization, several boundary value problems, and many others can be transformed into the equations in the form (1). For solving these equations, the most commonly used technique is iterative algorithms. A popular and widely accepted iterative procedure for solving (1) is Newton’s algorithm, which is expressed as:

$$ s_{k+1}=s_{k}-\left[G'\left(s_{k}\right)\right]^{-1}G\left(s_{k}\right),\ k \geq 0. $$

(2)

Some classical cubically convergent schemes like Chebyshev’s, Super-Halley’s, and the Halley’s algorithms are generated by substituting (β=0), (β=1) and ($\beta =\frac {1}{2}$) respectively in

$$ s_{k+1}=s_{k}-\left(1+\frac{1}{2}\left(1-\beta H_{G}\left(s_{k}\right)\right)^{-1} H_{G}(s_{k}) \right)\left[G'(s_{k})\right]^{-1}G\left(s_{k}\right), $$

(3)

where H_G(s_k)=G^′(s_k)⁻¹G^″(s_k)G^′(s_k)⁻¹G(s_k).

Convergence analysis is an important part in the study of iterative methods. This is generally categorized into two types, viz. semi-local and local convergence. “The semi-local convergence is based on the information around an initial point and gives criteria that ensures the convergence of iteration procedures”. “The local convergence analysis of iterative schemes is based on the information around a solution and provides the radius of convergence ball as well as the convergence domain”[1]. “The choice of the initial point is a shot in the dark, in general since the radius of convergence is not computed. The radius of convergence is useful even in cases when the solution is not known. Also, the local analysis provides computable error bounds as well as the uniqueness of the solution” [2]. Numerous researchers [1, 3–5] studied the local convergence analysis of different variants of Chebyshev-Halley type schemes including modified Halley-like, deformed Halley, and improved Chebyshev-Halley type methods. In addition to that, the local convergence study for Newton-type, Jarratt-type, Weerakoon-type, etc., is studied in Banach spaces in [2, 6–15]. In this study, our primary focus is to enhance the applicability of a third-order scheme using Lipschitz continuity condition only on first-order Fréchet derivative in Banach spaces.

In [16], the authors derived a third-order algorithm applying quadrature rule for obtaining the solutions of nonlinear systems. The scheme is given as:

$$ \begin{aligned} t_{k}&=s_{k}-[G'(s_{k})]^{-1}G(s_{k}) \\ s_{k+1} &=s_{k}-4\left[G'(s_{k}) + 3G'\left(\frac{1}{3}(s_{k}+2t_{k})\right)\right]^{-1}G(s_{k}) \end{aligned} $$

(4)

To execute this scheme, we require the computation of the first-order Fréchet derivative. However, the analysis of convergence is established by applying the Taylor series approach based on higher-order derivatives. These techniques, which need the higher-order derivatives, limit the applicability of the algorithm. As an illustration, define a function G on $\Omega =\left [-\frac {1}{2}, \frac {5}{2}\right ]$ by

$$G(s)= \left\{\begin{array}{lc} {s^{3} \log(s^{2}) + s^{5} -s^{4}}, & {if\ \ \ \ s \neq 0} \\ 0, & {if\ \ \ \ s=0}\\ \end{array}\right. $$

It is important to note that G^‴ is unbounded on Ω. Therefore, the theory based on higher-order derivatives [16] fail to solve the above problem. Also, one can get no idea about the domain of convergence in [16]. The local convergence study gives valuable information regarding the radius of convergence ball. For the algorithm (4), we discuss the local convergence by following the approach based on G^′ to stay away from the evaluation of higher-order Fréchet derivatives. In particular, we consider that the first-order Fréchet derivative belongs to the Lipschitz class. This analysis enlarges the utility of the method (4) by solving such problems for which earlier studies can not be used due to the computation higher-order Fréchet derivatives.

We arrange the manuscript as follows: the “Local convergence analysis” section deals with the local convergence study of the algorithm (4). The “Numerical examples” is dedicated to the numerical applications of our analytical results. The “Conclusions” section is placed in the last section.

Local convergence analysis

We describe the local convergence analysis of the algorithm (4) in this segment. Let the notations for the closed and open balls with center c and radius ρ>0 be $\bar {B}(c, \rho)$ and B(c,ρ) respectively. BL(Y,X) is the notation for the set of all bounded linear operators from Y to X. k₀, k₁ be two positive parameters with k₀≤k₁. In Theorems 1 and 2, we provide the local convergence analysis of the scheme (4).

Local convergence analysis of the method (4) under Lipschitz continuity condition

We define the function Φ₁ on the interval $\left [0, \frac {1}{k_{0}}\right)$ by

$$ \Phi_{1}(u)=\frac{k_{1}u}{2(1-k_{0} u)} $$

(5)

and the parameter

$$\eta_{1}=\frac{2}{2k_{0} + k_{1}} < \frac{1}{k_{0}}. $$

It is easy to observe that Φ₁(η₁)=1. Again, we define the functions Φ₂ and Ψ₂ on $\left [0, \frac {1}{k_{0}}\right)$ by

$$ \Phi_{2}(u)= \frac{1}{3}\left(1+2\Phi_{1}(u)\right) $$

(6)

and

$$\Psi_{2}(u)=\Phi_{2}(u)-1. $$

Now, $\Psi _{2}(0)=-\frac {2}{3}<0$ and ${ \underset {u \to \left (\frac {1}{k_{0}}\right)^{-}}{\lim } \Psi _{2}(u)=+\infty }$. The intermediate value theorem confirms the existence of the zeros of the function Ψ₂(u) in $\left (0,\frac {1}{k_{0}}\right)$. We denote the smallest zero of Ψ₂(u) in $\left (0, \frac {1}{k_{0}}\right)$ as η₂. Again, we define Φ₃ and Ψ₃ on $\left [0, \frac {1}{k_{0}}\right)$ by

$$ \Phi_{3}(u)= \frac{k_{0}}{4}(1+3\Phi_{2}(u))u $$

(7)

and

$$\Psi_{3}(u)=\Phi_{3}(u)-1. $$

Now, Ψ₃(0)=−1<0 and ${ \underset {u \to \left (\frac {1}{k_{0}}\right)^{-}}{\lim } \Psi _{3}(u)=+\infty }$. So, The zeros of the function Ψ₃(u) lies in $(0, \frac {1}{k_{0}})$. We denote the smallest zero of Ψ₃(u) in $\left (0, \frac {1}{k_{0}}\right)$ as η₃. Lastly, we define Φ₄ and Ψ₄ on [0,η₃) by

$$ \Phi_{4}(u)= \frac{k_{1}\left[1+\frac{3}{2}\Phi_{2}(u)\right]u}{2(1-\Phi_{3}(u))} $$

(8)

and

$$\Psi_{4}(u)=\Phi_{4}(u)-1. $$

Now, Ψ₄(0)=−1<0 and ${ \underset {u \to \eta _{3}^{-}}{\lim } \Psi _{4}(u)=+\infty }$. Let η₄ be the notation for the smallest zero of Ψ₄(u) in (0,η₃). The existence of η₄ is guaranteed by the intermediate value theorem. We choose

$$ R'=min\{\eta_{1}, \eta_{2}, \eta_{4}\} $$

(9)

to confirm the followings.

$$\begin{array}{*{20}l} &0\leq \Phi_{1}(u) <1, \end{array} $$

(10)

$$\begin{array}{*{20}l} &0\leq \Phi_{2}(u) <1, \end{array} $$

(11)

$$\begin{array}{*{20}l} &0\leq \Phi_{3}(u) <1 \end{array} $$

(12)

and

$$ 0\leq \Phi_{4}(u) <1, $$

(13)

for each u∈[0,R^′). Also, we use the following assumptions on the Fréchet differentiable operator G:Ω⊆X→Y.

$$\begin{array}{*{20}l} G(s^{*})=0,\ G'(s^{*})^{-1} \in BL(Y, X), \end{array} $$

(14)

$$\begin{array}{*{20}l} ||G'\left(s^{*}\right)^{-1} (G'(s)-G'\left(s^{*}\right))|| \leq k_{0} ||s-s^{*}||,\ \forall s \in \Omega \end{array} $$

(15)

and

$$ ||G'\left(s^{*}\right)^{-1} \left(G'(s)-G'(t)\right)|| \leq k_{1} ||s-t||,\ \forall s, t \in \Omega. $$

(16)

Now, we discuss the local convergence analysis of the algorithm (4) in Theorem 1.

Theorem 1

Let s^∗∈Ω. Suppose the Fréchet differentiable operator G:Ω⊆X→Y obeys (14)–(16) and

$$ \bar{B}\left(s^{*}, R'\right) \subseteq \Omega, $$

(17)

where R^′ is given in (9). Starting from s₀∈B(s^∗,R^′) the scheme (4) produces the sequence {s_k} which is well defined, {s_k}_k≥0∈B(s^∗,R^′) and converges to s^∗. Also, the followings hold ∀k≥0

$$ ||t_{k}-s^{*}|| \leq \Phi_{1}\left(||s_{k}-s^{*}||\right)||s_{k}-s^{*}||<||s_{k}-s^{*}||<R', $$

(18)

$$ \left|\left|\left[G'(s_{k}) + 3G'\left(\frac{1}{3}(s_{k}+2t_{k})\right)\right]^{-1}G'(s^{*})\right|\right| \leq \frac{1}{4\left(1-\Phi_{3}\left(||s_{k}-s^{*}||\right)\right)} $$

(19)

and

$$ ||s_{k+1}-s^{*}|| \leq \Phi_{4}(||s_{k}-s^{*}||)||s_{k}-s^{*}||<||s_{k}-s^{*}||<R', $$

(20)

where the functions Φ₁, Φ₃, and Φ₄ are provided in (5), (7), and (8) respectively. For $\delta \in [R', \frac {2}{k_{0}})$, the equation G(s)=0 has only one solution s^∗ in $\bar {B}\left (s^{*}, \delta \right) \cap \Omega$.

Proof

It follows from (9), (15) and the assumption s₀∈B(s^∗,R^′) that

$$ ||G'\left(s^{*}\right)^{-1} \left(G'(s_{0})-G'\left(s^{*}\right)\right)|| \leq k_{0} ||s_{0}-s^{*}||< k_{0}R'<1. $$

(21)

Now, Banach Lemma on invertible operators [17–21] ensures that G^′(s₀)⁻¹∈BL(Y,X) and

$$ ||G'\left(s_{0}\right)^{-1}G'\left(s^{*}\right)|| \leq \frac{1}{1-k_{0} ||s_{0}-s^{*}||} < \frac{1}{1-k_{0}R'}. $$

(22)

Therefore, t₀ is well defined. Again,

$$ {}\begin{aligned} t_{0}-s^{*} &=s_{0}-s^{*}-G'\left(s_{0}\right)^{-1}G\left(s_{0}\right) \\ &=-\left[G'(s_{0})^{-1}G'(s^{*})\right]\! \left[\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)\,-\,G'\left(s_{0}\right)\right) \left(s_{0}\!-s^{*}\right)\ d\theta \right]. \end{aligned} $$

(23)

Using (5), (9), (10), (16), (22) and (23), we find

$$ {}{\begin{aligned} ||t_{0}-s^{*}|| & \leq \left[ ||G'(s_{0})^{-1}G'\left(s^{*}\right)|| \right] \left[ \left|\left|\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)-G'\left(s_{0}\right)\right) \left(s_{0}-s^{*}\right)\ d\theta \right|\right| \right] \\ & \leq \frac{k_{1} ||s_{0}-s^{*}||}{2\left(1-k_{0}||s_{0}-s^{*}||\right)}||s_{0}-s^{*}|| \\ & = \Phi_{1}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}|| < ||s_{0}-s^{*}|| < R'. \end{aligned}} $$

(24)

So, (18) holds for k=0. Now,

$$ \begin{aligned} \left|\left|\frac{1}{3}(s_{0}+2t_{0})-s^{*}\right|\right| & \leq \frac{1}{3} \left|\left|(s_{0}+2t_{0})-3s^{*}\right|\right| \\ & \leq \frac{1}{3} \left(||s_{0}-s^{*}|| + 2||t_{0}-s^{*}||\right) \\ & \leq \frac{1}{3} \left(||s_{0}-s^{*}|| + 2\Phi_{1}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}||\right) \\ & = \frac{1}{3} \left(1 + 2\Phi_{1}\left(||s_{0}-s^{*}||\right)\right) ||s_{0}-s^{*}|| \\ & = \Phi_{2}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}|| < ||s_{0}-s^{*}|| < R'. \end{aligned} $$

(25)

So, $\frac {1}{3}\left (s_{0}+2t_{0}\right) \in B\left (s^{*}, R'\right)$. Then, our claim is $\left [G'(s_{0}) + 3G'\left (\frac {1}{3}(s_{0}+2t_{0})\right)\right ]^{-1} \in BL(Y, X)$. The equations (7), (9), (12), (15), (24), and (25) are used to deduce

$${}\begin{aligned} & \left|\left|(4G'(s^{*}))^{-1} \left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)-4G'\left(s^{*}\right)\right]\right|\right| \\ & \leq \frac{1}{4}\left[||G'\left(s^{*}\right)^{-1}\left(G'(s_{0})-G'\left(s^{*}\right)\right)||+ 3\left|\left|G'\left(s^{*}\right)^{-1}\left(G'\left(\frac{1}{3}\left(s_{0}+2t_{0}\right)\right)-G'(s^{*})\right)\right|\right|\right] \\ & \leq \frac{k_{0}}{4}\left[||s_{0}-s^{*}||+3\left|\left|\frac{1}{3}(s_{0}+2t_{0})-s^{*}\right|\right|\right] \\ & \leq \frac{k_{0}}{4}\left[||s_{0}-s^{*}||+3\Phi_{2}\left(||s_{0}-s^{*}||\right)||s_{0}-s^{*}||\right] \\ & = \frac{k_{0}}{4}\left[1+3\Phi_{2}\left(||s_{0}-s^{*}||\right)\right] ||s_{0}-s^{*}||\\ & = \Phi_{3}\left(||s_{0}-s^{*}||\right)<\Phi_{3}\left(R'\right)<1. \end{aligned} $$

Now, we obtain $\left [G'(s_{0}) + 3G'\left (\frac {1}{3}(s_{0}+2t_{0})\right)\right ]^{-1} \in BL(Y, X)$ using Banach Lemma on invertible operators. Also,

$$ \left|\left|\left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)\right]^{-1}G'(s^{*})\right|\right| \leq \frac{1}{4(1-\Phi_{3}(||s_{0}-s^{*}||))}. $$

(26)

Hence, s₁ is well defined. We use (8) (9), (13), (16), (25), and (26) to derive

$$ {}\begin{aligned} ||s_{1}-s^{*}|| & \leq \left(\left|\left|\left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)\right]^{-1}G'\left(s^{*}\right)\right|\right| \right) \\ & \left(\left|\left|\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'(s_{0})-G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)\right) \left(s_{0}-s^{*}\right)\ d\theta \right|\right| \right. \\ & \left. + 3\left|\left|\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)-G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)\right) \left(s_{0}-s^{*}\right)\ d\theta \right|\right| \right) \\ & \leq \frac{\frac{k_{1}}{2} ||s_{0}-s^{*}||^{2} + 3k_{1} \int_{0}^{1}\left(||\frac{1}{3}(s_{0}+2t_{0})-s^{*}-\theta\left(s_{0}-s^{*}\right)||\right) d\theta ||s_{0}-s^{*}||}{4\left(1-\Phi_{3}(||s_{0}-s^{*}||)\right)} \\ & \leq \frac{\frac{k_{1}}{2} ||s_{0}-s^{*}||^{2} + 3k_{1} \left(||\frac{1}{3}(s_{0}+2t_{0})-s^{*}||+\frac{||s_{0}-s^{*}||}{2}\right) ||s_{0}-s^{*}||}{4(1-\Phi_{3}(||s_{0}-s^{*}||))} \\ & \leq \frac{\frac{k_{1}}{2} ||s_{0}-s^{*}||^{2} + 3k_{1} \left[\Phi_{2}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}||+\frac{||s_{0}-s^{*}||}{2}\right] ||s_{0}-s^{*}||}{4(1-\Phi_{3}(||s_{0}-s^{*}||))} \\ & = \frac{(2k_{1} ||s_{0}-s^{*}|| + 3k_{1}\Phi_{2}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}||) ||s_{0}-s^{*}||}{4(1-\Phi_{3}(||s_{0}-s^{*}||))} \\ & = \frac{k_{1}\left[\left(1 + \frac{3}{2}\Phi_{2}\left(||s_{0}-s^{*}||\right)\right)||s_{0}-s^{*}||\right] ||s_{0}-s^{*}||}{2(1-\Phi_{3}(||s_{0}-s^{*}||))} \\ & = \Phi_{4}(||s_{0}-s^{*}||)||s_{0}-s^{*}|| < ||s_{0}-s^{*}||<R'. \end{aligned} $$

(27)

Therefore, we prove that (20) is true for k=0. We find the estimates (18)-(20) by substituting s_k, t_k and s_k+1 in place of s₀, t₀ and s₁ respectively in the earlier estimations. From the inequality ||s_k+1−s^∗||≤Φ₄(R^′)||s_k−s^∗||<R^′, we have s_k+1∈B(s^∗,R^′) and ${{\lim }_{k \to \infty } s_{k}=s^{*}}$. Now, we have to show the uniqueness part. Let t^∗(≠s^∗)∈B(s^∗,δ) be such that 0=G(t^∗). Consider $B=\int _{0}^{1} G'\left (t^{*}+\theta \left (s^{*}-t^{*}\right)\right)\ d\theta$. From Eq. (15), we get

$$\begin{aligned} ||G'(s^{*})^{-1} (B-G'(s^{*}))|| & \leq \int_{0}^{1} k_{0} ||t^{*}+\theta(s^{*}-t^{*})-s^{*}||\ d\theta \\ & \leq \frac{k_{0}}{2}||s^{*}-t^{*}|| \\ & \leq \frac{k_{0} \delta}{2} <1. \end{aligned} $$

Applying Banach Lemma, we confirm that B⁻¹∈BL(Y,X). The identity 0=G(s^∗)−G(t^∗)=B(s^∗−t^∗) implies that s^∗=t^∗. □

Local convergence analysis of the method (4) under Hölder continuity condition

There are numerous problems for which the technique based on Lipschitz condition fails to solve without using higher-order derivatives. As an illustration, we consider the following problem given in [6].

$$G(s)(x)=s(x)-3 \int_{0}^{1} G_{1}(x, y)\ s(y)^{\frac{5}{4}}\ dy, $$

where s(x)∈C[0,1] and G₁(x,y) is Green’s function defined on [0,1]×[0,1] by

$$G_{1}(x, y)= \left\{\begin{array}{cc} (1-x)y, & \text{if~~ \(y\leq x\)} \\ x(1-y), & \text{if~~ \(x\leq y\)} \end{array}\right. $$

Then,

$$||G'(s)-G'(t)|| \leq \frac{15}{32} ||s-t||^{\frac{1}{4}}. $$

It is important to note that G^′ does not obey Lipschitz condition. However, G^′ is Hölder continuous. So, we discuss the local convergence of the method (4) with Hölder continuous first-order Fréchet derivative. This analysis also generalizes the local convergence analysis presented in the previous section.

For q∈(0,1], we define Θ₁ on $\left [0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$ by

$$ \Theta_{1}(w)=\frac{k_{1}w^{q}}{(q+1)(1-k_{0} w^{q})} $$

(28)

and the parameter

$$\xi_{1}=\left(\frac{(q+1)}{(q+1)k_{0} + k_{1}}\right)^{\frac{1}{q}} < \left(\frac{1}{k_{0}}\right)^{\frac{1}{q}}. $$

Observe that Θ₁(ξ₁)=1. Again, we define functions Θ₂ and Γ₂ on $\left [0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$ by

$$ \Theta_{2}(w)= \frac{1}{3}(1+2\Theta_{1}(w)) $$

(29)

and

$$\Gamma_{2}(w)=\Theta_{2}(w)-1. $$

Now, $\Gamma _{2}(0)=-\frac {2}{3}<0$ and ${ \underset {w \to \left (\left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)^{-}}{\lim }\Gamma _{2}(w)=+\infty }$. The intermediate value theorem confirms the existence of the zeros of the function Γ₂(w) in $\left (0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$. We denote the smallest zero of Γ₂(w) in $\left (0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$ as ξ₂. Again, we define Θ₃ and Γ₃ on $\left [0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$ by

$$ \Theta_{3}(w)= \frac{k_{0}}{4}\left[1+3\Theta_{2}(w)^{q}\right] w^{q} $$

(30)

and

$$\Gamma_{3}(w)=\Theta_{3}(w)-1. $$

Now, Γ₃(0)=−1<0 and ${\underset {w \to \left (\left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)^{-}} {\lim } \Gamma _{3}(w)=+\infty }$. So, the zeros of the function Γ₃(w) lies in $\left (0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$. We denote the smallest zero of Γ₃(w) in $\left (0, \left (\frac {1}{k_{0}}\right)^{\frac {1}{q}}\right)$ as ξ₃. Lastly, we define Θ₄ and Γ₄ on [0,ξ₃) by

$$ \Theta_{4}(w)= \frac{ k_{1} \left(\frac{4}{q+1} + 3 \Theta_{2}(w)^{q} \right) w^{q}}{4(1-\Theta_{3}(w))} $$

(31)

and

$$\Gamma_{4}(w)=\Theta_{4}(w)-1. $$

Now, Γ₄(0)=−1<0 and ${\underset {w \to \xi _{3}^{-}}{\lim } \Gamma _{4}(w)=+\infty }$. Let ξ₄ be the notation for the smallest zero of Γ₄(w) in (0,ξ₃). The existence of ξ₄ is guaranteed by the intermediate value theorem. We choose

$$ R=min\left\{\xi_{1}, \xi_{2}, \xi_{4}\right\} $$

(32)

to confirm the followings.

$$\begin{array}{*{20}l} &0\leq \Theta_{1}(w) <1, \end{array} $$

(33)

$$\begin{array}{*{20}l} &0\leq \Theta_{2}(w) <1, \end{array} $$

(34)

$$\begin{array}{*{20}l} &0\leq \Theta_{3}(w) <1 \end{array} $$

(35)

and

$$ 0\leq \Theta_{4}(w) <1, $$

(36)

for each w∈[0,R). Also, we use the following assumptions on the Fréchet differentiable operator G:Ω⊆X→Y.

$$\begin{array}{*{20}l} G(s^{*})=0,\ G'(s^{*})^{-1} \in BL(Y, X), \end{array} $$

(37)

$$\begin{array}{*{20}l} ||G'(s^{*})^{-1} (G'(s)-G'(s^{*}))|| \leq k_{0} ||s-s^{*}||^{q},\ \forall s \in \Omega \end{array} $$

(38)

and

$$ ||G'\left(s^{*}\right)^{-1} (G'(s)-G'(t))|| \leq k_{1} ||s-t||^{q},\ \forall s, t \in \Omega. $$

(39)

Theorem 2

Let s^∗∈Ω. Suppose the Fréchet differentiable operator G:Ω⊆X→Y obeys (37)–(39) and

$$ \bar{B}(s^{*}, R) \subseteq \Omega, $$

(40)

where R is given in (32). Starting from s₀∈B(s^∗,R), the scheme (4) yields the sequence {s_k} which is well defined, {s_k}_k≥0∈B(s^∗,R) and converges to s^∗. Also, the following holds ∀k≥0

$$\begin{array}{*{20}l} ||t_{k}-s^{*}|| \leq \Theta_{1}\left(||s_{k}-s^{*}||\right)||s_{k}-s^{*}||<||s_{k}-s^{*}||<R, \end{array} $$

(41)

$$\begin{array}{*{20}l} \left|\left|\left[G'(s_{k}) + 3G'\left(\frac{1}{3}\left(s_{k}+2t_{k}\right)\right)\right]^{-1}G'(s^{*})\right|\right| \leq \frac{1}{4(1-\Theta_{3}(||s_{k}-s^{*}||))} \end{array} $$

(42)

and

$$ ||s_{k+1}-s^{*}|| \leq \Theta_{4}\left(||s_{k}-s^{*}||\right)||s_{k}-s^{*}||<||s_{k}-s^{*}||<R, $$

(43)

where the functions Θ₁, Θ₃, and Θ₄ are provided in (28), (30), and (31) respectively. For $\Delta \in \left [R, \left (\frac {q+1}{k_{0}} \right)^{\frac {1}{q}} \right)$, the solution s^∗ is the only solution of G(s)=0 in $\bar {B}(s^{*}, \Delta) \cap \Omega$.

Proof

It follows from (32), (38) and the assumption s₀∈B(s^∗,R) that

$$ ||G'\left(s^{*}\right)^{-1} \left(G'\left(s_{0})-G'(s^{*}\right)\right)|| \leq k_{0} ||s_{0}-s^{*}||^{q}< k_{0}R^{q}<1. $$

(44)

Now, Banach Lemma on invertible operators [17–21] ensures that G^′(s₀)⁻¹∈BL(Y,X) and

$$ ||G'(s_{0})^{-1}G'\left(s^{*}\right)|| \leq \frac{1}{1-k_{0} ||s_{0}-s^{*}||^{q}} < \frac{1}{1-k_{0}R^{q}}. $$

(45)

Therefore, t₀ is well defined. Again,

$$\begin{array}{*{20}l} {}t_{0}-s^{*} &=s_{0}-s^{*}-G'(s_{0})^{-1}G(s_{0}) \\ &=\!-\left[G'(s_{0})^{-1}G'(s^{*})\right] \left[\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)-G'\left(s_{0}\right)\right) (s_{0}-s^{*})\ d\theta \right]. \end{array} $$

(46)

Using (28), (32), (33), (39), (45) and (46), we find

$$ {}{\begin{aligned} ||t_{0}-s^{*}|| & \leq \left[ ||G'(s_{0})^{-1}G'(s^{*})|| \right] \left[ \left|\left|\int_{0}^{1} G'(s^{*})^{-1} (G'(s^{*}+\theta(s_{0}-s^{*}))-G'(s_{0})) (s_{0}-s^{*})\ d\theta \right|\right| \right] \\ & \leq \frac{k_{1} ||s_{0}-s^{*}||^{q}}{(q+1)(1-k_{0}||s_{0}-s^{*}||^{q})}||s_{0}-s^{*}|| \\ & = \Theta_{1}(||s_{0}-s^{*}||) ||s_{0}-s^{*}|| < ||s_{0}-s^{*}|| < R. \end{aligned}} $$

(47)

So, (41) holds for k=0. Now,

$$ \begin{aligned} \left|\left|\frac{1}{3}(s_{0}+2t_{0})-s^{*}\right|\right| & \leq \frac{1}{3} \left|\left|(s_{0}+2t_{0})-3s^{*}\right|\right| \\ & \leq \frac{1}{3} \left(||s_{0}-s^{*}|| + 2||t_{0}-s^{*}||\right) \\ & \leq \frac{1}{3} \left(||s_{0}-s^{*}|| + 2\Theta_{1}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}||\right) \\ & = \frac{1}{3} \left(1 + 2\Theta_{1}\left(||s_{0}-s^{*}||\right)\right) ||s_{0}-s^{*}|| \\ & = \Theta_{2}\left(||s_{0}-s^{*}||\right) ||s_{0}-s^{*}|| < ||s_{0}-s^{*}|| < R. \end{aligned} $$

(48)

So, $\frac {1}{3}(s_{0}+2t_{0}) \in B(s^{*}, R)$. Then, our claim is $\left [G'(s_{0}) + 3G'\left (\frac {1}{3}(s_{0}+2t_{0})\right)\right ]^{-1} \in BL(Y, X)$. The Eqs. (30), (32), (35), (38), (47), and (48) are used to deduce

$$\begin{aligned} & \left|\left|(4G'(s^{*}))^{-1} \left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)-4G'(s^{*})\right]\right|\right| \\ & \leq \frac{1}{4}\left[||G'(s^{*})^{-1}(G'(s_{0})-G'(s^{*}))||+ 3\left|\left|G'(s^{*})^{-1}\left(G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)-G'(s^{*})\right)\right|\right|\right] \\ & \leq \frac{k_{0}}{4}\left[||s_{0}-s^{*}||^{q}+3\left|\left|\frac{1}{3}(s_{0}+2t_{0})-s^{*}\right|\right|^{q}\right] \\ & \leq \frac{k_{0}}{4}\left[||s_{0}-s^{*}||^{q}+3\Theta_{2}(||s_{0}-s^{*}||)^{q}||s_{0}-s^{*}||^{q}\right] \\ & = \frac{k_{0}}{4}[1+3\Theta_{2}(||s_{0}-s^{*}||)^{q}] ||s_{0}-s^{*}||^{q}\\ & = \Theta_{3}(||s_{0}-s^{*}||)<\Theta_{3}(R)<1. \end{aligned} $$

Now, we obtain $\left [G'(s_{0}) + 3G'\left (\frac {1}{3}(s_{0}+2t_{0})\right)\right ]^{-1} \in BL(Y, X)$ using Banach Lemma on invertible operators. Also,

$$ \left|\left|\left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)\right]^{-1}G'(s^{*})\right|\right| \leq \frac{1}{4(1-\Theta_{3}(||s_{0}-s^{*}||))}. $$

(49)

Hence, s₁ is well defined. We use (31) (32), (36), (39), (48), and (49) to derive

$$ {}\begin{aligned} ||s_{1}-s^{*}|| & \leq \left(\left|\left|\left[G'(s_{0}) + 3G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)\right]^{-1}G'(s^{*})\right|\right| \right) \\ & \left(\left|\left|\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(s_{0}\right)-G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)\right) \left(s_{0}-s^{*}\right)\ d\theta \right|\right| \right. \\ &\left. + 3\left|\left|\int_{0}^{1} G'\left(s^{*}\right)^{-1} \left(G'\left(\frac{1}{3}(s_{0}+2t_{0})\right)-G'\left(s^{*}+\theta\left(s_{0}-s^{*}\right)\right)\right) (s_{0}-s^{*})\ d\theta \right|\right| \right) \\ & \leq \frac{\frac{k_{1}}{q+1} ||s_{0}-s^{*}||^{q+1} + 3k_{1} \int_{0}^{1}\left(||\frac{1}{3}\left(s_{0}+2t_{0}\right)-s^{*}-\theta\left(s_{0}-s^{*}\right)||^{q}\right)\ d\theta\ ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & \leq \frac{\frac{k_{1}}{q+1} ||s_{0}-s^{*}||^{q+1} + 3k_{1} \left(||\frac{1}{3}(s_{0}+2t_{0})-s^{*}||^{q}+\frac{||s_{0}-s^{*}||^{q}}{q+1}\right) ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & \leq \frac{\frac{k_{1}}{q+1} ||s_{0}-s^{*}||^{q+1} + 3k_{1} \left[\Theta_{2}(||s_{0}-s^{*}||)^{q} ||s_{0}-s^{*}||^{q}+\frac{||s_{0}-s^{*}||^{q}}{q+1}\right] ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & = \frac{ \left(\frac{k_{1}}{q+1} ||s_{0}-s^{*}||^{q} + 3k_{1} \left[\Theta_{2}(||s_{0}-s^{*}||)^{q} ||s_{0}-s^{*}||^{q}+\frac{||s_{0}-s^{*}||^{q}}{q+1}\right] \right) ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & = \frac{ \left(\frac{4k_{1}}{q+1} ||s_{0}-s^{*}||^{q} + 3k_{1} \left[\Theta_{2}(||s_{0}-s^{*}||)^{q} ||s_{0}-s^{*}||^{q}\right] \right) ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & = \frac{ k_{1} \left(\frac{4}{q+1} ||s_{0}-s^{*}||^{q} + 3 \left[\Theta_{2}(||s_{0}-s^{*}||)^{q} ||s_{0}-s^{*}||^{q}\right] \right) ||s_{0}-s^{*}||}{4(1-\Theta_{3}(||s_{0}-s^{*}||))} \\ & = \Theta_{4}(||s_{0}-s^{*}||)||s_{0}-s^{*}|| < ||s_{0}-s^{*}||<R. \end{aligned} $$

(50)

Thus, we prove that (43) holds for k=0. We find the estimates (41)-(43) by substituting s_k, t_k, and s_k+1 in place of s₀, t₀, and s₁ respectively in the preceding estimations. From the inequality ||s_k+1−s^∗||≤Θ₄(R)||s_k−s^∗||<R, we have s_k+1∈B(s^∗,R) and ${{\lim }_{k \to \infty } s_{k}=s^{*}}$. Now, we have to show the uniqueness part. Let t^∗(≠s^∗)∈B(s^∗,Δ) be such that 0=G(t^∗). Consider $A=\int _{0}^{1} G'(\theta s^{*}+t^{*}(1-\theta))\ d\theta$. From Eq. (38), we get

$$\begin{aligned} ||G'(s^{*})^{-1} (A-G'(s^{*}))|| & \leq \int_{0}^{1} k_{0} ||t^{*}+\theta\left(s^{*}-t^{*}\right)-s^{*}||^{q}\ d\theta \\ & \leq \frac{k_{0}}{q+1}||s^{*}-t^{*}||^{q} \\ & \leq \frac{k_{0} \Delta^{q}}{q+1} <1. \end{aligned} $$

Applying Banach Lemma, we confirm that A⁻¹∈BL(Y,X). The identity 0=G(s^∗)−G(t^∗)=A(s^∗−t^∗) implies that s^∗=t^∗. □

Numerical examples

In this section, we compute the radii of convergence balls for standard numerical problems. Also, we compare the convergence radii with that of the third order convergent midpoint method obtained by the technique of Argyros and George provided in [10]. We obtain better results using our technique in all cases.

Example 1

[6] Let G is defined on $\bar {B}(0, 1)$ for (s₁,s₂,s₃)^t by

$$G(s)=\left(e^{s_{1}}-1, \frac{e-1}{2}s_{2}^{2} + s_{2}, s_{3}\right)^{t} $$

We have s^∗=(0,0,0)^t, q=1, k₀=e−1 and k₁=e. We compute the value of the radius R employing ^′′Θ^′′ functions (Table 1).

Table 1 Parameters for example 1

Example 2

[6] Define G on Ω=[−1,1] by

$$G(s)=\sin (s) $$

We have s^∗=0, q=1 and k₀=k₁=1. R is obtained using ^′′Θ^′′ functions (Table 2).

Table 2 Parameters for example 2

Example 3

[7] Consider the nonlinear Hammerstein type integral equation given by

$$G(s)(x)=s(x)-5 \int_{0}^{1} x\ y\ s(y)^{3}\ dy, $$

where G(s)∈C[0,1]. We have s^∗=0, q=1, k₀=7.5 and k₁=15. We use the definitions of ^′′Θ^′′ functions to compute the value of R (Table 3).

Table 3 Parameters for example 3

Example 4

[6] Define G on $\Omega =\left [-\frac {1}{2}, \frac {5}{2}\right ]$ by

$$G(x)= \left\{\begin{aligned} &{{s}^{3}} \log\left({{s}^{2}}\right) + {{s}^{5}} -{{s}^{4}}, {if} s \neq 0 \\ &0, {if}s=0 \end{aligned}\right. $$

We have s^∗=1, q=1 and k₀=k₁=96.6628. The radius R is calculated from the definitions of ^′′Θ^′′ functions.

Thus, we confirm the convergence of the scheme (4) with radius R=0.005613 (Table 4).

Table 4 Parameters for example 4

Example 5

[6] Consider the nonlinear Hammerstein type integral equation given by

$$G(s)(x)=s(x)-5 \int_{0}^{1} x\ y\ s(y)^{\frac{3}{2}}\ dy, $$

where s(x)∈C[0,1]. We have s^∗=0. Also, q=0.5 and $k_{0}=k_{1}=\frac {15}{4}$. We use ^′′Θ^′′ functions to compute the radius R (Table 5).

Table 5 Parameters for example 5

Example 6

[6] Consider the nonlinear integral equation given by

$$G(s)(x)=s(x)-3 \int_{0}^{1} G_{1}(x, y)\ s(y)^{\frac{5}{4}}\ dy, $$

where s(x)∈C[0,1] and G₁(x,y) is Green’s function. We have s^∗=0. Also, q=0.25 and $k_{0}=k_{1}=\frac {15}{32}$. The radius R is computed using ^′′Θ^′′ functions.

Thus, we guarantee the convergence of the algorithm (4) for example 6 with radius R=0.598292 (Table 6).

Table 6 Parameters for example 6

Conclusions

Local convergence analysis of a higher-order convergent nonlinear system solver (4) is discussed. For expanding the applicability of the method, this study is provided under the only condition that the first-order Fréchet derivative is Lipschitz continuous. This technique is applicable in solving such problems for which previous studies can not be applied. Also, the generalization of this analysis using Hölder condition is studied. At last, standard examples are solved to show the convergence of the scheme.