\(^{1}\) Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma 73505-6377, USA, e-mail: ioannisa@cameron.edu.

\(^{2}\) College of Information and Engineering, Hangzhou Polytechnic, Hangzhou 311402, Zhejiang, PR China, e-mail: rhm65@126.com.

1 Introduction

Newton’s method is one of the most fundamental tools in computational analysis, operations research, and optimization [1, 2, 4, 7–11]. One can find applications in management science; industrial and financial research; data mining; linear and nonlinear programming. In particular interior point algorithms in convex optimization are based on Newton’s method.

The basic idea of Newton’s method is linearization. Suppose \(F:\mathbb {R}\rightarrow \mathbb {R}\) is a differentiable function, and we would like to solve equation

Starting from an initial guess, we can have the linear approximation of \(F(x)\) in the neighborhood of \(x_0: F(x_0+h)\approx F(x_0)+F^\prime (x_0)h\), and solve the resulting linear equation \(F(x_0)+F^\prime (x_0)h=0\), leading to the recurrent method

This is Newton’s method as proposed in 1669 by I. Newton (for polynomial only). It was J. Raphson, who proposed the usage of Newton’s method for general functions \(F\). That is why the method is often called the Newton-Raphson method. Later in 1818, Fourier proved that the method converges quadratically in a neighborhood of the root, while Cauchy (1829, 1847) provided the multidimensional extension of Newton’s method (1.2). In 1948, L. V. Kantorovich published an important paper [7] extending Newton’s method for functional spaces (the Newton-Kantorovich method (NKM)). That is \(F:D\subseteq X\rightarrow Y\), where \(X,Y\) are Banach spaces, and \(D\) is a open convex set [1, 7]. Ever, since thousands of papers have been written in a Banach space setting for the (NKM) as well as Newton-type methods, and their applications. We refer the reader to the publications [1–11] for recent results (see also, the references there).

It is stated in the (KT) that Newton’s method (1.2) converges provided the famous for its simplicity and clarity Kantorovich hypotheses (KH) (see \((C_6)\)) is satisfied. (KH) uses the information \((x_0,F,F^\prime )\). Any successful attempt for weakening (KH) under the same information is extremely important in computational mathematics, since that will imply the extension of the applicability of (NKM). We have already provided conditions weaker than (KH) [1, 2] by introducing the center Lipichitz condition, which is a special case of the Lipschitz condition.

In this study we present a new proof of the (KT) for Newton’s method using telescopic majorizing sequences. Our proof is simpler than the corresponding one provided by Kantorovich [7]. Section 2 contains the semilocal convergence of Newton’s method (1.2). Numerical examples where our results apply to solve nonlinear equations are provided in Section 3.

2 Semilocal convergence of Newton-Kantorovich method (NKM)

Let us assume that \(F^\prime (x_0)^{-1}\in L(Y,X)\) at some \(x_0\in D\), and the following conditions hold:

\((C_1)\) \(0{\lt}\| F^\prime (x_0)^{-1}\| \le \beta \),

\((C_2)\) \(0{\lt}\| F^\prime (x_0)^{-1}F(x_0)\| \le \eta \),

and Lipschitz continuity condition

\((C_3)\) \(\| F^\prime (x)-F^\prime (y)\| \le L\| x-y\| \)  for some  \(L{\gt}0\), and for all \(x,y\in D\).

It is convenient for us to define for \(a_0=b_0=1\),

and scalar sequences

We shall find a connection between (NKM) \(\{ x_n\} \), and scalar sequences \(\{ a_n\} \), \(\{ b_n\} \), \(\{ c_n\} \). Sequences \(\{ a_n\} \), \(\{ b_n\} \) and \(\{ c_n\} \) have been used by Kantorovich [7]. However, we present a new proof for the (KT).

We shall use induction to show items \((I_n)-(III_n)\). \((I_0)\) and \((II_0)\) follow immediately from the initial conditions. To show \((III_0)\), we use (1.2) for \(n=0\), \((II_0)\), and \((C_3)\) to obtain

\(\Rightarrow \)

If \(x_{k+1}\in D\; (k\le n)\), then it follows from \((C_3)-(C_5)\) and the induction hypotheses:

In view of (2.7), and the Banach lemma on invertible operators [1, 2, 7] \(F^\prime (x_{k+1})^{-1}\in L(Y,X)\), so that

which shows \((I_n)\) for all \(n\ge 0\).

As in (2.5), we have

and

which shows \((III_n)\) for all \(n\ge 0\). Moreover, we get by (1.2), (2.8) and (2.10):

That completes the induction for \((II_n)\) and the proof of the lemma.

We need to show the convergence of sequence \(\{ x_n\} \), which is equivalent to proving that \(\{ b_n\} \) is a Cauchy sequence. To this effect we need the following auxiliary results:

(a) We shall show using induction that \(\{ a_n\} \), \(\{ b_n\} \) and \(\{ c_n\} \) are positive sequences.

In view of the initial conditions, \(a_0\), \(b_0\), \(c_0\) are positive, and \(1-\gamma a_0b_0{\gt}0\). Assume \(a_k\), \(b_k\), \(c_k\) and \(1-\gamma a_k b_k \) are positive for \(k\le n\). By (2.2), \(a_{k+1}{\gt}0\), consequently \(b_{k+1}{\gt}0\). Moreover, \(1-\gamma a_{k+1}b_{k+1}{\gt}0\) by \((C_6)\) (see also [2], [7]). The induction is completed.

Solving (2.2) for \(b_n\), we obtain

By telescopic sum, we have:

or

But \(1-\gamma \sum ^{n-1}_{k=0}b_k\) decreases, so \(\{ a_n\} \) given by (2.15) increases. Note also that \(a_n\ge a_0=1\).

(b) By (a) \(a_n\ge 1\) is increasing, so that

Therefore, \(\{ \frac{1}{a_n}\} \) is monotonic on the compact set \([0,1]\) and as such it converges to some limit \(\overline{a}\). By letting \(n\rightarrow \infty \) in (2.13), we get

(c) It follows from (b) and (2.14) that there exists \(r=\sum ^\infty _{k=0}b_k\). The value of \(r\) is well known to be given by (2.12) [2], [7].

(d) We have \(\| x_1-x_0\| \le b_0\eta =\eta \Rightarrow x_1\in U(x_0,r\eta )\). Assume \(x_k\in U(x_0,r\eta )\subseteq D\) for all \(k\le n\). Then, we have by Lemma 2.1 in turn that

\(\Rightarrow \)

That completes the proof of the lemma.

We can show the semilocal convergence result for (NKM) (1.2).

where, \(\)r\( is given by {\rm (2.12)}.\)\(\)

Then, sequence \(\{ x_n\} \) generated by (NKM) (1.2) is well defined, remains in \(\overline{U}(x_0,r\eta )\) for all \(n\ge 0\) and converges to a solution \(x^\star \in \overline{U}(x_0,r\eta )\) of equation \(F(x)=0\). Moreover, the following estimate holds:

Furthermore, \(x^\star \) is the only solution of equation \(F(x)=0\) in \(D_0\bigcap D=D_1\ \text{where,}\ D_0=U(x_0,\frac{1}{\beta L_0})\).,

It follows from Lemmas 2.1 and 2.2 (see also \((II_n)\)) that \(\{ x_n\} \) is a Cauchy sequence in a Banach space \(X\) and as such it converges to some \(x^\star \). We have \(\lim _{n\rightarrow \infty }b_n=0\), which implies by (2.3) that \(\lim _{n\rightarrow \infty }c_n=0\). By letting \(n\rightarrow \infty \) in (2.10) and using the continuity of operator \(F\), we obtain \(F(x^\star )=0\).

By \((C_7)\), we obtain

\(\Rightarrow x_{n+1}\in \overline{U}(x_0,r\eta )\)

\(\Rightarrow x^\star =\lim _{n\rightarrow \infty }x_n\in \overline{U}(x_0,r\eta )\) (since \(\overline{U}(x_0,r\eta )\) is a closed set).

Let \(m{\gt}n\). Then, we have

By letting \(m\rightarrow \infty \) in (2.20), we obtain (2.18). Finally, to show uniqueness, let \(y^\star \in D_0\) be a solution of equation \(F(x)=0\). Define linear operator

We have using \((C_3)\):

It follows from (2.22), and the Banach lemma on invertible operators that \(M^{-1}\in L(Y,X)\). In view of (2.21), we get

which implies \(x^\star =y^\star \). That completes the proof of the theorem.

3 Applications

In the first example we show that the Kantorovich hypothesis (see (3.2)) is satisfied with the bigger uniqueness ball of solution than before [2], [7].

In the second example we apply Theorem 2.4 to a nonlinear integral equation of Chandrasekhar-type.