On some iterative methods for solving nonlinear equations

In the papers [3], [4] and [5] are studied nonlinear equations having the from:

where, $f,g:X\rightarrow X,X$ is a Banach space, $f$ is a differentiable operator and $g$ is continuous but nondifferentiable. For this reason the Newton’s method, i.e. the approximation of the solution $x^{*}$ of the equation (1) by the sequence $\left(x_{n}\right)_{n\geq 0}$ given by
(2) $x_{n+1}=x_{n}-\left(f^{\prime}\left(x_{n}\right)+g^{\prime}\left(x_{n}\right)\right)^{-1}\left(f\left(x_{n}\right)+g\left(x_{n}\right)\right),\quad n=1,2,\ldots,x_{0}\in X$, cannot be applied.

In the mentioned papers the following Newton-like methods are then considered:

or

where $A$ is a linear operator approximating $f^{\prime}$. It is shown that, under certain conditions, these sequences are converging to the solution of (1).

In the present paper, for solving equation (1), we propose the following method:

where by $[x,y;g]$ we have denoted the first order divided difference of $g$ at the points $x,y\in X$.

So, the proposed method is obtained by combining the Newton’s method with the method of chord. The r-convergence order of this method, denoted by $p$, is the same as for the method of chord (where $p=\frac{1+\sqrt{5}}{2}\approx 1.618$ ), which is greater than the r-order of the methods (3) and (3’) (see also the numerical example), but is less than the r-order of Newton’s method (where usually $p=2$ ).

But, unlike the method of chord, the proposed method has a better rate of convergence, because the use of $f^{\prime}\left(x_{n}\right)$ instead of $\left[x_{n-1},x_{n};f\right]$, as it is in the method of chord, does not affect the coefficient $c_{2}$ from the inequalities of the type:

which we shall obtain in the following.

We shall use, as in 1 and 2 the known definitions for the divided differences of an operator.

Definition 1. An operator belonging to the space $\mathcal{L}(X,X)$ (the Banach space of the linear and bounded operators from $X$ to $X$ ) is called the first order divided difference of the operator $g:X\rightarrow X$ at the points $x_{0},y_{0}\in X$ if the following properties hold:
a) $\left[x_{0},y_{0};g\right]\left(y_{0}-x_{0}\right)=g\left(y_{0}\right)-g\left(x_{0}\right)$, for $x_{0}\neq y_{0}$;
b) if $g$ is Fréchet differentiable at $x_{0}\in X$, then

Definition 2. An operator belonging to the space $\mathcal{L}(X,\mathcal{L}(X,X))$, denoted by $\left[x_{0},y_{0},z_{0};g\right]$ is called the second order divided difference of the operator $g:X\rightarrow X$ at the points $x_{0},y_{0},z_{0}\in X$ if the following properties hold:
$\left.\mathrm{a}^{\prime}\right)\left[x_{0},y_{0},z_{0};g\right]\left(z_{0}-x_{0}\right)=\left[y_{0},z_{0};g\right]-\left[x_{0},y_{0};g\right]$ for the distinct points $x_{0},y_{0},z_{0}\in X$;
$\mathrm{b}^{\prime}$ ) if $g$ is two times differentiable at $x_{0}\in X$, then

We shall denote by $B_{r}\left(x_{1}\right)=\left\{x\in X\mid\left\|x-x_{1}\right\|<r\right\}$ the ball having the center at $x_{1}\in X$ and the radius $r>0$.

Concerning the convergence of the iterative process (4) we shall prove the following result.

Theorem 3. If there exist the elements $x_{0},x_{1}\in X$ and the positive real numbers $r,l,M,K$ and $\varepsilon$ such that the conditions
i) the operator $f$ is Fréchet differentiable on $B_{r}\left(x_{1}\right)$ and $f^{\prime}$ satisfies

ii) the operator $g$ is continuous on $B_{r}\left(x_{1}\right)$,
iii) for any distinct points $x,y\in B_{r}\left(x_{1}\right)$ there exists the application $\left(f^{\prime}(y)+[x,y;g]\right)^{-1}$ and the inequality

is true;
iv) for any distinct points $x,y,z\in B_{r}\left(x_{1}\right)$ we have the inequality

v) the elements $x_{0},x_{1}$ satisfy

vi) the following relations hold:

where $\left(u_{k}\right)_{k\geq 0}$ is the Fibonacci’s sequence $u_{k+1}=u_{k}+u_{k-1},k\geq 1$, $u_{0}=u_{1}=1;$
are fulfilled, then the sequence $\left(x_{n}\right)_{n\geq 0}$ generated by (4) is well defined, all its terms belonging to $B_{r}\left(x_{1}\right)$.

Moreover, the following properties are true:
j) the sequence $\left(x_{n}\right)_{n\geq 0}$ is convergent;
jj) let $x^{*}=\lim_{n\rightarrow\infty}x_{n}$. Then $x^{*}$ is a solution of the equation (1);
jjj) we have the a priori error estimates:

Proof. We shall prove first by induction that, for any $n\geq 2$,

For $n=2$, from $v$ ) and $vi)$ we infer the above relations.
Let us suppose now that relations (5), (6) and (7) hold for $n=2,3,\ldots,k$, where $k\geq 2$. Since $x_{k},x_{k-1}\in B_{r}\left(x_{1}\right)$, we can construct $x_{k+1}$ from (4), whence, using $iii)$, we have
$\left\|x_{k+1}-x_{k}\right\|=\left\|\left(f^{\prime}\left(x_{k}\right)+\left[x_{k-1},x_{k};g\right]\right)^{-1}\left(f\left(x_{k}\right)+g\left(x_{k}\right)\right)\right\|\leq M\left\|f\left(x_{k}\right)+g\left(x_{k}\right)\right\|$.
For the estimation of $\left\|f\left(x_{k}\right)+g\left(x_{k}\right)\right\|$ we shall rely on the equality

(easily obtained from Definition 1 and Definition 2), which imply, using iv),

and on the inequality

valid because of the assumptions $i)$ concerning $f$.
For $n=k-1$, by (4), we get

whence

The above relation, together with (8), (9) and (6) for $n=k$ imply

From the hypothesis of the induction we have on one hand that

that is, (6) for $n=k+1$, and, on the other hand

that is, (7) for $n=k+1$.
The fact that $x_{k+1}\in B_{r}\left(x_{1}\right)$ results from:

Now we shall prove that $\left(x_{n}\right)_{n\geq 0}$ is a Cauchy sequence, whence $j$ ) follows.
It is obvious that

for $k\geq 1$.

So, for any $k\geq 1,m\geq 1$ we have

Using Bernoulli’s inequality, it follows

Hence

and $\left(x_{n}\right)_{n\geq 0}$ is a Cauchy sequence.
It follows that $\left(x_{n}\right)_{n\geq 0}$ is convergent, and let $x^{*}=\lim_{n\rightarrow\infty}x_{n}$. For $n\rightarrow\infty$ in (4) we get that $x^{*}$ is a solution of (1). For $m\rightarrow\infty$ in the above $n$ equality we obtain the very relation $jjj$ ).

The theorem is proved.

Given the system

we shall consider $X+\left(\mathbb{R}^{2},\|\cdot\|_{\infty}\right),\|x\|_{\infty}=\left\|\left(x^{\prime},x^{\prime\prime}\right)\right\|_{\infty}=\max\left\{\left|x^{\prime}\right|,\left|x^{\prime\prime}\right|\right\},f=\left(f_{1},f_{2}\right),g=\left(g_{1},g_{2}\right)$. For $x=\left(x^{\prime},x^{\prime\prime}\right)\in\mathbb{R}^{2}$ we take $f_{1}\left(x^{\prime},x^{\prime\prime}\right)=3\left(x^{\prime}\right)^{2}x^{\prime\prime}+\left(x^{\prime\prime}\right)^{2}-1,f_{2}\left(x^{\prime},x^{\prime\prime}\right)=\left(x^{\prime}\right)^{4}+x^{\prime}\left(x^{\prime\prime}\right)^{3}-1,g_{1}\left(x^{\prime},x^{\prime\prime}\right)=\left|x^{\prime}-1\right|,g_{2}\left(x^{\prime},x^{\prime\prime}\right)=\left|x^{\prime\prime}\right|$.

We shall take $[x,y;g]\in M_{2\times 2}(\mathbb{R})$ as

Using method (3) with $x_{0}=(1,0)$ we obtain

Using the method of chord with $x_{0}=(5,5),x_{1}=(1,0)$, we obtain

Using method (4) with $x_{0}=(5,5),x_{1}=(1,0)$, we obtain

It can be easily seen that, given these data, method (4) is converging faster than (3) and than the method of chord.

[1] G. Goldner, M. Balázs, On the method of the chord and on a modification of it for the solution of nonlinear operator equations, Stud. Cerc. Mat., 20 (1968), pp. 981-990 (in Romanian).
[2] G. Goldner, M. Balázs, Remarks on divided differences and method of chords, Rev. Anal. Numer. Teoria Aproximaţiei, 3 (1974) no. 1, pp. 19-30 (in Romanian). ©
[3] T. Yamamoto, A note on a posteriori error bound of Zabrejko and Nguen for Zincenko’s iteration, Numer. Funct. Anal. Optimiz., 9 (1987) 9&10, pp. 987-994. ©

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