Let $X$ be a Banach space and $D\subseteq X$. Given $x_0\in D$, we consider the ball $S(x_0,\rho )=\{x\in X\mid \Vert x-x_0\Vert \le \rho \},$with $\rho \in ℝ$, $\rho >0$ and we assume that $S(x_0,\rho )\subseteq D$.

In this paper we shall study certain aspects of the problems regarding the existence, uniqueness, and the approximation of the solution of an operatorial equation of the form

where $f:D\to X.$

Let$A:X\to X$ be a linear and continuous operator. For the approximation of the solution $\overline{x}$ of equation (1) we shall consider the sequence ${(x_n)}_{n\ge 0}$ given by (2)

In the following we shall obtain conditions under which equation (1) has a unique solution in the ball $S(x_0,\rho ),$ and, moreover, the sequence ${(x_n)}_{n\ge 0}$ generated by (2) is convergent and $lim{x}_n=\overline{x}.$

We shall apply the obtained results to the unified study of the convergence of some sequences of the form (2), namely generated by the modified Newton method, and the modified chord method. For the modified Newton method we shall retrieve in this way a known result, which for the modified chord method the result is new.

Regarding the existence and uniqueness of solution of equation (1) in the ball $S(x_0,\rho ),$ and the convergence of sequence (2), the following result holds:

In this sense we notice that $h$ may be written as

By (8) and (3) for $y=x_0$ and by ${\mathrm{𝐢𝐯}}_1.$ we get

for all $x\in S(x_0,\rho ).$ By (9) it follows that for all $x\in S(x_0,\rho ),$ $h(x)$ given by (7) belongs to $S(x_0,\rho ).$

Consider now $y_1,$ $y_2\in S.$ By (7) we get:

whence, taking into account (3) it follows

i.e., $h$ is a contraction on $S(x_0,\rho ).$ Consequently, $h$ has a unique fixed point $\overline{x}\in S(x_0,\rho ).$ Since $A$ is a linear invertible operator by (3) it follows that equation (1) has the unique solution $\overline{x}\in S(x_0,\rho ).$

Properties ${\mathrm{𝐣𝐣}}_1$ and ${\mathrm{𝐣𝐣𝐣}}_1$ are immediate consequences of the fact that $X$ is a Banach space and $h$ is a contraction on the invariant ball $S(x_0,\rho )$ (see [1], [4]).Denoting $\rho =\Vert Af(x_0)\Vert$ then it can easily seen that

These relations are obtained from (8) for $x=\overline{x}$ and $\overline{x}=h(\overline{x}).$

From (10), if we replace $x_0=x_n$ and $p=\Vert Af(x_n)\Vert$ we obtain lower and upper error bounds at the $n$-th step:

The last relations offer a posteriori error bounds.  

We consider now the divided differences of $f$ at some given points in $D$. Let $ℒ(X)$ denote the set of linear operators from $X$ to $X.$

Definition [3]. The linear operator $\alpha (x,y)\in L(X)$ is called first order divided difference of $f$ at the points $x,y\in X$ if

a)

$\alpha (x,y)(x-y)=f(x)-f(y);$

b)

if $f$ is Fréchet differentiable at $y$ then $\alpha (y,y)=f^{\prime }(y).$

We shall use in the following the usual notation $\alpha (x,y)=[y,x;f].$ In order to approximate the solution of equation (1) we consider the sequence ${(x_n)}_{n\ge 0}$ given by

We obtain the following result.

The definition of the divided differences, leads to the equality

whence, by (13), it follows for $A={[x_0,{\overline{x}}_0;f]}^{-1}$,

It is obvious that Theorem 2 is a consequence of Theorem 1 and of relations (10) and (11).  

Assume that $f$ is Fréchet differentiable at all points $x\in D.$ Let $x_0\in D$ and $f^{\prime }(x_0)\in ℒ(X).$ Assume that $f^{\prime }(x_0)$ is invertible.

In order to approximate $\overline{x}$ we shall consider the sequence ${(x_n)}_{n\ge 0}$ generated by the modified Newton method.

Concerning the convergence of this sequence the following result holds:

Let $g:S(x_0,\rho )\to X$ be an operator differentiable at all points $x\in S(x_0,\rho ).$ Then it is known (see, e.g.,[2]) that for all $x,y\in S(x_0,\rho )$ one has

Let now take $g$ given by

then

If we take into account (25) and (26), by (24) we get

which shows that (3) is verified for $A=f^{\prime }{(x_0)}^{-1.}$ Theorem 3 is a consequence of Theorem 1, and therefore properties ${𝐣}_3$–${\mathrm{𝐣𝐣𝐣}}_3$are true.