Consider the equation

where $f:[a,b]\to ℝ,$ $a,b\in ℝ,$

Consider also the following two equations, both equivalent to (1):

In order to approximate a root $x^{\ast }$ of (1), we consider the sequence ${(x_n)}_{n\ge 0},$ generated by the following relations:

where $[x,y;f]$ denotes the first order divided difference of $f$ on $x$ and $y.$

Relation (3) suggests an Aitken type method. If we assume that $f\in C^3[a,b]$ and $f^{\prime }(x)\ne 0$ for all $x\in [a,b],$ denoting $F=f([a,b]),$ it is known that there exists $f^{-1}:F\to [a,b]$ and $f^{-1}\in C^3([a,b]).$ Moreover, the following relation holds

where $y=f(x)$and $x\in [a,b]$ (see [2], [3], [5], [8]).

In [4] was studied the convergence of a Steffensen type method, analogous to (3). In order to obtain a control of the error at each iteration step, we have studied in [7] the cases when the generalized Steffensen type method leads to monotone approximations. Under some reasonable conditions of monotonicity and convexity (concavity) on $f,$ in [7] were obtained monotone sequences which yield bilateral approximations to the solution $x^{\ast }$ of (1). A condition such that the Steffensen type method studied in [7] to lead to monotone approximations is that the sign of the expression

to be negative: $\forall x\in [a,b].$

In this note we shall show that if $E_f(x)>0$ $\forall x\in [a,b],$ one may construct functions $g_1$ and $g_2,$ and on may choose the initial approximations $x_0\in [a,b]$ such that, under some reasonable hypotheses on $f,$ the sequence (3) converges monotonically.

In an analogous fashion as in [7], we can easily show that for a given $x_n\in [a,b],$ then exists ${\xi }_n\in ]c{}_n,d{}_n[,$ where $[c_n,d_n]$ is the smallest interval containing the points $x_n,$ $g_1(x_n),$ $g_2(x_n),$ $x^{\ast },$ such that

We consider the following hypotheses on the functions $f_1,g_1,$ $g_2,$ and on the initial approximation $x_0\in [a,b]:$

• i.

  $f\in C^3[a,b]$ and $E_f(x)>0,$ $\forall x\in [a,b]$ where $E_f$ is given by (5);

• ii.

• iii.

  $f^{\prime }(x)>0,$ $\forall x\in [a,b];$

• iv.

  $f^{\prime \prime }(x)\ge 0,$ $x\in [a,b];$

• v.

  $g_1$ and $g_2$ are continuous functions, decreasing on $[a,b];$

• vi.

  equation (1) has a solution $x^{\ast }\in ]a,b[;$

• vii.

  $g_1(x_0)\le b,g_2(x_0)\le b;$

• viii.

  equations (1) and (2) are equivalent.

The following result holds:

The basic conditions on $g_1$ and $g_2$ on v. and vii.

We shall consider a function $g:[a,b]\to ℝ$ given by

where $\lambda \in ℝ$.

If $g^{\prime }(x)\le 0,$ then $g$ is decreasing. This attracts

i.e.,

But since $f^{\prime }(x)>0$ and $f^{\prime \prime }\ge 0,$ we obtain

or

By (9), it is obvious that if $\lambda >\frac{1}{f^{\prime }(a)}$ then (8) is verified for every $x\in [a,b],$ from which $g$ is decreasing.

In order to obtain $g(x_0)\le b,$ we need that$x_0-\lambda (x_0)\le b,$ which leads to $\lambda \le \frac{x_0-b}{f(x_0)}$ which in turn holds if $x_0$ is sufficiently close to $x^{\ast }.$

Let ${\lambda }_i,$ $i=1,2,$ ${\lambda }_1\ne {\lambda }_2$ be two numbers such that

then functions $g_i,$ $i=1,2$ given by

verify hypotheses v. and vii.

Consider the equation

with $x\in [\frac{\pi }{6},\frac{\pi }{2}].$

Obviously, $f^{\prime }(x)>0,$ for all $x\in [\frac{\pi }{6},\frac{\pi }{2}],$ and $f^{\prime \prime }(x)>0,$ $x\in [\frac{\pi }{6},\frac{\pi }{2}].$ If in (7) we take ${\lambda }_1=0.5,$ ${\lambda }_2=0.6$ and $x_0=\frac{\pi }{6},$ then $g_1$ and $g_2$ verify v. and vii.

We have:

Obviously, and Function $E_f$ is given by

In the table below we have obtained the following results.

The numerical results confirm that inequalities

hold.