\(^\ast \)Cameron University, Department of Mathematics Sciences, Lawton, OK 73505, USA, e-mail: iargyros@cameron.edu.

\(^\S \)Poitiers University, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France, e-mail: said.hilout@math.univ-poitiers.fr.

1 Introduction

In this study we are concerned with the problem of approximating a locally unique solution \(x^\star \) of equation

where \(F\) is a Fréchet–differentiable operator defined on a convex subset \(\mathcal{D}\) of a Banach space \(\mathcal{X}\) with values in a Banach space \(\mathcal{Y}\).

The field of computational sciences has seen a considerable development in mathematics, engineering sciences, and economic equilibrium theory. For example, dynamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. For the sake of simplicity, assume that a time–invariant system is driven by the equation \(\dot{x} =T(x)\), for some suitable operator \(T\), where \(x\) is the state. Then the equilibrium states are determined by solving equation (1.1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative–when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework. We note that in computational sciences, the practice of numerical analysis for finding such solutions is essentially connected to variants of Newton’s method.

We consider the secant method (SM) in the form

where, \(\delta F(x,y) \in \mathcal{L}(\mathcal{X},\mathcal{Y})\) \((x,y\in \mathcal{D})\) is a consistent approximation of the Fréchet–derivative of \(F\) [ 4 ] , [ 11 ] . \( \mathcal{L}(\mathcal{X}, \mathcal{Y})\) denotes the space of bounded linear operators from \(\mathcal{X}\) into \(\mathcal{Y}\). (SM) is an alternative method of Newton’s method (NM)

Dennis [ 8 ] , Potra [ 14 ] – [ 16 ] , Bosarge and Falb [ 7 ] , Hernández, M.J. Rubio and J.A. Ezquerro [ 9 ] , Argyros [ 4 ] , and others [ 10 ] , [ 13 ] , [ 20 ] , have provided sufficient convergence conditions for (SM) based on Lipschitz–type conditions on \(\delta F\) (see, also relevant works in [ 1 ] , [ 2 ] , [ 5 ] , [ 12 ] , [ 17 ] , [ 18 ] , [ 19 ] , [ 21 ] ).

In the previous mentioned references, the conditions usually associated with the semilocal convergence of secant method (1.2) are:

1. \(F\) is a nonlinear operator defined on a convex subset \(\mathcal{D}\) of a Banach space \(\mathcal{X}\) with values in a Banach space \(\mathcal{Y}\);

2. \(x_{-1}\) and \(x_0\) are two points belonging to the interior \(\mathcal{D}^0\) of \(\mathcal{D}\) and satisfying the inequality

   \[ \parallel x_0 -x_{-1} \parallel \leq c; \]

3. \(F\) is Fréchet–differentiable on \(\mathcal{D}^0\), and there exists an operator
   \(\delta F \, \colon \, \mathcal{D}^0 \times \mathcal{D}^0 \to \mathcal{L}(\mathcal{X},\mathcal{Y})\), such that the linear operator \(A \! =\! \delta F(x_{-1} ,x_0)\) is invertible, its inverse \(A^{-1}\) is bounded, and:

   \[ \parallel A^{-1} \, F(x_0) \parallel \leq \eta ; \]
   \[ \parallel A^{-1} \, ( \delta F(x,y) -F'(z)) \parallel \leq \ell \, (\parallel x-z\parallel +\parallel y -z\parallel ), \quad {\rm for \, \, all} \quad x,y,z\in \mathcal{D}; \]
   \[ \overline{U} (x_0,r)= \{ x\in \mathcal{X}\, : \, \parallel x- x_0 \parallel \leq r \} \subseteq \mathcal{D}^0 , \]

   for some \(r{\gt}0\) depending on \(\ell \), \(c\), and \(\eta \); and

   \begin{equation} \label{1.3} \ell \, \, c + 2 \, \, \sqrt{\ell \, \, \eta } \leq 1 . \end{equation}

   1.3

The sufficient convergence condition (1.3) is easily violated. Indeed, let \(\ell =1\), \(\eta = .18\), and \(c=.185\). Then, (1.3) does not holds, since

Moreover, our recently found corresponding conditions are also violated [ 6 ] (see, Remark 4(c)). Hence, there is not guarantee that equation (1.1) under the information (\(\ell , c, \eta \)) has a solution that can be found using (SM). In this study we are motivated by optimization considerations, and the above observation.

Here, using a combination of Lipschitz and center–Lipschitz conditions, we provide a semilocal convergence analysis for (SM). Our error bounds are tighter, and our convergence conditions hold in cases where the corresponding hypotheses in earlier references [ 8 ] , [ 9 ] , [ 12 ] – [ 14 ] , [ 16 ] , [ 19 ] – [ 21 ] are violated. Applications and examples are also provided in this study.

2 Semilocal convergence analysis of (SM)

We need the following result on majorizing sequences for (SM).

We shall show using mathematical induction on \(k\geq 0\)

By (2.5) for \(k=0\), we must show

which is true from (2.1), and the choice of \(\delta \geq \delta _0\).

Let assume that (2.9) holds for \(k\leq n+1\). The induction hypothesis gives

Moreover, we have:

We prove now (2.9). By (2.5), we can have estimate

or

In view of (2.13), we are motivated to define (for \(\delta =s\)) the functions for \(k\geq 1\)

We need the relationship between two consecutive functions \(f_k\). Using (2.14), we obtain

Note that \(\delta \) is the unique positive root of polynomial \(f_1\).

Instead of (2.13), we shall show

Estimate (2.16) holds for \(k=1\), as equality. Using (2.15), we get in turn

since,

Assume (2.16) holds for \(m \leq k\). Then, again by (2.15), we have:

which completes the induction for (2.16).

Moreover, define function \(f_{\infty }\) on \([0,1)\) by

Using (2.16), and (2.17), we obtain

Hence, we showed sequence \(\{ t_n\} \) (\(n\geq -1\)) is non–decreasing and bounded above from by \(t^{\star \star } \), so that (2.8) holds. It follows that there exists \(t^\star \in [0, t^{\star \star } ]\), so that \(\displaystyle \lim _{n\longrightarrow \infty } t_n= t^\star \).

That completes the proof of Lemma 1.

We shall study (SM) for triplets \((F, x_{-1},x_0)\) belonging to the class
\(\mathcal{C}(\ell ,\ell _0 ,\eta ,c, \delta )\) defined as follows:

The semilocal convergence theorem for (SM) is as follows.

by (\(\mathcal{A}_5\)).

Using the Banach Lemma on invertible operators [ 4 ] , [ 11 ] , and (2.22), \(L\) is invertible and

By (\(\mathcal{A}_3\)), we have

We can write the identity

then, for all \(x,y,u,v\in \mathcal{D}^0\), we obtain

and

By a continuity argument (2.24)–(2.27) remain valid if \(x\) and/or \(y\) belong to \(\mathcal{D}_c\).

Now we show (2.19). If (2.19) holds for all \(n \leq k\) and if \(\{ x_n\} \) \((n\geq 0)\) is well defined for \(n = 0,1,2,\cdots ,k\), then

That is (1.2) is well defined for \(n =k+1\). For \(n =-1\), and \(n=0\), (2.19) reduces to \(\parallel x_{-1} -x_0\parallel \leq c\), and \(\parallel x_0 -x_1\parallel \leq \eta \). Suppose (2.19) holds for \(n = -1,0, 1,\cdots ,k\) \((k\geq 0)\). By (2.23), (2.27), and

we obtain the following estimate

and the induction for (2.19) is completed. It follows from (2.19), and Lemma 1 that\(\{ x_n\} \) \((n\geq -1)\) is Cauchy in a Banach space \(\mathcal{X}\), and as such it converges to some \(x^\star \in \overline{U} (x_0,t^\star )\) (since \(\overline{U} (x_0 ,t^\star )\) is a closed set). By letting \(k\to \infty \) in (2.30), we obtain \(F(x^\star ) =0\). Estimate (2.20) follows from (2.19) by using standard majoration techniques [ 4 ] , [ 5 ] , [ 11 ] .

Finally, for showing the uniqueness in \( \overline{U} (x_0, t^\star ) \), let \(y^\star \in \overline{U} (x_0, t^\star ) \) be a solution (1.1). Set

It then follows by (\(\mathcal{A}_3\)), and (\(\mathcal{A}_5\)):

It follows from (2.31), and the Banach lemma on invertible operators that \(\mathcal{M}^{-1}\) exists on \({U} (x_0, t^\star )\).

Using the identity:

we deduce \(x^\star = y^\star \). Finally, we shall show uniqueness in \({U} (x_0, R)\). As in (2.31), we arrive at

by (2.21)–(2.23).

That completes the proof of Theorem 3.

3 Examples

In this section, we present some numerical examples.

Conclusion

We provided new sufficient convergence conditions for (SM) to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center–Lipschitz conditions on the divided difference operator, we obtained the semilocal convergence analysis of (SM). Our error bounds are more precise than earlier ones, and under our convergence hypotheses we can cover cases where earlier conditions are violated [ 8 ] , [ 9 ] , [ 12 ] – [ 14 ] , [ 16 ] , [ 19 ] – [ 21 ] . Applications, and numerical examples are also provided in this study.