\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA; iargyros@cameron.edu

\(^\bullet \)Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine; stepan.shakhno@lnu.edu.ua; yuriy.shunkin@lnu.edu.ua

\(^\circ \)Department of Computational Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine; halyna.yarmola@lnu.edu.ua

1 Introduction

Consider the operator equation

where \(F\) and \(Q\) are nonlinear operators, defined on an subset \(D\) of a Banach space \(E_1\) with values in a Banach space \(E_2\). It is known, that \(F\) is a differentiable by Frèchet operator, \(Q\) is a continuous operator, whose differentiability in general is not required.

A plethora of problems from diverse disciplines can be converted to equation ?? via mathematical modelling [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 ] . Therefore, the task of computing a solution \(x_*\) is of extreme importance. We resort to iterative solvers, since closed form solutions can be obtained in rare cases.

The Newton solver [ 2 ] can not be used to find a solution of equation ??, because of the nondifferentiable \(Q\). However, in this case the Newton-type solver [ 3 ] , or one of combined iterative processes may be applicable [ 3 ] – [ 18 ] .

The special case of ?? is the equation \(F(x)=0\). Usually, to find the solution Newton’s solver is used

whose convergence order is quadratic [ 19 , 20 ] . Hence, one can use the difference solvers. These solvers use only a nonlinear operator, and do not require analytical derivatives. One of these solvers has \(R\)-convergence order \(1.839\ldots \)

where \(F(u,v)\) is a divided difference of order one. This solver was proposed by J. Traub for solving one nonlinear equation [ 21 ] , later it was generalized to Banach spaces by F. Potra [ 22 ] , and investigated under different conditions in [ 23 , 24 ] .

In the paper [ 25 ] a combined iterative process was proposed, which is built based Newton’s and Potra’s solvers

where \(Q(x,y)\) is a divided difference of order one, to be defined later.

This solver was studied in [ 26 ] under weak \(\omega \)-conditions. In this work we continue the study of a local and a semi-local convergence of solver ??. It is established that the convergence order of the combined iterative process ?? is similar to the convergence order of the Potra solver. But also it is important to extend the convergence region in particular without requiring an additional hypotheses. This fact will extend the number of initial approximations. By applying a new approach we achieve fewer iterations to obtain a result with predetermined accuracy, at least as many initial points, and same or less computational cost.

The rest of the paper is structured as follows: In Section 2, we present the local convergence analysis of the solver ?? and a Corollary. Section 3 contains the proofs of semi-local convergence and uniqueness of solution. In Section 4, we provide the numerical example. The article ends with some conclusions.

2 Local convergence of solver ??

Note that we used the classic Lipschitz conditions for the derivative of first order of operator \(F\) and for divided differences of order one and two of operator \(Q\). The following theorem present the convergence radius and the convergence speed of iterative process ??. Although we assume, that \(Q\) is differentiable by Fréchet operator.

Set \(U=U(x_*,r_*)=\{ x: \| x-x_*\| {\lt}r_*\} \). Let \(x,y,z \in D\).

By the definition of \(r_*\) ??, we get

Then, by the Banach Lemma on invertible operators [ 2 ] , \(A\) is invertible and

Suppose, that \(x_{n-2},x_{n-1},x_n \in U\). Then the operator

is invertible. Next, we can write

In view of ??–??, we get

Then, from ?? and ??, we obtain the estimate ??. Moreover, from inequalities ??, ?? we have in a turn

Hence, iterative process ?? is well defined, generated sequence \(\{ x_n\} _{n \ge 0}\) is in \(U\), and converges to the solution \(x_*\). Finally, by the last inequality, and estimate ?? we get, that \(\lim \limits _{n\to \infty }\| x_n-x_*\| =0\).

Hence, the \(R\)-convergence order of solver ?? is the unique positive root of nonlinear equation \(t^3-t^2-t-1=0\) [ 22 ] , which is \(1.839\ldots \)

3 Semi-local convergence of solver ??

Set \(U_0(x_0,r)=\{ x: \| x-x_0\| \le r\} \). The semi-local convergence of the combined Newton-Potra solver ?? is presented in what follows.

is nonnegative and decreasing converging to some \(t_* \in \mathbb {R}\), such that

Then the iterative process ?? is well defined, remains in \(U_0(x_0,r_1)\) and converges to a solution \(x \in U_0(x_0,r_1)\) of equation \(F(x)+Q(x)=0\), Moreover, the following estimates are true

Using ?? and \(k=0\), we obtain

so ??–?? are true for \(k=0\).

Suppose, that estimates ??–?? are satisfied for each \(k \le n\). Then, for \(k = n\) we have the following

Hence, that \(\{ t_n\} _{n \geq -2}\) is decreasing, nonnegative sequence which converges to some \(t_*\geq 0\). Next, we show, that iterative process ?? is well defined, and following estimate is true for each \(n\geq -2\)

In view of Lipschitz conditions ??–??, for \(k=n+1\), we obtain

Hence, \(A_{n+1}\) is invertible and

Taking into account the definition of the divided difference and conditions ??–?? we get in a turn

Hence

That is, iterative process ?? is well defined for each \(n\). Moreover

so the sequence \(\{ x_n\} _{n \ge 0}\) is fundamental, and as such convergent in the Banach space \(E_1\). By letting \(k \rightarrow \infty \) in ??, we get ??.

Let us show, that \(x_*\) is a root of equation \(F(x)+Q(x)=0\).

for \(n \rightarrow \infty \). Hence, \(F(x_*)+Q(x_*)=0\).

Next we will show the uniqueness of solution \(x_*\). Suppose, that \(x_{**} \in U_0(x_0,r_1)\), exists \(x_{**} \neq x_*\) and \(H(x_{**})=0\). Denote

Then \(P(x_{**}-x_*)=H(x_{**})-H(x_*)\). In case operator \(P^{-1}\) is invertible, we obtain, that \(x_{**}=x_*\).

Hence, \(P^{-1}\) exists.

Hence, \(\int _0^1F'(x_*+t(x_n-x_*))dt+Q(x_n,x_*)\) is invertible and

Using the estimation

we obtain estimate ??.

4 Numerical experiments

In order to demonstrate the results of iterative solver ??, we carried out numerical experiments on test cases with nondifferentiable operator. The calculations are performed for different initial approximations with accuracy \(\varepsilon = 10^{-10}\). The iterative process was performed until following conditions are satisfied:

Additional initial approximations were chosen by the following formula:

To compare the convergence speed of the combined Newton-Potra solver with a basic solvers, the number of iterations, required to obtain a solution of systems of nonlinear equations, are presented in a table. The Newton-type solver for equation ?? has the form [ 3 ] :

and the Potra solver [ 22 ] :

Consider the system of two equations

Consider the system of three equations.

5 Conclusions

Based on the obtained results we showed the advantages of combined solver ?? over basic solvers, in particular, over Potra solver ??, even the theoretical convergence order of both solvers are the same. Moreover the convergence region of iterative solvers in general is small, which limits the choice of initial points. So by using the new Lipschitz constants we get at least as many initial points and fewer iterations to achieve predetermined accuracy, without any additional cost. This technique can be applied to extend the applicability of other iterative solvers. Therefore, the proposed combined solver ?? is an effective alternative for solving nonlinear equations with nondifferentiable operator.