Consider the matrix

where $a_{ij\geq 0},\ i,j=1,\ldots,m$. Let $f_{i}:[0,\infty)\rightarrow[0,\infty),i=1,\ldots,m,$ be Lipschitz functions, i.e.,

where $l>0$ is a given constant.

Systems of equations of the form

were investigated in several papers (see [1, 2, 3, 4, 5, 6, 7, 8, 9], [13]-[15] and the references therein). The existence and the uniqueness of a solution $(x_{1},\ldots,x_{m})\in[0,\infty)^{m}$ were established using, among other results, Brower’s theorem and the iterative monotonic convergence method. Such systems appear frequently in applications.

Several real-world problems can be attacked using systems with the above characteristics. The corresponding mathematical models involve also second order Dirichlet problems, Dirichlet problems for partial difference equations, equations with periodic solutions, numerical solutions for differential equations, all of them with important applications to economics. Details can be found in the papers mentioned in our bibliography and in the references therein.

In this paper we consider systems of the form

In order to study the existence and the uniqueness of a solution we use Perov’s theorem.

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. In this paper we use the terminologies and notations from [12]. For the convenience of the reader we shall recall some of them.

Denote by $A^{0}:=Id_{X},\ A^{1}:=A,\ A^{n+1}:=A\circ A^{n},\ n\in\mathbb{N}$, the iterate operators of the operator $A$ and by $F_{A}:=\left\{x\in X|\ A(x)=x\right\}$ the fixed point set of $A.$

As concerns matrices which are convergent to zero, we mention the following equivalent characterizations:

The matrices convergent to zero were used by Perov [10] to generalize the contraction principle in the case of generalized metric spaces with the metric taking values in the positive cone of $\mathbb{R}^{m}.$

Consider again the matrix $A$ with $a_{ij}\geq 0,i,j=1,\ldots,m$ and the functions $f_{i}:[0,\infty)\rightarrow[0,\infty),i=1,\ldots,m,$ satisfying the Lipschitz condition (1.1). Denote

Then $x\in[0,\infty)^{m},\ G(x)\in[0,\infty)^{m}.$ The system (1.3) can be written as

For $x,y\in[0,\infty)^{m},$ let

Then $d$ is a generalized metric and $[0,\infty)^{m}$ is a complete generalized metric space.

Now let us consider the system of equations

where, as before, $a_{ij}\geq 0,\ p_{i}\geq 0,\ i,j=1,\ldots,m.$

Let $x\in[0,\infty)^{m}$ and

Then the system (3.2) can be written as

With the same distance $d$ as before we can state

Consider the system of equations

where $x_{1},\ x_{2}\geq 0$. Then $m=2,\ f_{1}(t)=\log(t+2),\ f_{2}(t)=\log(t+3),\ t\geq 0$. Since $f_{i}^{\prime}(t)\leq\tfrac{1}{2},\ i=1,2,t\geq 0,$ we can take $l=\tfrac{1}{2}$. Moreover, the system is of the form (3) with

and $A$ has the eigenvalues $\mu_{1}=1,\ \mu_{2}=-\tfrac{4}{15}$. Consequently, the matrix

has eigenvalues $\tfrac{1}{2}$ and $-\tfrac{2}{15}$; according to Theorem 2.1, $Q$ is convergent to zero.

With $x=\left(\begin{array}[c]{c}x_{1}\\ x_{2}\end{array}\right),$ the operator $G$ has the form

According to Theorem 3.1, $G$ is a $Q$-contraction and its unique fixed point $x^{\ast}$ is the unique solution of the system (4.1).

Let

be given. Let $x^{(1)}=G(x^{(0)}),\ x^{(2)}=G(x^{(1)}),\ldots$.

Then $x^{(n)}=G^{n}(x^{(0)})$ and

In our case,

and this gives an estimate of the rate of convergence in

From $x^{(n+1)}=G(x^{(n)}),\ n\geq 0$, we have

Choosing different values for $x_{1}^{(0)}$ and $x_{2}^{(0)}$ we get in Fig. 1, Fig 2 and Fig 3 the iterations and the representation of solutions.

Our paper is devoted to a specific family of algebraic systems, having significant applications to real-world problems. Several papers from the literature are concerned with finding approximate solutions to them. Our approach is based on the Perov’s theorem. This allows to estimate componentwise the rate of convergence.

We intend to return to this topic in order to compare our results with other existent ones and to find new applications.