In the last few decades, much attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in control theory correspond to the maximal deviation of the regulated quantity. A classical example is that of an electric generator. In this case, the mechanism becomes active when the maximum voltage variation that is permitted is reached in an interval of time $I_{t}=[t-h,t]$ with $h$ a positive constant. The equation which describes the action of this regulator has the form

where $\delta$ and $p$ are constants that are determined by the characteristic of system, $V(t)$ is the voltage and $F(t)$ is the effect of the perturbation that appears associated to the change of voltage [1].

The use of the Perov’s fixed point theorem [10, 11] generates an efficient technique to approach systems of functional-differential equations [5, 14]. In the study of existence and uniqueness of the solution of an operatorial equation, the notions of Picard and weakly Picard operators are very useful [11, 13], [15]-[17]. Some applications of the theory of weakly Picard operators can be found in [13]-[17], [3, 4] and [6]-[9]. Some problems concerning differential equations with maxima were studied in [1, 5, 8, 9] and in the monograph [2]. In [8] we have obtained conditions for existence and uniqueness, inequalities of Čaplygin type and data dependence for the solutions of functional-differential equations with maxima while in [9] we apply the technique of weakly Picard operators for the second order functional-differential equations with maxima, of mixed type. Here we continue the work from [8] and [9] with the study of systems of functional-differential equations with maxima, of mixed type.

We consider the following functional-differential system

with the “boundary” conditions

Suppose that:

• (C₁)

  $h_{1},h_{2},\ a\$and $b\in\mathbb{R},\ a<b,\ h_{1}>0,\ h_{2}>0;$

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• (C₂)

  $f\in C([a,b]\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m},\mathbb{R}^{m})$;

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• (C₃)

  there exists a matrix $L_{f}\in M_{m\times m}(\mathbb{R}_{+})$ such that

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  $$\left|f(t,u^{1},u^{2},u^{3})-f(t,v^{1},v^{2},v^{3})\right|\leq L_{f}{\left(\begin{array}[c]{c}\underset{1\leq i\leq 3}{\max}\left|u_{1}^{i}-v_{1}^{i}\right|\\ \vdots\\ \underset{1\leq i\leq 3}{\max}\left|u_{m}^{i}-v_{m}^{i}\right|\end{array}\right)},$$

  for all $t\in[a,b]$ and $u^{i}=(u_{1}^{i},\ldots,u_{m}^{i}),v^{i}=(v_{1}^{i},\ldots,v_{m}^{i})\in\mathbb{R}^{m},i=1,2,3$ where

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  $$\left|w\right|:=\left(\begin{array}[c]{c}\left|w_{1}\right|\\ \vdots\\ \left|w_{m}\right|\end{array}\right);$$

• (C₄)

  $\varphi\in C([a-h_{1},a],\mathbb{R}^{m})$ and $\psi\in C([b,b+h_{2}],\mathbb{R}^{m}).$

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Let $G$ be the Green function of the following problem

where $\chi\in C([a,b],\mathbb{R})$. Denoting

the problem (1)–(2), with smoothness condition $x\in C([a-h_{1},b+h_{2}],\mathbb{R}^{m})\cap C^{2}([a,b],\mathbb{R}^{m})$, is equivalent to the following equation

$x\in C([a-h_{1},b+h_{2}],\mathbb{R}^{m}).$

The equation (1) is equivalent to

$x\in C([a-h_{1},b+h_{2}],\mathbb{R}^{m}).$

In what follows we consider the operators:

defined by $B_{f}(x)(t):=$ the right hand side of (3) and $E_{f}(x)(t):=$ the right hand side of (4). Let $X:=(C[a-h_{1},b+h_{2}],\mathbb{R}^{m})$ and $X_{\varphi,\psi}:=\{x\in X|\ x|_{[a-h_{1},a]}=\varphi,\ x|_{[b,b+h_{2}]}=\psi\}$. It is clear that $X=\underset{\varphi,\psi}{\bigcup}X_{\varphi,\psi}\$is a partition of $X.$

The following result is known.

Let $M_{G}:=(\left\|G_{ij}\right\|)_{i,j=\overline{1,m}}\in M_{m\times m}(\mathbb{R}_{+})$, where $\left\|G_{i,j}\right\|=\max\{\left|G_{i,j}(x,s)\right|:$ $(x,s)\in[a,b]\times[a,b]\},\ i,j=\overline{1,m}$ and

The following is a synopsis of the paper. In Section 2 we introduce notation, definitions and results from weakly Picard operator theory. In Section 3 we obtain existence and uniqueness result using Perov’s fixed point theorem and the weakly Picard operator technique. Sections 4 and 5 present inequalities of Čaplygin type and data dependence results.

In this section, we introduce notation, definitions, and preliminary results which are used throughout this paper (see [12]-[17]). Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We shall use the following notations:

$F_{A}:=\{x\in X\mid A(x)=x\}$ - the fixed point set of $A$;

$I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\}$ - the family of the nonempty invariant subset of $A$;

$A^{n+1}:=A\circ A^{n},\;A^{0}=1_{X},\;A^{1}=A,\;n\in\mathbb{N}$.

Throughout this paper we denote by $M_{m\times m}(\mathbb{R}_{+})$ the set of all $m\times m$ matrices with positive elements and by $I$ the identity $m\times m$ matrix. A square matrix $Q$ with nonnegative elements is said to be convergent to zero if $Q^{k}\rightarrow 0$ as $k\rightarrow\infty$. It is known that the property of being convergent to zero is equivalent to any of the following three conditions (see [12]):

• (a)

  $I-Q$ is nonsingular and $(I-Q)^{-1}=I+Q+Q^{2}+\cdots$ (where $I$ stands for the unit matrix of the same order as $Q$);

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• (b)

  the eigenvalues of $Q$ are located inside the unit open disc of the complex plane;

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• (c)

  $I-Q$ is nonsingular and $(I-Q)^{-1}$ has nonnegative elements.

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We finish this section by recalling the following fundamental result (see, e.g., [10]).

Let us consider the problem (1)–(2). We obtain the following existence and uniqueness theorem.

The problem (1)–(2) is equivalent to the fixed point equation

From the condition $(C_{3})$ we have, for any $t\in[a,b]$

Then $\left\|B_{f}(x)-B_{f}(y)\right\|\leq Q\left\|x-y\right\|,\text{ for all }x,y\in X$ and by (ii), the operator $B_{f}$ is $Q$-contraction. From the Perov’s fixed point theorem we have that the operator $B_{f}$ is PO and has a unique fixed point $x^{\ast}=(x_{1}^{\ast},\ldots,x_{m}^{\ast})\in X$. Since $f$ is continuous, we have that $x^{\ast}\in C^{2}([a,b],\mathbb{R}^{m})$ is the unique solution for the problem (1)–(2).  

In order to establish the Čaplygin type inequalities we need the following abstract result.

Now we consider the operators $E_{f}$ and $B_{f}$ on the ordered Banach space $(C([a-h_{1},b+h_{2}],\mathbb{R}^{m}),\left\|\cdot\right\|,\leq)$ where we consider the following order relation on $\mathbb{R}^{m}$: $x\leq y\Leftrightarrow x_{i}\leq y_{i},\ i=\overline{1,m}$.

In the terms of the operator $E_{f},$ we have $x=E_{f}(x)$ and $y\leq E_{f}(y)$. From Remark 3.2 we have that $E_{f}$ is WPO. On the other hand, from the condition (c) and Lemma 4.1 we get that the operator $E_{f}^{\infty}$ is increasing. Hence $y\leq E_{f}(y)\leq E_{f}^{2}(y)\leq\ldots\leq E_{f}^{\infty}(y)=E_{f}^{\infty}(\widetilde{w}(y))\leq E_{f}^{\infty}(\widetilde{w}(x))=x$. So, $y\leq x$.  

In order to study the monotony of the solution of the problem (1)–(2) with respect to $\varphi,\ \psi$ and $f$, we need the following result from the WPOs theory.

From this abstract result we obtain the following result:

Consider the boundary value problem (1)–(2) and suppose that the conditions of the Theorem 3.1 are satisfied with the same Lipshitz constants. Denote by $x^{\ast}(\cdot;\varphi,\psi,f)\$the solution of this problem. We get the data dependence result.

Additionally

We have