Functional-differential equations with delay arise when modeling biological, physical, engineering, etc., processes whose rate of change of state at any moment of time $t$ is determined not only by the present state, but also by past state.

The description of certain phenomena in physics has to take into account that the rate of propagation is finite. For example, oscillation in a vacuum tube can be described by the following equation in dimensionless variables [4], [9]

In this equation, time delay is due to the fact that the time necessary for electrons to pass from the cathode to the anode in the tube is finite. The same equation has been used in the theory of stabilization of ships [9]. The dynamics of an autogenerator with delay and second-order filter was described in [1] by the equation

The model of ship course stabilization under conditions of uncertainty may be described by the following equation [3],

with $x(t)$ being the angle of the deviation from course, $\Psi(t)$ the turning angle of the rudder and $\xi_{0}(t)$ the stochastic disturbance. In the process of mathematical modeling often small delays are neglected, that’s why sometimes it appears false conclusions. As an example we can give the following equation [4],

which is asymptotically stable for $h=0$, but unstable for arbitrary $h>0$. Here $a>1$. If $h=0$ the above system is asymptotically stable. The characteristic equation is $\Delta(z)=(z^{2}+z+1)(1-ae^{-hz})=0$ has the solutions with $\Delta$ the positive real part of $h>0$. So the trivial solution is unstable for any $h>0$.

In this paper we continue the research in this field and develop the study of the following general functional differential equation with delay

Existence, uniqueness, data dependence (monotony, continuity, differentiability with respect to parameter) and Ulam-Hyers stability results of solution for the Cauchy problem are obtained. Our results are essentially based on Perov’s fixed point theorem and weakly Picard operator technique, which will be presented in Section 2. More results about functional and integral differential equations using these techniques can be found in [2], [5]-[7]. The problem (1.1) is equivalent to the following system

with the initial conditions

By a solution of the system (1.2) we understand a function $\binom{x}{z}\in\linebreak C([a-h,b],\mathbb{R}^{2})\cap C^{1}([a,b],\mathbb{R}^{2})$ that verifies the system.

We suppose that

1. (C₁)

   $a<b,\ h>0;$

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2. (C₂)

   $f\in C([a,b]\times\mathbb{R}^{4},\mathbb{R}),\varphi\in C^{1}([a-h,a],\mathbb{R});$

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3. (C₃)

   there exists $L_{1},L_{2}>0$ such that $\forall t\in[a,b],u_{i},v_{i},\widetilde{u}_{i},\widetilde{v}_{i}\in\mathbb{R},\linebreak i\mathbb{=}1,2,$ we have

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   	$\displaystyle\left|f(t,u_{1},v_{1},u_{2},v_{2})-f(t,\widetilde{u}_{1},\widetilde{v}_{1},\widetilde{u}_{2},\widetilde{v}_{2})\right|\leq$
   	$\displaystyle\leq L_{1}\max\left(\left|u_{1}-\widetilde{u}_{1}\right|,\left|u_{2}-\widetilde{u}_{2}\right|\right)+L_{2}\max\left(\left|v_{1}-\widetilde{v}_{1}\right|,\left|v_{2}-\widetilde{v}_{2}\right|\right).$

If $\binom{x}{z}\in C([a-h,b],\mathbb{R}^{2})\cap C^{1}([a,b],\mathbb{R}^{2})$ is a solution of the problem (1.2)-(1.3) then $\binom{x}{z}$ is a solution of the following integral system

If $\binom{x}{z}\in C([a-h,b],\mathbb{R}^{2})$ is a solution of (1.4) then $\binom{x}{z}\in C([a-h,b],\linebreak\mathbb{R}^{2})\cap C^{1}([a,b],\mathbb{R}^{2})$ and $\binom{x}{z}\$is a solution of (1.2)-(1.3).

Moreover, the system (1.2) is equivalent to the functional integral system

We consider the operators $B,E:C([a-h,b],\mathbb{R}^{2})\rightarrow C([a-h,b],\mathbb{R}^{2}),\ B=\binom{B_{1}\binom{x}{z}}{B_{2}\binom{x}{z}},\ E=\binom{E_{1}\binom{x}{z}}{E_{2}\binom{x}{z}}$ defined by $B\binom{x}{z}(t):=$the right hand side of (1.4), for $t\in[a-h,b]$ and $E\binom{x}{z}(t):=$the right hand side of (1.5), for $t\in[a-h,b].$

In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper, see [10]–[18]. 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 points set of $A$;

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

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

The following concept is important for our further considerations.

Now we have

Another result from the WPO theory is the following (see, e.g., [12]).

Throughout this paper we denote by $M_{mm}(\mathbb{R}_{+})$ 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 each of the following three conditions (see [10], [11]):

• (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 open unit 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 [8], [10]).

In this section we present existence, uniqueness and data dependence (monotony, continuity, differentiability with respect to parameter) results of solution for the Cauchy problem (1.2)-(1.3).