^{1}

^{2}

^{1}

^{2}

Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.

Functional-differential equations with delay arise when modeling biological, physical, engineering, and other processes whose rate of change of state at any moment of time

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 [

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

We suppose that

there exists

If

If

Moreover, the system (

We consider the operators

In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper; see [

Let

the sequence

Let

If

It is clear that

Let

The following concept is important for our further considerations.

Let

Now we have the following.

If

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

Let

the operator

there exists

if

Throughout this paper we denote by

the eigenvalues of

We finish this section by recalling the following fundamental result (see [

Let

Then

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

Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (

One supposes that

the conditions (C_{1})–(C_{3}) are satisfied;

Then,

the problem (

for all

the operator

the operator

Consider on the space

Let

On the other hand, for

Now we establish the Čaplygin type inequalities.

One supposes that

the conditions (a), (b), and (c) in Theorem

Let

We have that

In this subsection, we study the monotony of the solution of the problem (

Let

Let

Then

We consider the operators

Consider the problem (

Let

there exists

there exists

Then

Consider the operators

Thus,

Consider the following differential system with parameter:

Suppose that the following conditions are satisfied:

there exists

for

Then, from Theorem

For this we consider the system

Consider the problem (_{1})–(C_{5}) hold. Then,

Equations (

The problem (

Let

It is clear, from the proof of the Theorem _{1})–(C_{5}), the operator

Let

Supposing that there exists

This relation suggests that we consider the following operator:

In this way, we have the triangular operator

From Theorem

If we take

By induction we prove that

From a Weierstrass argument we get that there exists

We start this section by presenting the Ulam-Hyers stability concept (see [

The system (

One supposes that

the conditions (C_{1})–(C_{3}) are satisfied;

Then the system (

The system (

From Theorem

Another proof for the above theorem can be done using Gronwall lemma [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work of the first author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094.