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
Then the operator
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 (C1)–(C3) 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 ( Equations (
The problem (
Let
It is clear, from the proof of the Theorem
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 (C1)–(C3) 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.