Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay

Abstract

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.

1. Introduction

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 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 [1, 2]: 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 [2]. The dynamics of an autogenerator with delay and second-order filter was described in [3] by the equation The model of ship course stabilization under conditions of uncertainty may be described by the following equation [4]: with being the angle of the deviation from course, the turning angle of the rudder, and the stochastic disturbance. In the process of mathematical modeling, often small delays are neglected; that is why sometimes false conclusions appear. As an example we can give the following equation [1]: which is asymptotically stable for but unstable for arbitrary . Here . If the above system is asymptotically stable. The characteristic equation is and has the following zeros with positive real part if , , . So the trivial solution is unstable for any .

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, and 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 [5–8]. The problem (5) is equivalent to the following system: with the initial conditions By a solution of the system (6) we understand a function that verifies the system.

We suppose that(C₁), ;(C₂), ;(C₃)there exists such that , , , we have

If is a solution of the problem (6)-(7), then is a solution of the following integral system:

If is a solution of (9), then and is a solution of (6)-(7).

Moreover, the system (6) is equivalent to the functional integral system

We consider the operators , , defined by the right hand side of (9), for and the right hand side of (10), for .

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper; see [9–17]. Let be a metric space and an operator. We will use the following notations: —the fixed points set of ; —the family of the nonempty invariant subset of ; , , , .

Definition 1. Let be a metric space. An operator is a Picard operator (PO) if there exists such that(i); (ii)the sequence converges to for all .

Definition 2. Let be a metric space. An operator is a weakly Picard operator (WPO) if the sequence converges for all and its limit (which may depend on ) is a fixed point of .

Definition 3. If is a weakly Picard operator, then we consider the operator defined by

Remark 4. It is clear that .

Definition 5. Let be a weakly Picard operator and . The operator is -weakly Picard operator if

The following concept is important for our further considerations.

Definition 6. Let be a metric space and an operator. The fixed point equation is Ulam-Hyers stable if there exists a real number such that for each and each solution of the inequation there exists a solution of (13) such that

Now we have the following.

Theorem 7 (see [17]). If is -WPO, then the equation is Ulam-Hyers stable.

Another result from the WPO theory is the following (see, e.g., [11]).

Theorem 8 (fibre contraction principle). Let and be two metric spaces and , , a triangular operator. One supposes that(i) is a complete metric space;(ii)the operator is Picard operator;(iii)there exists such that is a -contraction, for all ;(iv)if , then is continuous in .Then the operator is Picard operator.

Throughout this paper we denote by the set of all matrices with positive elements and by the identity matrix. A square matrix with nonnegative elements is said to be convergent to zero if as . It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [9, 10]):(a) is nonsingular and (where stands for the unit matrix of the same order as );(b)the eigenvalues of are located inside the open unit disc of the complex plane;(c) is nonsingular and has nonnegative elements.

We finish this section by recalling the following fundamental result (see [9, 18]).

Theorem 9 (Perov’s fixed point theorem). Let with be a complete generalized metric space and an operator. One supposes that there exists a matrix , such that(i), for all ;(ii) as .
Then(a);(b)  as  , ;(c).

3. Main Results

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 (6)-(7).

3.1. Existence and Uniqueness

Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (6)-(7).

Theorem 10. One supposes that(i)the conditions (C₁)–(C₃) are satisfied;(ii) as , where .
Then,(a)the problem (6)-(7) has a unique solution ;(b)for all , the sequence defined by converges uniformly to , for all , and (c)the operator is Picard operator in ;(d)the operator is weakly Picard operator in .

Proof. Consider on the space the norm which endows with the uniform convergence.
Let , for . Then is a partition of , and from [12] we have(1); (2).
On the other hand, for whence is a contraction in with . Applying Perov's theorem we obtain (a), (b), and (c). Moreover, the operator is -PO and is -WPO with .

3.2. Inequalities of Čaplygin Type

Now we establish the Čaplygin type inequalities.

Theorem 11. One supposes that(i)the conditions (a), (b), and (c) in Theorem 10 are satisfied;(ii),  ,   imply that
Let be a solution of (6) and a solution of the system Then

Proof. We have that From Theorem 10, (c), is a weakly Picard operator. From condition (ii), we obtain that is increasing [11]. So where .

3.3. Data Dependence: Monotony

In this subsection, we study the monotony of the solution of the problem (6)-(7) with respect to and .

Theorem 12 (comparison theorem). Let ,  , be as in Theorem 10. One supposes that(i); (ii) is increasing, .
Let be a solution of the system
Then imply that

Proof. We consider the operators corresponding to each system (25). The operators , are weakly Picard operators. Taking into consideration the condition (ii), is increasing. From (i) we have . On the other hand, we have that where . The proof follows from the abstract comparison Lemma (see [11]).

3.4. Data Dependence: Continuity

Consider the problem (6)-(7) with the dates ,  and suppose that satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.

Theorem 13. Let , ,  , be as in Theorem 10. One supposes that(i)there exists such that (ii)there exists such that
Then where denote the unique solution of (6)-(7).

Proof. Consider the operators , . From Theorem 10, it follows that Additionally,
Thus, and since , as implies that , we finally obtain

3.5. Data Dependence: Differentiability

Consider the following differential system with parameter: with the initial conditions where is a compact interval.

Suppose that the following conditions are satisfied:(C₁), , a compact interval;(C₂); (C₃); (C₄)there exists , such that (C₅)for , we have as .

Then, from Theorem 10, we have that the problem (6)-(7) has a unique solution, . We prove that , .

For this we consider the system with .

Theorem 14. Consider the problem (38)–(36) and suppose the conditions (C₁)–(C₅) hold. Then,(i)Equations (38)–(36) have a unique solution , in ;(ii), .

Proof. The problem (38)–(36) is equivalent with the following functional-integral system: for and Now let us take the operator , defined by
Let .
It is clear, from the proof of the Theorem 10, that in the condition (C₁)–(C₅), the operator is Picard operator.
Let be the unique fixed point of .
Supposing that there exists , from (39)-(40), we obtain that for all , .
This relation suggests that we consider the following operator: where for , , and where for . Here we use the notations , , , and .
In this way, we have the triangular operator , , where is Picard operator and is -contraction with .
From Theorem 8 the operator is Picard operator; that is, the sequences , converge uniformly, with respect to , to , for all .
If we take then
By induction we prove that So, , as , and , as .
From a Weierstrass argument we get that there exists

3.6. Ulam-Hyers Stability

We start this section by presenting the Ulam-Hyers stability concept (see [15, 16]). For , , and , , we consider the system and the following inequations

Definition 15. The system (51) is Ulam-Hyers stable if there exists a real number such that for each and for each solution of (52) there exists a solution of (51) with

Theorem 16. One supposes that(i)the conditions (C₁)–(C₃) are satisfied;(ii) as , where .
Then the system (6) is Ulam-Hyers stable.

Proof. The system (6) is equivalent with the functional integral system (10). We consider the operator , defined by the right hand side of (10), for . So
From Theorem 10, is -WPO with . Applying Theorem 7 we obtain that (6) is Ulam-Hyers stable.

Remark 17. Another proof for the above theorem can be done using Gronwall lemma [8, 14–17].

Conflict of Interests

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

Acknowledgment

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.