Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay
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 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 . The dynamics of an autogenerator with delay and second-order filter was described in  by the equation The model of ship course stabilization under conditions of uncertainty may be described by the following equation : 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 : 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
Moreover, the system (6) is equivalent to the functional integral system
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 ). If is -WPO, then the equation is Ulam-Hyers stable.
Another result from the WPO theory is the following (see, e.g., ).
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.
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
3.1. Existence and Uniqueness
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  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.
3.3. Data Dependence: Monotony
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 ).
3.4. Data Dependence: Continuity
Proof. Consider the operators , . From Theorem 10, it follows that
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 .
For this we consider the system with .
Proof. The problem (38)–(36) is equivalent with the following functional-integral system:for and
Now let us take the operator , defined by
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
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.
Conflict of Interests
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.
V. Kolmanovskiĭ and A. Myshkis, Applied Theory of Functional-Differential Equations, Kluwer Academic Publishers Group, Dordrecht, Germany, 1992.View at: MathSciNet
E. Pinney, Ordinary Difference-Differential Equations, University of California Press, Berkeley, Calif, USA, 1958.View at: MathSciNet
V. V. Guljaev, A. S. Dmitriev, and V. E. Kislov, “Strange attractors in the circle: selfoscillating systems,” Doklady Akademii Nauk SSSR, vol. 282, no. 2, pp. 53–66, 1985.View at: Google Scholar
V. B. Kolmanovskiĭ and V. R. Nosov, Stability of Functional-Differential Equations, Academic Press, London, UK, 1986.View at: MathSciNet
I. A. Rus, Principles ans Applications of the Fixed Point Theory, Dacia, Cluj-Napoca, Romania, 1979, Romanian.
I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, Romania, 2001.View at: MathSciNet
I. A. Rus, “Ulam stability of the operatorial equations,” in Functional Equations in Mathematical Analysis, T. M. Rassias and J. Brzdek, Eds., chapter 23, Springer, 2011.View at: Google Scholar