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.


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 ( ) + 2 ( ) + ( ) = ( ( − ℎ)) .
In this paper, we continue the research in this field and develop the study of the following general functional differential equation with delay: ( ) = ( , ( ) , ( ) , ( − ℎ) , ( − ℎ)) , 2 The Scientific World Journal 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][6][7][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 ( ) ∈ ([ −ℎ, ], R 2 )∩ 1 ([ , ], R 2 ) that verifies the system. We suppose that (C 1 ) < , ℎ > 0; is a solution of the problem (6)-(7), then ( ) is a solution of the following integral system:

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 > 0.
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 > 0 such that for each > 0 and each solution * of the inequation there exists a solution * of (13) such that Now we have the following.
Another result from the WPO theory is the following (see, e.g., [11]).
Then the operator is Picard operator.
Throughout this paper we denote by (R + ) 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 → 0 as → ∞. It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [9,10]): (c) − is nonsingular and ( − ) −1 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 ( , ) ∈ R be a complete generalized metric space and : → an operator. One supposes that there exists a matrix ∈ (R + ), such that Then

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). Proof. Consider on the space := ([ − ℎ, ], R 2 ) the norm

Existence and
which endows with the uniform convergence. Let is a partition of , and from [12] we have ( whence is a contraction in ( , ‖ ⋅ ‖) with = ( − ) ( 0 1

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

Data Dependence: Continuity.
Consider the problem (6)-(7) with the dates ∈ ([ , ]×R 4 , R), = 1, 2 and suppose that satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.
For this we consider the system   ) .
By induction we prove that ), as → ∞.
From a Weierstrass argument we get that there exists