Advances in Mathematical Physics
Volume 2015 (2015), Article ID 365328, 11 pages
http://dx.doi.org/10.1155/2015/365328

Research Article

Stability Criteria for Volterra Integrodynamic System

Nusrat Yasmin, Awais Younus, Usman Ali, and Safia Mirza

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60000, Pakistan

Received 28 July 2015; Revised 16 October 2015; Accepted 21 October 2015

Academic Editor: Ming  Mei

Copyright © 2015 Nusrat Yasmin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study conditions under which the solutions of linear Volterra integrodynamic system of the form are stable on certain time scales. We construct a number of Lyapunov functionals on time scales from which we obtain necessary and sufficient conditions for stability of Volterra integrodynamic system and also we prove several results concerning qualitative behavior of this system.

1. Introduction

The theory of Volterra integrodifferential equation (VIDE) has been studied extensively by several researchers [1–5]. In [1], Becker investigates the variation of parameters formula for a VIDE and its adjoint. The interesting and useful aspects of VIDE have been shown in different articles of Burton and Mahfoud (see [2, 3, 6]). Elaydi, in [5], discusses the periodicity and stability of linear Volterra difference systems.

Time scale theory, introduced by Hilger [7] at the end of the twentieth century as a means to unify the discrete and the differential calculus, is now a well established subject. For an excellent introduction to the calculus and to the theory of dynamic equations on time scales we recommend the books [8, 9]. Volterra and Fredholm type equations (both integral and integrodynamic) on time scales become a new field of interest; for example, see [5, 10–22]. In [13], Adivar contributes to principle matrix and variation of parameter formula for VIDE. Recently, Lupulescu et al. [21] discuss the resolvent asymptotic stability and boundedness of VIDE and show that principle matrix and resolvent are equivalent to linear VIDE on time scales.

Constructing Lyapunov functionals for integrodifferential equations has been a very challenging task, even in the continuous case. Burton was the first one to construct such functionals and utilize them to qualitatively analyze solutions of integrodifferential equations. The study of integrodynamic equations on time scales provides deeper and comprehensive understanding of traditional integrodifferential equations and Volterra difference equations. This paper generalize some results of [4] for the continuous case (i.e., ) and all of them are new for the other time scales. We begin by stating some important facts and properties of time scales that we will be using during our analysis.

2. Preliminaries

Let be the space of -dimensional column vectors with a norm . Also, with the same symbol we will denote the corresponding matrix norm in the space of matrices. If , then we denote by its conjugate transpose. We recall that and the following inequality holds for all and .

By a time scale we mean any closed subset of . Since a time scale is not connected in general, we need the concept of jump operators. The forward jump operator is defined by , while the backward jump operator is defined by . In this definition, we put and . If , we say is a right-scattered point, while if , we say is a left-scattered point. Points that are right-scattered and left-scattered at the same time will be called isolated points. A point such that and is called a right-dense point. A point such that and is called a left-dense point. Points that are right-dense and left-dense at the same time will be called dense points. The set is defined to be if has a left-scattered maximum ; otherwise . The graininess function is defined by . Given a time scale interval , denotes the interval if and denotes the interval if . In fact, . Also, for , we define . If is a bounded time scale, then can be identified with .

Throughout this work, we assume that with bounded graininess; that is, .

If and , then we define the following neighborhoods of .

Let us consider some examples of time scales (see [8]).

Example 1. (i) If , is a time scale. Then we have , and for all . Hence, each point is isolated, for all , and .
(ii) Let . Then

Definition 2 (see [8]). A function is called regulated if its right-sided limits exist (finite) at all right-dense points in and its left-sided limits exist (finite) at all left-dense points in . A function is called rd-continuous if it is continuous at all right-dense points in and its left-sided limits exist (finite) at all left-dense points in .

Obviously, a continuous function is rd-continuous, and a rd-continuous function is regulated [8, Theorem ].

Definition 3 (see [8]). Let and . We define (provided it exists) with the property that, for every , there exists such that

for all . We call the delta derivative (-derivative for short) of at . Moreover, we say that is delta differentiable (-differentiable for short) on provided that exists for all .

Now we recall some properties of the exponential function on time scales. For definition of the exponential function on time scales, see [8, Definition ]. A function is called positively regressive if for all . If is a positively regressive function and , then (see [8, Theorem ]) the exponential function is the unique solution of the initial value problem:

The following properties of the exponential function holds:

(i) and .(ii).(iii).(iv).

In particular, if is such that for all , we have if and if with .

For more details, see [8]. Clearly, never vanishes.

The next definitions are about shift and convolution of functions on time scales (see [23]).

Definition 4. For a given function , the solution of shifting problem

is denoted by and it is called shift (or delay) of .

Example 5. For the regressive , satisfy the shift problem (4).

Definition 6. For given functions , their convolution is defined by

Furthermore, reader can see the existence and uniqueness of shift problem (4) and its properties in [23, Section ]. Next two results are needed for our proofs.

Lemma 7 (see [8, Theorem ]). Let , and assume that is continuous at , where with . Also assume that is rd-continuous on . Suppose that, for each , there exists a neighborhood of , independent of , such that

Then one has

Lemma 8 (see [8, Theorem ]). Let . For rd-continuous functions , one has the following inequality:

Let us consider a linear Volterra integrodynamic system of the form

where is continuous and a regressive matrix on , and is continuous and a regressive matrix on .

We develop the results about stability by using the Lyapunov second (direct) method. The principal idea of the second method is contained in the following physical reasoning: if the rate of change, , of the energy of an isolated physical system is negative for every possible state , except for a single equilibrium state (i.e., ), then the energy will continually decrease until it finally assumes its minimum value . In other words, a system that is perturbed from its equilibrium state will always return to it. This is intuitive concept of stability. The mathematical counterpart of the preceding statement is the following.

A solution of (9) will be denoted by if no confusion should arise. In the remainder of this paper when we say the zero solution of (9) we mean the zero solution of (9) with .

Definition 9. The zero solution of (9) is stable, if for every there exist such that, for any solution of (9), the inequality implies for .

Definition 10. The zero solution of (9) is asymptotically stable, if it is stable and attractive (i.e., if for any solution of (9), there exist such that implies as ).

Definition 11. The zero solution of (9) is unstable if there exists such that for any , if , then, for any solution of (9), there is with .

Let us consider new functions as follows:

assuming of course that exists for .

Definition 12 (see [14]). A dynamic system is stable if and only if there exists a “Lyapunov functional,’’ that is, some scalar function of the state with the properties:

(a), , when .(b), when .

3. The Scalar Equation

In this section, we consider the scalar case of Volterra equation (9); that is, and .

Theorem 13. Suppose that, for some ,

Then the zero solution of (9) is stable if and only if .

Proof. Suppose that and consider the functional

The derivative of along solution of (9) satisfies

Using Lemma 8, it follows that

Now by (11) we have the following estimation:

As is positive definite and , it follows that is stable.
Conversely, suppose on the contrary that ; that is, .
For for assumption (11) does not hold.
For consider the following functional:

Similar to the previous calculation, we obtain

which implies

Now, any , with and so that if is a solution of (9), then we have

Hence,

As , which is the contradiction to the fact that the zero solution is stable. Hence, which completes our proof.

Corollary 14. If (11) holds and and bounded, then the zero solution of (9) is asymptotically stable.

Proof. By Theorem 13, it follows that . This implies that is in and is bounded. It follows from (11) and (9) that is bounded. Thus, as . The proof is complete.

Remark 15. Notice that condition (11) would not hold if is allowed to vanish at some point . Therefore, Theorem 13 cannot apply unless for all . In next result we consider (9) where may vanish at any .

We select a continuous function with

and let

so that (9) may be written as

Theorem 16. Suppose that (21) holds and there are constants , , , and with such that

(i),(ii),(iii),

for . Furthermore, suppose that there is a continuous function with and as . Then the zero solution of (23) is stable if and only if .

Proof. Suppose that and consider the functional

The derivative of along a solution of (23) satisfies

Using (i) and (iii), we may write

Let ; we have to find so that if , this implies for all . Since for all , then

Hence,

On the other hand, (24) yields

Combining (28) and (29) we obtain

Now, as long as , we have

for all provided that . Since (9) and (23) are the same equation, then the zero solution of (9) is stable.
Conversely, suppose on the contrary that ; that is, .
For for all assumption does not hold.
For consider the following functional:

Then it follows that

Now, given any , with and so that if is a solution of (9), we have

We will show that is unbounded. If is bounded, then as is bounded, we have that is bounded; this implies that is in . Using Lemma 8, we have

The last integral is the convolution of function with a function tending to zero. Thus, the integral tends to zero as and hence

Since

then, for sufficiently large , for some and all . This contradicts the fact that is in . Thus, is unbounded and the zero solution of (9) is unstable, which is the contradiction to the fact that the zero solution is stable. Hence, which completes our proof.

Example 17. For , , , consider the following Volterra integrodynamic equation:

where

Let us consider the following function:

It is easy to see that and

so that (38) may be written as equation

It follows from (41) that

Furthermore, for every

Since for all , by using (45), we obtain that

If we take for , then all the assumptions of Theorem 16 satisfy so that the zero solution of (38) is stable.

Our next result is about asymptotic stability of (9). Assume for the present time that and are continuously differentiable and that both

Theorem 18. Suppose that (21) holds and there are constants , , , , and with such that, for , one has

(i),(ii) ,(iii) ,(iv) ,

and then the zero solution of (9) is asymptotic stable. Furthermore, every solution of (9) is in .
In addition, if

(v)   ,(vi) , , and are bounded, then as and is in .

Proof. Stability of the zero solution is clear from Theorem 16; hence,

for all . As , we have in . To show that the zero solution is asymptotically stable, we first observe from (iv) that is bounded and since is bounded, hence is bounded.
Now (9) yields that

and, using (iv), that is bounded. Since is in and is bounded, it follows that as . Hence, the zero solution is asymptotically stable.
To show that is in and differentiate (23), we obtain

Let , and consider the functional

The derivative along a solution of (23) for satisfies

and, substituting the value of from (23), we obtain

That implies

By (v), is bounded. Thus, by choosing and sufficiently large, we obtain

As , we have in . To show , we differentiate (9) and obtain

Since

we have

Thus, by (v) and (vi), is bounded and hence, by (49) and (56), is bounded; therefore, is bounded. As is in , we have as . This completes the proof.

4. The Vector Equation

We now extend the results of Section 3 to the system of Volterra equations; that is, . Owing to the greater complexity of systems over scalars, it seems preferable to reduce the generality of and .

Consider the system

in which is constant matrix and is matrix of functions continuos on

We suppose that there is a symmetric matrix which satisfies the equation

Theorem 19. Suppose that (60) holds for some symmetric matrix and that there is a constant such that

Then the zero solution of (59) is stable if and only if is positive definite.

Proof. We consider the functional

The derivative of along a solution of (59) satisfies

Using (60), we may write

By Lemma 8, we obtain

where . Now, if is positive definite, then for all and hence is positive definite and is negative definite and it follows that the zero solution of (59) is stable.
Conversely, suppose that the zero solution of (59) is stable but is not positive definite. Then there is such that .
If , then, along the solution of (59), we have

It follows that, for some , . Thus, .
Hence, is not always positive for , and we may suppose that there is such that .
Let . Since the zero solution of (59) is stable, then there is and such that and for all . We may choose such that and .
Moreover, we have

We have to show that is bounded away from zero. Suppose that it is not true; then, there is a sequence tending to infinity monotonically such that , a contradiction to . Thus, there is with so that , implying that as . This contradicts and completes the proof.

We now prove a complete instability result for the zero solution of (59). Suppose that is a positive definite symmetric matrix satisfying

Theorem 20. Suppose that (68) holds for some symmetric matrix and that there is a constant such that

Then the zero solution of (59) is completely unstable.

Proof. Consider the functional

The derivative of along a solution of (59) satisfies

Similarly, as in Theorem 19, we obtain the following estimate:

where . Choose so that . Then

for all . As , we have

If and , then

This is complete instability. The proof is now complete.

Conflict of Interests

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

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