Asymptotic properties for Volterra integro-dynamic systems

Using the resolvent matrix, a comparison principle and a useful equivalent system, we investigate the asymptotic behavior of linear Volterra integro-dynamic systems on time scales.


Introduction
Infectious diseases have long been recognized as a major cause of mortality in human and other populations.The spread of an infectious disease involves not only disease-related factors such as the infectious agent, mode of transmission, latent period, infectious period, but also social, demographic and geographic factors [18].Most of the work in the literature in modeling infectious disease epidemics is mathematically inspired and based on integro-differential systems [15].
Classical topics in the qualitative theory of integro-differential equations are asymptotic equivalence and asymptotic behavior of systems [7,12].Two systems of integro-differential equations are said to be asymptotically equivalent if, corresponding to each solution of one system, there exists a solution of the other system such that the difference between these two solutions tends to zero.If we know that two systems are asymptotically equivalent, and if we also know the asymptotic behavior of the solutions of one of the system, then we can obtain information about the asymptotic behavior of the solutions of the other system.
Morchalo [21] and Nohal [22] established asymptotic equivalence between linear integrodifferential systems and their perturbations by using the dominated convergence theorem and the Hölder inequality.In [10] Choi et al. studied the asymptotic property of linear integrodifferential systems by means of the resolvent matrices and useful equivalent systems.For Corresponding author.Email: awaissms@yahoo.com 2 S. Mirza, D. O'Regan, N. Yasmin and A. Younus asymptotic properties of linear Volterra difference systems we refer the reader to [8,9].The uniform asymptotic stability of recurrent neural networks (RNNs) is analyzed by comparing RNNs to linear Volterra integro-differential systems in [19] and discrete analogs for a class of continuous-time recurrent neural networks are discussed in [20].The results in this paper generalize some known properties concerning asymptotic equilibrium from the continuous and discrete cases [8][9][10] to the time scale situation.
Time scales theory was introduced by Hilger [14] to unify discrete and continuous differential calculus; see the books [4,5].We refer the reader to [1-3, 16, 17] for results on Volterra and Fredholm type equations (both integral and integro-dynamic) on time scales.For example in [3] Adivar discusses the principle matrix and a variation of parameter formula.Lupulescu et al. [17] discussed the resolvent asymptotic stability, boundedness and show that the principle matrix and resolvent are equivalent for certain linear problems on time scales.
In this paper we assume the reader is familiar with the basic calculus of time scales.Let R n be the space of n-dimensional column vectors x = col(x 1 , x 2 , ...x n ) with a norm • .We will use the same symbol • to denote the corresponding matrix norm in the space M n (R) of n × n matrices.We recall that A := sup{ Ax ; x ≤ 1} and the following inequality Ax ≤ A x holds for all A ∈ M n (R) and x ∈ R n .A time scale, denoted by T, is an arbitrary, nonempty and closed subset of real numbers.The operator σ : T → T called the forward jump operator is defined by σ(t) := inf{s ∈ T, s > t}.The step size function µ : T → R + is given by µ(t) := σ(t) − t.We say a point t ∈ T is right dense if µ(t) = 0, and right scattered if µ(t) > 0. Furthermore, a point t ∈ T is said to be left dense if ρ(t) := sup{s ∈ T, s < t} = t and left scattered if ρ(t) < t.If T has a right-scattered minimum m, then T k = T −{m}; otherwise set T k = T.If T has a left-scattered maximum M, then T k = T −{M}; otherwise set T k = T. Throughout this work, we assume that sup T = ∞ with bounded graininess, i.e., µ(t) < ∞.Moreover, the delta derivative of a function f : T → R at a point t ∈ T k is defined by A function f is called rd-continuous provided that it is continuous at right dense points in T, and has finite limit at left-dense points, and the set of rd-continuous functions are denoted by C rd (T, R).The set of functions C 1 rd (T, R) includes the functions f whose derivative is in C rd (T, R) too.For s, t ∈ T and a function f ∈ C rd (T, R), the ∆-integral is defined to be It should be noted that the ∆-integral by means of the Riemann sum is also introduced in [13].
Let E ⊆ T be a ∆-measurable set and let p ∈ R be such that p ≥ 1 and let f : E → R n be a ∆-measurable function.We say f belongs to L p (E) provided that E f (t) p ∆t < ∞.
For more details concerning L p spaces we refer the reader to [23].
The set of regressive functions and the set of positively regressive functions are denoted by R(T, R) and R + (T, R), respectively.
Let f ∈ R(T, R) and s ∈ T, then the generalized exponential function e f (•, s) on a time scale T is defined to be the unique solution of the following initial value problem and the exponential function can also be written in the form For f ∈ C rd (T, R) and µ f 2 ∈ R(T, R), the trigonometric functions cos f and sin f are defined by For further details about these notions we refer the reader to [4,5].
Let T 1 and T 2 be two given time scales and put , which is a complete metric space with the metric (distance) d defined by x 0 right-dense or maximal and y 0 right-dense or maximal, then f is continuous at (x 0 , y 0 ); (iv) if x 0 and y 0 are both left-dense, then the limit of f (x, y) exists (finite) as (x, y) approaches (x 0 , y 0 ) along any path in {(x, y) ∈ T 1 × T 2 : x < x 0 , y < y 0 }.
A brief introduction into the two-variable time scales calculus can be found in [6].
Let us consider the Volterra integro-dynamic equation and the corresponding homogeneous equation where A is an n × n matrix function, f is a n-vector function, which are continuous on T 0 := T ∩ [0, ∞), and K is an n × n matrix function, which is continuous on Definition 1.1.The principle matrix solution of (1.2) is the n × n matrix function Z(t, s) defined by where x i (t, s)(i = 1, 2, . . ., n) are the linearly independent solutions of (1.2).The principle matrix Z(t, s) is called the transition matrix if Z(τ, τ) = I.
Therefore, the transition matrix of (1.2) at initial time τ is the unique solution of the matrix initial value problem and x(t) = Z(t, τ)x 0 is the unique solution of system (1.2).The principle matrix is the unique solution of (1.4) Under continuity conditions on A and K, there is a unique solution of the initial value problem (see [17, Theorem 2.2]) (1.5) Both the principle matrix and the resolvent of the linear Volterra integro-dynamic equation are equivalent (see, [17,Theorem 2.7]).Then the unique solution y(t, t 0 , y 0 ) of (1.1) satisfying y(t 0 , t 0 , y 0 ) = y 0 is given by [3,17] y(t, t 0 , In the next section, we investigate the asymptotic property of (1.2) and its perturbation (1.1) by means of the resolvent matrix R(t, s).With results concerning the asymptotic equilibrium we investigate asymptotic equivalence between two linear Volterra systems in Section 3. In the last section, we use a useful equivalent system from [17,Theorem 3.1] to study the asymptotic property of (1.1) and (1.2).

Asymptotic property
In this section we investigate the asymptotic property of the linear Volterra integro-dynamic system (1.1) and (1.2).We need the following integral inequality.
Lemma 2.1.Suppose that u, f ∈ C rd (T, R) are nonnegative functions, and c is a nonnegative constant.Assume that k(t, s) is a nonnegative and rd-continuous function for s, t ∈ T with s ≤ t.Then Proof.The proof is similar to [11,Theorem 3.13].
Let p, v : T 0 → R be nonnegative functions.The Hardy-Littlewood symbols O and o have the usual meaning: z(n) = O(p(t)) means that there exists c > 0 such that z(t) ≤ cp(t) for large t, and z(t) = o(p(t)) means that there exists v(t) such that z(t) ≤ p(t)v(t) and lim t→∞ v(t) = 0. Definition 2.2.A linear Volterra integro-dynamic system (1.2) is said to have asymptotic equilibrium if there exist a unique ζ ∈ R n and r > 0 such that any solution x(t) of (1.2) satisfies and conversely, for every ζ ∈ R n there exists a solution x(t) of (1.2) with x 0 < r such that (2.1) is satisfied.
Our next result give necessary and sufficient conditions for (1.2) to have asymptotic equilibrium via the resolvent matrix R(t, s).Theorem 2.3.System (1.2) has asymptotic equilibrium iff lim t→∞ R(t, t 0 ) exists and is invertible for each t ≥ t 0 ≥ 0.
Conversely, let ζ ∈ R n be any vector.Then there exists a solution x(t, t 0 , x 0 ) of (1.2) with This completes the proof.
Proof.Let x(t) be the solution of (1.2).We can write (1.2) in an equivalent form Let us take u(t) = R(t, t 0 ) and and we have the estimate Using Lemma 2.1, we obtain v(t) ≤ e p (t, t 0 ), where p(s) = A(s) + s t 0 K(s, τ) ∆τ.Thus there exists a constant M > 0 with It is easy to see that u(t) ≤ v(t) for each t ≥ t 0 and v(t) is increasing and bounded.Furthermore, for any t ≥ t 1 ≥ t 0 , we have This implies that, given any ε > 0, we can choose a t 1 > 0 sufficiently large so that Hence R(t, t 0 ) converges to a constant n × n matrix R ∞ (t 0 ) as t → ∞.
Next there exists a constant N > 0 such that R(t, t 0 ) < N for each t > t 0 .Since then for a given t 0 > 0, we obtain Let us take By taking norms, we have the estimate Using (2.4), we obtain lim t→∞ q(t, t 0 ) < 1.
Let us consider the Volterra integro-dynamic equation and the corresponding homogeneous equation Definition 2.9.The two Volterra integro-dynamic systems (2.8) and (2.9) are said to be asymptotically equivalent if, for every solution x(t) of (2.9), there exists a solution y(t) of (2.8) such that x(t) = y(t) + o(1) as t → ∞ (2.10) and conversely, for every solution y(t) of (2.8), there exists a solution x(t) of (2.9) such that the asymptotic relationship (2.10) holds.
Proof.Let x(t) be the solution of (2.9) with the initial value x 0 .Then there exists a solution y(t) of (2.8) with initial condition y(t where p ∞ = lim t→∞ t t 0 R(t, σ(τ)) f (τ)∆τ.Conversely, let y(t) be the solution of (2.8) with the initial value y 0 .Then there exists a solution x(t) of (2.9) with initial condition x(t 0 ) = y This completes the proof.

Asymptotic property via equivalent system
In this section we use a useful equivalent system to study the asymptotic property of ( The solution z(t) of (4.1) with initial condition z(t 0 ) = x 0 is given by where Φ B (t, t 0 ) is a fundamental matrix solution of z ∆ (t) = B(t)z(t).
It follows from P ∞ (t 0 ) < 1 that I + P ∞ (t 0 ) is invertible and Q ∞ is also invertible.Hence (3.2) has an asymptotic equilibrium by Theorem 2.3.In a similar manner we can obtain the converse.Our next result is about asymptotic equivalence between linear systems (3.1) and (3.2).In addition to the assumptions of Theorem 3.1, suppose that (3.1) has an asymptotic equilibrium.Then (3.1) and (3.2) are asymptotically equivalent.