The limits of solutions of a linear delay integral equation

In this paper we classify the limits of solutions of a linear integral equation with finite delay. In particular, if the solution tends to a point or a periodic orbit, we establish the explicit expressions depending on given initial functions by using analysis of characteristic roots and the formal adjoint theory. Our results also present a necessary and sufficient condition for the exponential stability of the equation.


1.
Introduction. In the last half century, qualitative theories of Volterra integral equations and Volterra integro-differential equations have undergone developments rapidly. It has been strongly promoted by many applications that these theories have found in physics, engineering, and biology. For the general background of Volterra integral equations, one can refer to some books [2,3,5].
We consider a linear integral equation with finite delay x(s)ds, t ≥ 0 (1.1) with the initial condition x(ξ) = ϕ(ξ), −h ≤ ξ ≤ 0, (1.2) where A is a real m × m constant matrix, h is a positive constant, and the initial function ϕ belonging to L 1 ([−h, 0], R m ) is given arbitrarily. Recently, integral equations with finite or infinite delay are studied by several authors; one can refer to [1,4,6,7,8,9] and the references therein. The purpose of this paper is to completely classify the limits of solutions of (1.1) in terms of eigenvalues of A and the delay parameter h, and to present asymptotic formulae of the solutions. More precisely, if the solution of (1.1) tends to a point or a periodic orbit, we establish the explicit expressions depending on the initial function ϕ. Our research is motivated by the following stability result (Theorem A) and asymptotic formula of solutions (Theorem B) for the equation of convolution type where the kernel G is a measurable real m × m matrix valued function on [0, h].
Throughout this paper we will use the Euclidean norm for vectors denoted by |·| and the induced norm for matrices denoted by · .
Then the zero solution of (1.3) is exponentially stable, that is, every solution of (1.3) converges to 0 exponentially as t → ∞. where γ = h 0 tG(t)dt. Judging from these theorems above, we have planned the classification of the limits of solutions of (1.1). By the transformation x(t) = P y(t) with a nonsingular matrix P , Eq. (1.1) can be written as Consequently, we may assume without loss of generality that the matrix A is the Jordan canonical form. Moreover, throughout this paper, we treat the essential case where A is the 2 × 2 matrix given in the following three cases: where a 1 , a 2 , a and θ are real numbers and 0 < |θ| ≤ π/2. In fact, one can easily extend our results to higher dimensional cases.
Our main results are stated as follows.
HereΦ 3 (t) and c 3 = c 3 (ϕ) are given bỹ By virtue of Theorems 1.1, 1.2 and 1.3, one can immediately obtain the following stability criterion for (1.1) (Corollary 1.1) and asymptotic formula of solutions of (1.1) with m = 1 (Corollary 1.2), which may be considered as some improvements of Theorems A and B for (1.3), respectively, in the case where G(t) is constant. Corollary 1.1. Let a j e iθj (j = 1, 2, . . . , m) be the eigenvalues of the real m × m matrix A, where a j and θ j (possibly 0) are real numbers and |θ j | ≤ π/2. Then the zero solution of (1.1) is exponentially stable if and only if Corollary 1.2. Suppose that m = 1, and A = a is a real number. Let x(t; ϕ) be a solution of (1.1) with (1.2) where ϕ ∈ L 1 ([−h, 0], R). Then the following statements hold: (i) If ah < 1, then x(t; ϕ) converges to 0 exponentially as t → ∞. Here c = c(ϕ) is given by Note that, to the best of our knowledge, these corollaries above are fundamental but unknown results for linear integral equations with finite delay. This paper is outlined as follows. In section 2, we summarize preparatory results on linear integral equations to establish an explicit asymptotic formula of solutions of (1.1). In section 3, we study the distributions of the characteristic roots of (1.1) in detail, which is an important role in our research. Finally in section 4, we give proofs of main results.

2.
Preliminaries. In this section we will introduce the decomposition theory and the formal adjoint theory for linear autonomous integral equations with delay developed in [7,8].
Let ρ be a fixed positive constant and let R − be the set of nonpositive real numbers. Denote by X the function space defined by Let us consider the linear integral equation with infinite delay where the kernel K is a measurable m × m matrix valued function on [0, ∞) with complex components satisfying the conditions ∞ 0 K(t) e ρt dt < ∞ and ess sup{ K(t) e ρt | t ≥ 0} < ∞.
Eq. (2.1) can be formulated as an autonomous equation on X of the form where L is a bounded linear operator defined by L(ϕ) = 0 −∞ K(−ξ)ϕ(ξ)dξ for ϕ ∈ X. For any ϕ ∈ X, a function x : (−∞, b) → C m , b > 0, is called a solution of (2.2) (or (2.1)) with the initial function ϕ if x satisfies the following conditions: , and x(t) = L(x t ) for t ∈ (0, b). It is known by [7, Proposition 3] that there exists a unique global solution x : R → C m of (2.2) such that x 0 = ϕ on R − , which is called the solution of (2.2) through (0, ϕ), and denoted by x(t; ϕ).
For any t ≥ 0 and ϕ ∈ X, the solution operator T (t) : X → X is defined by the relation T (t)ϕ = x t (·; ϕ). Since the family {T (t)} t≥0 is a strongly continuous semigroup of linear operators on X, the generator A 0 of {T (t)} t≥0 is characterized as follows: Denote by σ(A 0 ), P σ (A 0 ) and ess(A 0 ) the spectrum, the point spectrum and the essential spectrum of A 0 , respectively. Let C −ρ = {z ∈ C | Re z > −ρ}, and introduce the characteristic equation for (2.1) defined by where E m is the m × m unit matrix. We call λ ∈ C −ρ such that det ∆(λ) = 0 a characteristic root of (2.1). Then the spectra of A 0 are characterized as Then Σ cu ∩ess(A 0 ) = ∅ and Σ cu is a finite set, and hence X is decomposed as a direct sum where X cu and X s are closed subspaces satisfying the following properties: Denote by Π cu the projection from X onto X cu . Then we have the following result on the asymptotic behavior of solutions of (2.1).
The following statements hold: Let λ 0 be a characteristic root of (2.1), and let M λ0 In particular, X is decomposed as Next we introduce the formal adjoint theory for (2.1) to give the explicit form of x(t; Π cu ϕ). Let R + be the set of nonnegative real numbers and let C m * be the space of all m-dimensional row vectors. Consider the function space X defined by We call the operator A 0 the formal adjoint operator of A 0 . Furthermore, let us consider the bilinear form ·, · on X × X defined by (2.4) Then for ϕ ∈ D(A 0 ) and ψ ∈ D(A 0 ), the dual relation ψ, A 0 ϕ = A 0 ψ, ϕ is satisfied. For λ ∈ C −ρ and k ∈ N, the function ϕ belongs to R((A 0 − λI) k ) if and only if ψ, ϕ = 0 for all ψ ∈ N ((A 0 − λI) k ).
For λ ∈ C −ρ and k ∈ N, we introduce functions w k (λ) : are characterized as follows.

Proposition 2.4. ([8, Theorems 3.1 and 3.2])
Let Φ and Ψ be a basis vector for X cu and N , respectively. LetΦ be the canonical prolongation of Φ defined on R.
Then the matrix Ψ, Φ is nonsingular, and the projection Π cu is given by Moreover, the solution x * (t) of (2.1) through (0, Π cu ϕ) is expressed as In particular, for any ϕ ∈ X, the solution x(t; ϕ) of (2.1) satisfies the relation 3. Analysis of characteristic roots. In this section we will investigate the characteristic roots of (1.1). Since Eq. (1.1) can be regarded as Eq. (2.1) with where O m is the m×m zero matrix, the characteristic equation for (1.1) with m = 2 becomes Then we obtain the following results on the distributions of the roots of (3.1).   (i) If (|θ| − π)/ sin |θ| < ah < θ/ sin θ, then all the roots of (3.1) have negative real parts. (ii) If ah = θ/ sin θ, then (3.1) has a pair of simple roots ±iω 0 and the remaining roots have negative real parts. (iii) If ah = (|θ| − π)/ sin |θ|, then (3.1) has a pair of simple roots ±iω −1 and the remaining roots have negative real parts. (iv) If ah > θ/ sin θ or ah < (|θ| − π)/ sin |θ|, there exists a root of (3.1) with a positive real part.
First, we will treat the roots of (3.1) when the matrix A has real eigenvalues. For simplicity, let If the matrix A is of the form (I), we have Also if the matrix A is of the form (II), we get Hence, to prove Theorems 3.1 and 3.2, it suffices to verify the following proposition. To prove Proposition 3.1, we will prepare some lemmas. Note that since f (z; a) is an analytic function of z and a, one can regard the root z = z(a) of f (z; a) = 0 as a continuous function of a.   namely, sin ωh = ω/a and cos ωh = 1. Therefore we must have ω/a = 0, which contradicts ω = 0. wherez denotes the complex conjugate of any complex z. Note that if λ is a root of det ∆(z) = 0, thenλ is also a root of det ∆(z) = 0. Indeed, g(λ; a) = 0 implies g(λ; a) = g(λ; a) = 0. Consequently, to prove Theorem 3.3, it suffices to verify the following proposition.
Remark 3.1. The definition of a n leads to the relation .
Proof of Proposition 3.2. Let a = 0. There are four cases to consider.
(i) a −1 < a < a 0 . Lemma 3.5 shows that if |a| < 1/h, there exist no roots of g(z; a) = 0 which have positive real parts. Also, Lemma 3.6 shows that if a = a n for n ∈ Z, there exist no purely imaginary roots of g(z; a) = 0. By virtue of relation (3.2) and the continuity of the roots with respect to a, we conclude that if 0 < a < a 0 or a −1 < a < 0, all the roots of g(z; a) = 0 have negative real parts.
(iv) a > a 0 or a < a −1 . Let λ 1 (a) and λ 2 (a) be the branches of the roots of g(z; a) = 0 satisfying λ 1 (a 0 ) = iω 0 and λ 2 (a −1 ) = iω −1 , respectively. Lemma 3.7 shows Re λ 1 (a) > 0 for all sufficiently small a − a 0 > 0 and Re λ 2 (a) > 0 for all sufficiently small a −1 − a > 0. Since λ 1 (a) cannot cross the imaginary axis from right to left as a increases from a 0 by Lemma 3.7, one can obtain Re λ 1 (a) > 0 for all a > a 0 . Similarly, since λ 2 (a) cannot cross the imaginary axis from right to left as a decreases from a −1 by Lemma 3.7, one can obtain Re λ 2 (a) > 0 for all a < a −1 .
(ii) a 1 h = 1 and a 2 h < 1. Theorem 3.1 (ii) shows Σ cu = {0} and the root 0 is simple, which yield that the ascent of 0 is equal to 1 by Proposition 2.2. Thus we get X cu = N (A 0 ). Notice that This, together with (4.1), yields that LetΦ be the canonical prolongation of Φ, that is,Φ(t) = col 1 0 for t ∈ R. By applying Proposition 2.4, we therefore obtaiñ 1 , 0) exponentially as t → ∞. (iii) a 1 h < 1 and a 2 h = 1. The proof can be carried out by almost the same argument as in the proof of Theorem 1.1 (ii); so we omit the proof.
(iv) a 1 h = a 2 h = 1. Theorem 3.1 (iii) shows Σ cu = {0} and the root 0 is double, which yields that the ascent of 0 is equal to 1 or 2 by Proposition 2.2. Hence we get X cu = N (A 0 ) or X cu = N (A 2 0 ). Also, by Proposition 2.2, we know that From Proposition 2.3 (i), we find that which lead to X cu = N (A 0 ). Similarly, from Proposition 2.3 (ii), we have N = N (A 0 ) = span 1 0 , 0 1 . Therefore basis vectors Φ for X cu and Ψ for N are expressed as This, together with (4.1), yields that LetΦ be the canonical prolongation of Φ, that is,Φ(t) = E 2 for t ∈ R. By applying Proposition 2.4, we therefore obtaiñ and x(t; ϕ) converges to c 1 exponentially as t → ∞.
(v) a 1 h > 1 or a 2 h > 1. Theorem 3.1 (iv) shows there exists a root of (3.1) with a positive real part, which implies that there exists an unbounded solution of (1.1) with an initial function in X cu .
(ii) ah ≥ 1. If ah > 1, Theorem 3.2 (iii) shows there exists a root of (3.1) with a positive real part, which implies that there exists an unbounded solution of (1.1) with an initial function in X cu .