On positiveness of the fundamental solution for a linear autonomous differential equation with distributed delay

We present necessary and sufficient conditions for the nonoscillation of the fundamental solutions to a linear autonomous differential equation with distributed delay. The conditions are proposed in both the analytic and geometric forms.


Introduction
One of the significant features of differential equations with aftereffect (in contrast to ordinary differential equations) is that solutions to linear equations of first order can have alternating sign.This fact assigns a meaning to the problem of obtaining efficient conditions of oscillation and fixed sign of solutions, to the question on the number of zeros, to the estimation of interval of nonoscillation, and so on.In this paper we establish a number of facts equivalent to the positiveness of the fundamental solution for autonomous equations with aftereffect.
Apparently, there is no known criterion of nonoscillation of the fundamental solution for ẋ(t) + ax(t) + t 0 x(t − s) dr(s) = 0 yet.Until now nonautonomous equations have usually been studied.Conditions of nonoscillation for them are only sufficient, and in the case of autonomous equations they are far from sharp conditions.
Corresponding author.Email: TSabatulina@gmail.com 2 T. Sabatulina and V. Malygina Furthermore, authors of many papers (see [4,14]) obtain conditions of the oscillation of the fundamental solution instead of conditions of nonoscillation.Since our purpose is to obtain necessary and sufficient conditions, obtaining conditions of oscillation and that of nonoscillation are in fact the same problem.
Consider a linear autonomous differential equation with bounded distributed delay where a 0 ∈ R, ω > 0, the function r is defined on the segment [0, ω] and does not decrease, r(0) = 0, and the function f is locally summable.Without loss of generality it can be assumed [2, pp.9-10] that x(ξ) = 0 for all ξ < 0. Denote the total variation of the function r by ρ = ω 0 dr(ξ) = r(ω).We say that a solution of equation (2.1) is an absolutely continuous function satisfying equation (2.1) almost everywhere.
It is known [2, p. 84, Theorem 1.1] that for every given x 0 ∈ R there exists a unique solution of equation (2.1) such that x(0) = x 0 .It has the form (2.3) Thus, the fundamental solution is the main subject of our study.It follows from (2.3) that the fundamental solution of a homogeneous equation is positive if and only if all solutions have fixed sign.For a nonhomogeneous equation, if the kernel of the integral operator is positive, then the operator is isotone.This presents the possibility of fine two-way estimates of a solution.

Properties of the characteristic function
According to the statement of the problem and the estimation |X(t)| e (|a 0 |+ρ)t for t ∈ R + (see [3, p. 94, Property 2]) we see that the conditions from [13, pp. 212-213] hold.Hence, the Laplace transform can be applied to the left and right-hand sides of the equation from problem (2.2).
The Laplace image of the fundamental solution X is X 0 (p) = 1 g(p) , where g is the characteristic function, g(p) = p + a 0 + ω 0 e −pξ dr(ξ), p ∈ C. Denote the real part of p by p, and the imaginary part by p. Below we present a number of lemmas on properties of the function g. (i) g is an analytic function; (ii) there exists λ 0 ∈ R such that g has no zeros for p > λ 0 ; (iii) the set of roots of g is finite in every vertical band on the complex plane.
Thus, ω 0 e −pζ dr(ζ) can be represented in the form of a series that converges for all p ∈ C. Therefore g is an analytic function.
(ii) Note that Hence the function g has a finite set of roots in every vertical band.The function g(p) − p is bounded for p = µ.Further, Proof.For sufficiently large y we put p = x + iy, and estimate We have Now, add the function ce pt p to and subtract it from the subintegral function e pt g(p) , put p = x + iy, and estimate the modulus of the expression for sufficiently large |y|.
where z j are zeros of g inside ABCD, and s is the number of zeros.Further, For y → +∞ we have Proof.The characteristic function g has a finite number of zeros p i with the maximal real parts p i = α max .By Lemma 3.1 there exists α 0 < α max such that the function g has a finite number (denote it by s) of zeros p i with p i α 0 .Then where i, i 0 ∈ N, α i , β i ∈ R, A i , B i are polynomials.From Lemma 3.2 we obtain |ε(t)| Ne α 0 t , N ∈ R. From (3.1) we have Assume that the function g has no real zeros.Then Denote the power of the polynomials A i , B i by m 0 .Denote coefficients at t m 0 in the polynomials A i , B i by a i and b i respectively.Then

Clearly, ε(t)
t m 0 e αmaxt , ε ai (t), ε bi (t) → 0 as t → +∞.Hence ε ab (t) → 0 as t → +∞.Without loss of generality we can assume that where k is a multiple of 4: For sufficiently great k we have R 1 . Then for any n ∈ N we get where θ n = 1 β 1 arccos a 1 R 1 + 2πn .Hence the function ψ (k) has a sequence of zeros t n , n ∈ N, such that lim n→+∞ t n = +∞.
Consider ψ (k−1) .From Lagrange's theorem there exist t * n in the intervals [θ n , Hence the function ψ (k−1) has a sequence of zeros t n , n ∈ N, such that lim n→+∞ t n = +∞.Likewise, we obtain that ψ oscillates.Hence y oscillates.Therefore x oscillates.

A theorem on differential inequalities and the positiveness of the fundamental solution
In this section we apply a theorem on differential inequalities (see [3, p. 57 Assume that there exists t 0 such that X(t 0 ) = 0. Then This fact is impossible because v is positive.Hence the fundamental solution of equation (2.1) is positive.Sufficiency.
From here by Lemma 4.1 we obtain that the fundamental solution of equation (2.1) is positive.
The following lemmas provide some properties of the function P. Proof.This follows immediately from the definition of a derivative.1.The fundamental solution of equation (2.1) is positive on the semiaxis [0, +∞).
2. The function P has at least one real zero.
where α, k ∈ R. Clearly, for k 0 the fundamental solution of equation (4.2) is positive on the semiaxis [0, +∞).Let k > 0. Using Theorem 4.6 we obtain that the fundamental solution of equation (4.2) is positive on the semiaxis [0, +∞) if and only if αω q(kω 2 ), where the function u = q(v) is defined by The region of positiveness contains all points below and on the curve on Fig. 4.1.
The product and sums in the expressions for the functions ϕ, ϕ , w converge for all ζ.
Notice that the function w has the following properties: The approximate behavior of the function w is represented on Fig.

Geometric representation
It is known [12, p. 22, Theorem I.14] that the function r is represented as a sum of a jump function, a continuous component, and a singular component.That is We shall obtain the domain of positivity of the fundamental solution X in terms of the parameters (a 0 , b 1 , b 2 , a 1 , a 2 , . ..) for fixed κ, σ, r i , i ∈ N.
Let F be a set of points (u 0 , u 1 , u 2 , . ..) ∈ l 1 such that (5.1) Take A = (a 0 , 0, . ..) ∈ l 1 , M = (a 0 , b 1 , b 2 , a 1 , a 2 , a 3 , . ..) ∈ l 1 .Then parametric equations define the ray AM.Combining (5.2) and (5.1), we get Clearly, the ray AM intersects the set F if and only if the equation We say that the set F is a surface, and the point M is below the surface

Lemma 3 . 1 .
The characteristic function g has the following properties:

Lemma 3 . 2 .
Let g(p) have no zeros for p = µ.Then for all t 0 there holds the estimationµ+i∞ µ−i∞ e pt g(p) dp Ne µt ,where N is a positive real number.Proof.Add the function e pt p to and subtract it from the subintegral function, and estimate the integral

Lemma 3 . 4 .
Consider the function G(p) = 1 g(p) − 1 p .It is easily shown that G(p) is the Laplace image of some function [13, p. 231, Theorem 8.5].Therefore 1 g(p) is the Laplace image of some function as it is the sum of analytic functions.Thus one can apply the inverse Laplace transform for X.For the fundamental solution X of equation (2.1) there exists the estimation |X(t)| Mt n e ζ 0 t for all t 0, where M ∈ R + , ζ 0 is the maximal real part of zeros of the function g, n is the maximal multiplicity of zeros with real part ζ 0 .Proof.Consider a rectangle ABCD in C containing zeros of the function g with the maximal real part.Here the leg AB is to the right of these zeros and lies on the line p = λ > ζ 0 .The leg CD is on the line p = µ < ζ 0 < λ.Real parts of other zeros are less than µ.The legs AD and BC are on the lines p = −y < 0 and p = y > 0 respectively, where y is a positive real number.The integral ABCD e pt g(p) dp is represented as the sum of the integrals Using Lemma 3.3 for I 2 and I 4 we obtain lim y→+∞ |I 2 | = 0 and lim y→+∞ |I 4 | = 0.For I 3 Lemma 3.2 holds.In view of [7, p. 79] we get

Lemma 4 . 3 .
The fundamental solution of equation (2.1) is positive on the semiaxis [0, +∞) if and only if the function P has at least one real zero.Proof.Necessity.The converse contradicts Lemma 3.5.

Lemma 4 . 5 .Theorem 4 . 6 .
The function P has at least one real zero if and only if it follows from P (ζ * ) = 0, where ζ * ∈ R, that P(ζ * ) 0. Proof.Note that P (ζ) > 0 for all ζ ∈ R. Hence the function P increases on the whole axis.Moreover, lim ζ→−∞ P (ζ) = −1 and lim ζ→+∞ P (ζ) = +∞.Therefore the function P has a unique minimum point ζ * .If P(ζ * ) > 0 then the function P has no zeros.If P(ζ * ) = 0 then ζ * is the unique zero of P. If P(ζ * ) < 0 then the function P has exactly two zeros (to the right and to the left from ζ * ).By Lemma 4.3 and Lemma 4.5, we obtain the following theorem.The following conditions are equivalent.
e z j t Ne µt .Mt n e ζ 0 t .If all zeros of the function g with the maximal real part are not real then the fundamental solution of equation (2.1) oscillates.
, Lemma 2.4.3],[1,p.356, Theorem 15.6]) to obtain a criterion of the positiveness of the fundamental solution.Below we formulate this theorem in the form suitable for us, and prove it.