On the Asymptotic Behavior of the Pantograph Equations

where a(t) is a nonnegative continuous scalar function on R+ := [0;1) and 0 0 and ’(t) is a given continuous function on [pt0;t0] then the solution x(t) with x(s) = ’(s) for s 2 [pt0;t0] is dened for t ! 1 and it diers from any constant solution if ’ is not constant. Equation (1.1) can be transformed to the equation


Introduction
Our aim is studing the asymptotic behaviour of the solutions of the equation ẋ(t) = −a(t)x(t) + a(t)x(pt) , where a(t) is a nonnegative continuous scalar function on R + := [0 , ∞) and 0 < p < 1 is a constant.This equation is a special case of the so called pantograph equations arising in industrial applications [5,11].The only solution of equation (1.1) with initial data x(0) = x 0 is x(t) ≡ x 0 .However, if t 0 > 0 and ϕ(t) is a given continuous function on [pt 0 , t 0 ] then the solution x(t) with x(s) = ϕ(s) for s ∈ [pt 0 , t 0 ] is defined for t → ∞ and it differs from any constant solution if ϕ is not constant.

Supported by the
T. Krisztin [9] investigated the equation with infinite delay.The application of his result for (1.1) gives that if t pt a(s) ds is bounded on R + then all solution of (1.1) tends to a constant as t → ∞. [3,4] studied linear scalar equation

N. G. De Bruijn
From his results it can be proved that if a(u) du} = ∞ then every solution of (1.1) has a finite limit if t → ∞.On the other hand if a(t) is twice continuously differentiable and there exists a continuous nonincreasing positive function Φ such that ∞ 1 Φ(s) ds < ∞ and for w(t) := 1/a(t) the conditions w(t), |w (t)|, |w (t)| < e t Φ(t) hold, then there exists a continuous periodic function ψ of period 1 and a positive constant c such that The scalar equation ẋ(t) = −ax(t) + bx(pt) , (where a and b are constants, a > 0) is also studied.The exact asymptotic behaviour of the solutions as t → ∞ is known [1,2,8].In the special case a = b the following assertion is proved.For any solution x(t) there exists an infinitely many times differentiable, periodic function ψ of period 1 such that We give an extension of the last results for (1.1).We need some light monotonicity like conditions for a(t) that restrict too fast changes of a(t).Our condition works for and give conditions such that (or a similar) estimate is true.Estimation of such type was given by J. Kato [6,7] and his results were sharpened by T. Krisztin [10].However these results and our ones in this paper cannot be compared and methods are different, too.

An asymptotic estimate of the solutions
Let us consider the equation where a(t), b(t), p(t) are continuous functions on R + , p(t) ≤ t and lim t→∞ p(t) = ∞.
Let us define the function m(t) := inf{s : p(s) > t} on R + .Then t ≤ m(t), p(m(t)) = t and m(t) is increasing.Let be given t 0 ≥ 0 such that p(t 0 ) < t 0 and introduce the qualities and the intervals Moreover we also have p(I n+1 ) ⊂ ∪ n k=0 I k .For a given function ρ : R + → (0, ∞) having bounded differential on finite intervals let us introduce the numbers and a(t) is nonnegative, we get for t ∈ I n that 0 ≤ρ(t) ds . (2.3)

4)
and Proof.Introduce the function Then y(t) satisfies the equation which is equivalent to .
EJQTDE, 1998 No. 2, p. 5 where Proof.First of all we remark that the product in the definition of C exists since So, Theorem 1 implies the assertion.

The asymptotic behavior
Consider the equation where 0 < p < 1.
Let c(t) be nonnegative, continuously differentiable on R + .Then the solutions are twice differentiable and y(t) = ẋ(t) satisfies the equation Now, we apply the above results to this equation.
Theorem 3. Suppose that c(t) is continuously differentiable on R + and there exist where Theorem 4. Suppose that there exist Let x(t) be a solution of (3.1) on [pt 0 , ∞) then where C is the same as in Theorem 3 and Proof.Use Corollaries 1 and 3 as in Theorem 3.
Note that it is only a technical detail that we estimate the derivative on the interval [t 0 /p, ∞) in Theorems 3 and 4. If we choose an initial function so that the solution is continuously differentiable at the point t 0 , then we can prove a similar estimate using M as the supremum of the appropriate function on the interval [pt 0 , t 0 ) and a little bit different C's.We will use this comment later.
Note also, that it is easy to see that if x is a solution of (3.1), M 0 = sup t∈[pt 0 ,t 0 ) x(t) Definition.We say that the function x(t) is asymptotically logarithmically periodic, if x(e t ) is asymptotically periodic, i.e. there is a periodic function φ(t) such that |x(e t ) − φ(t)|→0 as t→∞.
Theorem 5. Suppose that all the conditions of Theorem 3 are satisfied and k > α.
Then all the solutions of equation (3.1) are asymptotically logarithmically periodic.
Proof.Let φ : [pt 0 , t 0 ]→R be given and consider x(t) = x(t, t 0 , φ), the solution of (3.1) starting at t 0 with the initial function φ.To simplify our notation let us assume that t 0 = 1, for other t 0 's the proof is similar.Let M and C be the constants appearing in Theorem 3 and hence we have | ẋ(t)| ≤ CM/t k on the interval [1/p, ∞).
Let us transform the equation by replacing t = e s and x(t) = y(ln(t)) = y(s).
where h = − ln(p) (here we use that t 0 = 1 and hence the solution y corresponding to x starts from ln(t 0 ) = 0).We also have ẋ(t) = ẏ(ln(t))/t for t ≥ 1/p and hence Hungarian National Foundation for Scientific Research with grant numbers T/016367 and F/016226, and by the Foundation of the Hungarian Higher Education and Research.EJQTDE, 1998 No. 2, p. 1

|Theorem 6 . 10 [ 1 ,
ẏ(s)| ≤ CM e s(1−k) for s ≥ h.Then we use the equation to have |y(s) − y(s − h)| ≤ CM e −sk c(e s ) ∀s ≥ h EJQTDE, 1998 No. 2, p. 9 Using that k > α it is easy to prove that the sequence e −(s+ih)k c(e s+ih ) ≤ e −(s+ih)k e (s+ih)α /m ≤ e −ih(k−α) /m (s ∈ [0, h]) is summable.Therefore |y(s + lh) − y(s + nh)| ≤ CM l i=n+1 e −(s+ih)k c(e s+ih ) ∀s ∈ [0, h] and l ≥ n ≥ 0 (3.3) and hence the function sequence z n (s) := y nh (s) = y(s + nh) (for s ∈ [−h, 0]) is a Cauchy-sequence in the supremum norm.Thus it converges to a function χ.Consider χ(s) to be an h-periodic function, and then we have |y(s)−χ(s)|→0 as s→∞.Therefore all solutions of (3.2) are asymptotically h-periodic, which means that all solutions of (3.1) are asymptotically logarithmically periodic.Suppose that all the conditions of Theorem 4 are satisfied.Then all the solutions of equation (3.1) are asymptotically logarithmically periodic.Proof.The proof is very similar to that of Theorem 5.The only difference is that after the transformation we have |y(s) − y(s − h)| ≤ CM e −sk ∀s ≥ 0 since the c(t) in the estimate on ẋ(t) and the c(e s ) coming from (3.2) cancel each other.The rest of the proof is the same.In this section we established conditions under which we can prove asymptotical logarithmical periodic behavior of the solutions of equation (3.1).Both the conditions of Theorem 3 and 4 are reasonable, they require c(t) not to be too small or decrease too fast.Clearly, all constant functions are solutions of equation (3.1), and asymptotic logarithmic periodicity includes the special case of the solutions being asymptotically constant.We now show by an example that there is an equation of the form (3.1) which has an asymptotically non-constant solution.Let c(t) = 1, t 0 = 1, k = 1, m = 1, α = 0 in Theorem 6.Let φ : [p, 1]→R be given (it will be specified later, but it satisfies the condition that the solution is continuously differentiable at t 0 and hence we have an estimate on the derivative on the interval EJQTDE, 1998 No. 2, p. ∞)).We do the same transformation as we did in the proof of Theorem 6.Then we have |y(s) − y(s − h)| ≤ CM e −s ∀s ≥ 0. By induction we get |y(s + lh) − y(s − h)| ≤ CM e −s 1 − e −h .Let ψ(s) := φ(e s ) = φ(t) for t ∈ [p, 1].Define s max and s min so that ψ(s max ) is a maximum and ψ(s min ) is a minimum of ψ in the interval [−h, 0].The above inequality gives (as a special case) that y(s max + lh) ≥ y(s max ) − CM e −(s max +h) 1 − e −h and y(s min + lh) ≤ y(s min ) + CM e −(s min +h) 1 − e −h .Putting these together we obtain y(s max + lh) − y(s min + lh) ≥ (y(s max ) − y(s min )) − CM e −(s max +h) + e −(s min +h) 1 − e −h Now we define φ a little more precisely.Let φ be strictly increasing on the interval [p, (1 + p)/2] and decreasing on [(1 + p)/2, 1], hence s min = 0, s max = ln((1 + p)/2) and ψ(s max ) − ψ(s min ) > 0. Then we have e −s max + e −s min 1 − e −h = 3 + p 1 − p 2 and hence y(s max + lh) − y(s min + lh) ≥ ψ(s max ) − ψ(s min ) − CM p(3 + p) 1 − p 2 ≥ γ > 0 if we choose p small enough.This shows that y at the shifts of s max and s min differs by a fixed positive constant and hence y cannot tend to a constant.Since x(t) = y(ln(t)), we also proved that x does not tend to a constant.EJQTDE, 1998 No. 2, p. 11 = t α if α > −1, or for the function a(t) = a + sin bt, if the constants a, b satisfies |b| < (a − b) 2 .We show by an example that ψ may be non-constant function.In the proof of the results we need to know the decay rate of solutions.This argument works for more general equation.Therefore in the second part of the article Let m n := max t∈I n |y(t)| n = 0, 1, 2, 3, . . .and M n := max{m 0 , m 1 , . . ., m n }.Since |y(p(t))| ≤ M n for t ∈ I n+1 , we have 1 by the definition of k.The relations p(m(t)) = t and p 1 t ≤ p(t) ≤ p 2 t imply that