Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 2 1-6; http://www.math.u-szeged.hu/ejqtde/

UNIQUENESS FOR RETARDED DELAY DIFFERENTIAL EQUATIONS WITHOUT LIPSCHITZ CONDITION

Mária Bartha and József Terjéki

Bolyai Institute, University of Szeged Aradi vértanúk tere 1, H-6720 Szeged, Hungary E-mails: bartham@math.u-szeged.hu, terjeki@math.u-szeged.hu

Abstract

Consider the equation ẋ(t) = f(t, x(t), x(t − r(t))) with the initial condition x_{0} = ϕ. Here f is a continuous real function, but it does not satisfy other regularity conditions. We prove that the initial value problem has a unique solution under the following monotonicity conditions:

(x − y)f(t, x, y) ≤ 0 for all t, x, y ∈ IR,

f(t, x_{1}, y) ≥ f(t, x_{2}, y) for all t, y ∈ IR, and x_{1} < x_{2}, and

if there is t_{0} ≥ 0 such that r(t_{0}) = 0, then the function t_{0} − t + r(t) does not change sign on an interval [t_{0}, t_{0} + δ). We show an example that the result cannot be applied in the state de-

pendent case.

1. Introduction

It is well-known that the stability of the solution of the delay differential equation

ẋ(t) = G(t, x_{t})

through a continuous function ϕ implies the uniqueness of this solution. Consider the retarded differential equation

(0)  ẋ(t) = −g(x(t)) + g(x(t − r(t))),

where g and r are continuous real functions, r(t) ≥ 0, and g is monotone increasing. According to a result of Razumikhin [4] the constant solution of Eq. (0) is stable, therefore uniqueness holds for this solution. For some interesting uniqueness results we refere the interested reader to [2], [3].

Our aim is to show uniqueness for every solution of Eq. (0) provided that r(t) satisfies the following condition: if there is t_{0} ≥ 0 so that r(t_{0}) = 0, then there exists δ = δ(t_{0}) > 0 such that the function t ↦ t_{0} − t + r(t) does not change sign on the interval [t_{0}, t_{0} + δ). Note that the above assumption is

This paper is in final form and no version of it is submitted for publication elsewhere. common for several equations which arise in applications and it is satisfied, for example, when r(t) > 0 or the function t − r(t) is monotone increasing.

We prove our uniqueness result for an equation more general than Eq.(0), that is

ẋ(t) = f(t, x(t), x(t − r(t)))

under certain monotonicity assumptions on f. Here f is continuous, but it does not satisfy other regularity conditions.

Our result cannot be applied when r depends on x(t). We show that by an example.

2. Uniqueness result

Consider the initial value problem (IVP)

(1)  ẋ(t) = f(t, x(t), x(t − r(t))), x_{0} = ϕ,

where f : IR^{3} → IR, r : IR → [0, ∞) and ϕ : (−∞, 0] → IR are continuous. The solution segment x_{t} : (−∞, 0] → IR is given by x_{t}(s) = x(t + s), s ≤ 0.

Theorem 2.1. Assume that

(i) (x − y)f(t, x, y) ≤ 0 for all x, y ∈ IR, t ≥ 0, (ii) f(t, x_{1}, y) ≥ f(t, x_{2}, y) for all y ∈ IR, x_{1}, x_{2} ∈ IR, x_{1} < x_{2}, and t ≥ 0, (iii) if there is t_{0} ≥ 0 so that r(t_{0}) = 0, then there exists δ = δ(t_{0}) > 0 such

that t_{0} − t + r(t) ≤ 0 or t_{0} − t + r(t) ≥ 0 for all t ∈ [t_{0}, t_{0} + δ). Then IVP (1) has a unique solution.

Proof. Suppose by way of contradiction that there are two solutions x_{1}(t) and x_{2}(t) of IVP (1) on an interval [0, A), A ∈ IR such that x_{1}(t) = x_{2}(t) = ϕ(t) for all t ≤ 0, and there is \overline{t} > 0 such that x_{1}(\overline{t}) ≠ x_{2}(\overline{t}).

Set H = {s ∈ (0, A) : x_{1}(s) ≠ x_{2}(s)} and t_{0} = inf H. Since t_{0} ∉ H, it follows x_{1}(t) = x_{2}(t) for all t ≤ t_{0}.

Let r(t_{0}) > 0 or r(t_{0}) = 0 with t_{0} − t + r(t) ≥ 0 for all t ∈ [t_{0}, t_{0} + δ), where δ is defined in assumption (iii). In both cases t − r(t) ≤ t_{0} for all t ∈ [t_{0}, t_{0} + δ_{1}) with some δ_{1} ∈ (0, δ].

The definition of t_{0} implies that there is a sequence (t_{n}) in H so that t_{n} > t_{0}, t_{n} → t_{0} and x_{1}(t_{n}) ≠ x_{2}(t_{n}) for all n ∈ IN. We can assume without loss of generality that x_{1}(t_{n}) < x_{2}(t_{n}) for all n ∈ IN.

Define the functions

z(t) = x_{2}(t) − x_{1}(t) and u(t) = max_{t_{0}≤s≤t} z(s) for all t ∈ [t_{0}, A).

Clearly, we have u(t_{0}) = 0, u(t) is monotone increasing on [t_{0}, A) and u(t) > 0 for all (t_{0}, A). Further, define the function

D^{+}u(t) = lim sup_{h→0+} \frac{u(t + h) − u(t)}{h} for all t ∈ (t_{0}, A).

According to Theorem 2.3 (Appendix) [5] there is τ ∈ (t_{0}, t_{0} + δ_{1}) such that D^{+}u(τ) > 0. The definition of u(t) gives z(τ) ≤ u(τ). We claim that z(τ) = u(τ). Obviously, if z(τ) < u(τ), then u(t) is constant in a neighbourhood of τ, therefore D^{+}u(τ) = 0, and this is a contradiction. Consequently, z(τ) = u(τ).

Next we will show that ż(τ) > 0 and ż(τ) ≤ 0 at the same time, and this will prove the result in the studied case.

Since D^{+}u(τ) > 0, there is a constant K > 0 such that K < D^{+}u(τ), and there exists a sequence (h_{n}), h_{n} > 0, h_{n} → 0 so that

0 < K < \frac{u(τ + h_{n}) − u(τ)}{h_{n}}, n ∈ IN.

It is easy to see, that there is a sequence (\overline{h}_{n}), 0 < \overline{h}_{n} ≤ h_{n} such that u(τ + h_{n}) = z(τ + \overline{h}_{n}). Indeed, the definition of u(t) yields u(τ + h_{n}) = max(max_{t_{0}≤s≤τ} z(s), max_{τ≤s≤τ+h_{n}} z(s)) = max(u(τ), max_{τ≤s≤τ+h_{n}} z(s)). As u(τ) = z(τ), we infer u(τ + h_{n}) = max_{τ≤s≤τ+h_{n}} z(s). Th continuity of z(s) on [τ, τ + h_{n}] gives max_{τ≤s≤τ+h_{n}} z(s) = z(τ + \overline{h}_{n}), where 0 < \overline{h}_{n} ≤ h_{n}. Thus, u(τ + h_{n}) = z(τ + \overline{h}_{n}). These facts lead to the following estimations:

0 < K < \frac{u(τ + h_{n}) − u(τ)}{h_{n}} ≤ \frac{z(τ + \overline{h}_{n}) − z(τ)}{\overline{h}_{n}}.

Letting \overline{h}_{n} → 0, we conclude ż(τ) > 0. Now, we show that ż(τ) ≤ 0. Clearly, ż(τ) = ẋ_{2}(τ) − ẋ_{1}(τ). Being x_{1}(t) and x_{2}(t) solutions of IVP (1), we obtain ż(τ) = f(τ, x_{2}(τ), x_{2}(τ −r(τ)))−f(τ, x_{1}(τ), x_{1}(τ −r(τ))). Since τ −r(τ) ≤ t_{0} and x_{1}(t) = x_{2}(t) for t ≤ t_{0}, we infer x_{1}(τ − r(τ)) = x_{2}(τ − r(τ)). As u(τ) = z(τ) = x_{2}(τ) − x_{1}(τ) and u(τ) > 0, we find x_{1}(τ) < x_{2}(τ). Finally, assumption (ii) implies f(τ, x_{1}(τ), x_{1}(τ − r(τ))) ≥ f(τ, x_{2}(τ), x_{2}(τ − r(τ))), that is ż(τ) ≤ 0.

It remains to consider case r(t_{0}) = 0 and t_{0} − t + r(t) ≤ 0 for [t_{0}, t_{0} + δ). The definition of t_{0} implies the existence of a sequence (t_{n}) in H so that

t_{n} > t_{0}, t_{n} → t_{0} and x_{1}(t_{n}) ≠ x_{2}(t_{n}) for all n ∈ IN. We have x_{1}(t_{n}) ≠ x_{1}(t_{0}) or x_{2}(t_{n}) ≠ x_{2}(t_{0}) for all n ∈ IN. We can assume without loss of generality that x_{2}(t_{n}) ≠ x_{2}(t_{0}) for all n ∈ IN. Define the functions

u(t) = max_{t_{0}≤s≤t} |x_{2}(s) − x_{2}(t_{0})| and

D^{+}u(t) = lim sup_{h→0+} \frac{u(t + h) − u(t)}{h} for all t ∈ (t_{0}, A).

Obviously, u(t_{0}) = 0, u(t) is monotone increasing on [t_{0}, A) and u(t) > 0 for all (t_{0}, A). According to Theorem 2.3 (Appendix)[5] there is τ ∈ (t_{0}, t_{0} + δ) such that D^{+}u(τ) > 0. Arguing similarly as in the previous case, we obtain u(τ) = |x_{2}(τ) − x_{2}(t_{0})|. u(τ) > 0 yields x_{2}(τ) − x_{2}(t_{0}) ≠ 0.

Suppose x_{2}(τ) − x_{2}(t_{0}) > 0. We can choose δ > 0 in assumption (iii) so that x_{2}(s) − x_{2}(t_{0}) > 0 for all s ∈ (τ − δ, τ + δ). We will show that ẋ_{2}(τ) > 0 and ẋ_{2}(τ) ≤ 0 at the same time, and this contradiction will prove the result when x_{2}(τ)−x_{2}(t_{0}) > 0. Since D^{+}u(τ) > 0, it follows that there is a constant K > 0 such that K < D^{+}u(τ), and there is a sequence (h_{n}), h_{n} > 0, h_{n} → 0 so that

0 < K < \frac{u(τ + h_{n}) − u(τ)}{h_{n}}, n ∈ IN.

It is easy to see, using the definition of u(t) and the continuity of x_{2}(s)−x_{2}(t_{0}) on [τ, τ + h_{n}], that there is a sequence (\overline{h}_{n}), 0 < \overline{h}_{n} ≤ h_{n} such that u(τ + h_{n}) = x_{2}(τ + \overline{h}_{n}) − x_{2}(t_{0}). These facts lead to the following estimations:

0 < K < \frac{u(τ + h_{n}) − u(τ)}{h_{n}} ≤ \frac{x_{2}(τ + \overline{h}_{n}) − x_{2}(τ)}{\overline{h}_{n}}.

Letting \overline{h}_{n} → 0, we conclude ẋ_{2}(τ) > 0. Now, we prove ẋ_{2}(τ) ≤ 0. Since t_{0} ≤ τ − r(τ) ≤ τ, the monotone increasing property of u implies u(τ − r(τ)) ≤ u(τ). Therefore |x_{2}(τ − r(τ)) − x_{2}(t_{0})| ≤ x_{2}(τ) − x_{2}(t_{0}). Hence x_{2}(τ − r(τ)) ≤ x_{2}(τ). By assumption (i) we get f(τ, x_{2}(τ), x_{2}(τ − r(τ))) ≤ 0, that is ẋ_{2}(τ) ≤ 0.

If x_{2}(τ) − x_{2}(t_{0}) < 0, arguing similarly as above, we show that ẋ_{2}(τ) < 0 and ẋ_{2}(τ) ≥ 0 at the same time using assumption (i). The proof of Theorem 2.1 is complete.

Remark. In case r(t_{0}) = 0 and t_{0} − t + r(t) ≤ 0 for all t ≥ t_{0}, the unique solution of IVP (1) is the constant solution x(t) = x(t_{0}) for all t ≥ t_{0}.

Note that modifying slightly assumption (iii) of Theorem 2.1 and assuming condition (ii) of Theorem 2.1, we obtain the following result.

Theorem 2.2. Suppose that

a) f(t, x_{1}, y) ≥ f(t, x_{2}, y) for all y ∈ IR, x_{1}, x_{2} ∈ IR, x_{1} < x_{2}, and t ≥ 0,

b) for all t_{0} ≥ 0 there is δ = δ(t_{0}) > 0 such that t_{0} − t + r(t) ≥ 0 for all t ∈ [t_{0}, t_{0} + δ),

then IVP (1) has a unique solution.

We mention that Theorem 2.2 is a generalization of Ding’s result [1] for scalar equations.

3. Example

Consider the functions

f(x, y) = − ^{3} \sqrt{x} + ^{3} \sqrt{y}, (x, y) ∈ IR^{2}, and r(u) = N|u|^{\frac{1}{α}} + r_{0}, u ∈ IR,

where r_{0}, N and α are positive constants. Clearly, assumptions (i), (ii) and (iii) of Theorem 2.1 are satisfied.

Consider the IVP

(2)  ẋ(t) = − ^{3} \sqrt{x(t)} + ^{3} \sqrt{x(t − r(x(t)))}, x_{0} = ϕ,

where

ϕ(t) = \{ \begin{matrix}|t + r_{0}|^{α}, & t ≤ −r_{0} \\ 0, & −r_{0} < t ≤ 0. \end{matrix}

Our aim is to find two solutions of IVP (2) of form x(t) = M t^{α}, namely we propose to choose two different sets of positive constants α, N, M and r_{0} such that x(t) = M t^{α} is a solution, and hence IVP (2) is not uniquely solved.

The definition of r and the form of x imply

t − r(x(t)) = (1 − N M^{\frac{1}{α}})t − r_{0} for all t > 0.

We may assume that 1 − N M^{\frac{1}{α}} < 0. Then (1 − N M^{\frac{1}{α}})t − r_{0} < −r_{0} < 0.

\begin{aligned}x(t − r(x(t))) & = ϕ(t − r(x(t))) = |t − r(x(t)) + r_{0}|^{α}. \\ x(t − r(x(t))) & = (N M^{\frac{1}{α}} − 1)^{α} t^{α}. \end{aligned}

Being x(t) = M t^{α} a solution of IVP (2), it follows

(3)  α M t^{α−1} = −M^{\frac{1}{3}} t^{\frac{α}{3}} + (N M^{\frac{1}{α}} − 1)^{\frac{α}{3}} t^{\frac{α}{3}} for all t > 0.

Obviously, α − 1 = \frac{α}{3}, that is α = \frac{3}{2}. We deduce from (3) that

3M + 2M^{\frac{1}{3}} = 2(N M^{\frac{2}{3}} − 1)^{\frac{1}{2}} .

Set N = A^{2} and M^{\frac{−2}{3}} = X, where A > 0 and X > 0. Therefore

(4)  \frac{3}{X} + 2 = 2(A^{2} − X)^{\frac{1}{2}} .

Comparing the graphs of the functions (0, A^{2}] ∋ X ↦ \frac{3}{X} + 2 and (0, A^{2}] ∋ X ↦ 2(A^{2} − X)^{\frac{1}{2}} , it follows that, if 5 < 2(A^{2} − 1)^{\frac{1}{2}} , that is A > \frac{\sqrt{29}}{2}, then there are two solutions X_{1}, X_{2} ∈ (0, A^{2}) of (4). Then 1 − N M^{\frac{1}{α}} < 0, because 1 − N M^{\frac{1}{α}} = 1 − A^{2} M^{\frac{2}{3}} = 1 − A^{2} X^{−1} < 0. Let A = 4 > \frac{\sqrt{29}}{2}. From the definition of A and X, we obtain N = 16,

M_{1} = X1^{\frac{−3}{2}} , M_{2} = X2^{\frac{−3}{2}} . Consider IVP (2), where

r(u) = 16|u|^{\frac{2}{3}} + r_{0}, r_{0} > 0,

and

ϕ(t) = \{ \begin{matrix}|t + r_{0}|^{\frac{3}{2}} , & t ≤ −r_{0} \\ 0, & −r_{0} < t ≤ 0. \end{matrix}

As we have shown, there are two positive constants M_{1} ≠ M_{2} such that x_{1}(t) = M_{1} t^{\frac{3}{2}} and x_{2}(t) = M_{2} t^{\frac{3}{2}} are two different solutions of IVP (2).

References

[1] T. R. Ding, Asymptotic behavior of solutions of some delay differential equations, Chinese Sci., 8 (1981), 939–949.

[2] J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential delay equations with state-dependent time lags: I, Arch. Rational Mech. Anal. 120 (1992), 99–146.

[3] J. Mierczynski, Uniqueness for quasimonotone systems with strongly monotone first integral, Nonlinear Analysis (3), 30 (1997), 1905–1909.

[4] B. S. Razumikhin, Application of Liapunov’s method to problems in the stability of systems with a delay. [Russian] Automat. i Telemeh. 21 (1960), 740–749.

[5] N. Rouche, P. Habets, M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New York -Heidelberg-Berlin, 1977.

(Received July 25, 2007)