Introduction.

We are concerned here with the oscillatory character of certain solutions of scalar differential equations of the form

$$\dot x(t)= f(t,x(t),x(t-r(t))),$$ (1)

where $t-r(t)$ is a given strictly increasing map defined for $t\geq t_o$ for some $t_o\in {\rm I\kern -.2em R}.$ We also assume that $r(t_o)=0$ and $0<r(t)\leq t-t_o$ for $t\geq t_o.$ Eq. (1) is a particular form of a ``retarded differential equation", the delay in time being provided by the argument $t-r(t)$ and it is a special kind of a functional differential equation of the type

$$\dot x(t)= g(t,x_t),$$ (2)

where $x_t$ is the ``tail" map (as it is given, for instance, in [6]), $x_t(\theta)=x(t+\theta),\,\, \theta \in [-r(t),0].$ A particular instance of the more general situation to be investigated in this paper, was studied in [2], namely, the ``scaled differential equation"

$$\dot x(t)= -ax(t)+ax(pt),$$ (3)

where $a>0$ and $0<p<1$ are given real numbers. Here, we have $t_o=0$ and $r(t)=t-pt.$ The nomenclature for Eq. (3) is due to the change of scale in time of the argument $pt.$ Variations of this equation have been used in some mathematical models for pantograph equipment ([3,4,5,9,10,13,14,15]). Eq. (1) has an interesting feature: two different kinds of initial value problem (IVP) can be attached to it, namely, the punctual IVP ``$x(t_o)=x_o,$" or the functional IVP `` $x(t)=\psi(t)'',$ where $x_o$ is arbitrarily chosen in ${\rm I\kern -.2em R}$ or $\psi$ is arbitrarily chosen in ${\cal C}_\tau=:{\cal C}([\tau',\tau],{\rm I\kern -.2em R}),$ and $\tau>t_o$ is an arbitrary real constant, with $\tau'=\tau-r(\tau).$ Here, as usual, ${\cal C}([a,b],{\rm I\kern -.2em R})$ stands for the space of the continuous maps from $[a,b]$ into ${\rm I\kern -.2em R},$ equipped with the supremum norm $\vert\vert\phi\vert\vert=\sup\{\vert\phi(t)\vert:t\in[a,b]\}.$ One often refers to either $(t_o,x_o)$ or $(\tau,\,\psi)$ as an ``initial condition". This duality of IVPs directly leads to the phenomenon of collapse of backward continuation ([2]), responsible for a wild kind of oscillatory behavior of solutions of Eq. (1), as we shall see. In order to make notations simpler, we extend the above nomenclature of $\tau'$ by putting $\tau''=(\tau')'=\tau'-r(\tau')$ and so on.

By a ``solution" of the punctual IVP we mean a differentiable map $x(t),$ defined in an interval $[t_o,b]$ which, of course, satisfies Eq. (1) for $t\in [t_o,b]$ and $x(t_o)=x_o.$ The derivative at $t_o$ means, of course, the right hand derivative while the derivative at $b$ is the left hand one. As usual, the notations $\dot x(t+)$ and $\dot x(t-)$ denote the right-hand and the left-hand derivative of $x$ at $t,$ respectively. Similarly, a ``solution" of the functional IVP is a continuous map $x:[\tau',b]\to {\rm I\kern -.2em R},$ such that $b>\tau,$ $x$ is differentiable in $[\tau,b]$ and, of course, satisfies $x(t) =\psi(t),$ $t\in [\tau',\tau],$ $\dot x(t) = f(t,x(t),x(t-r(t)),$ $t\in[\tau,b]$ with the derivative at $\tau$ being the right-hand one, etc.. To emphasize the dependence on $\tau$ and $\psi$ of this solution we shall denote it by `` $x(\,.\,,\tau,\psi).$" The solution of the punctual IVP will be denoted simply by ``$x(\,.\,,x_o).$" We assume from now the following conditions:

Hypothesis (H): (i)- $f(t,x,y)$ is continuous in $t$ and $C^\infty$ with respect to $x,\,y$ for $t\geq t_o$ (this condition implies that $f(t,\,.\,,\,.\,)$ is locally Lipschitz, i.e., for each compact rectangle $Q=[a,b]\times [c,d]$ and $t\in{\rm I\kern -.2em R}$ there is a constant $M_t^Q>0$ such that

$$\vert f(t,x,y)-f(t,u,v)\vert\leq M_t^Q\max\{\vert x-u\vert,\vert y-v\vert:\,(x,y), (u,v)\in Q \},$$

where $\vert.\vert$ denotes the modulus or absolute value map); (ii)- $D_yf(t,x,y)\neq 0$ for all $(t,x,y),\, t\geq t_o$ ((ii) implies that $f(t,x,\,.\,)$ is strictly monotone); (iii)- $f(t,x,\,.\,)$ is surjective; (iv)- $f(t,0,0)=0, \,t\geq t_o$ and (v)- $r$ is $C^{\infty}$. Assumption (iv) implies that $x(t)\equiv 0$ is a solution of Eq. (1), which is called ``the trivial (or null) solution" and it is the unique solution for both the punctual IVP ``$x(t_o)=0$" and the functional IVP ``$x_\tau =0''$ as it will become clear in the sequel. This equilibrium is denoted by $x(\,.\,,0).$

Note that (ii), (iii) and (iv), together, imply that the sets ${{\cal A}_t}^+=\{y: f(t,0,y)>0\}$ and ${{\cal A}_t}^-=\{y: f(t,0,y)<0\}$ are nonempty, disjoint and ${A_t}^+\cup{{\cal A}_t}^-\cup\{0\}={\rm I\kern -.2em R},$ $t\geq t_o.$ To fix the ideas, we may think, without loss of generality, that:

Hypothesis (G): $\ {{\cal A}_t}^+={\rm I\kern -.2em R}^+=\{y:y>0\}\$ and $\ {{\cal A}_t}^-={\rm I\kern -.2em R}^-=\{y:y<0\}.$

Note that since $f$ is continuous, it follows that $\{ y\,:\, f(t,0,y)=0\} =\{0\}$ for each $t\ge t_o$.

If we compare the above conditions with the simpler ones in [2], where the authors made use of the stronger semigroup property which holds for linear delay equations, we see that the present conditions are not yet sufficient to guarantee backward continuation of a large set of initial conditions, as it is in the linear case. In this paper we use a special property called piling property to be defined, as well as applications of implicit function theorems, to extend the results to a much more general class of problems.