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UNIQUENESS FOR RETARDED DELAY DIFFERENTIAL EQUATIONS

Consider the equation u x(t) = f(t,x(t),x(t r(t))) with the initial condition x0 = �. 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 2 IR, f(t,x1,y) � f(t,x2,y) for all t, y 2 IR, and x1 < x2, and if there is t0 � 0 such that r(t0) = 0, then the function t0 t + r(t) does not change sign on an interval [t0,t0 + �). We show an example that the result cannot be applied in the state dependent case.