Lower Bounds and Positivity Conditions for Green’s Functions to Second Order Differential-Delay Equations

We consider the Cauchy problem on the positive half-line for the differential-delay equation ü(t) + 2c0(t)u̇(t) + c1(t)u̇(t − h) + d0(t)u(t) + d1(t)u(t − h) + d2(t)u(t − 2h) = 0 where ck(t), dj(t) (t ≥ 0; k = 0, 1; j = 0, 1, 2) are continuous functions. Conditions providing the positivity of the Green function and a lower bound for that function are derived. Our results are new even in the case of ordinary differential equations. Applications of the obtained results to equations with nonlinear causal mappings are also discussed. Equations with causal mappings include ordinary differential and integro-differential equations. In addition, we establish positivity conditions for solutions of functional differential equations with variable and distributed delays.