Nonoscillation and exponential stability of the second order delay differential equation with damping

Abstract

Abstract For a delay differential equation x ¨ ( t ) + a ( t ) x ˙ ( t ) + ∑ k = 1 m b k ( t ) x ( g k ( t ) ) = 0 , g k ( t ) ≤ t , $$\ddot{x}(t) + a(t)\dot{x}(t)+ \displaystyle\sum\limits_{k=1}^m b_k(t)x(g_k(t))=0, g_k(t)\leq t, $$ a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation.