Lyapunov type inequalities for second order forced mixed nonlinear impulsive differential equations

Abstract

In this paper, we present some new Lyapunov and Hartman type inequalities for second order forced impulsive differential equations with mixed nonlinearities: x ''(t) + p(t)vertical bar x(t)vertical bar(beta-1)x(t) + q(t)vertical bar x(t)vertical bar(gamma-1)x(t) = f(t), t not equal theta(i); Delta x'(t) + p(i)vertical bar x(t)vertical bar(beta-1)x(t) + q(i)vertical bar x(t)vertical bar(gamma-1) x(t) = f(i), t = theta(i), where p, q, f are real-valued functions, {p(i)}, {q(i)}, {f(i)} are real sequences and 0 < gamma < 1 < beta < 2. No sign restrictions are imposed on the potential functions p, q and the forcing term f and the sequences {p(i)}, {q(i)}, {f(i)}. The inequalities obtained generalize and complement the existing results for the special cases of this equation in the literature. (C) 2016 Elsevier Inc. All rights reserved.