On the Fundamental Analyses of Solutions To Nonlinear Integro-Differential Equations of Second Order

Abstract

In this paper, sufficient criteria of stability, asymptotically stability (AS) of trivial solution, integrability of square of derivatives of solutions as well as boundedness of solutions for second order integro-differential equation (IDE) x ''+ phi(t, x x') + b(t)g(x') + f (x) = integral(t)(0) k(t, t - s)h(x'(t -s))ds + psi(t, x x') respectively. The obtained criteria are applicable to a wide class of nonlinear IDEs. In particular cases of both non-perturbed and perturbed IDEs, two examples for the mentioned fundamental properties of solutions are provided along with trajectories of their solutions. The proofs of the main results depend on the construction of a suitable Lyapunov-Krasovskii functional (LKF). The main results of this paper are novel and less restrictive, and the advantages of the derived explicit criteria over the existing ones are demonstrated with examples. Finally, we follow that the established conditions improve some of earlier results on the topic of interest.