stability and boundedness for a certain second order integro-differential equation

In theory applications of second order integro-differential equations, it is crucial to investigate qualitative features of solutions such as stability, boundedness, and etc. there is an extensive literature regarding these qualitative behaviors of solutions of second order ordinary differential equations. these qualitative properties of solutions of second order ordinary differential equations have been extensively studied in the literature. despite this case, the literature on these qualitative aspects of second-order integro-differential equations is somewhat limited. in this paper, we obtained some new criteria for the stability and boundedness of solutions to a certain second order nonlinear integro-differential equation (ide). by defining and then using a suitable lyapunov function (lf), we are able to establish the asymptotic stability and boundedness of solutions of that ide. in particular cases of the ide, two examples on the stability and boundedness of solutions are given. the results of this study extend and improve some recent results of the literature and have new contributions to the qualitative theory of ides.