Existence, boundedness and integrability of global solutions to delay integro-differential equations of second order

Abstract

The paper deals with the problems of the global existence, boundedness and integrability, and solutions of an integro-differential equation (IDE) of second order with n-constant retardations. Three new theorems, Theorems 2.1–2.3 and a corollary, Corollary 2.1, have been proved, which consist of new sufficient criteria on global existence, stability, asymptotic stability, boundedness of solutions and square integrability of the first derivatives of solutions of the considered nonlinear retarded IDE of second order (RIDE). The techniques used in the proofs are based on the construction of a suitable Lyapunov–Krasovskii functional (LK functional). The results of this work include and get better results that are available in the literature. The given criteria have very simple forms, and they can be easily verified with applications. A numerical example is given and the orbits of the exact solutions are drawn with assigned initial conditions to demonstrate the effectiveness of the obtained results.

Keywords

• ride
• second order
• multiple retardations
• globally existence
• boundedness
• square integrability
• lk functional