The Method of Lyapunov Functionals and the Boundedness of Solutions and Their First and Second Derivatives for a Third-Order Linear Equation of the Volterra Type on the Half-Line

Sufficient conditions are established for the boundedness of all solutions and their first two derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line. To this end, using a method proposed by the first author in 2006, first, we reduce the equation under consideration to an equivalent system consisting of one first-order differential equation and one second-order Volterra integro-differential equation. Then a new generalized Lyapunov functional is proposed for this system, the nonnegativity of this functional on solutions of this system is proved, and an upper bound is given for the derivative of this functional via the original functional. The resulting estimate is an integro-differential inequality whose solution gives an estimate of the functional.