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 twoderivatives 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 equationunder consideration to an equivalent system consisting of one first-order differential equation andone second-order Volterra integro-differential equation. Then a new generalized Lyapunovfunctional is proposed for this system, the nonnegativity of this functional on solutions of thissystem is proved, and an upper bound is given for the derivative of this functional via the originalfunctional. The resulting estimate is an integro-differential inequality whose solution gives anestimate of the functional.