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
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引用次数: 0
Abstract
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.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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