On Positivity Conditions for the Cauchy Function of Functional-Differential Equations
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We study how the statements on estimates of solutions to linear functional-differential equations, analogous to the Chaplygin differential inequality theorem, are connected with the positivity of the Cauchy function and the fundamental solution. We prove a comparison theorem for the Cauchy functions and the fundamental solutions to two functional-differential equations. In the theorem, it is assumed that the difference of the operators corresponding to the equations (and acting from the space of absolutely continuous functions to the space of summable ones) is a monotone totally continuous Volterra operator. We also obtain the positivity conditions for the Cauchy function and the fundamental solution to some equations with delay as long as those of neutral type.
Keywordsfunctional-differential equation Cauchy function Chaplygin differential inequality theorem linear Volterra operator
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- 1.Chaplygin, S. A. Foundation of a New Method of Approximate Integration of Differential Equations (Moscow, 1919) [in Russian].Google Scholar
- 2.Selected Works of N. V. Azbelev, Ed. by V. P. Maksimov and L. F. Rakhmatullina (Inst. Comp. Issl., Moscow–Izhevsk, 2012) [in Russian].Google Scholar
- 3.Azbelev, N. V.,Maksimov,V. P., Rakhmatullina, L. F. Introduction to the Theory of Functional Differential Equations (Nauka,Moscow, 1991) [in Russian].Google Scholar
- 4.Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F. Elements of the Modern Theory of Functional Differential Equations. Methods and Applications (Inst. Comp. Issl., Moscow–Izhevsk, 2002) [in Russian].Google Scholar
- 5.Maksimov, V. P. Problems of the General Theory of Functional Differential Equations. Selected Works (Perm, 2003) [in Russian].Google Scholar
- 13.Myshkis, A. D. LinearDifferential EquationswithDelay Arguments (Nauka,Moscow, 1972) [inRussian].Google Scholar