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On Positivity Conditions for the Cauchy Function of Functional-Differential Equations


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.

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Correspondence to E. S. Zhukovskii.

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Original Russian Text © E.S. Zhukovskii, K. M. T. Tahir, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 11, pp. 75–81.

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Zhukovskii, E.S., Tahir, K.M.T. On Positivity Conditions for the Cauchy Function of Functional-Differential Equations. Russ Math. 62, 67–71 (2018).

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  • functional-differential equation
  • Cauchy function
  • Chaplygin differential inequality theorem
  • linear Volterra operator