# Institute of Mathematics

## The periodic problem for the second order integro-differential equations with distributed deviation

#### Sulkhan Mukhigulashvili, Veronika Novotná

###### Received April 19, 2019.   Published online June 12, 2020.

Abstract:  We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation $u"(t)=p_0(t)u(t)+\int_0^{\omega}p(t,s)u(\tau(t,s)) {\rm d}s+ q(t)$, and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.
Keywords:  linear integro-differential equation; periodic problem; distributed deviation; solvability
Classification MSC:  34K06, 34K13, 34B15
DOI:  10.21136/MB.2020.0061-19

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Affiliations:   Sulkhan Mukhigulashvili, Institute of Mathematics of the Czech Academy of Sciences, Žižkova 22, 616 62 Brno, Czech Republic, e-mail: smukhig@gmail.com; Veronika Novotná, Faculty of Business and Management, Brno University of Technology, Kolejní 2906/4, 612 00 Brno, Czech Republic, e-mail: novotna@fbm.vutbr.cz

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