Mathematica Bohemica, Vol. 146, No. 2, pp. 167-183, 2021

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

[1] E. Bravyi: On solvability of periodic boundary value problems for second order linear functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 5, 18 pages. DOI 10.14232/ejqtde.2016.1.5 | MR 3462810 | Zbl 1363.34211
[2] E. I. Bravyi: On the best constants in the solvability conditions for the periodic boundary value problem for higher-order functional differential equations. Differ. Equ. 48 (2012), 779-786; Translation from Differ. Uravn. 48 (2012), 773-780. DOI 10.1134/S001226611206002X | MR 3180094 | Zbl 1259.34052
[3] K.-S. Chiu: Periodic solutions for nonlinear integro-differential systems with piecewise constant argument. Sci. World J. 2014 (2014), Article ID 514854, 14 pages. DOI 10.1155/2014/514854
[4] L. H. Erbe, D. Guo: Periodic boundary value problems for second order integrodifferential equations of mixed type. Appl. Anal. 46 (1992), 249-258. DOI 10.1080/00036819208840124 | MR 1167708 | Zbl 0799.45007
[5] G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). MR 0944909 | Zbl 0634.26008
[6] S. Hu, V. Lakshmikantham: Periodic boundary value problems for second order integro-differential equations of Volterra type. Appl. Anal. 21 (1986), 199-205. DOI 10.1080/00036818608839591 | MR 0840312 | Zbl 0569.45011
[7] I. T. Kiguradze: Boundary-value problems for systems of ordinary differential equations. J. Sov. Math. 43 (1988), 2259-2339 Translated from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30 (1987), 3-103. DOI 10.1007/BF01100360 | MR 0925829 | Zbl 0782.34025
[8] I. Kiguradze, B. Půža: On boundary value problems for functional-differential equations. Mem. Differ. Equ. Math. Phys. 12 (1997), 106-113. MR 1636865 | Zbl 0909.34054
[9] S. Mukhigulashvili, N. Partsvania, B. Půža: On a periodic problem for higher-order differential equations with a deviating argument. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 3232-3241. DOI 10.1016/ | MR 2793558 | Zbl 1225.34073
[10] J. Nieto: Periodic boundary value problem for second order integro-ordinary differential equations with general kernel and Carathéodory nonlinearities. Int. J. Math. Math. Sci. 18 (1995), 757-764. DOI 10.1155/S0161171295000974 | MR 1347066 | Zbl 0837.45006

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

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