We prove the theorem on solvability of the Cauchy problem for the complete Volterra integrodifferential linear equations of the second order in a Hilbert space. For a complete equation, we consider three main classes of equations depending on the ordering of operator coefficients.
Similar content being viewed by others
References
N. D. Kopachevskii, Volterra Integrodifferential Equations in Hilbert Spaces. A Special Course of Lectures [in Russian], FLP “Bondarenko O. A.”, Simferopol’ (2012).
V. V. Vlasov, D. A. Medvedev, and N. A. Rautian, “Functional-differential equations in Sobolev spaces and their spectral analysis,” Sovr. Probl. Mat. Mekh., Mat., 8, Issue 1, 8–306 (2011).
V. Poblete, “Solutions of second-order integrodifferential equations on periodic Besov spaces,” Proc. Edinburgh Math. Soc., 50, 477–492 (2007).
R. Alikhani, F. Bahrami, and A. Jabbari, “Existence of global solutions to nonlinear fuzzy Volterra integrodifferential equations,” in: Nonlin. Anal.: Theory, Meth. Appl., 75, 1810–1821 (2012).
S. Trostorff, “On integrodifferential inclusions with operator-valued kernels,” Math. Meth. Appl. Sci. (2014), doi: 10.1002/mma.3111.
V. Vlasov and N. Rautian, “Spectral analysis and representations of solutions of abstract integrodifferential equations in Hilbert space,” Operator Theory: Adv. Appl., 236, 517–535 (2014).
T. Diagana, “Existence results for some damped second-order Volterra integrodifferential equations,” Appl. Math. Comput., 237, 304–317 (2014).
S. G. Krein (editor), Functional Analysis [in Russian], Nauka, Moscow (1972).
M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces, [in Russian], 2nd Edn., Lan’, St.-Petersburg (2010).
S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).
S. Ya. Yakubov, Linear Differential-Operator Equations and Their Applications [in Russian], Élm, Baku (1985).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 11, pp. 1485–1497, November, 2014.
Rights and permissions
About this article
Cite this article
Kopachevskii, N.D., Semkina, E.V. Complete Volterra Integrodifferential Equations of the Second Order Unsolved with Respect to the Higher Derivative. Ukr Math J 66, 1665–1679 (2015). https://doi.org/10.1007/s11253-015-1043-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1043-y