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Zbl 1146.45001
Ashyraliyev, Maksat
A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space. (English)
[J] Numer. Funct. Anal. Optim. 29, No. 7-8, 750-769 (2008). ISSN 0163-0563; ISSN 1532-2467/e

Summary: The initial-value problem for integral-differential equations of the hyperbolic type in a Hilbert space $H$ is considered. The unique solvability of this problem is established and the stability estimates for its solution are obtained. The difference scheme approximately solving this problem is presented and the stability estimates for its solution are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented and the stability estimates for its solutions are obtained.

MSC 2000:: *45K05 Integro-partial differential equations
26D10 Inequalities involving derivatives, diff. and integral operators
47G20 Integro-differential operators

Keywords: integral-differential equation of the hyperbolic type; integral inequalities

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