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A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space. (English) Zbl 1146.45001
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:
45K05 Integro-partial differential equations
26D10 Inequalities involving derivatives and differential and integral operators
47G20 Integro-differential operators
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