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Dynamical systems and functional equations related to boundary value problems for hyperbolic differential operators. (English. Russian original) Zbl 1138.35051
Dokl. Math. 72, No. 3, 949-953 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 405, No. 5, 598-603 (2005).
The paper at hand deals with two problems:
(1) It studies Cauchy-type functional equations
$F(\delta_1(t)+\delta_2(t))- F(\delta_1(t))-F(\delta_2(t))= H(t)$
with a $$Z$$-configuration $$\delta_1,\delta_2$$ [cf. the author, in: Proceedings of the 3rd international conference, Karmiel, Israel, June 19–22, 2001. Israel Mathematical Conference Proceedings Contemp. Math. 364, 205–223 (2004; Zbl 1063.39504)], where $$H$$ and $$F$$ are, respectively, given and unknown functions defined on real intervals. Conditions for the smoothness and number of solutions are deduced.
(2) Secondly, these results are applied in order to study boundary problems on certain bounded domains for hyperbolic differential operators in the plane, i.e.,
$P(\delta_x,\delta_y)u=f$
on $$D$$, subject to the boundary condition $$u=g$$. For a third-order strictly hyperbolic operator $$P(\delta_x,\delta_y)$$ it is shown that the above problem has a unique solution in $$C^2(\overline D)$$, provided $$f\in C(\overline D)$$ and $$g\in C^2(\overline D)$$.

##### MSC:
 35L35 Initial-boundary value problems for higher-order hyperbolic equations 39B12 Iteration theory, iterative and composite equations 39B22 Functional equations for real functions