Geodesic
On the Classical and Generalized Solutions of Boundary-Value Problems for Difference-Differential Equations with Variable Coefficients
Matematičeskie zametki, Tome 94 (2013) no. 5, pp. 702-719.

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The first boundary-value problem for second-order difference-differential equations with variable coefficients on a finite interval $(0,d)$ is considered. The following question is studied: Under what conditions will the boundary-value problem for a difference-differential equation have a classical solution for an arbitrary continuous right-hand side? It is proved that a necessary and sufficient condition for the existence of a classical solution is that certain coefficients of the difference operators on the orbits generated by the shifts be equal to zero.
Keywords: difference-differential equation, first boundary-value problem, difference operator, Sobolev space. 
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D. A. Neverova; A. L. Skubachevskii. On the Classical and Generalized Solutions of Boundary-Value Problems for Difference-Differential Equations with Variable Coefficients. Matematičeskie zametki, Tome 94 (2013) no. 5, pp. 702-719. http://geodesic.mathdoc.fr/item/MZM_2013_94_5_a6/

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