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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     author = {D. A. Neverova and A. L. Skubachevskii},
     title = {On the {Classical} and {Generalized} {Solutions} of {Boundary-Value} {Problems} for {Difference-Differential} {Equations} with {Variable} {Coefficients}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {702--719},
     publisher = {mathdoc},
     volume = {94},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_94_5_a6/}
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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/