Geodesic
The Dirichlet problem for an ordinary differential equation of the second order with the operator of distributed~differentiation
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 29 (2019) no. 4, pp. 48-57

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In this paper, we study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation, and for him we study the two-point boundary value problem by the Greens function method. A special function is introduced, in terms of which the Green function of the Direchle problem is constructed and the main properties are proved. Sufficient conditions on the kernel of the operator of continuously distributed differentiation are determined that guarantee the fulfillment of the solvability condition for the Dirichlet problem. In the case when the homogeneous Dirichlet problem for the homogeneous equation under consideration has a nontrivial solution, an analog of the Lyapunov inequality is obtained for the kernel of a continuously distributed ifferentiation operator.
Keywords: fractional integro-differentiation operator, operator of continuously distributed differentiation, fundamental solution, Cauchy problem, Dirichlet problem, analog of Lyapunov's inequality. 
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     author = {B. I. Efendiev},
     title = {The {Dirichlet} problem for an ordinary differential equation of the second order with the operator of distributed~differentiation},
     journal = {Vestnik KRAUNC. Fiziko-matemati\v{c}eskie nauki},
     pages = {48--57},
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B. I. Efendiev. The Dirichlet problem for an ordinary differential equation of the second order with the operator of distributed~differentiation. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 29 (2019) no. 4, pp. 48-57. http://geodesic.mathdoc.fr/item/VKAM_2019_29_4_a5/