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Русская версия

**B. P. Harlamov**

Institute of Problems of Mechanical Engineering, RAS, Saint-Petersburg, Russia

An ordinary differential equation of the second order is considered. Coefficients of the equation are assumed to provide a unique Dirichlet problem solution on every finite interval. A consistency condition with respect to convolution of operators Dirichlet family of this equation is defined. The Dirichlet problem solutions representing in a kernel form are investigated. In one-dimensional case such a sub-probability kernel is reduced to two functions (left and right) corresponding to two boundaries of the interval. The left (right) boundary of the initial interval is said to be unattainable if the the left (right) sub-probability is equal to zero. In terms of the initial differential equation some sufficient conditions are derived for the left (right) boundary to be unattainable. One can control these conditions by a constructive method. As interpretation of obtain results unattainable boundaries of a diffusion semi-Markov process are demonstrated

- diffusion semi-Markov process
- Dirichlet problem
- integral equation
- ordinary differential equation of the second order
- sub-probability kernel

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