Geodesic
A nonlocal problem for a~mixed-type equation whose order degenerates along the line of change of type
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2013), pp. 57-65

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We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of a solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.
Keywords: mixed-type equation, nonlocal problem, fractional integro-differentiation operators, singular integral equation with Cauchy kernel, Fredholm equation, regularizer, Dirichlet problem, Cauchy problem. 
@article{IVM_2013_8_a5,
     author = {O. A. Repin and S. K. Kumykova},
     title = {A nonlocal problem for a~mixed-type equation whose order degenerates along the line of change of type},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {57--65},
     publisher = {mathdoc},
     number = {8},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_8_a5/}
}
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O. A. Repin; S. K. Kumykova. A nonlocal problem for a~mixed-type equation whose order degenerates along the line of change of type. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2013), pp. 57-65. http://geodesic.mathdoc.fr/item/IVM_2013_8_a5/