This article is cited in 10 papers

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

O. A. Repin^a, S. K. Kumykova<sup>b

a</sup> Chair of Mathematical Statistics and Econometrics, Samara State Economic University, 141 Sovetskoi Armii str., Samara, 443090 Russia
^b Chair of Function Theory and Functional Analysis, Kabardino-Balkarian State University, 173 Chernyshevskogo str., Nalchik, 360004 Russia

Abstract: 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.

UDC: 517.946

Received: 12.04.2012

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2013, 57:8, 49–56

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