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


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 the 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 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.

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Correspondence to O. A. Repin.

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Original Russian Text © O.A. Repin and S.K. Kumykova, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 8, pp. 57–65.

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Repin, O.A., Kumykova, S.K. A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type. Russ Math. 57, 49–56 (2013).

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Keywords and phrases

  • mixed-type equation
  • nonlocal problem
  • fractional integro-differentiation operators
  • singular integral equation with Cauchy kernel
  • Fredholm equation
  • regularizer
  • Dirichlet problem
  • Cauchy problem