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In this article, the issue of the unique solvability of a problem with shift in an unbounded domain is investigated; the elliptical part of the domain is a horizontal strip. The uniqueness of the theorem is proven by the method of energy integrals under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition. Problem in the elliptical part of the mixed domain, which is a horizontal strip, is solved by the Green's function method. By reduction to the Fredholm equation of the second kind, the existence of a solution to the problem is proved; the unconditional solvability to the problem follows from the uniqueness of the solution to the problem. [ABSTRACT FROM AUTHOR]

Важным этапом в становлении теории краевых задач стали предложенные А.М. Нахушевым в 1969 году нелокальные задачи нового типа, впоследствии названные у нас краевыми задачами со смещением, а за рубежом — задачами (проблемами) Нахушева. Они являются обобщением задачи Трикоми, а так же содержат широкий класс корректных самосопряженных задач. Эти задачи сразу вызвали большой интерес многих авторов. За последние годы исследования задач со смещением для уравнений смешанного типа ведутся особенно интенсивно. Но в этих работах краевые условия, как правило, содержат классические операторы, в то время как нелокальным краевым задачам, содержащим операторы более сложной структуры и операторы дробного интегро-дифференцирования. Настоящая статья посвящена исследованию вопроса однозначной разрешимости краевых задач со смещением для уравнения смешанного эллиптико-гиперболического типа. Сформулированы корректные краевые задачи со смещением для уравнения смешанного типа. В данной работе исследованы вопросы однозначной разрешимости задач со смешением для уравнения смешанного эллиптико-гиперболического типа в смешанной области Ω, ограниченной в полуплоскости y>0 гладкой кривой Жордана, а в полуплоскости y

For a degenerate first-order hyperbolic equation of the second order containing a term with a lower derivative, we study two boundary value problems with an offset that generalize the well-known first and second Darboux problems. Theorems on an existence of the unique regular solution of problems are proved under certain conditions on given functions and parameters included in the formulation of the problems under study. The properties of all regular solutions of the equation under consideration are revealed, which are analogues of the mean value theorems for the wave equation.

We consider the reduced spectral problem for the Cauchy-Riemann operator with nonlocal boundary conditions to Fredholm linear integral equation of the second kind with a continuous kernel. The corresponding Fredholm determinant is defined for all spectral parameters, excepting the points: two and one. Finding zeros of the Fredholm determinant recorded in this form is inefficient, because it is not an entire function of the spectral parameter and the main part of the determinant is not separated. Moreover, we study the structure of the kernel by the method shown above, and establish that the linear Fredholm integral equation could not be solved exactly. Therefore, for its approximate solution the results of I. Akbergenov have been applied, where the estimates of the magnitude of the difference between the exact and approximate solutions of the integral equation are given, main part of the kernel is separated. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. Moreover, the design of the approximated solution of the nonhomogeneous boundary value problem is given.

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function. [ABSTRACT FROM AUTHOR]