Nonlocal Dezin Problem for a Mixed Type Equation of the Second Kind

For the equation $$u_{{xx}}+yu_{{yy}}+\alpha u_y=0 $$ with parameter $$\alpha \leq -1/2 $$ given in the mixed domain that is the rectangle $$[0,1]\times [-\beta ,\gamma ]$$ , where $$\beta >0 $$ and $$\gamma >0 $$ , we investigate Dezin’s problem in which a periodicity condition is set on the vertical rectangle sides, the values of the unknown function are specified on the top side, matching conditions are given on the singular line, and a nonlocal condition relating the values of the unknown function on the singular line to the values of the normal derivative of this function is specified on the bottom side. The solution of the problem is constructed in the form of a series. Sufficient conditions ensuring the existence of a solution are found for the given functions and the parameters $$\beta $$ and $$\gamma $$ . A uniqueness criterion is established.