On a Problem for a Nonlocal Mixed-Type Equation of Fractional Order with Degeneration
Abstract
The present work is devoted to the study of the solvability questions for a nonlocal problem with an integro-differential conjugation condition for a nonlocal fourth-order mixed type equation with degeneration. The case of a $$0<\alpha<1$$ — order Gerasimov–Caputo type fractional operator was considered. The solution of the nonlocal mixed type differential equation was studied in the class of regular functions. The Fourier series method was used and a countable system of ordinary differential equations was obtained. When conditions of smoothness are fulfilled, then using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series was proved. If these conditions are violated, then the problem will have an infinite set of non-trivial solutions. The stability of unknown function $$u(t,x)$$ of the considering problem with respect to the nonlocal condition was studied.