Averaging of abstract parabolic equations with multipoint integral boundary conditions

A multipoint boundary value problem for an abstract parabolic equation with a rapidly time-oscillating nonlinear part is considered in the time interval. The operator $-A$, where $A$ is the senior stationary linear operator of the equation, is positive. The hypotheses are formulated in terms of the theory of semigroups and fractional powers of the operator $-A$. Multipoint boundary conditions on a time interval contain integral terms. For the specified problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the space of continuous vector functions over a time interval is justified. Thus, the Krylov–Bogolyubov averaging method is justified for abstract parabolic equations with multipoint boundary conditions. The results obtained are applicable to parabolic equations in a limited spatial domain with multipoint boundary conditions over a time interval and some other problems of mathematical physics.