On stability of difference schemes in fractional spaces

The Cauchy problem εv′(t)+A(t)v(t)=f(t)(0≤t≤T),v(0)=φ in fractional spaces EαEα with the strongly positive operators A(t)A(t) with domain D(A(t))=D(A(0))D(A(t))=D(A(0)) and with an arbitrary εε positive parameter multiplying the derivative term is considered. The first order of accuracy single-step uniform difference scheme for the solution of this problem is presented. The stability and convergence estimates for the solution of this difference scheme in fractional spaces are obtained. The theorem on the well-posedness of this difference scheme in fractional spaces is established. Convergence estimates for the solution of the Cauchy problem for multi-dimensional parabolic equations with an arbitrary εε positive parameter multiplying the derivative term are obtained.