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Nonlocal boundary value problems for elliptic-parabolic differential and difference equations. (English) Zbl 1168.35397
Summary: The abstract nonlocal boundary value problem -d 2 u(t)/dt 2 +Au(t)=g(t), 0<t<1, du(t)/dt-Au(t)=f(t), -1<t<0, u(1)=u(-1)+μ for differential equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well-posedness of this problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of boundary value problems for elliptic-parabolic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this nonlocal boundary value problem is presented. The well-posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.
35M10PDE of mixed type
34G10Linear ODE in abstract spaces
35A05General existence and uniqueness theorems (PDE) (MSC2000)