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Uniform well-posedness of boundary value problems for abstract equations with the Keldysh-Feller operator. (English. Russian original) Zbl 0863.34061

Differ. Equations 31, No. 8, 1373-1381 (1995); translation from Differ. Uravn. 31, No. 8, 1419-1425 (1995).
The paper is devoted to studying some boundary value problems for the linear second order abstract differential equation \(Q(t)u(t)+ Au(t)=0\), where \(A\) is a linear (in particular, unbounded) operator acting in a Banach space \(B\), \(Q(t)= a(t)u''(t)+ b(t)u'(t)\) is the Keldysh-Feller operator that means either \(b(0)+ma'(0)<0\) for some positive integer \(m\) or \(b(0)-a'(0)>0\), and the coefficient \(a(t)\) may vanish as \(t\to 0\). For the case where \(A\) is the generator of a strongly continuous semigroup in \(B\) the author proves the uniform well-posed solvability of three boundary value problems for the differential equation with the Keldysh-Feller in a class of generalized solutions. All proofs are based on the finite difference method of the solutions to these boundary value problems.

MSC:

34G10 Linear differential equations in abstract spaces
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
65N06 Finite difference methods for boundary value problems involving PDEs
65J05 General theory of numerical analysis in abstract spaces
34B05 Linear boundary value problems for ordinary differential equations
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