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On solvability and nonsolvability of a variant of boundary value problems with a free boundary for abstract second order differential equations. (Russian) Zbl 0763.34051

The paper deals with the second order abstract differential equation (1) \(\ddot u=Fu\), \(t\in[0,T]\), where \(X\) is a Banach space, \(F: W_ p^ 2([0,T];X)\to{\mathcal L}_ p([0,T];X)\) is a given operator. The author finds a couple \((u(t),\tau)\) such that \(\tau\in(0,T]\), and \(u\in W_ p^ 2([0,\tau];X)\) satisfies (1) for \(t\in[0,\tau]\) and boundary conditions (2) \(u(0)=a\), \(u(\tau)=b\), \(\|\dot u(0)\|=v\), where \(a,b\in X\), \(v\in(0,\infty)\). He establishes conditions for the existence and uniqueness of solutions \((u(t),\tau)\) to (1), (2). Further he shows effective existence, uniqueness and non-existence conditions in the special case \(F=A\), where \(A\) is a positive definite selfadjoint operator in a Hilbert space \(H\).

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
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