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Fredholm solvability of inverse boundary problems for second order abstract differential equations. (English. Russian original) Zbl 0799.34064

Differ. Equations 28, No. 4, 562-569 (1992); translation from Differ. Uravn. 28, No. 4, 687-697 (1992).
Let \(A\) be a closed linear operator in a Banach space \(X\), \(F(t)\) [resp. \(\Phi(t)\)] a function with values in \(X\) [resp. with values in the space \(L(X)\) of linear bounded operators in \(X\)]. The author considers the problem of finding \(p\in X\) such that the solution \(u(\cdot)\) of the initial value problem (1) \(u''(t)= Au(t)+ \Phi(t)p+ F(t)\), \(u(0)= u_ 0\), \(u'(0)= u_ 1\) \((0\leq t\leq T)\) satisfies \(u(0)= u_ 2\), where \(u_ 0\), \(u_ 1\), \(u_ 2\) are given elements of \(X\). If \(A\) is the infinitesimal generator of a cosine function then this problem can be reduced to an integral equation applying the variation-of-constant formula to (1). The main result is an alternative theorem that, among other things, requires \((-A)^{1/2}\) to generate a compact semigroup.

MSC:

34G10 Linear differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations
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