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The Bessel operator function. (English. Russian original) Zbl 0965.34052
Dokl. Math. 55, No. 1, 103-105 (1997); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 352, No. 5, 587-589 (1997).
From the introduction: Assume that \(\mathbb{A}\) is a closed operator in the Banach space \(\mathbb{E}\) whose domain of definition \(\mathbb{D}(\mathbb{A})\) is dense in \(\mathbb{E}\). For \(k>0\), consider the abstract Cauchy problem \[ u''(t)+{k\over t} u'(t)= \mathbb{A} u,\quad t>0,\tag{1} \] \[ u(0)= u_0,\quad u'(0)= 0.\tag{2} \] The author obtains necessary and sufficient conditions on the resolvent of the operator \(\mathbb{A}\) that ensure the uniform correctness of problem (1), (2) with \(k>0\). He establishes that the class of operators for which problem (1), (2) is uniformly correct is wider than the class of generators of the cosine operator function.

34G10 Linear differential equations in abstract spaces
47D09 Operator sine and cosine functions and higher-order Cauchy problems
34K30 Functional-differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
47N20 Applications of operator theory to differential and integral equations