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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.

##### MSC:
 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
##### Keywords:
Bessel operator; abstract Cauchy problem; cosine operator