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Study of a complete abstract differential equation of elliptic type with variable operator coefficients. I. (English) Zbl 1161.34028

The authors consider the abstract differential equation of second-order
\[ u''(t)+ p(t)u'(t)+ q(t)u(t)-\lambda u(t)= f(t),\quad t\in(0,\delta),\quad u(0)= \varphi,\;u'(\delta)=\psi, \]
where \(\lambda\) is a positive real number, \(q(t)\) are closed linear operators and \(p(t)\) are bounded linear operators. The existence, uniqueness and maximal regularity results under appropriate differentiability assumptions combining those of Yagi and Da Prato-Grisvard are obtained. An example is given.

MSC:

34G10 Linear differential equations in abstract spaces
34K10 Boundary value problems for functional-differential equations
34K30 Functional-differential equations in abstract spaces
35J25 Boundary value problems for second-order elliptic equations
44A45 Classical operational calculus
47D03 Groups and semigroups of linear operators
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