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Mild solutions of second-order differential equations on the line. (English) Zbl 0958.34043
The author considers differential equations of the type \[ Lu(t)= Au(t)+ f(t),\quad t\in\mathbb{R}, \] with \(Lu(t)\equiv u'(t)\) and \(Lu(t)\equiv u''(t)\), respectively, \(A\) is a closed linear operator on a Banach space, and \(f\) is a bounded uniformly continuous function. She gives necessary conditions on \(A\) for the existence and uniqueness of so-called mild solutions, i.e., bounded uniformly continuous functions \(u= u(t)\) satisfying a corresponding integral equation. It turns out that the solvability is related to the operator equations \(AX- XD= -\delta_0\) and \(AX- XD^2= -\delta_0\), respectively, where \(X\) is a bounded linear operator sought for, \(D\) is the generator of the shift group on the space of bounded uniformly continuous functions, and \(\delta_0\) is Dirac’s distribution concentrated in \(t= 0\).

34G10 Linear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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