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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 $$t= 0$$.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
##### Keywords:
existence; uniqueness; mild solutions
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