Geodesic
An~existence principle for a~periodic solution of a~differential equation in Banach space
Matematičeskie zametki, Tome 4 (1968) no. 1, pp. 105-111.

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The equation $d^2x/dt^2=Ax+f(t,x)$ is considered in a Banach space $E$, where $A$ is a fixed unbounded linear operator, and $f(t,x)$ is a nonlinear operator which is periodic in $t$ and satisfies a Lipschitz condition with respect to $x\in E$. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation. 
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     author = {N. V. Medvedev},
     title = {An~existence principle for a~periodic solution of a~differential equation in {Banach} space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {105--111},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_4_1_a12/}
}
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N. V. Medvedev. An~existence principle for a~periodic solution of a~differential equation in Banach space. Matematičeskie zametki, Tome 4 (1968) no. 1, pp. 105-111. http://geodesic.mathdoc.fr/item/MZM_1968_4_1_a12/