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An existence principle for a periodic solution of a differential equation in Banach space

N. V. Medvedev

Vladimir Pedagogical Institute

Abstract: 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.

UDC: 517.9

Received: 23.10.1967

 English version: Mathematical Notes, 1968, 4:1, 551–554

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