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Solutions of higher order inhomogeneous periodic evolutionary process. (English) Zbl 06958470

Summary: Let \(\{U(t,s)\}_{t\geq s}\) be a periodic evolutionary process with period \(\tau>0\) on a Banach space \(X\). Also, let \(L\) be the generator of the evolution semigroup associated with \(\{U(t,s)\}_{t\geq s}\) on the phase space \(P_{\tau}(X)\) of all \(\tau\)-periodic continuous \(X\)-valued functions. Some kind of variation-of-constants formula for the solution \(u\) of the equation \((\alpha I-L)^nu=f\) will be given together with the conditions on \(f\in P_{\tau}(X)\) for the existence of coefficients in the formula involving the monodromy operator \(U(0,-\tau)\). Also, examples of ODEs and PDEs are presented as its application.

MSC:

47A10 Spectrum, resolvent
47D06 One-parameter semigroups and linear evolution equations
35B10 Periodic solutions to PDEs
34C25 Periodic solutions to ordinary differential equations
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References:

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