SOLUTIONS OF HIGHER ORDER INHOMOGENEOUS PERIODIC EVOLUTIONARY PROCESS
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Let $\{U(t,s)\}_{t\ge s}$ 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 $\{U(t,s)\}_{t\ge s}$ 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.
Abstract
Let $\{U(t,s)\}_{t\ge s}$ 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 $\{U(t,s)\}_{t\ge s}$ 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.
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