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Naito Toshiki,Shin Jong-Son Korean Mathematical Society 2006 대한수학회보 Vol.43 No.2
In this paper we introduce a new concept, a 'periodicizing' function for the linear differential equation with the periodic forcing function. Moreover, we construct this function, which is closely related with the solution of a difference equation and an indefinite sum. Using this function, we can obtain a representation of solutions from which we see immediately the asymptotic behavior of the solutions.
Toshiki Naito,Jong Son Shin 대한수학회 2006 대한수학회보 Vol.43 No.2
In this paper we introduce a new concept, a \period-icizing" function for the linear dierential equation with the peri-odic forcing function. Moreover, we construct this function, whichis closely related with the solution of a dierence equation and anindenite sum. Using this function, we can obtain a representationof solutions from which we see immediately the asymptotic behaviorof the solutions.
SOLUTIONS OF HIGHER ORDER INHOMOGENEOUS PERIODIC EVOLUTIONARY PROCESS
김도한,Rinko Miyazaki,Toshiki Naito,신종손 대한수학회 2017 대한수학회지 Vol.54 No.6
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.
Normal eigenvalues in evolutionary process
김도한,Rinko Miyazaki,Toshiki Naito,Jong Son Shin 대한수학회 2016 대한수학회지 Vol.53 No.4
Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $C_0$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $P_{\tau}(X)$ associated with the $\tau$-periodic evolutionary process on a Banach space $X$, where $P_{\tau}(X)$ stands for the space of all $\tau$-periodic continuous functions mapping ${\mathbb R}$ to $X$.
SOLUTIONS OF HIGHER ORDER INHOMOGENEOUS PERIODIC EVOLUTIONARY PROCESS
Kim, Dohan,Miyazaki, Rinko,Naito, Toshiki,Shin, Jong Son Korean Mathematical Society 2017 대한수학회지 Vol.54 No.6
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.
NORMAL EIGENVALUES IN EVOLUTIONARY PROCESS
Kim, Dohan,Miyazaki, Rinko,Naito, Toshiki,Shin, Jong Son Korean Mathematical Society 2016 대한수학회지 Vol.53 No.4
Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $C_0$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $P_{\tau}(X)$ associated with the ${\tau}$-periodic evolutionary process on a Banach space X, where $P_{\tau}(X)$ stands for the space of all ${\tau}$-periodic continuous functions mapping ${\mathbb{R}}$ to X.