Semidiscrete approximations of semilinear periodic problems in Banach spaces. (English) Zbl 1126.34352
Let \(B\) be a Banach space. Consider in \(E\) the \(T\)-periodic problem
\[
u'= Au+ f(t,u),\quad u(t)= u(t+ T)\;\forall t\in R^+,\tag{\(*\)}
\]
where \(f\) is \(T\)-periodic in \(t\) and \(A\) generates an analytic compact semigroup. Together with \((*)\) consider the semidiscrete approximation
\[
u_n'= A_n u_n+ f_n(t, u_n),\quad u_n(t)= u_n(t+ T)\;\forall t\in R^+,\tag{\(**\)}
\]
where \(A_n\) generates an analytic semigroup in the Banach space \(E_n\) and \(f_n\) is \(T\)-periodic in \(t\).
Suppose \((*)\) to have a solution \(u^*\). The authors derive conditions such that \((**)\) has a \(T\)-periodic solution \(u^*_n\) for almost all \(n\) and \(u^*_n\) converges uniformly with respect to \(t\in [0,T]\) to \(u^*\).
Suppose \((*)\) to have a solution \(u^*\). The authors derive conditions such that \((**)\) has a \(T\)-periodic solution \(u^*_n\) for almost all \(n\) and \(u^*_n\) converges uniformly with respect to \(t\in [0,T]\) to \(u^*\).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
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