Asymptotic behavior of abstract difference equations. (English) Zbl 1227.43006
Summary: We consider the difference operator equations having the form
\[ \sum_{i=1}^k a_ix(t+r_i)= aAx(t)+f(t), \quad t\in\mathbb R, \]
where the free term \(f\) belongs to a closed subspace \(M\) of \(L^\infty(\mathbb R,X)\), \(A\) is the generator of a \(C_0\)-semigroup of operators defined on a Banach space \(X\), and \(a_i\), \(a\in\mathbb C\), \(r_i\in\mathbb R\), \(i=1,\dots,k\). We answer the following question: Under what conditions does every solution belong to \(M\)? Also certain conditions are imposed to insure that every bounded solution vanishes at \(\infty\).
\[ \sum_{i=1}^k a_ix(t+r_i)= aAx(t)+f(t), \quad t\in\mathbb R, \]
where the free term \(f\) belongs to a closed subspace \(M\) of \(L^\infty(\mathbb R,X)\), \(A\) is the generator of a \(C_0\)-semigroup of operators defined on a Banach space \(X\), and \(a_i\), \(a\in\mathbb C\), \(r_i\in\mathbb R\), \(i=1,\dots,k\). We answer the following question: Under what conditions does every solution belong to \(M\)? Also certain conditions are imposed to insure that every bounded solution vanishes at \(\infty\).
MSC:
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
