Bounded solutions of nonlinear Cauchy problems (Q1864348)

The author considers the equation \[ u'(t)+ A(u(t))+ \omega u(t)\ni f(t),\quad t\in\mathbb{R},\tag{1} \] in a real Banach space \(X\), where \(A\) is \(m\)-accretive in \(X\) and \(w\) is positive. Let \(\text{BUC}(\mathbb{R}, X)\) be a set of bounded and uniformly continuous functions from \(\mathbb{R}\) into \(X\) and \(Y\) a closed and translation-invariant linear subspace of \(\text{BUC}(\mathbb{R}, X)\). Assume that for \(h\in Y\) and \(\lambda> 0\), the function \(\{s\mapsto J_\lambda(h(s))\}\) is in \(Y\). It is proved that for \(f\in Y\) the integral solution \(u\) to the equation (1) is an element of \(Y\). This result can be applied to the particular cases \(Y= AP(\mathbb{R}, X)\), the space of almost periodic function and \(Y= W(\mathbb{R}, X)\), the space of Eberlein weakly almost periodic functions. This result is an extension of well-known results.