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Bounded solutions of nonlinear Cauchy problems. (English) Zbl 1048.47049
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.

MSC:
 47J35 Nonlinear evolution equations
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