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Almost-periodic mild solutions to a class of partial functional-differential equations. (English) Zbl 0981.34064

Here, the semilinear evolution equation \[ \frac{dx}{dt} = A x + B(t, x),\quad x\in X,\tag{1} \] is considered where \(X\) is a Banach space, \(A\) is the infinitesimal generator of a \(C_0\)-semigroup of linear operators, and \(B\) is an everywhere defined continuous operator from \(\mathbb{R} \times X\) to \(X.\) The authors give sufficient conditions for the existence of almost-periodic mild solutions to (1) as common fixed-points of some with (1) associated semigroup in the space of almost-periodic functions.
These results are applied to study the existence of almost-periodic mild solutions to abstract functional-differential equations of the form \[ \frac{dx}{dt} = A x + f(t, x, x_t), \quad x \in X, \tag{2} \] where \(f\) is an everywhere defined continuous operator from \(\mathbb{R} \times X \times C\) to \(X,\) \(C\) is the space of uniformly continuous and bounded functions from \((-\infty, 0]\) to \(X,\) \(x_t\) is the map \(x(t + \theta) = x_t(\theta)\), \(\theta \in (-\infty, 0],\) where \(x(\cdot)\) is defined on \((-\infty, a]\) for some \(a > 0.\) As an example a partial functional-differential equation is considered.

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K30 Functional-differential equations in abstract spaces
47H20 Semigroups of nonlinear operators
35R10 Partial functional-differential equations
35B15 Almost and pseudo-almost periodic solutions to PDEs
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