Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings

We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) , t ≥ 0 , u ⁢ ( 0 ) = u 0 , \frac{du}{dt}(t)\in A(t)u(t),\quad t\geq 0,\qquad u(0)=u_{0}, and their whole line analogues, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) {\frac{du}{dt}(t)\in A(t)u(t)} , t ∈ ℝ {t\in\mathbb{R}} , with a family { A ⁢ ( t ) } t ∈ ℝ {\{A(t)\}_{t\in\mathbb{R}}} of ω-dissipative operators A ⁢ ( t ) ⊂ X × X {A(t)\subset X\times X} in a general Banach space X. According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part. The second main object of the study – in the above context – is to determine the corresponding “dominating” part [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) {[A(\,\cdot\,)]_{a}(t)} of the operators A ⁢ ( t ) {A(t)} , and the corresponding “dominating” differential equation, d ⁢ u d ⁢ t ⁢ ( t ) ∈ [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) ⁢ u ⁢ ( t ) , t ∈ ℝ . \frac{du}{dt}(t)\in[A(\,\cdot\,)]_{a}(t)u(t),\quad t\in\mathbb{R}.