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

Advances in Nonlinear Analysis

article

Josef Kreulich¹ 

[1] Universität Duisburg–Essen

关键词: Evolution equations;    almost periodicity;    limiting equation;   

DOI  :  10.1515/anona-2016-0075

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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}.

【 授权许可】

CC BY