Abstract
We investigate the semi-linear, non-autonomous, first-order abstract differential equation x′(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),t∈R. We obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies the so-called Acquistapace–Terreni conditions and the homogeneous associated problem has an integrable dichotomy. A new composition theorem and further regularity theorems are given.
Highlights
Of concern in the present paper is the existence and uniqueness of (ω, c)-periodic mild solutions for a class of semi-linear, non-autonomous equations
The second main result (Theorem 18) shows that (1.1) has a unique (ω, c)-periodic mild solution under the hypothesis that the nonlinear term g satisfies the assumptions of composition Theorem 5 and a standard Lipschitz condition
(generated by A(·)), we say that the system (5) has an integrable dichotomy (λ, P) if there exists a function λ : R2 → (0, ∞) such that kΓ(t + τ, s + τ )k ≤ λ(t, s), for each τ ∈ R, and
Summary
Of concern in the present paper is the existence and uniqueness of (ω, c)-periodic mild solutions for a class of semi-linear, non-autonomous equations. Our first main result (Theorem 15) ensures that linear (Definition 14) possesses a unique (ω, c)-periodic mild solution under the hypothesis that the homogeneous problem has an integrable dichotomy. The second main result (Theorem 18) shows that (1.1) has a unique (ω, c)-periodic mild solution under the hypothesis that the nonlinear term g satisfies the assumptions of composition Theorem 5 and a standard Lipschitz condition. The fourth main result (Theorem 28) gives a unique (ω, c)periodic mild solution for the equation with constant delay (1.3) In this case, to achieve our goal, we use the translation invariant result of the space of (ω, c)-periodic functions.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
