Periodic solutions of abstract neutral functional differential equations
This paper deals with the existence of periodic solutions for the class of linear abstract neutral functional differential equations the form \[ \frac{d}{dt}(x(t)-Bx(t-r))=Ax(t)+L(x_t)+f(t),\quad t\in {\mathbb R}, \] where \(x(t)\in X\) for any \(t\in {\mathbb R}\) with \(X\) a given Banach space. \(x_t\) is a function defined by \(x_t(\theta)=x(t+\theta)\) for \(\theta\) in an appropriate domain. \(A: D(A)\subset X\to X\) and \(B: D(B)\subset X\to X\) are closed linear operators, \(L\) is a bounded linear map defined on an appropriate space, and \(f\) is a locally \(p\)-integrable and \(2\pi\)-periodic function for \(1\leq p<+\infty\). When the underlying space is a UMD space, the authors characterize the existence of a periodic solution for the above mentioned equation with finite and infinite delay. In the case of finite delay, the existence and uniqueness of a periodic strong \(L^p\) solution, a periodic weak solution and periodic mild solution are discussed, respectively. In the case of infinite delay, existence and uniqueness of a periodic strong \(L^p\) solution are studied.
abstract neutral functional differential equation
scientific article; zbMATH DE number 5962638
scientific article