Semi-Fredholm operators and periodic solutions for linear functional-differential equations
The authors deal with the linear functional-differential equation with infinite delay of the form \[ \frac{dx(t)} {dt}=Ax(t)+B(t,x_t)+F(t). \] Denote by \(E\) a Banach space with the norm \(|\cdot|\). Let \(\mathcal B\) be a Banach space, consisting of functions \(\psi : (-\infty, 0]\to E\), which satisfies some axioms demonstrated in the paper. Assume that \(A: \mathcal D(A)\subset E\to E\) is the infinitesimal generator of a \(C_0\)-semigroup \(T(t)\) on \(E\); \(B:\mathbb{R}\times\mathcal B\to E\) is continuous and \(B(t,\cdot):\mathcal B\to E\) is linear; \(F:\mathbb{R}\to E\) is continuous. The authors obtain several criteria on existence and uniqueness of periodic solutions to the equation in the general phase space \(\mathcal B\) and in the concrete phase space \(\mathcal B=UC_g\). The technique used is established on perturbation theory of semi-Fredholm operators.
linear functional-differential equations
scientific article; zbMATH DE number 1286238
scientific article