Periodic solutions of non-densely defined delay evolution equations
Summary: The authors study the finite delay evolution equation \[ x'(t)= Ax(t)+ F(t,x_t),\quad t\geq 0,\quad x_0= \varphi\in C([-r,0],E), \] where the linear operator \(A\) is nondensely defined and satisfies the Hille-Yosida condition. First, they obtain some properties of ``integral solutions'' for this case and prove the compactness of an operator determined by integral solutions. This allows them to apply Horn's fixed-point theorem to prove the existence of periodic integral solutions when integral solutions are bounded and ultimately bounded. This extends the study of periodic solutions for densely operators to the nondensely defined operators. An example is given.
scientific article; zbMATH DE number 1844034