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Periodic solutions of non-densely defined delay evolution equations. (English) Zbl 1018.34063
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.

34K13 Periodic solutions to functional-differential equations
34K30 Functional-differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
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