Bloch-periodic functions and some applications (Q2925596)

In this paper, the authors study the existence and uniqueness of mild solutions of the following semilinear integrodifferential equation in a complex Banach space \(X\) NEWLINE\[NEWLINEu'(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t, Cu(t))\,,NEWLINE\]NEWLINE where \(C:X\to X\) is a bounded linear operator, \(A\) is a closed linear (non necessarily bounded) operator defined in \(X\), \(a\in L^1_{\text{loc}}(\mathbb R^+)\) is a scalar kernel and \(f\) is a Bloch-periodic function.NEWLINENEWLINEFirst of all, the authors study the properties of Bloch-periodic and asymptotically Bloch-periodic functions in a Banach space and then, using these results and the Banach fixed point theorem, prove the existence and the uniqueness of Bloch-periodic mild solutions for the problem under consideration.