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Dichotomy and periodic solutions to partial functional differential equations. (English) Zbl 1401.34084

Summary: We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form \[ \dot{u}=A(t)u+F(t)(u_t)+g(t,u_t) \] on a Banach space \(X\) where the operator-valued functions \(t\mapsto A(t)\) and \(t\mapsto F(t)\) are \(1\)-periodic, the nonlinear operator \(g(t,\phi)\) is \(1\)-periodic with respect to \(t\) for each fixed \(\phi\in \mathcal{C}:=C([-r,0],X)\), and satisfying \(\|g(t,\phi_1)-g(t,\phi_2)\| \leq \varphi(t)\|\phi_1-\phi_2\|_{\mathcal{C}}\) for \(\phi_1, \phi_2\in \mathcal{C}\) with \(\varphi\) being a positive function such that \(\sup_{t \geq 0}\int_{t}^{t+1}\phi(\tau)d\tau<\infty\). We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family \((A(t))_{t\geq 0}\) generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

MSC:

34K30 Functional-differential equations in abstract spaces
34K19 Invariant manifolds of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K25 Asymptotic theory of functional-differential equations
35R10 Partial functional-differential equations
Full Text: DOI

References:

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