Almost automorphic solutions for semilinear boundary differential equations

Boulite, S.; Maniar, L.; N’Guérékata, G.

doi:10.1090/S0002-9939-06-08423-1

Almost automorphic solutions for semilinear boundary differential equations
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by S. Boulite, L. Maniar and G. M. N’Guérékata PDF

Proc. Amer. Math. Soc. 134 (2006), 3613-3624 Request permission

Abstract:

In this work, we use the extrapolation methods to study the existence and uniqueness of almost automorphic solutions to the semilinear boundary differential equation \begin{equation} \tag {SBDE} \left \{ \begin {aligned} x’(t) &= A_mx(t)+h(t,x(t)), && t\in \mathbb {R}, \\ Lx(t) &= g(t,x(t)), && t\in \mathbb {R}, \end{aligned} \right . \end{equation} where $A:=A_m|\ker L$ generates a hyperbolic $C_0$-semigroup on a Banach space $X$ and $h,g$ are almost automorphic functions which take values in $X$ and a “boundary space” $\partial X$, respectively. These equations are an abstract formulation of partial differential equations with semilinear terms at the boundary, such as population equations, retarded differential equations and boundary control systems. An application to retarded differential equations is given.

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