Weighted pseudo almost periodic solutions for some partial functional differential equations (Q1029399)

The authors give sufficient conditions for the existence and uniqueness of weighted pseudo almost periodic integral solutions for the functional differential equations \[ \frac{dx}{dt}\;(t)=Ax(t)+L(x_t)+f(t),\,\,\,t\in\mathbb{R}, \] \[ \frac{dx}{dt}\;(t)=Ax(t)+L(x_t)+h(t,x(t-r)),\,\,\,t\in\mathbb{R}. \] Here \(A:D(A)\to E\) is a linear operator on a Banach space \(E\), which satisfies the Hille-Yosida condition, \(L:C\to E\) is a bounded linear operator, \(C:=C([-r,0];E)\) is the Banach space of continuous functions from \([-r,0]\) to \(E\), the history function \(x_t\in C\) is defined by \(x_t(\theta)=x(t+\theta)\) for \(-r\leq\theta\leq 0\), \(f:\mathbb{R}\to E\) is a weighted pseudo almost periodic function, and the nonlinear function \(h:\mathbb{R}\times E\to E\) is weighted pseudo almost periodic in \(t\) with respect to the second argument. An application of the obtained results to a partial functional differential equation with diffusion is also presented.