On the existence of stationary solutions for some non-Fredholm integro-differential equations (Q651256)

Let \(\Omega\) be a domain in \({\mathbb R}^d\) (\(d=1,\, 2,\, 3\)) and assume that \(a\) is a nonnegative real number. This paper deals with the following nonlinear problem \[ u_t=\Delta u+au+\int_\Omega G(x-y)F(u(y),y)dy=0, \] where \(F:{\mathbb R}\times\Omega\rightarrow {\mathbb R}\) has a linear growth. The main result establishes the existence of stationary solutions of this equation. The proof is mainly based on fixed point theory techniques.