Applications of linking to Hammerstein integral equations
The critical point theory is used to establish existence results concerning Hammerstein integral equations of the form \[ u(x) = \lambda \int_{\Omega} k(x,y) u (y) dy - \int_{\Omega}k(x,y) f (y,u(y)) dy, \quad x \in \Omega \] where \(\lambda>0\) is a constant and \(\Omega\) is a closed bounded set in \(\mathbb R^n\). The main results are obtained using the concept of linking (both homotopical and (co)homological). The kernel \(k:\Omega\times\Omega\to\mathbb R\) satisfies the following conditions: (i) \(k\in C(\Omega\times\Omega,\mathbb R)\), (ii) \(k(x,y)=k(y,x)\) for \(x,y\in\Omega\), (iii) \(\int_{\Omega\times\Omega}k(x,y)u(x)u(y)dxdy\geq0\), for all \(u\in L^2(\Omega)\).
scientific article; zbMATH DE number 5862464