[For the entire collection see Zbl 0638.00025.]
Let A be a linear operator in \(H=L^ 2[a,b]\) with dense domain and closed range and B the Nemytskij operator defined by \(Bu(x)=g(x,u(x))\) for \(u\in H\), where g:[a,b]\(\times {\mathbb{R}}\to {\mathbb{R}}\) is a given function satisfying Carathéodory’s conditions. The author proves some existence results for the equation \(Au+Bu=h\), where \(h\in H\), g is bounded or satisfies the condition: there exists a constant \(C>0\) and a real valued function \(b\in H\) such that \(| g(x,y)| \leq C| s| +b(x)\) and the limits \(\beta_{\pm}(x)=\lim_{s\to \pm \infty}g(x,s)/s\) exist uniformly in \(x\in [a,b]\). Moreover the author gave some applications to order 3 ordinary differential problems such as \(x\prime''+g(t,x(t))=h(t)\), \(t\in [0,a]\) with conditions \(x''(0)=x''(a)=0\), \(2x(0)+x(a)=0\), or \(x(0)=x(1)=0\), \(x''(0)=x''(1)\). (The theorems are too long to quote them here.)