On the number of solutions of some ordinary periodic boundary value problems by their geometrical properties (Q1191379)

The authors describe some geometrical properties of the operator \(\varphi:C^ 2_{2\pi}\to C^ 0_{2\pi}\), \(\varphi(x)=-x''+f(x)\), where \(C^ 2_{2\pi}=\{x\in C^ 2(0,2\pi;R):x(0)=x(2\pi)\), \(x'(0)=x'(2\pi)\), \(x''(0)=x''(2\pi)\}\), \(C^ 0_{2\pi}=\{x\in C^ 0(0,2\pi;R):x(0)=x(2\pi)\}\), \(f:R\to R\) is regular and \(\lim_{s\to- \infty}f(s)=\lim_{s\to+\infty}f(s)=+\infty\), and obtain multiplicity results on the number of solutions to the equation \(\varphi(x)=y\) by the study of the behaviour of the operator \(\varphi\) near certain singular points.