Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities (Q689729)

We consider the existence of solutions of the scalar boundary value problem \(x''+f(t,x)=0\), \(x(0)=x(2\pi)\), \(x'(0)=x'(2\pi)\), in case \(f\) satisfies conditions such as \(a_ +(t)\leq\liminf_{x\to+\infty}{f(t,x)\over x}\leq\limsup_{x\to+\infty}{f(t,x)\over x}\leq b_ +(t)\), \(a_ - (t)\leq\liminf_{x\to-\infty}{f(t,x)\over x}\), with appropriate restrictions on \(a_ +\), \(b_ +\), \(a_ -\), which can be interpreted as nonresonance assumptions. For instance, existence is proved if \(a_ +(t)\geq A_ +\), \(b_ +(t)\leq B_ +\), \(a_ -(t)\geq A_ -\) and if the unbounded set \(\bigl\{(\mu,v)| A_ +\leq \mu\leq B_ +,\;A_ -\leq v\bigr\}\) lies between two consecutive curves of the Fučik spectrum. Our main result is more general however and allows the set \(\bigl\{(\mu,v)| a_ +(t)\leq \mu\leq b_ +(t),\;a_ -(t)\leq v\bigr\}\) to intersect Fučik curves.