Second order periodic problems with at most linear growth (Q1909721)

The paper deals with the existence of weak solutions for the periodic boundary value problem \(x'' = f(t,x,x')\) a.e. \(t \in [0,1]\), \(x(0) = x(1)\), \(x'(0) = x'(1)\), where \(f : [0,1] \times \mathbb{R}^{2n} \to \mathbb{R}^n\) is a Carathéodory function such that for each \(M > 0\) there are \(p \in L^2\) and \(q \in L^1\) with \(|f(t,x,y) |\leq p(t) |y |+ q(t)\) for \(|x |\leq M\) and a.e. \(t\). The proofs are based on the topological transversality theorem of Granas. In the scalar case \((n = 1)\), the author also obtains some existence and multiplicity results by means of the upper and lower solutions technique and of the additivity property of the Leray-Schauder degree.