Periodic solutions of Lagrange equations (Q1876359)

This paper deals with the existence of periodic solutions (in a suitable sense) to an equation of the form \({{d}\over {dt}} L_{x'}(t,x'(t))+V_x(t,x(t))=0\), where, among other assumptions, it is assumed that \(L\) is convex in the second variable. Sections 1 to 4 of the present paper are exactly the same as the corresponding ones in the previous paper of the authors [J. Math. Anal. Appl. 264, No. 1, 168--181 (2001; Zbl 0998.34033)], where a convexity assumption on \(V\) was required. In Section 5, an example is given for an equation of the form \(k(t)x''(t)+V_x(t,x(t))=0\). It is assumed, among other technical assumptions, that \(k\) is positive and \(V(t,\cdot)\) is continuous and nonnegative in the positive cone in \({\mathbb R}^n\); moreover, the function \(V(t,\cdot)\) is assumed to be convex in a suitable subset of \({\mathbb R}^n\).