The existence of periodic solution for superlinear second order ODEs by a new fixed point approach (Q6616066)

The authors consider the \(T\)-periodic problem associated with a scalar second order equation of the type \N\[\Nx''+f(t,x)+p(t,x,x')=0.\N\]\NThey assume:\N\N(i) \(\lim_{|x|\to\infty} f (t,x) = +\infty\), uniformly in \(t\in[0,T]\);\N\N(ii) There exist two positive constants \(C_p\), \(D_p\) and a positive function \(\gamma_p\in L^1 (0,T)\) such that \N\[\N|p(t,x,y)|\le\gamma_p(t)+C_p|x|+D_p|y|,\N\]\Nfor every \((x,y)\in\mathbb{R}^2\) and almost every \(t\in[0,T]\).\N\N(iii) There exists a nonempty open bounded set \(E\) containing the origin with the following property: any solution \(x(\cdot)\) starting with \((x(0),x'(0))\in\partial E\) exists in \([0,T]\) and \((x(t),x'(t))\neq (0,0)\) for all \(t \in [0, T ]\).\N\NUnder the above assumptions, they prove the existence of at least one \(T\)-periodic solution.\N\NTheir main result slightly improves the one in [\textit{P. Gidoni}, J. Differ. Equations 345, 401--417 (2023; Zbl 1512.34088)].