Existence of a pair of periodic solutions of an O.D.E. Generalizing a problem in nonlinear elasticity, via variational methods
It is known that the radial oscillations of an elastic spherical membrane subjected to an internal pressure p: \({\mathbb{R}}\to (0,+\infty)\), continuous and T-periocic, are governed by the scalar equation (1) \(\ddot x=\tilde f(t,x),\) where \(\tilde f:{\mathbb{R}}\times (0,+\infty)\to {\mathbb{R}}\) is given by (2) \(\tilde f(t,x)=p(t)x^ 2-x+x^{-5}.\) Here x(t) is the ratio of the deformed radius of the membrane at time t with respect to the undeformed radius. A more general equation than (1), i.e. (3) \(\ddot x=g(t,x),\) where \(g: {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) is continuous and T-periodic in t is considered in the article. The existence of two T- periodic continuous functions \(z_ i(i=1,2): {\mathbb{R}}\to {\mathbb{R}}\) such that \((i)\quad \max_{t\in {\mathbb{R}}}z_ 1(t)<\min_{t\in {\mathbb{R}}}z_ 2(t),\) (ii) if \(x\in (z_ 1(t),z_ 2(t))\) then \(g(t,x)<0\), and if \(g(t,x)\leq 0\) than \(x\in [z_ 1(t),z_ 2(t)]\) was assumed. Under these conditions the existence of at least two T-periodic solutions of the equation (3) by means of a variational method is proved.
scientific article; zbMATH DE number 4099716