Multiple positive solutions of second-order differential equations
Consider the periodic boundary value problem \[ x'' + f (t,x,x')=0, \quad x(0)= x(T),\quad x'(0)=x'(T),\tag{P} \] where \(f: \mathbb R\times \mathbb R\times \mathbb R \rightarrow \mathbb R\) is continuously differentiable, \(f(t,0,0)>0,\) \(T\)-periodic in \(t\) and satisfies some Nagumo condition. The author introduces the concept of strict lower and upper solutions to problem (P) and assumes that there exist \(n-1\) strict lower solutions \(\alpha_k (t)\) and \(n\) strict upper solutions \(\beta_k(t)\) satisfying \[ 0<\beta_1 (t) < \alpha_1 (t) <\beta_2 (t) < \dots <\alpha_{n-1} (t) < \beta_n (t). \] Then (P) has at least \(2n-1\) positive periodic solutions. The proof is based on the coincidence degree theory.
scientific article; zbMATH DE number 2183187