Positive solutions of even order periodic boundary value problems (Q658377)

The authors are concerned with the existence of positive solutions of the \(2m\)-th order periodic boundary value problem consisting of the equation \[ \sum_{i=0}^{m}(-1)^{m+i}C^i_m\rho^{2i}u^{(2m-2i)}=a(t)f(t,u) \] with the boundary condition \[ u^{(2i)}(0)=u^{(i)}(w),\quad i=0,1,\dots,m-1, \] where \(w>0\), \(\rho>0\), and \(m\in\mathbb{N}\). The coefficient \(a: [0,w]\to[0,\infty)\) is continuous and satisfies \(\int_0 a(s)ds>0\). The nonlinearity \(f: [0,w]\times[0,\infty)\to[0,\infty)\) is continuous. By applying the Krasnosel'skii fixed point theorem and the fixed point index theory, the authors establish a series of criteria for the problem to have a single, twin or even an infinite number of positive solutions. Also, some criteria of nonexistence are provided. Several examples of applications illustrate the applicability of the results obtained.