Multiple positive solutions for a nonlocal boundary-value problem with nonconvex vector-valued response (Q1869015)

In their recent paper (not yet published), the authors consider the nonlinear problem \[ {d\over dt}\bigl( k(t)x'(t)\bigr) +V_x\bigl(t,x(t) \bigr) =0,\text{ a.e. in }[0,T],\;x(0)=0, \] where \(T>0\) is arbitrary, \(V:\mathbb{R} \times\mathbb{R}^n \to\mathbb{R}\) is Gateaux differentiable in the second variable and measurable in \(t\), and \(k:[0,T] \to\mathbb{R}^+\) satisfies certain conditions. They prove that if the function \(V(t,\cdot)\) is convex then the problem admits a countable family of positive solutions. In the present paper, they show that the conclusion remains true if one requires only the convexity of the function \(x\to \int^T_0 V(t,x(t))dt\) in the set \(\{x\in L^2:u \gamma\leq x(t),\;t\in [t_0T]\), \(x(t)\leq \beta v\), \(x(t)\in P\), \(t\in [0,t]\}\) where \(P\) is a positive cone in \(\mathbb{R}^n\).