Notes on existence and uniqueness of solutions for second order periodic-integrable boundary value problems (Q714613)

Consider the boundary value problem \[ x''+ f(t,x)= 0\quad\text{for }0< t< 2\pi, \] \[ x(0)= x(2\pi),\quad \int^{2\pi}_0 x(s)\,ds= 0 \] under the assumptions (A1). \(f\in C^1((0,2\pi)\times \mathbb{R},\mathbb{R})\). (A2). There exist \(N\in\mathbb{Z}^+\) and \(\varepsilon> 0\) such that \[ N^2+\varepsilon\leq f_x(t,x)\leq (N+ 1)^2-\varepsilon\quad\forall(t,x)\in [0,2\pi]\times\mathbb{R}. \] The authors prove that under the assumptions (A1) and (A2) the considered boundary value problem has a unique solution.