A non-homogeneous Hill's equation (Q2570688)

The authors study the nonhomogeneous Hill equation \[ x^{\prime\prime}+p(t)x=f(t),\tag{1} \] where \(p\) and \(f\) are continuous periodic functions of period \(w>0\). The main result of this paper is the following Theorem: If there exist constants \(m\) and \(M\) such that \[ 0<m\leq p(t)\leq M\leq\frac{8}{w^{2}}, \] then equation (1) has a \(w\)-periodic solution. The proof is based on a fixed-point theorem. The authors also generalize this result to the system of differential equations \[ X^{\prime\prime}+A(t)X=F(t),\tag{2} \] where \(F:[0,w]\to{\mathbb R}^{n}\) and \(A:[0,w]\to M_{n}({\mathbb R})\) are continuous \(w\)-periodic functions. Theorem: Assume that the following conditions are fulfilled. \(0\leq m_{ii}\leq a_{ii}\leq M_{ii}\leq\frac{1}{w^{2}},\) \( i=1,2,\ldots,n,\) \( t\in[0,w]\). \(|a_{ij}|\leq M_{ij},\) \( i,j=1,2,\ldots,n,\) \( i\neq j,\) \( t\in[0,w]\). \(\max\{1+\frac{w^{2}}{12}\sum_{1\leq i\leq n; i\neq j}(M_{ij}-m_{jj})\mid j=1,2,\ldots,n\}\leq\alpha<1.\) Then, equation (2) possesses a solution satisfying the periodic boundary conditions \[ X(0)=x(w),\qquad X^{\prime}(0)=X^{\prime}(w). \]