Positive solutions for a second order impulsive BVP with bounded linear operator conditions (Q2857603)

The authors consider the following boundary value problem for second-order impulsive differential equations with bounded linear functional conditions of the form NEWLINE\[NEWLINE\begin{aligned} & x''(t)+f(t,x(t))=0, \,\, t\in [0,1]\setminus\{t_0,t_1,\dots,t_{m+1}\},\\ &\Delta x(t_k)=P_k(x(t_k)), \,\, \Delta x'(t_k)=Q_k(x(t_k)),\,\, k=1,2,\dots, m,\\ &x(0)=L_1(x), \,\, x(1)=L_2(x), \end{aligned} NEWLINE\]NEWLINE where \(\Delta x(t_k)=x(t_k^+)-x(t_k^-)\), \(\Delta x'(t_k)=x'(t_k^+)-x'(t_k^-)\) and \(L_1, L_2\) are bounded linear operators. The existence of at least one positive solution is proved via a recent functional-operator fixed point theorem, which is a generalization of the Leggett and Williams fixed point theorem.