On the sign of Green's function for an impulsive differential equation with periodic boundary conditions
According to the abstract of the paper, the author investigates ``Green's function of second order linear differential equations subject to linear impulse conditions and periodic boundary conditions. Sufficient conditions that ensure the positiveness of the Green's function are established.'' To be more precise, the author considers the following linear boundary value problem with impulse (BVPI): \[ (1):\quad -[p(x)y'(x)]'+q(x)y(x)=h(x), \quad x\in[a,c)\cup(c,b], \] \[ (2):\quad y(c^{-})=d_1y(c^{+}), \quad y^{[1]}(c^{-})=d_2y^{[1]}(c^{+}), \] \[ (3):\quad y(a)=y(b), \quad y^{[1]}(a)=y^{[1]}(b), \] where \(a, b, c\) are fixed real numbers, \(a<c<b,\) \(d_1, d_2\) are nonzero real numbers, \(p(x), q(x)\) and \(h(x)\) are real-valued measurable functions on \([a,b]\) such that \[ \int_{a}^{b}\frac{1}{|p(x)|}dx<\infty,\quad \int_{a}^{b}|q(x)|dx<\infty,\quad \int_{a}^{b}|h(x)|dx<\infty, \] and \(y^{[1]}=py'\) denotes the quasi-derivative of the solution \(y.\) When \(h=0,\) let \(u, v\) the solutions of the homogeneous problem (1)-(2) associated to (BVPI) satisfying the initial conditions \(u(a)=1, u^{[1]}(a)=0\) and \(v(a)=0, v^{[1]}(a)=1,\) respectively. Consider the value \(D=u(b)+v^{[1]}(b)-\frac{1}{d_1 d_2}-1.\) If \(D\neq 0,\) the author shows that the non-homogeneous problem (1)-(2)-(3) has a unique solution \(y(x)\) which can be represented (in terms of a Green function whose description is too involved to be described here), as \(y(x)=\int_{a}^{b}G(x,s)h(s)ds,\) for \(x\in[a,c)\cup(c,b].\) Moreover, if the following conditions are assumed, \((H1): p(x)>0, q(x)\geq 0, q(x)\neq 0 \) almost everywhere, and \((H2):\quad d_1>0, d_2>0, d_1+d_2\geq 1+d_1d_2,\) then the Green function is positive for \(x,s\in [a,c)\cup(c,b],\) in fact, \(\inf_{x,s}G(x,s)>0.\)
scientific article; zbMATH DE number 5522174