Green's function for second-order periodic boundary value problems with piecewise constant arguments
Let \[ (Ly)(t) = x''(t) + ax'(t) + bx(t) + cx'([t]) + dx([t]), \] where \([\cdot ]\) denotes the greatest integer function. Using the unique solutions, \(h_1\) and \(h_2\), of \[ (Ly)(t) = 0, \;t \in[n, n+1), \quad x(n^+) = v_n, x'(n^+) = v'_n, \] where \(n\) is a natural number, satisfying the initial conditions \(h_1(0) = 1\), \(h_2(0) = 0\) and \(h_1(0) = 0\), \(h_2(0) = 1\), respectively, the authors give a necessary and sufficient condition such that the periodic boundary value problem \[ (Ly)(t) = 0, \quad t \in \mathbb{R},\quad x(0) = x(T), \quad x'(0^-) = x'(T^+), \quad x'(s^+) = x'(s^-) + 1, \] has a unique solution, \(K(t,s)\). The authors then show that if \(\sigma(t)\) is either nonnegative or nonpositive, then a necessary condition for the nonnegativeness or nonpositiveness of solutions of \[ (Ly)(t) = \sigma(t),\quad t \in [0, T],\quad x(0) = x(T), \quad x'(0) = x'(T) + \lambda,\quad \lambda \in \mathbb{R}, \] is that \(b+d\) satisfies a strict sign condition. This paper will be of interest to anyone studying sign conditions of Green's functions for second-order periodic boundary value problems.
piecewise constant functional dependence
scientific article; zbMATH DE number 2157552