Systems of Functional Differential Equations on Finite Intervals

In Chap. 16, boundary value problems for system of linear functional differential equations are studied, and results on positivity of Green’s matrices are obtained. A necessary and sufficient condition for nonnegativity of all the entries of the Cauchy matrix for linear systems of ordinary differential equations of the first order (obtained by Wazewski) is nonpositivity of off-diagonal coefficients. In the first part of this chapter, extension of this result to various boundary value problems for functional differential equations is obtained, and all the results include an analogue of Wazewski’s sign condition. The main results of the chapter are based on the idea to construct a first-order functional differential equation for the r-th component of the solution vector. Even in the case of systems of ordinary differential equations, this equation is functional differential, with a sufficiently complicated operator. Next, the assertions on differential inequalities for first-order functional differential equations from Chap. 15 are used. This approach explains the importance of considering first-order equations in the operator form, which describes only properties of the operators and does not assume their concrete form. Assertions on differential inequalities for the r-th component of the solution vector or, in the other terminology, about positivity of the entries of the r-th row of Green’s matrices are obtained in a form of theorems on several equivalences, one of the equivalent properties is nonoscillation. It is demonstrated that many tests obtained from these general theorems for one of the solution’s components are unimprovable in a certain sense.