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The following boundary value problem (0.1)x′(t)=p(t)∫0σ(t)q(s)x(τ(s))ds, (0.2)x(t)=φ(t)fort∈[τ0,0),lim inft→+∞|x(t)

Nonoscillation plays an important role in the theory of ordinary differential equations, but for functional differential equations and their important class such as delay differential equations, nonoscillation is defined only as the existence of an eventually positive solution on the semiaxis and cannot be used in the analysis of boundary value problems. The use of Azbelev’s definition of homogeneous equations allows us to deal with the standard notion of the nonoscillation interval and to obtain results about the existence and uniqueness of the solutions for the interpolation boundary value problems and sign behavior of their Green’s functions.

We consider integro-differential system of the form A ( μ ) d x d t = B ( μ ) x + G ( t , μ , x ) + ∫ 0 t K ( t , s , μ ) φ ( x ( s , μ ) ) d s , where the coefficients A ( μ ) , B ( μ ) , G ( t , μ , x ) and K ( t , s , μ ) continuously depend on a vector parameter μ , and the matrix A ( μ ) is singular at μ = 0 . For analysis of this system, a method based on its reduction to a countable singular system of ordinary differential equations is proposed. The finite sections method and an application of Drazin’s inverse matrices to the integration of linear singular systems are used.

The basis of the Fourier method of variables’ separation and a special reduction procedure for the stability of solutions of partial boundary value problems is studied.

In this paper, distribution of zeros of solutions to functional equations in a space of functions of two variables is studied. It will be demonstrated that oscillation properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. An exact nonoscillation test will be obtained.