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In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients. [ABSTRACT FROM AUTHOR]

In this paper new oscillatory criteria for odd order linear functional differential equations of the type y (n) (t) + p (t) y (τ (t)) = 0 have been established. Deviating argument τ(t) is supposed to have dominating delay part. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of second-order nonlinear retarded difference equations. We present some new oscillation criteria via comparison with first-order equations whose oscillatory behavior are known. The results are generalized to be applicable to different kinds of neutral equations. An example is also given to demonstrate the applicability of the obtained conditions. [ABSTRACT FROM AUTHOR]

Consider the first-order linear differential equation with several non-monotone retarded arguments x ′ ⁢ (t) + ∑ i = 1 m p i ⁢ (t) ⁢ x ⁢ (τ i ⁢ (t)) = 0 {x^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0} , t ≥ t 0 {t\geq t_{0}} , where the functions p i , τ i ∈ C ⁢ ([ t 0 , ∞) , ℝ +) {p_{i},\tau_{i}\in C([t_{0},\infty),\mathbb{R}^{+})} , for every i = 1 , 2 , ... , m {i=1,2,\ldots,m} , τ i ⁢ (t) ≤ t {\tau_{i}(t)\leq t} for t ≥ t 0 {t\geq t_{0}} and lim t → ∞ ⁡ τ i ⁢ (t) = ∞ {\lim_{t\to\infty}\tau_{i}(t)=\infty}. New oscillation criteria which essentially improve the known results in the literature are established. An example illustrating the results is given. [ABSTRACT FROM AUTHOR]

In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1; the transmission coeficients 1 = γ1, δ1 = γ1, δ2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator. [ABSTRACT FROM AUTHOR]

In this paper we consider a kind of nonlinear neutral difference equations with continuous arguments. First, we obtain some sufficient conditions for the oscillation. When the coefficients in the equations are constants, these conditions are also necessary. Secondly, we derive a necessary and sufficient condition for the nonoscillation. Finally, we give a sufficient condition for the asymptotic behavior of the nonoscillatory solutions. [ABSTRACT FROM AUTHOR]