The second order nonlinear delay differential equation with periodic coefficients x ′′(t)+ p(t)x ′(t)+ q(t)x(t) = r(t)x ′(t − τ(t))+ f (t, x(t), x(t − τ(t))), t ∈ R is considered in this work. By using Krasnoselskii’s fixed point theorem and the contraction mapping principle, we establish some criteria for the existence and uniqueness of periodic solutions to the delay differential equation. c ...

Consider the first-order linear delay differential equation x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, and the second-order linear delay equation x′′(t) + p(t)x(τ(t)) = 0, t ≥ t0, where p and τ are continuous functions on [t0,∞), p(t) > 0, τ(t) is nondecreasing, τ(t) ≤ t for t ≥ t0 and limt→∞ τ(t) = ∞. Several oscillation criteria are presented for the first-order equation when

By using a fixed point theorem of strict-set-contraction, some criteria are established for the existence of positive periodic solutions of the following impulsive functional differential equations with a parameter <: ẋ(t) = −a(t)f(t, x(t))x(t) + λg(t, xt, x(t− τ(t, x(t))), t ∈ R, and t 6= tk, x(t+k )− x(t − k ) = Ik(tk, x(tk − τ(tk, x(tk)))), k ∈ Z.

In this work, we investigate the oscillation criteria for second order neutral delay differential equations of the form (r(t)[y(t)+ p(t)y(δ (t))]′)′ +q(t)G(y(τ(t))) = 0 and (r(t)[[y(t)+ p(t)y(δ (t))]′]α )′ +q(t)(yβ (τ(t))) = 0, where α and β are the ratio of odd positive integers. Mathematics subject classification (2010): 34C10, 34C15.

Oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form r(t)ψ(x(t)) ∣z′(t) ∣∣α−1 z′(t) + b ∫ a q(t, ξ)f(x(g(t, ψ)))dσ(ξ) = 0, t ≥ t0, where α > 0 and z(t) = x(t) + p(t)x(t − τ). Our results improve and extend some known results in the literature. Some illustrating examples are also provided to show the importance of our...

* Correspondence: tangxh@mail. csu.edu.cn School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410083, Hunan, P.R. China Full list of author information is available at the end of the article Abstract In this article, we establish some stability criteria for the polar linear Hamiltonian dynamic system on time scales x (t) = α(t)x(σ (t)) + β(t)y(t), y (t) ...

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t x τ1 t q2 t x τ2 t 0 on a time scale T, where z t x t p t x τ0 t , 0 < α < γ < β. Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main r...

In this paper, some new oscillation criteria are obtained for the secondorder quasi-linear neutral delay differential equation ( r(t)| ( x(t) + p(t)x(τ(t)) )′|α−1(x(t) + p(t)x(τ(t)))′)′+ f ( t, x(σ(t)) ) = 0, t ≥ t0 under the case when ∫∞ t0 1 r 1 α (t) dt <∞. Our results improve and supplement some known results in the literature. An example is also provided to illustrate the main results.

The two-phase test sample representation (TPTSR) was proposed as a useful classifier for face recognition. However, the TPTSR method is not able to reject the impostor, so it should be modified for real-world applications. This paper introduces a thresholded TPTSR (T-TPTSR) method for complex object recognition with outliers, and two criteria for assessing the performance of outlier rejection a...