In this paper, oscillattion and nonoscillation criteria are established for neutral differential equations with positive and negative coefficients. Our criteria improve and extend many results known in the literature.
Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation \[ \Delta (c_n\Delta (y_n+p_ny_{n-k}))+q_ny_{n+1-m}^\beta =0,\quad n\ge n_0 \] where $k$, $m$ are positive integers and $\beta $ is a ratio of odd positive integers are established, under the condition $\sum _{n=n_0}^{\infty }\frac{1}{c_n}<{\infty }.$.
Consider the difference equation ∆x(n) +∑m i=1 pi(n)x(τi(n)) = 0, n ≥ 0 [ ∇x(n) − ∑m i=1 pi(n)x(σi(n)) = 0, n ≥ 1 ] , where (pi(n)), 1 6 i 6 m are sequences of nonnegative real numbers, τi(n) [σi(n)], 1 6 i 6 m are general retarded (advanced) arguments and ∆ [∇] denotes the forward (backward) difference operator ∆x(n) = x(n + 1) − x(n) [∇x(n) = x(n) − x(n − 1)]. New oscillation criteria are established when the well-known oscillation conditions lim sup n→∞ ∑m i=1 ∑n j=τ(n) pi(j) > 1 [ lim sup n→∞ ∑m i=1 σ∑ (n) j=n pi(j) > 1 ] and lim inf n→∞ ∑m i=1 n∑−1 j=τi(n) pi(j) > 1⁄e [ lim inf n→∞ ∑m i=1 σ∑i(n) j=n+1 pi(j) > 1⁄e ] are not satisfied. Here τ (n) = max 1≤i≤m τi(n) [σ(n) = min 1≤i≤m σi(n)]. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
Recently there has been an increasing interest in studying p(t)-Laplacian equations, an example of which is given in the following form (|u ′ (t)| p(t)−2 u ′ (t))′ + c(t)|u(t)| q(t)−2 u(t) = 0, t > 0. In particular, the first study of sufficient conditions for oscillatory solution of p(t)-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with p(t)- Laplacian via Riccati method. The results obtained are new and rare, except for a work of Zhang (2007). We present more detailed results than Zhang (2007).
It was hypothesized that an oscillation of tissue oxygen index (TOI) determined by near-infrared spectroscopy during recovery from exercise occurs due to feedback control of adenosine triphosphate and that frequency of the oscillation is affected by blood pH. In order to examine these hypotheses, we aimed 1) to determine whether there is an oscillation of TOI during recovery from exercise and 2) to determine the effect of blood pH on frequency of the oscillation of TOI. Three exercises were performed with exercise intensities of 30 % and 70 % peak oxygen uptake (Vo2peak) for 12 min and with exercise intensity of 70 % Vo2peak for 30 s. TOI during recovery from the exercise was analyzed by fast Fourier transform in order to obtain power spectra density (PSD). There was a significant difference in the frequency at which maximal PSD of TOI appeared (Fmax) between the exercises with 70 % Vo2peak for 12 min (0.0039±0 Hz) and for 30 s (0.0061±0.0028 Hz). However, there was no significant difference in Fmax between the exercises with 30 % (0.0043±0.0013 Hz) and with 70 % Vo2peak for 12 min despite differences in blood pH and blood lactate from the warmed fingertips. It is concluded that there was an oscillation in TOI during recovery from the three exercises. It was not clearly shown that there was an effect of blood pH on Fmax., T. Yano, R. Afroundeh, K. Shirakawa, C.-S. Lian, K. Shibata, Z. Xiao, T. Yunoki., and Obsahuje bibliografii
We obtain sufficient conditions for every solution of the differential equation [y(t) − p(t)y(r(t))](n) + v(t)G(y(g(t))) − u(t)H(y(h(t))) = f(t) to oscillate or to tend to zero as t approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when G has sub-linear growth at infinity. Our results also apply to the neutral equation [y(t) − p(t)y(r(t))](n) + q(t)G(y(g(t))) = f(t) when q(t) has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.
In this paper we consider the nonlinear difference equation with several delays (axn+1 + bxn) k − (cxn) k + ∑m i=1 pi(n)x k n−σi = 0 where a, b, c ∈ (0, ∞), k = q/r, q, r are positive odd integers, m, σi are positive integers, {pi(n)}, i = 1, 2, . . . , m, is a real sequence with pi(n) ≥ 0 for all large n, and lim inf n→∞ pi(n) = pi < ∞, i = 1, 2, . . . , m. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.
Some new criteria for the oscillation of difference equations of the form \[ \Delta ^2 x_n - p_n \Delta x_{n-h} + q_n |x_{g_n}|^c \mathop {\mathrm sgn}x_{g_n} = 0 \] and \[ \Delta ^i x_n + p_n \Delta ^{i-1} x_{n-h} + q_n |x_{g_n}|^c \mathop {\mathrm sgn}x_{g_n} = 0, \ i = 2,3, \] are established.
We study oscillatory properties of solutions of systems \[ \begin{aligned} {[y_1(t)-a(t)y_1(g(t))]}^{\prime }=&p_1(t)y_2(t), y_2^{\prime }(t)=&{-p_2}(t)f(y_1(h(t))), \quad t\ge t_0. \end{aligned} \].