Interval Oscillation Criteria Research Articles

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities ( p ( t ) x ′ ( t ) ) ′ + q ( t ) x ( t - τ ) + ∑ i = 1 n q i ( t ) | x ( t - τ ) | α i sgn x ( t - τ ) = e ( t ) , where τ ⩾ 0 , p ( t ) ∈ C 1 [ 0 , ∞ ) , q ( t ) , q i ( t ) , e ( t ) ∈ C [ 0 , ∞ ) , p ( t ) > 0 , α 1 > ⋯ > α m > 1 > α m + 1 > ⋯ > α n > 0 ( n > m ⩾ 1 ) . Without assuming that q ( t ) , q i ( t ) and e ( t ) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.

Interval oscillation theorems for second order nonlinear partial delay differential equations

Using the integral averaging method and the generalized Riccati technique, we derive new interval oscillation criteria for second order nonlinear partial delay differential equations. These results are different from most known ones in the sense that they are based on information only on a sequence of subintervals of (0,∞), rather than on the whole (0,∞). Our results are of a high degree of generality and sharper than the existing results in literature.

INTERVAL OSCILLATION CRITERIA FOR SECOND-ORDER DELAY AND ADVANCED DIFFERENCE EQUATIONS

Interval oscillation criteria are established for second-orderdigerence equations in the form(k (n)x(n)) +p (n) x (g (n)) +q (n) jx (g (n))jn; n02 N = f0; 1; :::g;1x (g (n)) =e (n) ;(E ) wherenquences of real numbers;k (n) > 0is nondecreasing;g(n)is nondecreasing,limn 1g(n) = 1:Several oscillation criteria are given for equation(E )considered as to separate delay and advanced digerence equationswheng(n) nrespectively. Illustrative examples are included

Interval oscillation criteria for linear Hamiltonian systems

In this paper some new oscillation criteria of interval type for linear matrix Hamiltonian systems are established which are different from most known ones in the sense that they are based on the information only on a sequence of subinterval of [t0, ∞) rather than on the whole half line. Our results are shaper than previous results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Oscillation criteria for a forced second-order nonlinear dynamic equation

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation on a time scale 𝕋 where γ ≥ 1. As a special case when 𝕋 = ℝ our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233–240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123–125) but also improve these results. When 𝕋 = ℕ, 𝕋 = hℕ or 𝕋 = q N , i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples.

Comment on the paper: “Interval criteria for oscillation of second-order nonlinear neutral differential equations” [Comput. Math. Appl. 46 (5–6) (2003) 903–918

Comment on the paper: “Interval criteria for oscillation of second-order nonlinear neutral differential equations” [Comput. Math. Appl. 46 (5–6) (2003) 903–918

Oscillation criteria for forced second-order nonlinear differential equations with damping

In this paper, we are concerned with oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using a classical variational principle and an averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. An example is also given to illustrate the results.

Oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities

In the case of oscillatory potentials, we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div ( A ( x ) ∇ y ) + 〈 b ( x ) , ∇ y 〉 + C ( x , y ) = e ( x ) . When N = 1 , our results extend those of Kong [Q. Kong, Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. Appl. 229 (1999) 258–270] for certain second order linear ordinary differential equations, Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second order differential equations with mixed nonlinearities, Appl. Math. Comput. (2007), doi:10.10.1016/j.amc.2007.08.042 ] for certain second order differential equations with mixed nonlinearities, Yang [Q. Yang, Interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49–64] for a forced second order superlinear ordinary differential equation. When N ≥ 2 , our theorems are more general, and improve the oscillation criteria of Zhuang [R.-K. Zhuang, Annual oscillation criteria for second order nonlinear elliptic differential equations, J. Comput. Anal. Math. (2007), doi:10.1016/j.cam.2007.06.031 ] for a forced second order undamped elliptic differential equation.

Interval oscillation criteria for matrix differential systems with damping

Abstract This paper concerns the oscillation of solutions to Leighton type for the matrix differential system with damping: ( P ( t ) Y ′ ) ′ + R ( t ) Y ′ + Q ( t ) Y = 0 , under the hypothesis: P ( t ) = P ∗ ( t ) > 0 , Q ( t ) = Q ∗ ( t ) and R ( t ) = R ∗ ( t ) are n × n matrices of real valued continuous functions on the interval [ t 0 , ∞ ) .

Interval criteria for oscillation of nonlinear second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and an integral averaging technique. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Interval oscillation criteria for second order partial differential equations with delays

In this paper, we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0 , ∞ ) , rather than on whole half-line. Our results are sharper than some previous results and handles the cases which are not covered by known criteria.

Interval criteria for oscillation of second-order nonlinear differential equations

By employing the generalized Riccati technique and the integral averaging technique, new interval oscillation criteria are established for second-order nonlinear differential equations.

Interval oscillation criteria for a forced second-order differential equation with nonlinear damping

In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using an inequality due to Hardy et al., several new interval oscillation criteria for the equations are established. These criteria are different from most known ones in that they are based on the information only on a sequence of subintervals of [ t 0 , ∞ ) , rather than on the whole half-line. Our results are more general, and extend the oscillation criteria of Wong and Yang. Examples are also given to illustrate the results.

Interval criteria for oscillation of second-order functional differential equations

By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation, ( r ( t ) | x ′ ( t ) | α − 1 x ′ ( t ) ) ′ + F ( t , x ( t ) , x ( τ ( t ) ) , x ′ ( t ) , x ′ ( τ ( t ) ) ) = 0 , t ≥ t 0 that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [ t 0, ∞], rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional differential equations, our criteria implies that the τ(t) ≤ t delay and Gt( t) ≥ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.

Interval oscillation criteria for self-adjoint matrix Hamiltonian systems

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

Interval oscillation criteria for self-adjoint matrix Hamiltonian systems

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

Interval criteria for oscillation of second order half-linear differential equations with damping

Interval oscillation criteria are established for second order half-linear differential equations with damping that are different from most known ones in the sense that they are based only on a sequence of subintervals of $ [t_0, \infty)$, rather than on the whole half-line. The results extend the integral averaging technique and include earlier results.

Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping

The paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms. Several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results. In particular, it is shown that under some very mild conditions on k 1 , k 2 , and f the equation ( k 1 ( x , x ′ ) ) ′ + cos tk 2 ( x , x ′ ) x ′ + sin tf ( x ) = 0 is oscillatory, which extends the result obtained by Kwong and Wong (J. Math. Anal. Appl. 288 (2003) 15).

Interval Oscillation Criteria of Matrix Differential Systems with Damping

Using a generalized Riccati transformation, some new oscillation criteria of linear second order matrix differential system with damping are built by the method of integral average. These results are based on the information on a sequence of subintervals of 𝑡0, ).

Interval Oscillation Criteria for Second Order Nonlinear Delay Differential Equations

Interval Oscillation Criteria for Second Order Nonlinear Delay Differential Equations