Interval Oscillation Research Articles

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].

Forced Oscillation of Second‐Order Half‐Linear Dynamic Equations on Time Scales

We will establish a new interval oscillation criterion for second‐order half‐linear dynamic equation (r(t)[xΔ(t)] α) Δ + p(t)xα(σ(t)) = f(t) on a time scale 𝕋 which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second‐order half‐linear differential equation ([x′(t)] α) ′ + csintxα(t) = cos t, where α = p/q, p, q are odd positive integers.

Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities

Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form ( r ( t ) Φ α ( x Δ ( t ) ) ) Δ + p 0 ( t ) Φ α ( x ( τ 0 ( t ) ) ) + ∑ i = 1 n p i ( t ) Φ β i ( x ( τ i ( t ) ) ) = e ( t ) , t ∈ [ t 0 , ∞ ) T where T is a time scale, t 0 ∈ T a fixed number; [ t 0 , ∞ ) T is a time scale interval; Φ ∗ ( u ) = | u | ∗ − 1 u ; the functions r , p i , e : [ t 0 , ∞ ) T → R are right-dense continuous with r > 0 nondecreasing; τ k : T → T are nondecreasing right-dense continuous with τ k ( t ) ≤ t , lim t → ∞ τ k ( t ) = ∞ ; and the exponents satisfy β 1 > ⋯ > β m > α > β m + 1 > ⋯ β n > 0 . All results are new even for T = R and T = Z . Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives. Two examples are provided to illustrate one of the theorems.

Interval oscillation criteria for second order super‐half linear functional differential equations with delay and advanced arguments

AbstractSufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the formequation imagewhereϕδ(u) = |u|δ–1u;α> 0,β≥α, andγ≥αare real numbers;k,p,q,e,τ,σare continuous real‐valued functions;τ(t) ≤tandσ(t) ≥twith limt→∞τ(t) = ∞. The functionsp(t),q(t), ande(t) are allowed to change sign, provided thatp(t) andq(t) are nonnegative on a sequence of intervals on whiche(t) alternates sign.As an illustrative example we show that every solution ofequation imageis oscillatory provided that eitherm1orm2orr0is sufficiently large (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interval oscillation of second-order functional differential equations with oscillatory potentials

Interval oscillation criteria are established for second-order differential equations of the form ( E γ ) ( k ( t ) x ′ ( t ) ) ′ + p ( t ) x ( g ( t ) ) + q ( t ) | x ( g ( t ) ) | γ − 1 x ( g ( t ) ) = e ( t ) , t ≥ 0 where γ > 1 ; k , p , q , e and g are continuous real-valued functions; k ( t ) > 0 is nondecreasing; g ( t ) is nondecreasing, lim t → ∞ g ( t ) = ∞ . We consider g ( t ) ≤ t and g ( t ) ≥ t , separately. Illustrative examples are included.

Riccati techniques and oscillation for self-adjoint matrix Hamiltonian systems

The purpose of this paper is to develop a generalized matrix Riccati technique for the self-adjoint matrix Hamiltonian system U^'=A(x)U+B(x)V,V^'=C(x)U-A^*(x)V. Together with the integral averaging technique and monotone functionals, new oscillation and interval oscillation criteria are established for the system. These criteria extend, improve, complement a number of existing results, and handle some cases not covered by known criteria. In particular, two interesting examples are included to illustrate the versatility of our results.

Nonlinear oscillation of 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 the Young inequality. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Oscillation criteria for a forced mixed type Emden–Fowler equation with impulses

Some oscillation criteria for a forced mixed type Emden–Fowler equation with impulses are given. When the impulses are dropped, our results extend those of Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 15 (2008) 375–381], Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] for second-order forced ordinary differential equation with mixed nonlinearities, Nasr [A.H. Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Am. Math. Soc. 126 (1998) 123–125], Yang [Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49–64] for forced superlinear Emden–Fowler equation, Kong [Q. Kong, Interval criteria for oscillation of second-order linear differential equations, J. Math. Anal. Appl. 229 (1999) 483–492] for unforced second order linear differential equations, and Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999) 235–240] for forced second order linear differential equation.

Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects

In this paper, we derive new interval oscillation criteria for a forced super-half-linear impulsive differential equation having fixed moments of impulse actions. The results are extended to a more general class of nonlinear impulsive differential equations. Examples are also given to illustrate the relevance of the results.

New oscillation criteria of second-order nonlinear differential equations

By employing a class of new functions Φ = Φ ( t , s , l ) and a generalized Riccati technique, some new oscillation and interval oscillation criteria are established for the second-order nonlinear differential equation ( r ( t ) y ′ ( t ) ) ′ + Q ( t , y ( t ) , y ′ ( t ) ) = 0 . The obtained interval oscillation criteria can be applied to equations with forcing term. Two examples are also included to show the significance of our results.

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.

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 for second order sublinear differential equations with a damping term

We present new oscillation criteria for the second order nonlinear differential equation with a damping term (a(t)y′(t))′ + p(t)y′(t) + q(t)|y(t)|α−1y(t) = 0, t ≥ t0 where 0 < α ≤ 1. Our results here are different, generalise and improve some known results for oscillation of second order nonlinear differential equations that are different from most known ones in the sencse they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line and can be applied to extreme cases such as ∫t0∞ q(t)dt = −∞. Our results are illustrated with examples.

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.

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

We present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities: ( p ( t ) x ′ ) ′ + q ( t ) x + ∑ i = 1 n q i ( t ) | x | α i sgn x = e ( t ) , where p ( t ) , q ( t ) , q i ( t ) , e ( t ) ∈ C [ 0 , ∞ ) , p ( t ) is positive and differentiable, α 1 > ⋯ > α m > 1 > α m + 1 > ⋯ > α n . No restriction is imposed on the forcing term e ( t ) to be the second derivative of an oscillatory function. When n = 1 , our results reduce to those of El-Sayed [M.A. El-Sayed, An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc. 118 (1993) 813–817], Wong [J.S.W. Wong, Oscillation criteria for a forced second linear differential equations, J. Math. Anal. Appl. 231 (1999) 235–240], Sun, Ou and Wong [Y.G. Sun, C.H. Ou, J.S.W. Wong, Interval oscillation theorems for a linear second order differential equation, Comput. Math. Appl. 48 (2004) 1693–1699] for the linear equation, Nazr [A.H. Nazr, Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 (1998) 123–125] for the superlinear equation, and Sun and Wong [Y.G. Sun, J.S.W. Wong, Note on forced oscillation of nth-order sublinear differential equations, J. Math. Anal. Appl. 298 (2004) 114–119] for the sublinear equation.