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A correction on the oscillatory behavior of solutions of certain second order nonlinear differential equations
Abstract
In this paper, we correct a result stated in a recent paper related to the oscillatory behavior of solutions of certain second order nonlinear differential equations.
... ) is oscillatory. Also, there is a good amount of literature on oscillation of (1.2) (see [1] [2] [8] [9] [10] [12] [13] and the literature cited therein). In 1992, James S. W. Wong [12] proved the following extension of Cole's result [1] to the more general equation (1.2). ...
We obtain a new oscillation theorem for the nonlinear second-order differential equation via the generalization of Leighton's variational theorem.
This paper is imposed an oscillatory behavior of nonlinear delay differential equation (NDDE). In particular, the main results got improve those studied in the literature survey. A variety of oscillatory behaviors are given.
In this paper, we consider the asymptotic and oscillatory behavior of solutions of the nonlinear difference equation , where and p, q are odd integers. Several illustrative examples are given.
In this paper, we establish some sufficient conditions for the asymptotic behavior of all nonoscillatory solutions of the differential equation,{p(t)φ(y′(t))}′+q(t)f(y(t))=0{p(t)φ(y′(t))}′+q(t)f(y(t))=0,under suitable condition on p∈C([t0,∞);(0,∞)),q∈C([t0,∞),R)p∈C([t0,∞);(0,∞)),q∈C([t0,∞),R), and φ,f∈C(R;R)φ,f∈C(R;R).
In this paper, for second-order superlinear differential equations, we present new oscillation criteria. Our results involve comparison with related linear and half-linear second-order differential equations, so that the known oscillation theorems from literature can be employed directly.
Necessary and sufficient conditions are obtained for the existence of positive solutions of a nonlinear differential equation. Relations between this equation and an advanced type nonlinear differential equation are also discussed.
Classification schemes for positive solutions of a class of second order nonlinear differential systems are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also provided.
Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.
Some oscillation criteria for the second-order quasilinear functional differential equations are obtained. Our results generalize and improve some known results of both differential equations and delay differential equations.
Using the integral average method, we establish some oscillation criteria of Kamenev type and Yan type for the nonlinear system of differential equation ui'=|u3−i|λisgnu3−i+(−1)i−1bi(t)ui,(i=1,2) where the functions bi(t) (i = 1, 2) are nonnegative and summable on each finite segment of the interval Z0, ∞), λi > 0 (i = 1,2) with λ1 λ2 = 1.
In this paper, for all regular solutions of a class of second-order nonlinear perturbed differential equations, new oscillation criteria are established. Asymptotic behavior for forced equations is also discussed.
Sufficient conditions for oscillation of all solutions of a class of second-order quasilinear delay differential equations
with fixed moments of impulse effect are found.
In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations
{rcl}&&\left\{a(t)\Psi(x(t))\left[|(x(t)+p(t)x(\tau(t)))^{\Delta ^{n-1}}|^{\alpha-1}(x(t)+p(t)x(\tau(t)))^{\Delta^{n-1}}\right]^{\gamma}\right\}^{\Delta}\\[12pt]&&{}\quad +\lambda F(t,x(\delta(t)))=0,\begin{array}{rcl}&&\left\{a(t)\Psi(x(t))\left[|(x(t)+p(t)x(\tau(t)))^{\Delta ^{n-1}}|^{\alpha-1}(x(t)+p(t)x(\tau(t)))^{\Delta^{n-1}}\right]^{\gamma}\right\}^{\Delta}\\[12pt]&&{}\quad +\lambda F(t,x(\delta(t)))=0,\end{array}
on time scales, where α>0 is a constant, γ>0 is a quotient of odd positive integers and λ=±1. Our results in this paper not only extend and improve some known results but also present a valuable unified approach
for the investigation of oscillation and asymptotic behavior of nth-order nonlinear neutral delay differential equations and nth-order nonlinear neutral delay difference equations. Examples are provided to show the importance of our main results.
KeywordsOscillation-Asymptotic behavior-
nth-order nonlinear neutral delay dynamic equation-Time scale
Mathematics Subject Classification (2000)34K11-34K40
Some oscillation criteria for the second order retarded differential equation [r(t)¦u′(t)¦α−1 u′(t)]′ + p(t)¦u(τ(t))¦β−1u(τ(t)) = 0 are given, where α and β are positive constants, and satisfy some suitable conditions. Our results generalize those of Erbe, Ohriska and Lakshmikatham.
In this paper, oscillation criteria for the nonlinear second-order ordinary differential equation are given. The results extend the integral averaging technique and include earlier results. Our methodology is somewhat different from that of previous authors.
Using the generalized Riccati technique and the averaging technique, new oscillation criteria for certain even order delay differential equation of the form are established, where g∈C([t0,∞),R), F∈C([t0,∞)×R,R), and α>0 is a constant.
In this paper we consider the second-order nonlinear differential equation [a(t)(y'(t))(sigma)]' + q(t)f(y(t)) = 0, (*) where sigma > 0 is any quotient of odd integers, a is an element of C-1(R,(0,infinity)), q is an element of C(R,R), f is an element of C(R,R), xf(x)>0, f'(x)greater than or equal to 0 for x not equal 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples that dwell upon the importance of our results are also included. (C) 1998 Academic Press.
For the half-linear difference equation where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.
We present new interval oscillation criteria for certain classes of second-order nonlinear perturbed differential equations which are based on the information only on a sequence of subintervals of [to, ∞), rather than on the whole half-line. Our results are sharper than those known in the literature and deal with the cases which are not covered by the known criteria. In particular, our results can be applied to extreme cases such as η∞q(s)ds = −∞. Finally, two examples which dwell upon the sharpness of the conditions of our results are also illustrated.
Oscillation criteria for nth order differential equations with deviating arguments of the formare established, where g ∈ C([t0, ∞), ), F ∈ C([t0, ∞) × , ), and α > 0 is a constant.
Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]σ′i +p(t)[y′(t)]σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, aϵC(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [to, ∞), and fϵCl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.
Here, we shall study the oscillatory behavior of solutions of n(≥ 1)th order nonlinear neutral delay difference equations of the following form
(22.1)
where τ, σ are fixed non-negative integers, functions p, q are defined on N, q(k) ≥ 0, k ∈ N,and the continuous function f: ℝ →> ℝ satisfies (12.2).
We obtain some nonoscillatory theories of the second-order nonlinear difference equation Δ(r n (Δx n ) α )+f(n+1,x n+1 )=0,n∈ℕ where α is a quotient of positive odd integers, r n >0 for n∈ℕ and f∈C(ℕ×ℝ,ℝ).
A class of second-order nonlinear differential equations with damping term
(r(t)|x′(t)|x′(t))′+p(t)|x′(t)|x′(t)+q(t)f(x(t))=0
are investigated in this paper. By using a new method, we obtain some new sufficient conditions for
the oscillation of the above equation, and some references are extended in this paper. Examples are
inserted to illustrate this result.
The problem of oscillation of super- and sublinear Emden-Fowler equations is studied. Established are a number of oscillation theorems involving comparison with related linear equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations. 25 references.
Some new integral criteria for the oscillation of the nonlinear second order differential equation with damped term y” (t) + p(t)y’(t) + q(t)f(y(t)) = 0 are given.
A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″ ( t )+ a ( t ) f[x ( t ) ]=0 , where a ∈ C ( [t 0 ∞ ,)), f∈C ( R ) with yf ( y )>0 for y≠0 and and f is continously differentiable on R-{0} with f' ( y ) ≥0 for all y≠O. In the special case of the differential equation (γ > 1) this criterion leads to an oscillation result due to Wong [9].
New oscillation criteria are given for the differential equation u'' + a(t) |u|^\alpha \operatorname{sgn} u = 0, 0
The sublinear differential equation x’'(t) + a(t)f[x(t)] = 0, t ≥ to > 0 is considered, in which a ∈ C([t0,∞)), f ∈ C(R) with y f(y) > 0 for y ≠ 0 and ∫ ±1±0[1/f(y)dy < ∞, and f has a continuous derivative on R-{0} with f’(y) ≥ 0 for all y ≠ 0. No sign condition is assumed on a. Two new oscillation criteria are obtained. These criteria involve the average behavior of the integral of the coefficient a.
Recently, Kamenev (Differentsialńyl Uravneniya, 13 (1977), 2141–2148), has discussed the oscillatory property of regular solutions of the equation y″ + p(x) f(y) = 0 without the restriction p(x) > 0. The purpose of this paper is to show that Kamenev's results do hold equally well in the case of the more general second order nonlinear equation (r(x)y′)′ + p(x) f(y) = 0.
Best possible conditions are given here, under which all solutions of the equation y″(t) + p(t)f(y(t), y(g(t))) = 0 are oscillatory.
In this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation[formula]where 0<σ=p/qwithp, qodd integers, orpeven andqodd integers. Several examples which dwell upon the importance of our results are also included.
A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: limt → ∞ ∝t0a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that limt → ∞ ∝t0a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.