Oscillation of solutions of second-order nonlinear self-adjoint differential equations

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

On the oscillation of certain second-order nonlinear differential equations

In this paper, oscillation criteria for the nonlinear second-order ordinary differential equation

     

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.     

Oscillation of second-order nonlinear neutral differential equations

In this paper, some sufficient conditions for oscillation and nonoscillation are obtained for the second-order nonlinear neutral differential equation

(∗)     

Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.     

Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations

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.     

Oscillation of some nonlinear difference equations

We investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations. Several explicit oscillation criteria are established. Our main results are supported by illustrative examples.     

Limit-point type solutions of nonlinear differential equations

We are concerned with the nonexistence of L²-solutions of a nonlinear differential equation x″=a(t)x+f(t,x). By applying technique similar to that exploited by Hallam [SIAM J. Appl. Math. 19 (1970) 430-439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L²(t₀,∞) under milder conditions on the function a(t) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes [Nonlinear Anal. 7 (1983) 851-871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations.     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation

     

Oscillation criteria for a class of second-order neutral delay differential equations

In this article, we investigate oscillation and asymptotic behaviour of all solutions of a class of neutral delay differential equations of second-order with several positive and negative coefficients having the form

where R,P,Q are bounded beginning segments of positive integers, , , are delay functions and f is a continuous function. Our results improve and extend the recent results given in the papers [J. Manojlović, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second-order differential equations with positive and negative coefficients, Appl. Math. Comput. 181 (2006) 853–863] and [A. Weng, J. Sun, Oscillation of second order delay differential equations, Appl. Math. Comput. 198 (2) (2008) 930–935].     

The continuation of solutions for certain nonlinear differential equations and its applications

We give some sufficient conditions for the continuation of solutions for some nonlinear differential equations. As an application, we obtain a new criterion for the oscillation of solutions of the Liénard equation.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

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:

     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

New oscillation criteria for second-order nonlinear neutral delay differential equations

In this paper, several new oscillation criteria for the second-order nonlinear neutral delay differential equation

     

Oscillation criteria for second-order nonlinear neutral delay dynamic equations

In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation

(r(t)((y(t)+p(t)y(t−τ)Δ⁾γ⁾Δ⁾+f(t,y(t−δ))=0