Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments

This paper is devoted to the study of oscillatory behavior of solutions of nonlinear neutral hyperbolic equations with functional arguments by using the integral averaging method and generalized Riccati techniques. First, we establish oscillation results for nonlinear neutral hyperbolic equations by reducing the multidimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities. Second, we present oscillation results for nonlinear neutral hyperbolic equations by utilizing Riccati techniques.     

Multiple scales for two-parameter bifurcations in a neutral equation

Van der Pol??s equation with extended delay feedback is investigated as a neutral differential-difference equation. Normal forms near codimension two bifurcations, including Hopf?Cpitchfork and Hopf?CHopf bifurcation, are determined by the method of multiple scales. Through analyzing the associated amplitude equations, we obtain the detailed bifurcation sets and find some interesting phenomena such as quasi-periodic oscillations and strange attractor, which are confirmed by several numerical simulations.     

Generalized canonical transformation for second-order Birkhoffian systems on time scales

The theory of time scales, which unifies continuous and discrete analysis, provides a powerful mathematical tool for the study of complex dynamic systems. It enables us to understand more clearly the essential problems of continuous systems and discrete systems as well as other complex systems. In this paper, the theory of generalized canonical transformation for second-order Birkhoffian systems on time scales is proposed and studied, which extends the canonical transformation theory of Hamilton canonical equations. First, the condition of generalized canonical transformation for the second-order Birkhoffian system on time scales is established.Second, based on this condition, six basic forms of generalized canonical transformation for the second-order Birkhoffian system on time scales are given. Also, the relationships between new variables and old variables for each of these cases are derived. In the end, an example is given to show the application of the results.     

Oscillation criteria for second order nonlinear differential equation with damping

1IntroductionandProblem Thefollowingisthesecondordernonlineardifferentialequationwithdamping: [p(t)ψ(y)u(y′)]′ r(t)y′(t) q(t)f(y)g(y′)=0(t≥t0),(1) inwhich,p(t)∈C′([t0,∞),(0,∞)),r(t)∈C([t0,∞),(-∞,∞)),q(t)∈C([t0, ∞),[0,∞))withtheexistenceofT≥t0,q(t)≠0,t∈[T,∞),f(y),g(y),ψ(y),u(y) ∈C((-∞,∞),(-∞,∞)),andyf(y)>0,yu(y)>0,y≠0. Thesolutiony(t)ofEq.(1)iscallednormalsolutionify(t)isthenon_constantsolutionof Eq.(1)andsupt≥t0｜y(t)｜>0(refertoRef.[1]).Anormalsolutionisoscill…     

Oscillation behavior of solutions for even order neutral functional differential equations

Even order neutral functional differential equations are considered. Sufficient conditions for the oscillation behavior of solutions for this differential equation are presented. The new results are presented and some examples are also given.     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

Solvability of a class of second-order quasilinear boundary value problems

The second-order quasilinear boundary value problems are considered when the nonlinear term is singular and the limit growth function at the infinite exists. With the introduction of the height function of the nonlinear term on a bounded set and the consideration of the integration of the height function, the existence of the solution is proven. The existence theorem shows that the problem has a solution if the integration of the limit growth function has an appropriate value.     

Oscillation theorems for a second order nonlinear differential equation

In this paper, some new oscillation criteria for a second order nonlinear differential equation with clampings are established. These criteria improve and generalize the related results given in [1-4].     

Group analysis for a linear hyperbolic equation arising from a quasilinear reducible system

The aim of this paper is to carry out infinitesimal group analysis for a second order linear hyperbolic equation in canonical form. To such a kind of equation one could be led in considering the hodograph transformation for a first order reducible system. In the first part of the paper we characterize the most general expression for the generator of the Lie group, which has an infinite dimensional algebra. Then we calculate the invariant surfaces in some special cases of interest, and point out the corresponding ordinary differential equations whose integration allows us to determine possible classes of solutions for the equation we are considering.     

Blow-up of solutions to a quasilinear wave equation for high initial energy

This paper deals with blow-up solutions to a nonlinear hyperbolic equation with variable exponent of nonlinearities. By constructing a new control function and using energy inequalities, the authors obtain the lower bound estimate of the $L^2$ norm of the solution. Furthermore, the concavity arguments are used to prove the nonexistence of solutions; at the same time, an estimate of the upper bound of blow-up time is also obtained. This result extends and improves those of [1], [2].     

A second-order multivalued boundary-value problem and impulsive neutral functional differential inclusions in Banach spaces

In this paper, by using the new fixed-point theorem of O'Regan and Precup and a noncompact measure, we study the existence of solutions of a second-order multivalued boundary-value problem in Banach spaces and the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces. The compactness conditions and the upper-semicontinuity conditions for multivalued integral operators are weakened in this paper. Published in Neliniini Kolyvannya, Vol. 11, No. 2, pp. 191–207, April–June, 2008.     

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.     

The corner solution for quasilinear differential equation with two parameters

I.IntroductionWeconsiderinthispaperboundaryvalueproblemofquasilineardifferentialequationby# pi(x,y)y' g(x,y)=0,a相似文献   

Some integral inequalities on time scales

In this article, we study the reverse HSlder type inequality and HSlder inequality in two dimensional case on time scales. We also obtain many integral inequalities by using HSlder inequalities on time scales which give Hardy＇s inequalities as spacial cases.     

Oscillatory modes in a nonlinear second-order differential equation with delay

Solution properties of the nonlinear second-order delay-differential equation x(t)=–ax(t)+f[x(t–)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.