Disconjugacy, Disfocality, and Oscillation of Second Order Difference Equations

Chen, Shaozhu

doi:10.1006/jdeq.1994.1018

Cited by 50 publications

(21 citation statements)

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“…which has been extensively investigated by many authors [1,3,6], for results on oscillation, asymptotic behavior, boundary value problems, disconjugacy and disfocality. In [21], the periodic solutions of second-order self-adjoint difference equation…”

Section: Introductionmentioning

confidence: 99%

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Existence of periodic solutions for second-order nonlinear difference equations

Ren¹,

Li²,

Shi³

2016

J. Nonlinear Sci. Appl.

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By using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational technique. The problem is to solve the existence of periodic solutions of second-order nonlinear difference equations. One of our results obtained complements the result in the literature.

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Section: Introductionmentioning

confidence: 99%

“…which has been studied in [1,6,22] for results on oscillation, asymptotic behavior and the existence of positive solutions. Moreover, if q n u δ n + f (n, u n+1 , u n , u n−1 ) = q n g(u n ) + r n , (1.1) has been considered in [16] for oscillatory properties of its all solutions.…”

Section: Introductionmentioning

confidence: 99%

Existence of periodic solutions for second-order nonlinear difference equations

Ren¹,

Li²,

Shi³

2016

J. Nonlinear Sci. Appl.

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“…. (1.5) [3] Oscillations of interconnected systems 43 for any co > 0 for which //(«) by J # 0. Hence, if we define an operator g on Z fcodd |_ y=0 J then (1.5) is equivalent to the operator equation x + gn(x) = 0, on…”

Section: >{T)~-^4>exp(ina>t)mentioning

confidence: 99%

“…In this paper, we improve upon the old results and present new results, by providing conditions for the existence of limit cycles in interconnected systems with continuous nonlinearities which do not necessarily satisfy the Lipschitz condition. Using the method described in this paper, we are able to improve the oscillation result in [3] and discuss the existence of periodic solutions of second order difference equations.Of particular interest to the present discussion are some results in [1] and [9]. In this paper, we extend their results to a large class of interconnected systems.…”

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confidence: 99%

Oscillations of Interconnected Systems withC⁰Nonlinearities

He¹

1996

J. Aust. Math. Soc. Series B, Appl. Math

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In this paper we establish conditions which ensure the existence of self-excited oscillations in complex dynamical systems with nondifferentiable nonlinearities, by considering those types of systems which can be viewed as an interconnection of several simpler subsystems. We find that the nonlinear terms of the system in which we are interested do not need to satisfy the Lipschitz condition. IntroductionIn recent years, many researchers have concerned themselves with the qualitative analysis of large-scale dynamical systems. The analysis is in terms of the qualitative properties of the free subsystems and of the structure of the interconnecting system. Examples of this method can be found in [2,6,7,8,10,11]. However these results are not applicable to some systems, for example, when the nonlinearity does not satisfy the Lipschitz condition. In this paper, we improve upon the old results and present new results, by providing conditions for the existence of limit cycles in interconnected systems with continuous nonlinearities which do not necessarily satisfy the Lipschitz condition. Using the method described in this paper, we are able to improve the oscillation result in [3] and discuss the existence of periodic solutions of second order difference equations.Of particular interest to the present discussion are some results in [1] and [9]. In this paper, we extend their results to a large class of interconnected systems.

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“…Since the behaviour of a discrete system is sometimes sharply different to the behaviour of the corresponding continuous system (see, for example, [1,16,17]), many scholars have investigated discrete Hamiltonian systems for the disconjugacy, boundary value problems, oscillations and asymptotic behaviour of (1.1) [3,5,7,8,11]. Only very few papers deal with the existence of periodic solutions even to general difference equations, see for example [2,9,16,17].…”

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confidence: 99%