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1.
The following system considered in this paper:where \({p > 1, p^* > 1 (1/p + 1/p^* = 1)}\) and \({\phi_q(z) = |z|^{q-2}z}\) for q = p or q = p *. This system is referred to as a half-linear system. The coefficient f(t) is assumed to be bounded, but the coefficients e(t), g(t) and h(t) are not necessarily bounded. Sufficient conditions are obtained for global asymptotic stability of the zero solution. Our results can be applied to not only the case that the signs of f(t) and g(t) change like the periodic function but also the case that f(t) and g(t) irregularly have zeros. Some suitable examples are included to illustrate our results.
相似文献
$$x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,$$
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The system to be considered in this paper is
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O. Došlý A. Özbekler R. Šimon Hilscher 《Journal of Mathematical Analysis and Applications》2012,393(2):360-366
In this paper, we present an oscillation criterion for second order half-linear differential equations with periodic coefficients. The method is based on the nonexistence of a proper solution of the related modified Riccati equation. Our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by Sugie and Matsumura (2008). These two theorems provide a complete half-linear extension of the oscillation criterion of Kwong and Wong (2003) dealing with the Hill’s equation. 相似文献
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Jitsuro Sugie 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7151-7167
A necessary and sufficient condition is established for the equilibrium of the oscillator of half-linear type with a damping term, (?p(x′))′+h(t)?p(x′)+?p(x)=0 to be globally asymptotically stable. The obtained criterion is given by the form of a certain growth condition of the damping coefficient h(t) and it can be applied to not only the cases of large damping and small damping but also the case of fluctuating damping. The presented result is new even in the linear cases (p=2). It is also discussed whether a solution of the half-linear differential equation (r(t)?p(x′))′+c(t)?p(x)=0 that converges to a non-zero value exists or not. Some suitable examples are included to illustrate the results in the present paper. 相似文献
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We compute explicitly the oscillation constant for certain half-linear second-order differential equations involving periodic coefficients. If these periodic functions are constants, our results reduce to the well-known oscillation constants for half-linear Euler and Riemann–Weber differential equations. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2011,16(11):4451-4478
An almost periodic nonlinear predator–prey system with N prey species and M predator species is revisited in this paper. A set of essentially new criteria are obtained for the existence and global asymptotic stability of a unique positive almost periodic solution. The obtained results provide an answer to an open problem arising in a recent paper. Not only an important restriction of their theorem is removed, but also some other conditions are weakened in this paper. Moreover, a mistake in their assumption is corrected. By applying the obtained results to some special predator–prey systems and competitive systems, many earlier results are significantly improved. Finally, several illustrative examples and their simulations are provided to show the effectiveness of the obtained results. 相似文献
7.
V. L. Novikov 《Mathematical Notes》1974,16(5):1065-1071
Let x=A(t)x be a system of two linear ordinary differential equations with almost periodic coefficients. Then there exists for any positive ε an almost reducible system of equations x=B(t)x with almost periodic coefficients and such that sup ∥A(t)?B (t)∥<ε.-∞相似文献
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Yu. Yu. Klevtsova 《Journal of Applied and Industrial Mathematics》2009,3(2):234-245
Linear systems of ordinary differential equations with periodic coefficients are considered and some numerical algorithm is proposed for analyzing the asymptotic stability of the zero solution that is based on the Runge-Kutta method of the fourth order of accuracy. 相似文献
14.
In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed. 相似文献
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This paper deals with the question of persistence of differential equations with almost periodic coefficients which model predator-prey systems. It is shown that persistence does occur in many models that incorporate almost periodic time dependence in the prey equation 相似文献
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Yu. A. Mitropol'skii D. I. Martynyuk V. I. Tynnyi 《Ukrainian Mathematical Journal》1993,45(12):1869-1877
We establish sufficient conditions of the reducibility of the linear system of difference equationsx(t+1)=Ax(t) + P(t) x(t) with an almost periodic matrixP(t) to a system with a constant matrix.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1661–1667, December, 1993. 相似文献
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LU Wenlian & CHEN Tianping Key Laboratory of Nonlinear Mathematics Science of Ministry of Education Fudan University Shanghai China 《中国科学A辑(英文版)》2005,48(8):1015-1026
Research on delayed neural networks with variable self-inhibitions, interconnection weights, and inputs is an important issue. In this paper, we discuss a large class of delayed dynamical systems with almost periodic self-inhibitions, inter-connection weights, and inputs. This model is universal and includes delayed systems with time-varying delays, distributed delays as well as combination of both. We prove that under some mild conditions, the system has a unique almost periodic solution, which is globally exponentially stable. We propose a new approach, which is independent of existing theory concerning with existence of almost periodic solution for dynamical systems. 相似文献
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I. N. Blinov 《Mathematical Notes》1970,8(1):534-537
It is proved that among systems with almost periodic analytic coefficients (with an algebraic number as frequency base), there are systems which lose the property of reducibility on the boundary of some disk of values of the parameter.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 115–120, July, 1970.In conclusion the author wishes to express his gratitude to L. B. Danilov and M. G. Rabinovich for their comments on this work and also to D. V. Anosov for his suggestions. 相似文献
19.
Weak asymptotic stability of an equilibrium position for a periodic differential inclusion is studied. First the weak asymptotic stability for a discrete-time inclusion generated by the original differential inclusion is investigated with the help of first approximation techniques. Then using the results for discrete-time inclusions the weak asymptotic stability for the differential inclusion is derived from the properties of its first approximation. 相似文献
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Alexander O. Ignatyev Oleksiy A. Ignatyev 《Journal of Mathematical Analysis and Applications》2006,313(2):678-688
In this paper we consider the system of difference equations