• Title, Summary, Keyword: periodic solution

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PERIODIC SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS TO LOGISTIC EQUATION AND NEURAL NETWORKS

  • Li, Dingshi;Xu, Daoyi
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1165-1181
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    • 2013
  • In this paper, we consider a class of periodic It$\hat{o}$ stochastic delay differential equations by using the properties of periodic Markov processes, and some sufficient conditions for the existence of periodic solution of the delay equations are given. These existence theorems improve the results obtained by It$\hat{o}$ et al. [6], Bainov et al. [1] and Xu et al. [15]. As applications, we study the existence of periodic solution of periodic stochastic logistic equation and periodic stochastic neural networks with infinite delays, respectively. The theorem for the existence of periodic solution of periodic stochastic logistic equation improve the result obtained by Jiang et al. [7].

PERIODIC SOLUTIONS FOR DISCRETE ONE-PREDATOR TWO-PREY SYSTEM WITH THE MODIFIED LESLIE-GOWER FUNCTIONAL RESPONSE

  • Shi, Xiangyun;Zhou, Xueyong;Song, Xinyu
    • Journal of applied mathematics & informatics
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    • v.27 no.3_4
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    • pp.639-651
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    • 2009
  • In this paper, we study a discrete Leslie-Gower one-predator two-prey model. By using the method of coincidence degree and some techniques, we obtain the existence of at least one positive periodic solution of the system. By linalization of the model at positive periodic solution and construction of Lyapunov function, sufficient conditions are obtained to ensure the global stability of the positive periodic solution. Numerical simulations are carried out to explain the analytical findings.

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PSEUDO ALMOST PERIODIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS INVOLVING REFLECTION OF THE ARGUMENT

  • Piao, Daxiong
    • Journal of the Korean Mathematical Society
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    • v.41 no.4
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    • pp.747-754
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    • 2004
  • In this paper we investigate the existence and uniqueness of almost periodic and pseudo almost periodic solution for nonlinear differential equation with reflection of argument. For the case of almost periodic forced term, we consider the frequency modules of the solutions.

PERIODIC SOLUTIONS OF VOLTERRA EQUATIONS

  • Choi, Sung Kyu;Koo, Namjip;Yeo, Yun Hei;Yun, ChanMi
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.2
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    • pp.401-409
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    • 2010
  • We study the existence of periodic solutions of Volterra equations by using the limiting equations and contraction mappings.

NEW CONDITIONS ON EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS FOR BAM NEURAL NETWORKS WITH TIME-VARYING DELAYS

  • Zhang, Zhengqiu;Zhou, Zheng
    • Journal of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.223-240
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    • 2011
  • In this paper, the problem on periodic solutions of the bidirectional associative memory neural networks with both periodic coefficients and periodic time-varying delays is discussed. By using degree theory, inequality technique and Lyapunov functional, we establish the existence, uniqueness, and global asymptotic stability of a periodic solution. The obtained results of stability are less restrictive than previously known criteria, and the hypotheses for the boundedness and monotonicity on the activation functions are removed.

PERIODIC SOLUTIONS IN NONLINEAR NEUTRAL DIFFERENCE EQUATIONS WITH FUNCTIONAL DELAY

  • MAROUN MARIETTE R.;RAFFOUL YOUSSEF N.
    • Journal of the Korean Mathematical Society
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    • v.42 no.2
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    • pp.255-268
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    • 2005
  • We use Krasnoselskii's fixed point theorem to show that the nonlinear neutral difference equation with delay x(t + 1) = a(t)x(t) + c(t)${\Delta}$x(t - g(t)) + q(t, x(t), x(t - g(t)) has a periodic solution. To apply Krasnoselskii's fixed point theorem, one would need to construct two mappings; one is contraction and the other is compact. Also, by making use of the variation of parameters techniques we are able, using the contraction mapping principle, to show that the periodic solution is unique.

GLOBAL EXPONENTIAL STABILITY OF ALMOST PERIODIC SOLUTIONS OF HIGH-ORDER HOPFIELD NEURAL NETWORKS WITH DISTRIBUTED DELAYS OF NEUTRAL TYPE

  • Zhao, Lili;Li, Yongkun
    • Journal of applied mathematics & informatics
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    • v.31 no.3_4
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    • pp.577-594
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    • 2013
  • In this paper, we study the global stability and the existence of almost periodic solution of high-order Hopfield neural networks with distributed delays of neutral type. Some sufficient conditions are obtained for the existence, uniqueness and global exponential stability of almost periodic solution by employing fixed point theorem and differential inequality techniques. An example is given to show the effectiveness of the proposed method and results.

RANGE OF PARAMETER FOR THE EXISTENCE OF PERIODIC SOLUTIONS OF LI$\'{E}$NARD DIFFERENTIAL EQUATIONS

  • Lee, Yong-Hoon
    • Bulletin of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.271-279
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    • 1995
  • In 1986, Fabry, Mawhin and Nkashama [1] have considered periodic solutios for Lienard equation $$ (1_s) x" + f(x)x' + g(t,x) = s, $$ where s is a real parameter, f and g are continuous functions, and g is $2\pi$-periodic in t and have proved that if $$ (H) lim_{$\mid$x$\mid$\to\infty} g(t,x) = \infty uniformly in t \in [0,2\pi], $$ there exists $s_1 \in R$ such that $(1_s)$ has no $2\pi$periodic solution if $s< s_1$, and at least one $2\pi$-periodic solution if $s = s_1$, and at least two $2\pi$-periodic solutions if $s > s_1$.s_1$.

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QUALITATIVE ANALYSIS OF A GENERAL PERIODIC SYSTEM

  • Xu, Shihe
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.1039-1048
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    • 2018
  • In this paper we study the dynamics of a general ${\omega}-periodic$ model. Necessary and sufficient conditions for the global stability of zero steady state of the model are given. The conditions under which there exists a unique periodic solutions to the model are determined. We also show that the unique periodic solution is the global attractor of all other positive solutions. Some applications to mathematical models for cancer and tumor growth are given.