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ABSTRACT: This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation
xn + 1 - xn + Pn xn - kn = 0,n Î \mathbbZ (0)x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0), where P
n
is a sequence of nonnegative real numbers, and k
n
is a sequence of nonnegative integers with n − k
n
→∞ as n → ∞.
Science in China Series A Mathematics 04/2012; 46(6):884-892. · 0.70 Impact Factor
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ABSTRACT: In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction-diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number [Formula: see text] for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and [Formula: see text]. In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute [Formula: see text].
Journal of Mathematical Biology 12/2011; · 2.96 Impact Factor
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ABSTRACT: A fantastic resonance light scattering (RLS) enhancement phenomenon was found when the interaction between the metal ion Cu (II) and a natural antioxidant curcumin (C(21)H(20)O(6)) occurred in certain conditions. Based on this phenomenon, a novel and convenient assay of curcumin was developed and successfully applied on the determination of curcumin in human urine samples. This assay applied the RLS technique with a common metal ion Cu (II) as the spectral probe. In the pH range of 6.5-7.5, the interaction between Cu (II) and curcumin occurred and the weak RLS intensity of Cu (II) was greatly enhanced by curcumin. The maximum peak was located at 538.5 nm. Under the optimum conditions, the enhanced RLS intensity was proportional to the concentration of curcumin ranging from 0.4 to 60 microg ml(-1) with the detection limit of 0.07 microg ml(-1). The synthetic and human urine samples were determined satisfactorily. Good recoveries (98.8-102.5%) were obtained in the determination of urine samples, which proved that the assay proposed was reliable and applicable in the determination of curcumin in body fluid. In this work, the RLS and fluorescence spectral characteristics of the chemicals, the optimum conditions of the reaction and the influencing factors were investigated.
Spectrochimica Acta Part A Molecular and Biomolecular Spectroscopy 12/2008; 72(3):518-22. · 2.10 Impact Factor
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ABSTRACT: This article is concerned with the existence of solutions of boundary value problems for nonlinear fourth-order difference equations of the type , P n is nonzero and real value. The authors apply the Linking Theorem in the critical point theory and give new sufficient conditions for the existence of solutions of boundary value problems of the above equation.
Journal of Difference Equations and Applications 05/2006; 12(5):459-466. · 0.80 Impact Factor
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ABSTRACT: By using the Z
p
geometrical index theory, some sufficient conditions on the multiplicity results of periodic solutions to the second-order difference equations
D2xn-1+f(xn)=0\Delta^2x_{n-1}+f(x_n)=0 are obtained. By two examples, we show that our results are the best possible in the sense that the lower bound of the number of periodic solutions cannot be improved.
Journal of Dynamics and Differential Equations 01/2006; 18(4):943-960. · 0.73 Impact Factor
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Computers & Mathematics with Applications. 01/2006; 52:1639-1647.
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ABSTRACT: Using critical point theory, some sufficient conditions are obtained for the existence of nonconstant T -periodic solutions of a class of second order self-adjoint difference equations.
01/2005; 71:146-160.
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Proceedings of the Royal Society of Edinburgh Section A Mathematics 09/2004; 134(05):1013 - 1022. · 0.68 Impact Factor
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ABSTRACT: Using variational methods, we obtain some sufficient conditions on the existence of subharmonic solutions with prescribed minimal period pT for the second-order difference equation
Journal of Dynamics and Differential Equations 01/2004; 16(2):575-586. · 0.73 Impact Factor
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ABSTRACT: By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation $$\Delta^2 x_{n-1} +f(n, x_n)\,{=}\,0,$$ where $f\in C({\bf R}\times {\bf R}^m, {\bf R}^m), f(t+M,z)=f(t,z)$ for any $(t, z)\in {\bf R}\times{\bf R}^m$ and $M$ is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.
Journal of the London Mathematical Society 09/2003; 68(02):419 - 430. · 0.79 Impact Factor
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ABSTRACT: By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic
solutions for difference equations. For secord-order difference equations
D2 xn - 1 + f(n, xn ) = 0,\Delta ^2 x_{n - 1} + f(n, x_n ) = 0,
some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinity in z.
Science in China Series A Mathematics 06/2003; 46(4):506-515. · 0.70 Impact Factor
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ABSTRACT: In this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained.
Journal of Mathematical Analysis and Applications.
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ABSTRACT: In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory.
Journal of Differential Equations.
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ABSTRACT: In this paper, some new results are obtained for the existence of multiple periodic solutions and subharmonic solutions to discrete Hamiltonian systemsby using critical point theory, where x1, , .
Nonlinear Analysis: Theory, Methods & Applications.
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ABSTRACT: Several existence results of multiple periodic solutions for discrete Hamiltonian system are established, where the nonlinearity is either asymptotically linear or superlinear at infinity. The approach used here is based on Morse theory.
Nonlinear Analysis: Theory, Methods & Applications.
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ABSTRACT: In this paper, an existence theorem is obtained for periodic solutions of a second-order discrete Hamiltonian system with a change of sign in the potential by the minimax methods in the critical point theory.
Journal of Mathematical Analysis and Applications.
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ABSTRACT: In this paper, by using critical point theory, we establish some results for the existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems.
Anziam journal: The Australian & New Zealand industrial and applied mahtematics journal, ISSN 1446-1811, Vol. 47, Nº 1, 2005, pags. 89-102.
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ABSTRACT: The authors consider the fourth-order difference equation,Δ2(rn−2Δ2xn−2)+f(n,xn)=0,n∈Z,where f : Z × R → R is a continuous function in the second variable, f(n + T, z) = f(n, z), rn+T = rn, for all (n, z) ∈ Z × R, and T is a positive integer. By using linking theorem, the authors obtain some new criteria for the existence and multiplicity of periodic solutions of the fourth-order difference equation.
Computers & Mathematics with Applications.
