By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equationfor any (t, z) ∈ R × 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.

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Consider the second-order discrete systemfor any (t, Z) ∈ R × R m and M is a positive integer. By making use of critical-point theory, the existence of M -periodic solutions of ( * ) is obtained.

In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equationswithout any periodicity assumptions on p(t), q(t) and f , providing that f (t, x) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R, and satisfies some additional assumptions.

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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.

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 R₀ 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 R₀. In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute R₀.