On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems

We obtain a new quantitative deformation lemma, and then gain a new mountain pass theorem. More precisely, the new mountain pass theorem is independent of the functional value on the boundary of the mountain, which improves the well known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass theorem, new existence of nontrivial periodic solutions for some nonlinear second-order discrete systems is obtained, which greatly improves the result in \cite{Z04}.Comment: 11 page

Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories

Periodic q-difference equations

The concept of periodic functions defined on the real numbers or on the integers is a classical topic and has been studied intensively, yielding numerous applications in every kind of science. It is of importance that the real numbers and the integers are closed with respect to addition. However, for a number q \u3e 1, the so-called q-time scale, i.e., the set of nonnegative integer powers of q, is not closed with respect to addition, and therefore it was not possible to define periodic functions on the q-time scale in an obvious way. In this thesis, this important open problem has been resolved and the definition of periodic functions defined on the q-time scale is given. Using this new definition of periodic functions defined on the q-time scale, five distinct results involving periodic solutions of various kinds of q-difference equations are presented, namely as follows. First, Floquet theory for q-difference equations is established. Second, the Cushing-Henson conjecture is proved for periodic solutions of the Beverton-Holt q-difference equation, resulting in applications in the study of biology, in particular population models. Third, stability for Hamiltonian q-difference systems is investigated. Fourth, the existence of periodic solutions of a q-difference boundary value problem is examined by applying the well-known Mountain Pass theorem. Fifth, the existence of positive periodic solutions of higher-order functional q-difference equations is studied by applying the well-known fixed-point theorem in a cone. Besides these five research papers that are based on the newly introduced definition of periodic functions on the q-time scale, this thesis also contains an introduction, a section on time scales calculus, a section on quantum calculus, and a conclusion --Abstract, page v