Existence of infinitely many periodic solutions to the nonlinear Dirac-Klein-Gordon system

Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory.

The existence of periodic solutions for non-automous second order systems at resonance is investigated.With the sub-linear increase of the non-linear term,the periodic solutions of the system are converted into the critical points of a functional defined on a proper space,and the existence of periodic solutions is proved through the critical point theory.

Through the application of the critical point theory, the problem on the existence of periodic solutions for a class of second-order partial difference equations is investigated when the nonlinear terms are superlinear or asymptotically linear growth at the origin or infinity. Specifically, sufficient conditions for the existence of two nonconstant periodic solutions of this type of equations are shown without the Ambrosetti-Rabinowitz superlinear condition and the symmetry of nonlinearities. Our result not only improves and extends the existing ones in the literature, but also applies to show the existence of space-time periodic solutions of two-dimensional discrete Schrödinger equations with mixed nonlinearities.

where x ∈ En, S and A are constant matrices, S is in general a singular matrix, f(t, x, λ) is a 2π-periodic function of t, λ ∈ Em, λ is a parameter, t ∈ (−∞,∞), and Ep is the p-dimensional vector space. Further, we assume that f(t, x, λ) = C(t, x, λ) + D(t, x, λ), where C(t, x, λ) is a form of order s > 1 in the variables x and λ, D(t, x, λ) is a finite sum of forms of order > s in the same variables, C(t, 0, λ) ≡ 0, and D(t, 0, λ) ≡ 0. Consequently, x = 0 is a solution of system (1) for any λ. The problem on the existence of a periodic solution of system (1) for S = E (E is the identity matrix) was considered in [1–5]; an important requirement in this connection was that the generating solution is a nonzero solution. If the zero solution of system (1) is chosen as the generating solution, then the zero solution is the unique periodic solution whose existence was guaranteed in the above-mentioned papers. The same problem was considered in [6]. The existence and uniqueness theorem for a periodic solution was proved there by the method of successive approximations without using any properties of the principal solution matrix of the linear approximation system. The contraction mapping principle was used in [7] to obtain conditions for the existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial with a prescribed number of harmonics and a remainder term. The existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial and a series remainder was proved in [8, p. 375; 9, p. 202 of the Russian translation] by the method of successive approximations; the coefficients of the polynomial were found from an equation that is transcendental in general. Note that the papers [6–9] do not answer the question as to whether the periodic solution whose existence was proved in these papers for the case in which x = 0 is a solution of system (1) for any value of the parameter λ is nonzero. The problem on the existence of solutions and periodic solutions of system (1) was considered in [10–14] in the linear case under the assumption that S can be a singular matrix. In the present paper, we pose the problem of finding conditions for the existence of a nonzero periodic solution of system (1) in a neighborhood of the zero solution. The existence of a periodic solution in the form of a trigonometric series is proved by a fixed point method under a necessary condition for the existence of a solution of some finite-dimensional algebraic system of equations. We introduce the following notation: |u| = maxi {|ui|}, ‖Q‖ = max|u|≤1 |Qu|, u ∈ Ep, Q is a matrix, N is the set of positive integers, and N0 is the set of nonnegative integers. Let Mn be the set of series a0 + ∑∞ k=1 ak cos kt + bk sin kt, where a0 and ak, bk (the coefficients of the series) are n-dimensional vectors for any k ∈ N . The zero element of the set Mn is defined as the series with zero coefficients and is denoted by 0. Let x = a0 + ∑∞ k=1 ak cos kt + bk sin kt ∈ Mn and y = c0 + ∑∞ k=1 ck cos kt + dk sin kt ∈ Mn, let Q be an n× n matrix, and let γ be a real number. Then the operations x+ y, Qx, γx, and ẋ are defined by the formulas

Multiplicity results for periodic solutions to delay differential equations via critical point theory

In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.

Existence and numerical simulation of periodic traveling wave solutions to the Casimir equation for the Ito system

In this paper, the existence of periodic solutions is obtained for a kind of p-Laplacian systems by the minimax methods in critical point theory. Moreover, the existence of infinite periodic solutions is also obtained.

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation 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.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

Chapter 2 Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods

A novel hybrid model of tumor control, combining pulse surveillance with tumor size-guided therapies

The existence of periodic solutions to certain impulsive differential equations

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.

We obtain a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory.