Existence of periodic solutions of fourth-order nonlinear difference equations

Abstract

By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational technique. The problem is to solve the existence of periodic solutions of fourth-order nonlinear difference equations. Results obtained complement the existing one.

Existence of Periodic Solutions for a Class of Fourth-Order Difference Equation

“…In recent years, the theory of nonlinear difference equations has been widely used in the study of discrete models in the fields of economics, neural networks, ecology, etc. For the general background of difference equations, in particular, there are many authors who have discussed the existence and multiplicity of periodic solutions for discrete boundary value problems by exploiting various methods, including the method of upper and lower solutions, Leray-Schauder degree, fixed point theory, critical theory, and variational methods; see Bereanu and Mawhin [1], Cabada and Dimitrov [2], Graef et al [3,4], and Cai et al [5][6][7][8][9][10] and the references therein.…”

Multiple solutions of fourth-order difference equations with different boundary conditions

“…With the rapid development of the technique of computers and the theory of nonlinear difference equations, difference equations have been widely used to study discrete models in many fields such as finance insurance, computing, electrical circuit analysis, dynamical systems, physical field, and biology; see [4,5] and the references therein. Importantly, much literature and many monographs deal with problems of the existence and multiplicity of solutions by using various methods, such as critical point theory [6][7][8], topological degree theory [9], fixed-point index theory [10]. Recently, some literature [11,12] studied solutions of φ c -Laplacian difference equations.…”