Variational Methods and Their Applications to Systems of Nonlinear Differential-Difference Equations

DOI: https://doi.org/10.62517/jse.202611407

Author(s)

Zhang Peng

Affiliation(s)

School of Mathematics, Qilu Normal University,Shandong, China

Abstract

Nonlinear differential-difference equations and systems thereof arise in multiple disciplines at present, and the existence and multiplicity of their solutions constitute a hot research topic. This paper systematically sorts out the analytical approaches and major research achievements of variational methods applied to nonlinear differential-difference equations and systems. By virtue of critical point theory tools including the Mountain Pass Lemma, Linking Theorem, Nehari Manifold Method and Morse Theory, criteria for the existence and multiplicity of solutions are established. Derivative approaches such as the Variational Iteration Method also exhibit satisfactory performance in constructing approximate analytical solutions to nonlinear differential-difference equations. Variational methods provide a unified and powerful analytical framework for investigations on nonlinear differential-difference equations and systems, and the in-depth integration between variational methods and critical point theory remains a vital research direction in this field.

Keywords

Variational Methods; Nonlinear Differential-Difference Equations and Systems; Applications

References

[1] Fang Y, Yang Q. Research on Meromorphic Solutions of Nonlinear Complex Differential-Difference Equations[J/OL]. Journal of Neijiang Normal University, 1-7[2026-08-01]. [2] Ji X Y, Yang Q. Research on Finite-Order Meromorphic Solutions of Two Classes of Nonlinear Differential-Difference Equations[J]. Journal of Hengyang Normal University, 2025,46(6):62-68. [3] Fan Y P, Cao J L, Wang Z L, et al. Finite Difference Methods for Nonlinear Differential Equations with Double Caputo Fractional Derivatives[J]. Journal of Xiangtan University (Natural Science Edition), 2025,47(3):1-14. [4] Chen M F, Liang M L, Gao Z S. Existence of Meromorphic Solutions for a Class of Nonlinear Differential-Difference Equations[J]. Science China Mathematics, 2026,56(4):747-762. [5] Huang G N, Wang J Y, Wang M Q. Neural Network Solver for Nonlinear Partial Differential Equations Based on Improved Euler Method[J]. Journal of Fuzhou University (Natural Science Edition), 2024,52(4):396-403.