中文摘要：

应用临界点理论中的山路引理，研究了一类四阶非线性差分方程周期解和次调和解的存在性问题．通过把方程解的存在性转化为某个泛函临界点的存在性，获得了一类四阶非线性差分方程周期解和次调和解的存在性和多重性的一些充分条件，给出周期解和次调和解的存在性和多重性准则．

英文摘要：

By using the notable Mountain Pass Lemma of critical point theory, some sufficient condi-tions for the existence and multiplicity of periodic and subharmonic solutions to a class of fourth-order non-linear difference equations are obtained. The proof is based on the Mountain Pass Lemma in combination with variational technique. A practicable method to solve the existence and multiplicity of periodic and sub-harmonic solutions for fourth-order nonlinear forward and backward difference equations is given.