On the Weakly Almost Periodic Point and the Periodic Sequence Shadowing Property Under Strongly Uniform Convergence

The weakly almost periodic point and the periodic sequence shadowing property are studied under strongly uniform convergence. Some conclusions about them are obtained. First, let the sequence map f_n converge strongly uniformly to the equicontinuous map f and the sequence of pointsx_kbe the weakly almost periodic point of every map f_n. If \mathop \lim \limits_k \to \infty x_k = x, then the point x is the weakly almost periodic point of the map f. Second, if the sequence mapf_nconverges strongly uniformly to the equicontinuous map f, then limsup W(f_n)⊂W(f). Third, if f_n has the fine periodic sequence shadowing property, then f has periodic sequence shadowing property.