The authors consider the discrete equation
here is the difference operator, is a continuous map from into , and . A nontrivial solution of this equation is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The main result of the paper under review says that if and there exist and such that
then the above discrete equation has a bounded nonoscillatory solution. The authors notice that the latter condition can be replaced by the less restrictive condition
and the theorem remains valid. The results extend and correct the results of R. P. Agarwal and P. J. Y. Wong [Advanced topics in difference equations (1997; Zbl 0878.39001)].