A selection theorem is proved which contains as a particular case results of

*N. C. Yannelis* and

*N. D. Prabhakar* [J. Math. Econ. 12, 233-245 (1983;

Zbl 0536.90019)] and of

*X. P. Ding, W. K. Kim, K.-K. Tan* [Bull. Aust. Math. Soc. 46, 205-212 (1992;

Zbl 0762.47030)]. Applying this selection, a fixed point theorem, a nonempty intersection theorem, some existence theorems of solutions for generalized quasi-variational inequalities and some equilibrium existence theorems for abstract economies are proved.