Openness results for parametric set-valued mappings in Asplund spaces

In this paper, we first prove a lemma about the lower semicontinuity of the distance function in infinite dimensional spaces, which is crucial to openness results hereafter. Then we obtain some openness results in terms of Fréchet coderivatives for parametric set-valued mappings in Asplund spaces under mild conditions. The results of the paper generalize several corresponding results in the recent literature. Finally, we give two examples to illustrate our openness results.