Some new results on stability of Ekeland's ϵ-variational principle

In this paper, we give an example and point out that ϵ-solutions of Ekeland's variational principle are not always lower semicontinuous in infinite-dimensional Banach spaces, even with respect to the uniform metric. Further, the example shows that the ϵ-solutions need not be almost lower semicontinuous when the convergence of sequence of functions is weakened to Painlevé–Kuratowski epigraphical convergence. To provide some results of stability, we prove the almost lower semicontinuity of ϵ-solutions in a general framework.