The Bishop–Phelps theorem in complete random normed modules endowed with the -topology

In this paper, we adopt a new approach so that we can prove that the Bishop–Phelps theorem in complete random normed modules still holds under the ( ε , λ ) -topology, which solves an open problem posed in [T.X. Guo, Y.J. Yang, Ekelandʼs variational principle for an L ¯ 0 -valued function on a complete random metric space, J. Math. Anal. Appl. 389 (2012) 1–14].