title: ψ-Contraction and [Formula: see text]-contraction in Menger probabilistic metric space.
abstract: The purpose of this paper is to present the definition of [Formula: see text]-contractive mapping and to discuss the relation of [Formula: see text]-contractive mappings and [Formula: see text]-contractive mappings. Furthermore, the generalized [Formula: see text]-contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized [Formula: see text]-contraction mapping principle has been obtained by using an ingenious method.
authors.affiliation: Department of Mathematics, College of Science, Hebei North University, Zhangjiakou, 075000 People's Republic of China.
authors.forename: Pengcheng
authors.initials: P
authors.lastname: Ma
authors.affiliation: Department of Mathematics, College of Science, Hebei North University, Zhangjiakou, 075000 People's Republic of China.
authors.forename: Jinyu
authors.initials: J
authors.lastname: Guan
authors.affiliation: Department of Mathematics, College of Science, Hebei North University, Zhangjiakou, 075000 People's Republic of China.
authors.forename: Yanxia
authors.initials: Y
authors.lastname: Tang
authors.affiliation: Department of Mathematics, College of Science, Hebei North University, Zhangjiakou, 075000 People's Republic of China.
authors.forename: Yongchun
authors.initials: Y
authors.lastname: Xu
authors.affiliation: Department of Mathematics, College of Science, Hebei North University, Zhangjiakou, 075000 People's Republic of China.|Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387 People's Republic of China.
authors.forename: Yongfu
authors.identifier: 0000-0001-8656-471X
authors.initials: Y
authors.lastname: Su
keywords: Contraction mapping principle; Probabilistic metric spaces; [Formula: see text]-Contraction