Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings

Abstract

In this paper, the famous Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces,

Remarks on Caristi’s fixed point theorem

“…This is the direction taken by the authors in [9]. We believe that the subadditivity of η is very constraining.…”

Some theorems for a new type of multivalued contractivemaps on metric space

“…Later on, several studies were conducted on a variety of generalizations, extensions, and applications of this result of Nadler (see [1,3,6,7,13,14,16,18]). …”

Fixed point theorem for generalized set-valued contractions on partial metric space

“…In 1969, Nadler [32] extended the famous Banach contraction principle from single-valued mapping to set-valued mappings. The fixed point theory of set-valued contractions initiated by Nadler was developed in different directions by many authors, in particular, by Reich [41,1972], Mizoguchi-Takahashi [31,1989], Takahashi [46,1991], Azé-Penot [9,2006], Feng-Liu [21,2006], Benahmed-Azé [11,2010].…”