Caristi type fixed point theorems using Száz principle in quasi-metric spaces

Abstract

In this paper, we use Szaz maximum principle to prove generalizations of Caristi fixed point theorem in a ´ preordered K-complete quasi metric space. Examples are given to support our results.

On Caristi fixed point theorem for set-valued mappings

“…However, in order to study and discuss Penot's problem, we use the ideas of this direction. For further details, one can consult [1,7,2,16,18] and references therein.…”

“…It is not only a generalization of the Banach contraction principle [4] but it has also been proven to be equivalent to metric completeness [14,Theorem 6]. Moreover, it has been the subject of various generalizations and extensions (see e.g., [1,5,7] and the related references therein). For instance, in attempting to generalize Caristi's fixed point theorem, Kirk [12] raised the problem of whether a self-mapping T has a fixed point on a metric space (M, d) such that for all x ∈ M η(d(x, T x)) ≤ φ(x) − φ(T x), where η is a function from R + , the set of all nonnegative reals, into R + , having appropriate properties.…”

“…Let (M, d) be a complete metric space and φ ∶ M → R + a lower semi-continuous mapping. Let T ∶ M → C(M ) be a set-valued mapping with nonempty closed values such that (1) dist(x, T (x)) ≤ φ(x) − inf φ(T (x)), ∀x ∈ M.…”

On Generalized Caristi Type Satisfying Admissibility Mappings

“…In the literature, this theorem is known as the Caristi fxed point theorem (CFPT). New publications on the CFPT type are available: Aamri et al [16] obtained CFPTs using the Száz principle in quasi-MS, Altun et al [17] got CFPTs and some generalizations on M-MS, Aslantas et al [18] gave some CFPTs, and a study on CFPTs in MS with a graph was given by Chuensupantharat and Gopal [19]. Direct proof of CFP was shown by Du [20], Kuhlmann et al [21] showed the CFPTs from the point of view of ball spaces, and Hardan et al [22] surveyed CFPTs of contractive mapping with application.…”