Some generalizations of Caristi type fixed point theorem on partial metric spaces

Abstract

In the persent paper, we give Bae and Suzuki type generalizations of Caristi?s fixed point theorem on partial metric space.

Generalized Geraghty type mappings on partial metric spaces and fixed point results

“…Later on, Abdeljawad et al [1], Acar et al [2,3], Altun et al [4][5][6][7], Karapinar and Erhan [15], Oltra and Valero [17] and Valero [23] gave some generalizations of the result of Matthews. Also,Ćirić et al [8], Samet et al [21] and Shatanawi et al [22] proved some common fixed point results in partial metric spaces.…”

Weak condition for generalized f-weakly Picard mappings on partial metric spaces

“…Afterward, Acar et al [1,2], Altun et al [4,5,7,8], Karapinar and Erhan [17], Oltra and Valero [21], Romaguera [22,23] and Valero [31], gave some generalizations of the result of Matthews. Also, Ciric et al [13], Samet et al [27] and Shatanawi et al [30] proved some common fixed point results in partial metric spaces.…”

Refinements of Caristis fixed point theorem

“…We divide this proof into two parts: (1). There exists n such that d(x n+1 , x n ) > d(x n , x n−1 ).…”

“…However, there are still two problems in some practical cases: (1) the condition d(T x, x) ≤ · · · is too strong to be verified; (2) the backgrounds of some questions are only involving a part of points in metric spaces, that is, d(T x, x) ≤ · · · not hold for all points. To overcome the problems (1) and (2), we introduce a new Caristi-type fixed point theorems in the forms min {d(T x, T y), d(T x, x)} ≤ Dominated Function, (1.1) where the "Dominated Function" can be chosen as kd(x, y) + φ(x) − φ(T x), (1.2) or Φ(x)(f (x) − f (T x)), (1.3) or other corresponding forms under some advanced settings such as 'partial order', 'graph' and 'cyclic map', etc. To our knowledge, we provide all the possible conditions to make the Caristi-type fixed point theorem appropriately and applicably in most situations.…”

“…To our knowledge, we provide all the possible conditions to make the Caristi-type fixed point theorem appropriately and applicably in most situations. For known Caristi-type fixed point results in the literature, see [1,5,11,12,14,16,17]. This paper is organized as follows.…”