On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces

Abstract

Common coupled fixed point theorems are examined in this paper for comparable mappings ensuring nonlinear contraction in ordered partial metric spaces. Given theorems enlarge and universalize some conclusions of Gnana Bhaskar and Lakshmikantham (2006).

Imperative Remarks for “On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces” and an Answer to the Question: How to Smooth It Away

“…We first note that we can suggest further corrected forms for the paper [2]. We prefer Theorem 10 since it is the best possible corrected result inspired from the very defective main result in [2], that is, Theorem 2.…”

“…We prefer Theorem 10 since it is the best possible corrected result inspired from the very defective main result in [2], that is, Theorem 2. Secondly, we can list several consequences of Theorem 10, for instance, by taking = and/or by replacing ( ) = with 0 ≤ < 1.…”

“…(7) With respect to the previous remark, it is also necessary to point out that the contractivity condition (4) does not permit us to upper bound, for instance, the term ( 2 +1 , 2 +2 ). However, the authors affirmed in [2], page 3, equation (16), that "Similarly, we can obtain…”

“…(see [2], page 3, equations (26) and (27)). However, the authors did not justify neither why we can suppose that the subindices are even nor why (17), involving the partial subsequences { ( ) } and { ( ) }, can be deduced from (16), in which only { ( ) } and { ( ) } have a role.…”

“…Although the authors of [2] used the notation for a partial metric space, we prefer using in order to avoid confusion with the metric case. Every metric space is a partial metric space, but the converse is false.…”