Coincidence and Fixed Point of Nonexpansive Type Mappings in 2-Metric Spaces

Received: May 4, 2019; Accepted: June 6, 2019 2010 Mathematics Subject Classification: 47H10, 54H25.


Introduction and Preliminaries
The concept of 2-metric space was introduced by Gähler [2, 3, 4] whose abstract properties were suggested by the area function in Euclidean space. Employing various contractive conditions Iséki [5] setout the tradition of proving fixed point theorems in 2-metric spaces. Later on, Naidu and Prasad [6] contributed few fixed point theorems in 2-metric spaces introducing the concept of weak commutativity. Recently, Singh and Chandrashekhar [7] proved a fixed point theorem in 2-metric space for nonexpansive 192 type mappings. They obtained the following result: be a 2-metric space and X X T → : be a self mapping satisfying the following nonexpansive type condition: Tx  y  d  u  Ty  x  d  u  Ty  y  d  u  Tx  x  d  u  y  x  d  a  ,  ,  ,  ,  2   1  ,  ,  ,  ,  Our condition is an extension of that of Ćirić [1] (see also [8]). Also, we will show that our condition (2) includes the above condition (1). Now we give some definitions which are used frequently to prove our main results.

Definition 1.1. Gähler defined 2-metric space as follows:
A 2-metric on a set X with at least three points is a non-negative real-valued mapping satisfying the following properties: (1) To each pair of points a, b with b a ≠ in X there is a point X c ∈ such that ( ) if at least two of the points are equal, ( 1 ψ ) ψ is continuous and strictly increasing.
Φ be a set of all continuous functions satisfying the following conditions: Let Φ be set of all lower continuous functions , : In 2014, Ansari [10] introduced the concept of C-class functions which cover a large class of contractive conditions. Definition 1.5 [10]. Let R R → + 2 : F be a continuous mapping. Then it is called a C-class function if it satisfies the following conditions: Note for some F we have that ( ) We denote C-class functions as C .
Example 1.1 [10]. The following functions and is continuous, ( ) is a continuous function such that ( ) is a upper semicontinuous function such that ( ) , is a generalized Mizoguchi-Takahashi type function, ( ) Let Ψ be a set of all non-decreasing continuous functions , : Let Φ be a set of all continuous functions , : which satisfy the following conditions: (a) ϕ continuous; satisfying the following conditions: ( 1 ψ ) ψ is continuous and strictly increasing.
In this paper, we introduce a new class of self mappings satisfying the following nonexpansive type condition: for all , , ,

Main Results
be a 2-metric space. Let T, f be self mappings of X satisfying nonexpansive type condition (2).
(a) X is complete and f is surjective, or, X T Then f and T have a coincidence point in X. Further, the coincidence point is unique,   Tx  fx  hd  fx  Tx  fx  cd   fx  Tx  fx  d  fx  Tx  fx  d  fx  Tx  fx  d  b   fx  Tx  fx  d  fx  fx  fx  d  a   h  c  b  a   fx  Tx  fx  hd  fx  Tx  fx  cd   fx  Tx  fx  d  fx  Tx  fx  d  fx  Tx  fx On applying inequality (2) again and using triangular inequality and (3), we get         u  Tx  fx  d  u  Tx  fx  d  c   u  Tx  fx  d  u  Tx  fx  d  b   u  Tx  fx  d  u  Tx  fx  d  a h c b a n n n n n n n n n n n n (4) Suppose that, for some n, (  Tx  fx  hd  u  Tx  fx  cd   u  Tx  fx  d  u  Tx  fx  d  u  Tx  fx  d  b   u  Tx  fx  d  u  fx  fx  d  a   h  c  b  a   u  Tx  fx  hd  u  Tx  fx  cd   u  Tx  fx  d  u  Tx  fx  d  u  Tx  fx  d  b   u  Tx  fx  d  u  fx  fx  d n  n  n  n   ,  ,  ,  ,   ,  ,  ,  ,  ,  ,  ,  ,  max   ,  ,  ,  ,  ,  max  1   ,   ,  ,  ,  ,   ,  ,  ,  ,  ,  ,  ,  ,  max   ,  ,  ,  ,  ,  max  1 1 1  Tx  Tx  cd   u  Tx  Tx  bd  u  Tx  Tx  ad   h  c  b  a   u  Tx  Tx  cd   u  Tx  Tx  bd  u  Tx  Tx  On applying inequality (2) again and using (3), (5) and by triangular inequality, we get    Tx  d  u  Tx  Tx  d   u  Tx  Tx  d  Tx  Tx  Tx  d  c   u  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d  b   u  Tx  Tx  d   u  Tx  Tx  d  u  Tx  Tx  d   Tx  Tx  Tx  d   a   h  c  b  a  1   1  2  2  3   1  2  3   2  1  2   2  3  2  3   2  1  1  2  3   1   1  2  2  3   1  2 Tx  Tx  d  Tx  Tx  Tx  d   Tx  Tx  Tx  d  Tx  Tx  Tx  d  c   u  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d  b   u  Tx  Tx  d  u  Tx  Tx  d   u  Tx  Tx  d  Tx  Tx  Tx  d Tx  Tx  d  u  Tx  Tx  d   u  Tx  Tx  d  Tx  Tx  Tx  d  c   u  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d  b   u  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d Tx  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d   u  Tx  Tx  d  u  Tx  Tx  d  c   u  Tx  Tx  d  u  Tx  Tx  d  u  Tx  Tx  d  b   u  Tx  Tx  d  u  Tx  Tx  d are evaluated at ( ).  Tx  Tx  d  b  c  u  Tx  Tx  bd  u  Tx  Tx  ad   n  n  n  n  n  n   ,  ,  2  ,  ,  ,  ,   2  3  2  3  2 Tx  Tx  cd  b  u  Tx  Tx  d  c  b  a   n  n  n [7].