Turkish Journal of Analysis and Number Theory, 2016, Vol. 4, No. 6, 159-163
Available online at http://pubs.sciepub.com/tjant/4/6/2
©Science and Education Publishing
DOI:10.12691/tjant-4-6-2
New Approach of F-Contraction Involving Fixed Point
on a Closed Ball
Aftab Hussain
1,2,*
1
Department of Mathematics, International Islamic University, H-10, Islam-abad, Pakistan
2
Department of Mathematical Sciences, Lahore Leads University, Lahore, Pakistan
*Corresponding author: aftabshh@gmail.com
Abstract The article is written with a view to introducing the new idea of F-contraction on a closed ball and have
new theorems in a complete metric space. That is why this outcome becomes useful for the contraction of the
mapping on a closed ball instead of the whole space. At the same time, some comparative examples are constructed
which establish the superiority of our results. It can be stated that the results that have come into being give proof of
extension as well as substantial generalizations and improvements of several well known results in the existing
comparable literature.
Keywords: metric space, fixed point, F contraction, closed ball
Cite This Article: Aftab Hussain, “New Approach of F-Contraction Involving Fixed Point on a Closed Ball.”
Turkish Journal of Analysis and Number Theory, vol. 4, no. 6 (2016): 159-163. doi: 10.12691/tjant-4-6-2.
1. Introduction
Shoaib et al. [40] proved significant results concerning
the existence of fixed points of the dominated self
mappings satisfying some contractive conditions on a
closed ball in a 0-complete quasi-partial metric space.
Other results on closed ball can be seen in [5,6,7,8,25].
Over the years, Fixed Point Theory has been generalized
in different ways by several mathematicians (see
[2,3,4,9-15,17-21,23-27]).
For x X ∈ and 0, ε > ( ) ( ) { }
, : , Bx y X d xy ε ε = ∈ ≤
is a closed ball in ( ) , . Xd
Definition 1 [18] Let ( ) , Xd be a metric space. Let T be
self mappings and [ ) , : 0, X X αη × → +∞ be two
functions. T is called
α η − -continuous if for given
, x X ∈ and sequence { }
n
x with
( ) ( )
1 1
, , ,
.
n n n n n
n
x x as n x x x x
for all n Tx Tx
α η
+ +
→ →∞ ≥
∈ ⇒ →
In 2012, Wardowski [42] introduce a concept of
F-contraction as follows:
Definition 2 [33] Let ( ) , Xd be a metric space. A self
mapping T is said to be an F contraction if there exists
0 τ > such that
( )
( ) ( ) ( ) ( )
, , , 0
, , ,
x y d Tx Ty
F d Tx Ty Fd xy τ
∀ >
⇒ + ≤
(1.1)
where : F
+
→ is a mapping satisfying the following
conditions:
(F1) F is strictly increasing, i.e. for all , xy
+
∈ such
that , x y < ( ) ( ) ; F x F y <
(F2) For each sequence { }
1
n
n
α
∞
=
of positive numbers,
lim 0
n n
α
→∞
= if and only if ( ) lim ;
n n
F α
→∞
= −∞
(F3) There exists ( ) 0,1 κ ∈ such that
( ) lim 0 0.
k
F α α α
+
→ =
We denote by
F
∆ , the set of all functions satisfying
the conditions (F1)-(F3).
Furthermore, it was done by different investigators see
[1,3,15,16,23,27,28,29,31,32,33,34,37,38,39].
Hussain et al. [18] introduced the following family of
new functions.
Let
G
∆ denotes the set of all functions
4
: G
+ +
→
satisfying:
(G) for all
1 2 3 4
, , , t t t t
+
∈ with
1234
0 tttt = there
exists 0 τ > such that ( )
1 2 3 4
, , , . Gt t t t τ =
2. Banach Fixed Point Theorem for
F-Contraction on a Closed Ball
In this section, we introduce Banach fixed point
theorem for modified F-contraction on a closed ball in
complete metric spaces.
Now we state our main result.
Theorem 3 Let T be a continuous self mapping in a
complete metric space ( ) , Xd and
0
x be an arbitrary
point in , 0. Xr > Assume that 0 τ > and
F
F ∈∆ for all
( )
0
, , xy Bx r X ∈ ⊆ with ( ) , 0 d Tx Ty > such that
Received August 24, 2016; Revised November 20, 2016; Accepted November 28, 2016