Next Article in Journal
Whether an Enormously Large Energy Density of the Quantum Vacuum Is Catastrophic
Next Article in Special Issue
On Suzuki Mappings in Modular Spaces
Previous Article in Journal
Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle
Previous Article in Special Issue
Some Results for Split Equality Equilibrium Problems in Banach Spaces
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Fixed Point Results Using Ft-Contractions in Ordered Metric Spaces Having t-Property

1
Department of Mathematics, College of Education in Jubail, Imam Abdulrahman Bin Faisal University, P.O. 12020, Industrial Jubail 31961, Saudi Arabia
2
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
3
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
4
Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(3), 313; https://doi.org/10.3390/sym11030313
Submission received: 1 February 2019 / Accepted: 21 February 2019 / Published: 2 March 2019
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

Abstract

:
In this paper, we prove the existence of fixed points of F t -contraction mappings in partially ordered metric spaces not necessarily complete. We require that the ordered metric space has the t-property, which is a new concept introduced recently by Rashid et al. We also give some examples to illustrate the new concepts and obtained results.

1. Introduction and Preliminaries

Banach Contraction Principle is the most important result in metric fixed point theory. This result was due to Banach [1] in 1922. Banach contraction principle has been generalized by many researchers. Among the first generalizations in the setting of ordered metric spaces was proved by Ran-Reurings [2] in 2004. Many papers have been reported in ordered metric spaces (see [3,4,5,6,7,8,9,10,11,12,13,14,15]).
In 2012, Wardowski [16] generalized the Banach Contraction Principle by introducing a new type of contractions, called F-contractions. This concept attracted many researchers to contribute in this field. Many papers are reported on the existence of fixed points using F-contractions in different spaces (see [17,18,19,20,21,22,23,24,25,26]). For instance, let F be the set of all functions F : ( 0 , ) R satisfying the following conditions:
( F 1 ) F is strictly increasing, i.e., for all a , b ( 0 , ) with a < b , then F ( a ) < F ( b ) .
( F 2 ) For each sequence { α n } of positive numbers,
lim n α n = 0 , iff lim n F ( α n ) = .
( F 3 ) There exists k ( 0 , 1 ) such that lim a 0 + a k F ( a ) = 0 .
Consider F 1 ( α ) = ln α , F 2 ( α ) = ln α + α , F 3 ( α ) = 1 α and F 4 ( α ) = ln ( α 2 + α ) . We have that F n F for all n = 1 , 2 , 3 , 4 .
Definition 1.
[16]. Let ( X , d ) be a metric space and T : X X be a self-mapping. Then, T is said to be an F-contraction if for F F , there exists τ > 0 such that
x , y X w i t h [ d ( T x , T y ) > 0 τ + F ( d ( T x , T y ) ) F ( d ( x , y ) ) ] .
If we take F ( α ) = ln α , the previous inequality becomes
d ( T x , T y ) e τ d ( x , y ) , for all x , y X , T x T y .
In addition, for x , y X such that T x = T y , the inequality d ( T x , T y ) e τ d ( x , y ) holds. Hence, T is a contraction mapping where the Lipshitiz constant is λ = e τ . Thus, every contraction is also an F-contraction, but the converse is not true in general as it is proved in Example 2.5 of [16].
Definition 2.
[27]. A sequence { x n } in a partially ordered set ( X , ) is said to be increasing or ascending if for m < n , x m x n . It is said strictly increasing if x m x n and x m x n . We denote it as x m x n .
In most fixed point results (including the ones dealing with F-contractions of Wardowski [16]), the completeness hypothesis is essential to ensure the existence of a fixed point. Note that this hypothesis is strong and it would be interesting to obtain fixed point results without the set being complete. The aim of this paper goes in this direction, that is, we have strong results for weaker hypotheses. More precisely, our motivation is based on a very recent paper [28], where the authors introduced the concept of t-property (for partially ordered metric spaces) to ovoid the completeness hypothesis, that is, the metric space may be incomplete.
Definition 3.
[28]. Let ( X , d , ) be any ordered metric space. X has the t-property if every strictly increasing Cauchy sequence { x n } in X has a strict upper bound in X, i.e., there exists u X such that x n u .
We present the following examples illustrating Definition 3.
Example 1.
[28]. Let X = R , Q , ( a , b ] , a , b R be equipped with the natural ordering ≤ and the usual metric. Then, X has the t-property.
Example 2.
[28]. Let X = { ( x , y ) : x , y Q } . We define ⪯ in X by ( x 1 , x 2 ) ( y 1 , y 2 ) iff x 1 y 1 and x 2 y 2 . Let d be the Euclidean metric on X. Then, ( X , d , ) has the t-property.
Example 3.
[28]. Let X = C [ a , b ] be equipped with the metric d defined as d ( f , g ) = a b f g d x . Then, ( X , d ) is not a complete metric space. For f , g X , f g iff f ( x ) g ( x ) for each x [ a , b ] . Obviously, ( C [ a , b ] , d , ) has t-property.
In the following example, the increasing Cauchy sequence does not have any strict upper bound.
Example 4.
[28]. Let us consider X = { ( x , y , z ) : x , y , z Q w i t h max { x , y , z } < 2 } . Endow X with the Euclidean metric on R 3 . Define ⪯ in X by ( x 1 , y 1 , z 1 ) ( x 2 , y 2 , z 2 ) if x 1 x 2 , y 1 y 2 and z 1 z 2 . Consider x n = ( q n , q n , q n ) in X such that q 0 = 1 and { q n } is strictly increasing in Q . We have that q n < 2 for all n 0 . In addition, { x n } is a strictly increasing Cauchy sequence in X, but it does not have any strict upper bound in X.
In this paper, we prove some fixed point results for F t -contraction mappings (introduced in Definition 4) without requiring that the metric space is complete, but using the concept of the t-property. We give some examples to illustrate our obtained results.

2. Main Results

Definition 4.
Let ( X , d , ) be an ordered metric space and T : X X be a self-mapping. T is said an F t -contraction if for F F , there exists τ > 0 such that for all x , y X with x T x , y T y and x y , we have
τ + F ( d ( y , T ( y ) ) ) F ( d ( x , T ( x ) ) ) .
In the following example, the considered mapping T is not an F-contraction, but it is an F t -contraction.
Example 5.
Let X = Z be endowed by the usual metric of R and the natural ordering ≤. Define T : X X by
T ( x ) = 2 x , x < 0 , x , x 0 .
For x = 3 and y = 4 , we get d ( T x , T y ) > d ( x , y ) . Let F F . By ( F 1 ) , we have
τ + F ( d ( T x , T y ) ) > F ( d ( x , y ) ) , f o r τ > 0 .
Thus, T is not an F-contraction. Now, we show that T is an F t -contraction. Clearly, F ( α ) = ln ( α ) + α F for α ( 0 , ) . Set τ = 1 2 . We show that Equation (1) is satisfied. Let x , y X such that x T x , y T y and x < y . Then, y x 1 and x < y < 0 . Further, d ( x , T x ) = x and d ( y , T y ) = y . In addition,
F ( d ( x , T x ) ) F ( d ( y , T y ) ) = [ ln ( x ) x ] [ ln ( y ) y ] = ln ( x y ) + ( y x ) 1 > 1 2 = τ .
Thus, τ + F ( d ( y , T y ) ) F ( d ( x , T x ) ) . This shows that T is an F t -contraction.
Example 6.
Let A = { 0 , 1 , 2 , 3 , 4 , 5 } and B = ( 5 , 10 ) . Endow X = A B with the usual metric of R and the natural ordering ≤. Define T : X X by
T ( x ) = 2 x 5 + 7 , x A , x , x B .
Let F : R + R be defined by F ( α ) = ln ( α ) + α . Clearly, F F . It can be easily proved that T is an F t -contraction.
Our first fixed point result is:
Theorem 1.
Let ( X , d , ) be an ordered metric space having t-property. Let T : X X be an F t -contraction. Suppose that T is non-decreasing and there exists x 0 X such that x 0 T ( x 0 ) . Then, T has a fixed point in X.
Proof. 
By assumption, we have x 0 X such that x 0 T ( x 0 ) . If x 0 = T ( x 0 ) , the proof is completed. Otherwise, choose x 1 = T ( x 0 ) such that x 0 x 1 . By monotonicity of T, we have T ( x 0 ) T ( x 1 ) , that is, x 1 T ( x 1 ) . If x 1 = T ( x 1 ) , the proof is completed. Otherwise, choose x 2 = T ( x 1 ) such that x 1 x 2 . Again, by monotonicity of T, we have T ( x 1 ) T ( x 2 ) . Continuing this process, we get a strictly increasing sequence { x n } in X such that x n + 1 = T ( x n ) . As x 0 x 1 , by Equation (1), we have
τ + F ( d ( x 1 , T ( x 1 ) ) ) F ( d ( x 0 , T ( x 0 ) ) ) .
Again, since x 1 x 2 , by Equation (1), we have
τ + F ( d ( x 2 , T ( x 2 ) ) ) F ( d ( x 1 , T ( x 1 ) ) ) .
From Equations (2) and (3), we get
F ( d ( x 2 , T ( x 2 ) ) ) F ( d ( x 1 , T ( x 1 ) ) ) τ F ( d ( x 0 , T ( x 0 ) ) ) 2 τ .
Continuing in this process, we get
F ( d ( x n , T ( x n ) ) ) F ( d ( x n 1 , T ( x n 1 ) ) ) τ F ( d ( x 0 , T ( x 0 ) ) ) n τ .
Denote λ n = d ( x n , T ( x n ) ) for n N . From Equation (4), we obtain
F ( λ n ) F ( λ n 1 ) τ F ( λ 0 ) n τ .
We get lim n F ( λ n ) = . Using property ( F 2 ) ,
lim n λ n = 0 .
By ( F 3 ) , there exists k ( 0 , 1 ) such that
lim n λ n k F ( λ n ) = 0 .
By Equation (5), we have for all n
λ n k F ( λ n ) λ n k F ( λ 0 ) λ n k n τ 0 .
Letting n , we get
lim n n λ n k = 0 .
Thus, there exists n 1 N such that n λ n k 1 for all n n 1 , that is,
λ n 1 n 1 / k ,
for all n n 1 . Now, we show that { x n } is a Cauchy sequence. Let n , m N with n 1 n < m . Using Equation (6), one writes
d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + + d ( x m 1 , x m ) = d ( x n , T ( x n ) ) + d ( x n + 1 , T ( x n + 1 ) ) + + d ( x m 1 , T ( x m 1 ) ) = λ n + λ n + 1 + + λ m 1 = i = n m 1 λ i i = n λ i i = n 1 i 1 / k .
Taking n , we get lim n , m d ( x n , x m ) = 0 . Thus, { x n } is a strictly increasing Cauchy sequence in X, which has t-property. Therefore, there exists u X such that x n u . If T ( u ) = u , the proof is completed. Otherwise, by Equation (1), we have τ + F ( d ( u , T ( u ) ) ) F ( d ( x n , T ( x n ) ) ) . Using Equation (4), we get
F ( d ( u , T ( u ) ) ) F ( d ( x 0 , T ( x 0 ) ) ) ( n + 1 ) τ .
At the limit, F ( d ( u , T ( u ) ) ) = . By ( F 2 ) , we have T ( u ) = u . Thus, u is a fixed point of T in X. □
Now, we report some examples to illustrate our obtained result. The first example clarifies Theorem 1 where the mapping T is not an F-contraction.
Example 7.
Let A = { a n : a n + 1 = 4 a n + 1 f o r n 0 a n d a 0 = 1 } and B = ( 1 , 0 ] Q . Take X = A B , so X = { , 43 , 11 , 3 , 1 } B . Endow X with the usual metric on R and the natural ordering ≤. Clearly, ( X , d , ) is not complete but has the t-property. Define T : X X by
T ( x ) = 4 x + 1 , i f x A , x , i f x B .
Obviously, T is non-decreasing. Now, it remains to prove that T satisfies Equation (1). Letting x , y X with x < y , x T ( x ) and y T ( y ) , we have x < y 1 . Then, d ( x , T ( x ) ) = ( 3 x + 1 ) , d ( y , T ( y ) ) = ( 3 y + 1 ) and y x 2 . If we take F ( α ) = ln ( α ) + α F and τ = 1 > 0 . Then, τ + F ( d ( y , T ( y ) ) ) F ( d ( x , T ( x ) ) ) , i.e., T is an F t -contraction. Hence, all the conditions of Theorem 1 are satisfied. B is the set of fixed points of T.
Example 8.
Let X = R 2 be endowed with the Euclidean metric. Consider: ( x , y ) ( u , v ) iff x u and y v . Then, ( X , d , ) is an ordered metric space having t-property. Take A = { , 8 , 6 , 4 , 2 } . Let E X be defined by E = { ( a , b ) : a R a n d b A } . Clearly, for all ( a , b ) , ( c , d ) E such that ( a , b ) ( c , d ) , we have d b 2 . Define T : R 2 R 2 by
T ( x , y ) = ( x , 1 ) , i f   ( x , y ) E ( x , y ) , i f   ( x , y ) E c .
Clearly, T is non-decreasing. We show that T satisfies Equation (1). Let x = ( x 1 , y 1 ) , y = ( x 2 , y 2 ) X such that x y , y T ( y ) and x T ( x ) . Then, x , y E and y 2 y 1 2 . In addition,
d ( y , T ( y ) ) = d ( ( x 2 , y 2 ) , ( x 2 , 1 ) ) = 1 y 2 ,
and
d ( x , T ( x ) ) = d ( ( x 1 , y 1 ) , ( x 1 , 1 ) ) = 1 y 1 .
Since y 1 < y 2 < 0 , we have ( 1 y 1 ) > ( 1 y 2 ) . Take F ( α ) = ln ( α ) + α F and τ = 1 . We have
F ( d ( x , T ( x ) ) ) F ( d ( y , T ( y ) ) ) = [ ln ( 1 y 1 ) + ( 1 y 1 ) ] [ ln ( 1 y 2 ) + ( 1 y 2 ) ] = ln ( 1 y 1 1 y 2 ) + ( y 2 y 1 ) 2 > 1 = τ .
Hence, for all x , y X with x y , x T ( x ) and y T ( y ) , we have
τ + F ( d ( y , T ( y ) ) ) F ( d ( x , T ( x ) ) ) .
Thus, all the conditions of Theorem 1 are satisfied. Any element of E c is a fixed point of T.
The following example clarifies Theorem 1, where the space is not complete.
Example 9.
Let X = C [ 0 , 1 ] be equipped with the metric d defined as d ( f , g ) = a b f g d x . For f , g X , f g iff f ( t ) g ( t ) for each t [ 0 , 1 ] . Note that ( X , d , ) is an ordered metric space having t-property, but it is not complete. Let B = { f n ( t ) : f n + 1 ( t ) = 3 f n ( t ) + 1 f o r n 0 a n d f 0 ( t ) = 1 , f o r   e a c h t [ 0 , 1 ] } be a subset of X. Define T : X X by
T ( f ( t ) ) = 3 f ( t ) + 1 , f ( t ) B f ( t ) , f ( t ) B c ,
for t [ 0 , 1 ] . Clearly, T is non-decreasing. We prove that T is an F t -contraction. Let f = f ( t ) , g = g ( t ) X with f g , f T ( f ) and g T ( g ) . Then, f , g B and g ( t ) f ( t ) 1 for each t [ 0 , 1 ] . We have
d ( f , T ( f ) ) = ( 2 f ( t ) + 1 ) ,
and
d ( g , T ( g ) ) = ( 2 g ( t ) + 1 ) .
Consider F ( α ) = ln ( α ) + α and τ = 1 . We have
F ( d ( f , T ( f ) ) ) F ( d ( g , T ( g ) ) ) = [ ln { ( 2 f ( t ) + 1 ) } ( 2 f ( t ) + 1 ) ] [ ln { ( 2 g ( t ) + 1 ) } ( 2 g ( t ) + 1 ) ] = ln ( ( 2 f ( t ) + 1 ) ( 2 g ( t ) + 1 ) ) + 2 ( g ( t ) f ( t ) ) 2 > 1 = τ .
Thus, for all f , g X with f g , f T ( f ) and g T ( g ) , we have
τ + F ( d ( g , T ( g ) ) ) F ( d ( f , T ( f ) ) ) .
Hence, all the conditions of Theorem 1 are satisfied. B c is the set of fixed points of T.
Definition 5.
A function ψ : [ 0 , ) [ 0 , ) is said to be a sublinear altering distance function, if it satisfies the following:
1. 
ψ is monotonic increasing and continuous.
2. 
ψ ( t ) = 0 iff t = 0 .
3. 
ψ ( a + b ) ψ ( a ) + ψ ( b ) , for any a , b [ 0 , ) .
Example 10.
The map ψ : [ 0 , ) [ 0 , ) defined by f ( x ) = a x ( a > 0 ) is a sublinear altering function.
Example 11.
Let us define ψ : [ 0 , ) [ 0 , ) defined by f ( x ) = x . Then, ψ is a sublinear altering function.
Definition 6.
Let ( X , d , ) be an ordered metric space and T : X X be a self-mapping. T is said an ( ψ , ϕ , F t ) -contraction, if for F F , there exists τ > 0 such that for all x , y X with x T x , y T y and x y , we have
τ + F [ ψ ( d ( y , T ( y ) ) ] F [ ψ ( d ( x , T ( x ) ) ) ϕ ( d ( x , T ( x ) ) ) ] ,
where ψ is a sublinear altering function, ϕ : [ 0 , ) [ 0 , ) is such that ϕ ( t ) = 0 iff t = 0 and ψ ( t ) > ϕ ( t ) for each t > 0 .
Our second fixed point result is:
Theorem 2.
Let ( X , d , ) be an ordered metric space having t-property. Let T : X X be an ( ψ , ϕ , F t ) -contraction. Suppose that T is non-decreasing and there exists x 0 X such that x 0 T ( x 0 ) . Then, T has a fixed point in X.
Proof. 
Let x 0 X be such that x 0 T ( x 0 ) . If x 0 = T ( x 0 ) , the proof is completed. Otherwise, choose x 1 = T ( x 0 ) such that x 0 x 1 . Proceeding similarly as Theorem 1, we get a strictly increasing sequence { x n } in X such that x n + 1 = T ( x n ) . As x 0 x 1 , by Equation (7),
τ + F [ ψ ( d ( x 1 , T ( x 1 ) ) ] F [ ψ ( d ( x 0 , T ( x 0 ) ) ) ϕ ( d ( x 0 , T ( x 0 ) ) ) ] .
By a property of ϕ, we get
F [ ψ ( d ( x 1 , T ( x 1 ) ) ] F [ ψ ( d ( x 0 , T ( x 0 ) ) ) ] τ .
Since x 1 x 2 , by Equation (7), we have
τ + F [ ψ ( d ( x 2 , T ( x 2 ) ) ] F [ ψ ( d ( x 1 , T ( x 1 ) ) ) ϕ ( d ( x 1 , T ( x 1 ) ) ) ] .
Again,
F [ ψ ( d ( x 2 , T ( x 2 ) ) ] F [ ψ ( d ( x 0 , T ( x 0 ) ) ) ] 2 τ .
Continuing in the same way, we get
F [ ψ ( d ( x n , T ( x n ) ) ] F [ ψ ( d ( x 0 , T ( x 0 ) ) ) ] n τ .
Denote γ n = ψ ( d ( x n , T ( x n ) ) ) for n N . By (8), we obtain
F ( γ n ) F ( γ n 1 ) τ F ( γ 0 ) n τ .
We get lim n F ( γ n ) = . By ( F 2 ) , we have
lim n γ n = 0 .
Using ( F 3 ) , there exists k ( 0 , 1 ) such that
lim n γ n k F ( γ n ) = 0 .
By Equation (9), we have for all n
γ n k F ( γ n ) γ n k F ( γ 0 ) γ n k n τ 0 .
Letting n in Equation (12) and using Equations (10) and (11), we get
lim n n γ n k = 0 .
Hence, there exists n 1 N such that n γ n k 1 for all n n 1 , i.e.,
γ n 1 n 1 / k .
Now, we show that { x n } is a Cauchy sequence. Using the triangular inequality, properties of ψ and Equation (13), we have
ψ ( d ( x n , x m ) ) ψ [ d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + + d ( x m 1 , x m ) ] ψ [ d ( x n , T ( x n ) ) ] + ψ [ d ( x n + 1 , T ( x n + 1 ) ) ] + + ψ [ d ( x m 1 , T ( x m 1 ) ) ] = γ n + γ n + 1 + + γ m 1 = i = n m 1 γ i i = n γ i i = n 1 i 1 / k .
Letting n , we get lim n ψ [ d ( x n , x m ) ] = 0 . By properties of ψ, we get lim n , m d ( x n , x m ) = 0 . Thus, { x n } is a strictly increasing Cauchy sequence in X, which has the t-property, so there exists u X such that x n u . If T ( u ) = u , the proof is completed. Otherwise, by Equation (7), we have
τ + F [ ψ ( d ( u , T ( u ) ) ] F [ ψ ( d ( x n , T ( x n ) ) ) ϕ ( d ( x n , T ( x n ) ) ) ] ,
From Equation (8),
F [ ψ ( d ( u , T ( u ) ) ] F [ ψ ( d ( x 0 , T ( x 0 ) ) ) ( n + 1 ) τ .
Hence, lim n F [ ψ ( d ( u , T ( u ) ) ] = . By ( F 2 ) , we have lim n ψ ( d ( u , T ( u ) ) = 0 . This implies that d ( u , T ( u ) ) = 0 , i . e . , T ( u ) = u . Hence, u is a fixed point of T in X. □
Example 12.
Let A = { a n : a n + 1 = 4 a n + 1 f o r n 0 a n d a 0 = 1 } and B = ( 1 , 0 ] Q . Take X = A B , then X = { , 43 , 11 , 3 , 1 } B . Endow X with the usual metric on R and the natural ordering ≤. Clearly, ( X , d , ) has the t-property, but it is not complete. Define T : X X by
T ( x ) = 4 x + 1 , i f x A , x , i f x B .
Then, T is non-decreasing. Further, define ψ , ϕ : [ 0 , ) [ 0 , ) by ψ ( t ) = 4 t and ϕ ( t ) = 2 t . We prove that Equation (7) is satisfied. Take F ( α ) = ln ( α ) + α F and τ = 4 > 0 . Let x , y X such that x < y , x T ( x ) and y T ( y ) . Then, x < y 1 , d ( x , T ( x ) ) = ( 3 x + 1 ) , d ( y , T ( y ) ) = ( 3 y + 1 ) and 6 y 3 x 3 . In addition,
F [ ψ ( d ( x , T ( x ) ) ) ϕ ( d ( x , T ( x ) ) ) ] F [ ψ ( d ( y , T ( y ) ) ] = F [ 2 ( 3 x + 1 ) ] F [ 4 ( 3 y + 1 ) ] , = { ln [ 2 ( 3 x + 1 ) ] 2 ( 3 x + 1 ) } { ln [ 4 ( 3 y + 1 ) ] 4 ( 3 y + 1 ) } , = ln [ 2 ( 3 x + 1 ) 4 ( 3 y + 1 ) ] 6 x + 12 y + 2 , ( 6 y 3 x ) + 1 , 4 = τ .
Thus, all the conditions of Theorem 2 hold, so there exists a fixed point of T in X.

Author Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Funding

This research received no external funding.

Acknowledgments

The publication of this article was funded by the Qatar National Library.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Banach, S. Sure operations dans les ensembles abstraits et leur application aux equations integrals. Fund. Math. 1922, 3, 133–181. [Google Scholar] [CrossRef]
  2. Ran, A.C.M.; Reurings, M.C.B. A fixed point theorem in partial ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132, 1435–1443. [Google Scholar] [CrossRef]
  3. Abdeljawad, T.; Aydi, H.; Karapinar, E. Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces. Math. Probl. Eng. 2012, 2012. [Google Scholar] [CrossRef]
  4. Ansari, A.H.; Aydi, H.; Barakat, M.A.; Khan, M.S. On coupled coincidence point results in partially ordered metric spaces via generalized compatibility and C-class functions. J. Inequal. Spec. Funct. 2017, 8, 125–138. [Google Scholar]
  5. Aydi, H.; Karapinar, E.; Mustafa, Z. Coupled coincidence point results on generalized distance in ordered cone metric spaces. Positivity 2013, 17, 979–993. [Google Scholar] [CrossRef]
  6. Mukheimer, A. α-ψ-φ-contractive mappings in ordered partial b-metric spaces. J. Nonlinear Sci. Appl. 2014, 7, 168–179. [Google Scholar] [CrossRef]
  7. Berinde, V. Generalized coupled fixzed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74, 7347–7355. [Google Scholar] [CrossRef]
  8. Aydi, H.; Nashine, H.K.; Samet, B.; Yazidi, H. Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 2011, 74, 6814–6825. [Google Scholar] [CrossRef]
  9. Shah, M.H.; Hussain, N. Nonlinear contractions in partially ordered quasi b-metric spaces. Commun. Korean Math. Soc. 2012, 27, 117–128. [Google Scholar] [CrossRef]
  10. Aydi, H.; Damjanović, B.; Samet, B.; Shatanawi, W. Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model. 2011, 54, 2443–2450. [Google Scholar] [CrossRef]
  11. Mustafa, Z.; Karapinar, E.; Aydi, H. A discussion on generalized almost contractions via rational expressions in partially ordered metric spaces. J. Inequal. Appl. 2014. [Google Scholar] [CrossRef]
  12. Nieto, J.J.; Rodríguez-López, R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22, 223–239. [Google Scholar] [CrossRef]
  13. Samet, B.; Karapinar, E.; Aydi, H.; Rajić, V.C. Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013. [Google Scholar] [CrossRef]
  14. Turinici, M. Ran-Reurings, fixed point results in ordered metric spaces. Libertas Math. 2011, 31, 49–55. [Google Scholar]
  15. Turinici, M. Nieto-Lopez theorems in ordered metric spaces. Math. Student 2012, 81, 219–229. [Google Scholar]
  16. Wardowski, D. Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012. [Google Scholar] [CrossRef]
  17. Acar, Ö.; Durmaz, G.; Manak, G. Generalized multivalued F-contractions on complete metric spaces. Bull. Iran. Math. Soc. 2014, 40, 1469–1478. [Google Scholar]
  18. Altun, I.; Manak, G.; Dağ, H. Multivalued F-contractions on complete metric spaces. J. Nonlinear Complex Anal. 2015, 16, 658–666. [Google Scholar] [CrossRef]
  19. Manak, G.; Helvaca, A.; Altun, I. Ćirić type generalized F-contractions on complete metric spaces and fixed point results. Filomat 2014, 28, 1143–1151. [Google Scholar] [CrossRef]
  20. Al-Rawashdeh, A.; Aydi, H.; Felhi, A.; Sahmim, S.; Shatanawi, W. On common fixed points for αF-contractions and applications. J. Nonlinear Sci. Appl. 2016, 9, 3445–3458. [Google Scholar] [CrossRef]
  21. Aydi, H.; Karapinar, E.; Yazidi, H. Modified F-contractions via α-admissible mappings and application to integral equations. Filomat 2017, 31, 1141–1148. [Google Scholar] [CrossRef]
  22. Mustafa, Z.; Arshad, M.; Ullah Khan, S.; Ahmad, J.; Jaradat, M.M.M. Common fixed points for multivalued mappings in G-metric spaces with applications. J. Nonlinear Sci. Appl. 2017, 10, 2550–2564. [Google Scholar] [CrossRef]
  23. Secelean, N.A. Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 2013. [Google Scholar] [CrossRef]
  24. Shukla, S.; Radenović, S. Some common fixed point theorems for F-contraction type mappings in 0-complete partial metric spaces. J. Math. 2013, 2013. [Google Scholar] [CrossRef]
  25. Sgroi, M.; Vetro, C. Multi-valued F-contractions and the solution of certain functional and integral equations. Filomat 2013, 27, 1259–1268. [Google Scholar] [CrossRef]
  26. Shukla, S.; Radenović, S.; Kadelburg, Z. Some fixed point theorems for ordered F-generalized contractions in 0 − f-orbitally complete partial metric spaces. Theory Appl. Math. Comput. Sci. 2014, 4, 87–98. [Google Scholar]
  27. Turinici, M. Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 1986, 117, 100–127. [Google Scholar] [CrossRef]
  28. Rashid, T.; Khan, Q.H.; Aydi, H. t-property of metric spaces and fixed point theorems. Ital. J. Pure Appl. Math. 2019, 41, 1–12. [Google Scholar]

Share and Cite

MDPI and ACS Style

Aydi, H.; Rashid, T.; Haq Khan, Q.; Mustafa, Z.; Jaradat, M.M.M. Fixed Point Results Using Ft-Contractions in Ordered Metric Spaces Having t-Property. Symmetry 2019, 11, 313. https://doi.org/10.3390/sym11030313

AMA Style

Aydi H, Rashid T, Haq Khan Q, Mustafa Z, Jaradat MMM. Fixed Point Results Using Ft-Contractions in Ordered Metric Spaces Having t-Property. Symmetry. 2019; 11(3):313. https://doi.org/10.3390/sym11030313

Chicago/Turabian Style

Aydi, Hassen, Tawseef Rashid, Qamrul Haq Khan, Zead Mustafa, and Mohammed M. M. Jaradat. 2019. "Fixed Point Results Using Ft-Contractions in Ordered Metric Spaces Having t-Property" Symmetry 11, no. 3: 313. https://doi.org/10.3390/sym11030313

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop