Skip to main content

Fixed point results for Meir-Keeler-type ϕ-α-contractions on partial metric spaces

Abstract

The purpose of this paper is to study fixed point theorems for a mapping satisfying the generalized Meir-Keeler-type ϕ-α-contractions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

MSC:47H10, 54C60, 54H25, 55M20.

1 Introduction and preliminaries

Throughout this paper, by R + we denote the set of all nonnegative real numbers, while is the set of all natural numbers. In 1994, Mattews [1] introduced the following notion of partial metric spaces.

Definition 1 [1]

A partial metric on a nonempty set X is a function p:X×X R + such that for all x,y,zX,

( p 1 ) x=y if and only if p(x,x)=p(x,y)=p(y,y);

( p 2 ) p(x,x)p(x,y);

( p 3 ) p(x,y)=p(y,x);

( p 4 ) p(x,y)p(x,z)+p(z,y)p(z,z).

A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X.

Remark 1 It is clear that if p(x,y)=0, then from ( p 1 ) and ( p 2 ), x=y. But if x=y, p(x,y) may not be 0.

Each partial metric p on X generates a T 0 topology τ p on X which has as a base the family of open p-balls { B p (x,γ):xX,γ>0}, where B p (x,γ)={yX:p(x,y)<p(x,x)+γ} for all xX and γ>0. If p is a partial metric on X, then the function d p :X×X R + given by

d p (x,y)=2p(x,y)p(x,x)p(y,y)

is a metric on X.

We recall some definitions of a partial metric space as follows.

Definition 2 [1]

Let (X,p) be a partial metric space. Then

  1. (1)

    a sequence { x n } in a partial metric space (X,p) converges to xX if and only if p(x,x)= lim n p(x, x n );

  2. (2)

    a sequence { x n } in a partial metric space (X,p) is called a Cauchy sequence if and only if lim m , n p( x m , x n ) exists (and is finite);

  3. (3)

    a partial metric space (X,p) is said to be complete if every Cauchy sequence { x n } in X converges, with respect to τ p , to a point xX such that p(x,x)= lim m , n p( x m , x n );

  4. (4)

    a subset A of a partial metric space (X,p) is closed if whenever { x n } is a sequence in A such that { x n } converges to some xX, then xA.

Remark 2 The limit in a partial metric space is not unique.

Lemma 1 [1, 2]

  1. (1)

    { x n } is a Cauchy sequence in a partial metric space (X,p) if and only if it is a Cauchy sequence in the metric space (X, d p );

  2. (2)

    a partial metric space (X,p) is complete if and only if the metric space (X, d p ) is complete. Furthermore, lim n d p ( x n ,x)=0 if and only if p(x,x)= lim n p( x n ,x)= lim n p( x n , x m ).

In recent years, fixed point theory has developed rapidly on partial metric spaces, see [210].

In this study, we also recall the Meir-Keeler-type contraction [11] and α-admissible one [12]. In 1969, Meir and Keeler [11] introduced the following notion of Meir-Keeler-type contraction in a metric space (X,d).

Definition 3 Let (X,d) be a metric space, f:XX. Then f is called a Meir-Keeler-type contraction whenever, for each η>0, there exists γ>0 such that

ηd(x,y)<η+γd(fx,fy)<η.

The following definition was introduced in [12].

Definition 4 Let f:XX be a self-mapping of a set X and α:X×X R + . Then f is called α-admissible if

x,yX,α(x,y)1α(fx,fy)1.

The purpose of this paper is to study fixed point theorems for a mapping satisfying the generalized Meir-Keeler-type ϕ-α-contractions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

2 Main results

In the article, we denote by Φ the class of functions ϕ: R + 4 R + satisfying the following conditions:

( ϕ 1 ) ϕ is an increasing and continuous function in each coordinate;

( ϕ 2 ) for t R + {0}, ϕ(t,t,t,t)t, ϕ(t,0,0,t)t, ϕ(0,0,t, t 2 )t; and ϕ( t 1 , t 2 , t 3 , t 4 )=0 iff t 1 = t 2 = t 3 = t 4 =0.

We now state the new notions of generalized Meir-Keeler-type ϕ-contractions and generalized Meir-Keeler-type ϕ-α-contractions in partial metric spaces as follows.

Definition 5 Let (X,p) be a partial metric space, f:XX and ϕΦ. Then f is called a generalized Meir-Keeler-type ϕ-contraction whenever, for each η>0, there exists δ>0 such that

η ϕ ( p ( x , y ) , p ( x , f x ) , p ( y , f y ) , 1 2 [ p ( x , f y ) + p ( y , f x ) ] ) < η + δ p ( f x , f y ) < η .

Definition 6 Let (X,p) be a partial metric space, f:XX, ϕΦ and α:X×X R + . Then f is called a generalized Meir-Keeler-type ϕ-α-contraction if the following conditions hold:

  1. (1)

    f is α-admissible;

  2. (2)

    for each η>0, there exists δ>0 such that

    η ϕ ( p ( x , y ) , p ( x , f x ) , p ( y , f y ) , 1 2 [ p ( x , f y ) + p ( y , f x ) ] ) < η + δ α ( x , x ) α ( y , y ) p ( f x , f y ) < η .
    (2.1)

Remark 3 Note that if f is a generalized Meir-Keeler-type ϕ-α-contraction, then we have that for all x,yX,

α ( x , x ) α ( y , y ) p ( f x , f y ) ϕ ( p ( x , y ) , p ( x , f x ) , p ( y , f y ) , 1 2 [ p ( x , f y ) + p ( y , f x ) ] ) .

Further, if ϕ(p(x,y),p(x,fx),p(y,fy), 1 2 [p(x,fy)+p(y,fx)])=0, then p(fx,fy)=0. On the other hand, if ϕ(p(x,y),p(x,fx),p(y,fy), 1 2 [p(x,fy)+p(y,fx)])>0, then α(x,x)α(y,y)p(fx,fy)<ϕ(p(x,y),p(x,fx),p(y,fy), 1 2 [p(x,fy)+p(y,fx)]).

We now state our main result for the generalized Meir-Keeler-type ϕ-α-contraction as follows.

Theorem 1 Let (X,p) be a complete partial metric space, and ϕΦ. If α:X×X R + satisfies the following conditions:

( α 1 ) there exists x 0 X such that α( x 0 , x 0 )1;

( α 2 ) if α( x n , x n )1 for all nN, then lim n α( x n , x n )1;

( α 3 ) α:X×X R + is a continuous function in each coordinate.

Suppose that f:XX is a generalized Meir-Keeler-type ϕ-α-contraction. Then f has a fixed point in X.

Proof Let x 0 and let x n + 1 =f x n = f n x 0 for n=0,1,2, . Since f is α-admissible and α( x 0 , x 0 )1, we have

α(f x 0 ,f x 0 )=α( x 1 , x 1 )1.

By continuing this process, we get

α( x n , x n )1for all nN{0}.
(2.2)

If there exists n 0 N such that x n 0 + 1 = x n 0 , then we finished the proof. Suppose that x n + 1 x n for any n=0,1,2, . By the definition of the function ϕ, we have ϕ(p( x n , x n + 1 ),p( x n ,f x n ),p( x n + 1 ,f x n + 1 ), 1 2 [p( x n ,f x n + 1 )+p( x n + 1 ,f x n )])>0 for all nN{0}.

Step 1. We shall prove that

lim n p( x n , x n + 1 )=0,that is lim n d p ( x n , x n + 1 )=0.

By Remark 3 and ( p 4 ), using (2.2), we have

p ( x n + 1 , x n + 2 ) = p ( f x n , f x n + 1 ) α ( x n , x n ) α ( x n + 1 , x n + 1 ) p ( f x n , f x n + 1 ) < ϕ ( p ( x n , x n + 1 ) , p ( x n , f x n ) , p ( x n + 1 , f x n + 1 ) , 1 2 [ p ( x n , f x n + 1 ) + p ( x n + 1 , f x n ) ] ) = ϕ ( p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n + 1 , x n + 2 ) , 1 2 [ p ( x n , x n + 2 ) + p ( x n + 1 , x n + 1 ) ] ) ϕ ( p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n + 1 , x n + 2 ) , 1 2 [ p ( x n , x n + 1 ) + p ( x n + 1 , x n + 2 ) ] ) .
(2.3)

If p( x n , x n + 1 )p( x n + 1 , x n + 2 ), then

p ( x n + 1 , x n + 2 ) = p ( f x n , f x n + 1 ) < ϕ ( p ( x n + 1 , x n + 2 ) , p ( x n + 1 , x n + 2 ) , p ( x n + 1 , x n + 2 ) , p ( x n + 1 , x n + 2 ) ) p ( x n + 1 , x n + 2 ) ,

which implies a contradiction, and hence p( x n , x n + 1 )<p( x n 1 , x n ). From the argument above, we also have that for each nN,

p ( x n + 1 , x n + 2 ) = p ( f x n , f x n + 1 ) < ϕ ( p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n , x n + 1 ) ) p ( x n , x n + 1 ) .
(2.4)

Since the sequence {p( x n , x n + 1 )} is decreasing, it must converge to some η0, that is,

lim n p( x n , x n + 1 )=η.
(2.5)

It follows from (2.4) and (2.5) that

lim n ϕ ( p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n , x n + 1 ) , p ( x n , x n + 1 ) ) =η.
(2.6)

Notice that η=inf{p( x n , x n + 1 ):nN}. We claim that η=0. Suppose, to the contrary, that η>0. Since f is a generalized Meir-Keeler-type ϕ-contraction, corresponding to η use, and taking into account the above inequality (2.6), there exist δ>0 and a natural number k such that

η ϕ ( p ( x k , x k + 1 ) , p ( x k , x k + 1 ) , p ( x k , x k + 1 ) , p ( x k , x k + 1 ) ) < η + δ α ( x k , x k ) α ( x k + 1 , x k + 1 ) p ( f x k , f x k + 1 ) < η ,

which implies

p( x k + 1 , x k + 2 )=p(f x k ,f x k + 1 )α( x k , x k )α( x k + 1 , x k + 1 )p(f x k ,f x k + 1 )<η.

So, we get a contradiction since η=inf{p( x n , x n + 1 ):nN}. Thus we have that

lim n p( x n , x n + 1 )=0.
(2.7)

By ( p 2 ), we also have

lim n p( x n , x n )=0.
(2.8)

Since d p (x,y)=2p(x,y)p(x,x)p(y,y) for all x,yX, using (2.7) and (2.8), we obtain that

lim n d p ( x n , x n + 1 )=0.
(2.9)

Step 2. We show that { x n } is a Cauchy sequence in the partial metric space (X,p), that is, it is sufficient to show that { x n } is a Cauchy sequence in the metric space (X, d p ).

Suppose that the above statement is false. Then there exists ϵ>0 such that for any kN, there are n k , m k N with n k > m k k satisfying

d p ( x m k , x n k )ϵ.
(2.10)

Further, corresponding to m k k, we can choose n k in such a way that it is the smallest integer with n k > m k k and d( x 2 m k , x 2 n k )ϵ. Therefore

d p ( x m k , x n k 2 )<ϵ.
(2.11)

Now we have that for all kN,

ϵ d p ( x m k , x n k ) d p ( x m k , x n k 2 ) + d p ( x n k 2 , x n k 1 ) + d p ( x n k 1 , x n k ) < ϵ + d p ( x n k 2 , x n k 1 ) + d p ( x n k 1 , x n k ) .
(2.12)

Letting k in the above inequality and using (2.12), we get

lim n d p ( x m k , x n k )=ϵ.
(2.13)

On the other hand, we have

ϵ d p ( x m k , x n k ) d p ( x m k , x m k + 1 ) + d p ( x m k + 1 , x n k + 1 ) + d p ( x n k + 1 , x n k ) d p ( x m k , x m k + 1 ) + d p ( x m k + 1 , x m k ) + d p ( x m k , x n k ) + d p ( x n k , x n k + 1 ) + d p ( x n k + 1 , x n k ) .

Letting n, we obtain that

lim n d p ( x m k + 1 , x n k + 1 )=ϵ.
(2.14)

Since d p (x,y)=2p(x,y)p(x,x)p(y,y) and using (2.13) and (2.14), we have that

lim n p( x m k , x n k )= ϵ 2
(2.15)

and

lim n p( x m k + 1 , x n k + 1 )= ϵ 2
(2.16)

By Remark 3 and ( p 4 ), we have

p ( x m k + 1 , x n k + 1 ) = p ( f x m k , f x n k ) α ( x m k , x m k ) α ( x n k , x n k ) p ( f x m k , f x n k ) < ϕ ( p ( x m k , x n k ) , p ( x m k , f x m k ) , p ( x n k , f x n k ) , 1 2 [ p ( x m k , f x n k ) + p ( x n k , f x m k ) ] ) = ϕ ( p ( x m k , x n k ) , p ( x m k , x m k + 1 ) , p ( x n k , x n k + 1 ) , 1 2 [ p ( x m k , x n k + 1 ) + p ( x n k , x m k + 1 ) ] ) .
(2.17)

Since

p( x m k , x n k + 1 )p( x m k , x m k + 1 )+p( x m k + 1 , x n k + 1 )p( x m k + 1 , x m k + 1 )
(2.18)

and

p( x n k , x m k + 1 )p( x n k , x n k + 1 )+p( x n k + 1 , x m k + 1 )p( x n k + 1 , x n k + 1 ).
(2.19)

Taking into account the above inequalities (2.8), (2.17), (2.18) and (2.19), letting k, we have

ϵ 2 <ϕ ( ϵ 2 , 0 , 0 , ϵ 2 ) ϵ 2 ,

which implies a contradiction. Thus, { x n } is a Cauchy sequence in the metric space (X, d p ).

Step 3. We show that f has a fixed point ν in i = 1 m A i .

Since (X,p) is complete, then from Lemma 1, we have that (X, d p ) is complete. Thus, there exists νX such that

lim n d p ( x n ,ν)=0.

Moreover, it follows from Lemma 1 that

p(ν,ν)= lim n p( x n ,ν)= lim n , m p( x n , x m ).
(2.20)

On the other hand, since the sequence { x n } is a Cauchy sequence in the metric space (X, d p ), we also have

lim n d p ( x n , x m )=0.

Since d p (x,y)=2p(x,y)p(x,x)p(y,y), we can deduce that

lim n p( x n , x m )=0.
(2.21)

Using (2.20) and (2.21), we have

p(ν,ν)= lim n p( x n ,ν)= lim n p( x n k ,ν)=0.

Again, by Remark 3, ( p 4 ), and the conditions of the mapping α, we have

p ( x n + 1 , f ν ) = p ( f x n , f ν ) α ( x n , x n ) α ( ν , ν ) p ( f x n , f ν ) < ϕ ( p ( x n , ν ) , p ( x n , f x n ) , p ( ν , f ν ) , 1 2 [ p ( x n , f ν ) + p ( ν , f x n ) ] ) = ϕ ( p ( x n , ν ) , p ( x n , x n + 1 ) , p ( ν , f ν ) , 1 2 [ p ( x n , f ν ) + p ( ν , x n + 1 ) ] ) .
(2.22)

Letting n in (2.22), we get

p(ν,fν)<ϕ ( 0 , 0 , p ( ν , f ν ) , 1 2 p ( ν , f ν ) ) p(ν,fν),

a contradiction. So, we have p(ν,fν)=0, that is, fν=ν. □

We give the following example to illustrate Theorem 2.

Example 1 Let X=[0,1]. We define the partial metric p on X by

p(x,y)=max{x,y}.

Let α:[0,1]×[0,1] R + be defined as

α(x,y)=1+x+y,

let f:XX be defined as

f(x)= 1 16 x 2 ,

and, let ϕ: R + 4 R + denote

ψ( t 1 , t 2 , t 3 , t 4 )= 1 2 max { t 1 , t 2 , t 3 , 1 2 t 4 } .

Then f is α-admissible.

Without loss of generality, we assume that x>y and verify the inequality (2.1). For all x,y[0,1] with x>y, we have

α ( x , x ) α ( y , y ) p ( f x , f y ) 1 16 x 2 , p ( x , y ) = x , p ( x , f x ) = x , p ( y , f y ) = y and 1 2 [ p ( x , f y ) + p ( y , f x ) ] = 1 2 [ max { x , y 2 } + max { y , x 2 } ] 1 2 [ max { x , y } + max { y , x } ] < x ,

and hence ϕ(p(x,y),p(x,fx),p(y,fy), 1 2 [p(x,fy)+p(y,fx)])= 1 2 x. Therefore, all the conditions of Theorem 1 are satisfied, and we obtained that 0 is a fixed point of f.

If we let

α(x,y)=1for x,yX,

then it is easy to get the following theorem.

Theorem 2 Let (X,p) be a complete partial metric space and ϕΦ. Suppose that f:XX is a generalized Meir-Keeler-type ϕ-contraction. Then f has a fixed point in X.

References

  1. Mattews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer of Conference on General Topology and Applications 1994, 183–197.

    Google Scholar 

  2. Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36: 17–26.

    MathSciNet  Google Scholar 

  3. Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54(11–12):2923–2927. 10.1016/j.mcm.2011.07.013

    Article  MathSciNet  Google Scholar 

  4. Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40

    Google Scholar 

  5. Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 508730

    Google Scholar 

  6. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.

    MathSciNet  Google Scholar 

  7. Chi KP, Karapinar E, Thanh TD: A generalized contraction principle in partial metric spaces. Math. Comput. Model. 2012, 55(5–6):1673–1681. 10.1016/j.mcm.2011.11.005

    Article  MathSciNet  Google Scholar 

  8. Karapinar E: Weak ϕ -contraction on partial metric spaces. J. Comput. Anal. Appl. 2012, 14(1):206–210.

    MathSciNet  Google Scholar 

  9. Karapinar E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 4

    Google Scholar 

  10. Karapinar E, Erhan IM: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6: 239–244.

    MathSciNet  Google Scholar 

  11. Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6

    Article  MathSciNet  Google Scholar 

  12. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chi-Ming Chen.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Chen, CH., Chen, CM. Fixed point results for Meir-Keeler-type ϕ-α-contractions on partial metric spaces. J Inequal Appl 2013, 341 (2013). https://doi.org/10.1186/1029-242X-2013-341

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-341

Keywords