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Common fixed-point results for generalized Berinde-type contractions which involve altering distance functions

  • Fawzia Shaddad1Email author,
  • Mohd Salmi Md Noorani1 and
  • Saud M Alsulami2
Fixed Point Theory and Applications20142014:24

https://doi.org/10.1186/1687-1812-2014-24

Received: 22 July 2013

Accepted: 10 January 2014

Published: 31 January 2014

Abstract

In this paper, the existence and the uniqueness of a common fixed point for two self-mappings satisfying some generalized Berinde-type contractions which involve an altering distance function with λ [ 0 , 1 ] is established. Our theorems extend, unify, and generalize several existing results in the literature. An application of an integral equation is presented.

MSC:54H25, 54C60, 54E50.

Keywords

metric spacecommon fixed pointgeneralized Berinde-type contractionsUrysohn integral equations

1 Introduction and preliminaries

Fixed-point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. In 1922, Banach established the famous fixed-point theorem which is called the Banach contraction principle. This principle is a forceful tool in nonlinear analysis. It has many applications in solving nonlinear equations.

Later, Berinde [15] studied many interesting fixed-point theorems for many kinds of contraction mappings. In [3] and [4], he defined the almost contraction map as follows.

Definition 1.1 Let ( X , d ) be a metric space. A map f : X X is called an almost contraction if there exist a constant λ [ 0 , 1 ) and some L 0 such that
d ( f x , f y ) λ d ( x , y ) + L d ( y , f x )

for all x , y X .

Let ( X , d ) be a metric space. A map f : X X is said to satisfy ‘condition (B)’ if there exist a constant λ [ 0 , 1 ) and some L 0 such that for all x , y X .
d ( f x , f y ) λ d ( x , y ) + L min { d ( x , f x ) , d ( y , f y ) , d ( x , f y ) , d ( y , f x ) } .

Recently, Babu et al. [6] considered the class of mappings that satisfy ‘condition (B)’.

More recently, Abbas and Ilic in [7] introduced the following definition.

Definition 1.2 Let f and g be two self-maps on a metric space ( X , d ) . A map f is called generalized almost g-contraction if there exist a constant λ [ 0 , 1 ) and some L 0 such that
d ( f x , f y ) λ M ( x , y ) + L min { d ( g x , f x ) , d ( g y , f y ) , d ( g x , f y ) , d ( g y , f x ) }

for all x , y X , where M ( x , y ) = max { d ( g x , g y ) , d ( g x , f x ) , d ( f y , g y ) , 1 2 ( d ( f x , g y ) + d ( g x , f y ) ) } .

If g = I X , I X is the identity map on X, then they note that f satisfies ‘generalized condition (B)’.

Furthermore, Ćirić et al. [8] introduced the concept of the almost generalized contractive condition as follows.

Definition 1.3 Let f and g be two self-maps on a metric space ( X , d ) . They are said to satisfy almost generalized contractive condition if there exist a constant λ [ 0 , 1 ) and some L 0 such that
d ( f x , g y ) λ max { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , 1 2 ( d ( f x , y ) + d ( x , g y ) ) } + L min { d ( x , f x ) , d ( y , g y ) , d ( x , g y ) , d ( y , f x ) }

for all x , y X .

A new category of contractive fixed-point problems was addressed by Khan et al. [9]. In their study they introduced the notion of an altering distance function, which is a control function that alters the distance between two points in a metric space.

Definition 1.4 The function ψ : [ 0 , ) [ 0 , ) is called an altering distance function if the following properties are satisfied:
  1. (i)

    ψ is continuous and nondecreasing;

     
  2. (ii)

    ψ ( t ) = 0 t = 0 .

     

In the literature, there has been extensive research on common fixed points by using Berinde-type contractions, see [10, 11], by using an altering distance function, see [1214] and by using many other kinds of methods, see [1517].

The aim of this work is to prove that there is a unique common fixed point for two self-mappings satisfying some generalized Berinde-type contractions which involve an altering distance function with λ [ 0 , 1 ] . These results extend and generalize several well-known compatible recent and classical results in the literature. As an application, the existence of a solution for the Urysohn integral equation is presented.

2 Main results

Theorem 2.1 Let ( X , d ) be a complete metric space. Suppose ψ : [ 0 , ) [ 0 , ) is an altering distance function and φ : [ 0 , ) [ 0 , ) is a lower semi-continuous function with φ ( t ) = 0 if and only if t = 0 . Moreover, suppose that f and g are self-maps satisfying the inequality
ψ ( d ( f x , g y ) ) ψ ( λ u ( x , y ) ) φ ( λ u ( x , y ) ) + L N ( x , y ) ,
(2.1)
where
u ( x , y ) { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , 1 2 ( d ( f x , y ) + d ( x , g y ) ) }
and
N ( x , y ) = min { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , d ( f x , y ) , d ( x , g y ) } ,

with L 0 and 0 λ 1 . Then f and g have a unique common fixed point.

Proof We prove the theorem in several steps.

Step 1. Let x 0 X be an arbitrary point.

Taking x 1 = f x 0 and x 2 = g x 1 , then let x 3 = f x 2 and x 4 = g x 3 . Continue in this way, we can choose a sequence { x n } in X so that
x 2 n + 1 = f x 2 n
and
x 2 n + 2 = g x 2 n + 1
for all n = 0 , 1 , 2 ,  . Let
d n = d ( x n , x n + 1 ) .
(2.2)

We will prove that { d n } is a decreasing sequence which converges to 0.

If n is an even number, put x = x n and y = x n 1 in (2.1). We get
ψ ( d ( x n + 1 , x n ) ) = ψ ( d ( f x n , g x n 1 ) ) ψ ( λ u ( x n , x n 1 ) ) φ ( λ u ( x n , x n 1 ) ) + L N ( x n , x n 1 ) ,
(2.3)
where
u ( x n , x n 1 ) { d ( x n , x n 1 ) , d ( x n , f x n ) , d ( x n 1 , g x n 1 ) , 1 2 ( d ( f x n , x n 1 ) + d ( x n , g x n 1 ) ) }
and
N ( x n , x n 1 ) = min { d ( x n , x n 1 ) , d ( x n , f x n ) , d ( x n 1 , g x n 1 ) , d ( f x n , x n 1 ) , d ( x n , g x n 1 ) } ,
i.e.,
N ( x n , x n 1 ) = 0 .
Thus, we have
ψ ( d ( x n , x n + 1 ) ) ψ ( λ u ( x n , x n 1 ) ) φ ( λ u ( x n , x n 1 ) ) ,
where
u ( x n , x n 1 ) { d ( x n 1 , x n ) , d ( x n , x n + 1 ) , 1 2 d ( x n 1 , x n + 1 ) } .

If x n = x n + 1 for some n 0 , then the proof will be finished. Therefore, we suppose that x n x n + 1 for all n 0 . Now, we shall show that d ( x n , x n + 1 ) d ( x n 1 , x n ) . Arguing by contradiction, we assume d ( x n , x n + 1 ) > d ( x n 1 , x n ) . Therefore, we have three cases.

Case 1: u ( x n , x n 1 ) = d ( x n 1 , x n ) . Then
ψ ( d ( x n , x n + 1 ) ) ψ ( λ d ( x n 1 , x n ) ) φ ( λ d ( x n 1 , x n ) ) < ψ ( λ d ( x n 1 , x n ) ) .

Since ψ is increasing, we have d ( x n , x n + 1 ) < λ d ( x n 1 , x n ) , which is a contradiction.

Case 2: u ( x n , x n 1 ) = d ( x n , x n + 1 ) . Then
ψ ( d ( x n , x n + 1 ) ) ψ ( λ d ( x n , x n + 1 ) ) φ ( λ d ( x n , x n + 1 ) ) < ψ ( λ d ( x n , x n + 1 ) ) .

Since ψ is increasing, we have d ( x n , x n + 1 ) < λ d ( x n , x n + 1 ) , which is impossible.

Case 3: u ( x n , x n 1 ) = 1 2 d ( x n 1 , x n + 1 ) . Then
ψ ( d ( x n , x n + 1 ) ) ψ ( λ 2 d ( x n 1 , x n + 1 ) ) φ ( λ 2 d ( x n 1 , x n + 1 ) ) ψ ( λ 2 d ( x n 1 , x n + 1 ) ) .
Since ψ is increasing, we have
d ( x n , x n + 1 ) λ 2 d ( x n 1 , x n + 1 ) λ 2 ( d ( x n 1 , x n ) + d ( x n , x n + 1 ) ) ,
which leads to
d ( x n , x n + 1 ) λ 2 λ d ( x n 1 , x n ) ,
but d ( x n , x n + 1 ) > d ( x n 1 , x n ) , therefore
d ( x n , x n + 1 ) < λ 2 λ d ( x n , x n + 1 ) ,

which is impossible since λ / ( 2 λ ) 1 . Hence, from the above we obtain d ( x n , x n + 1 ) d ( x n 1 , x n ) .

Similarly, we can prove that d ( x n , x n + 1 ) d ( x n 1 , x n ) also in the case when n is an odd number.

Therefore, we find that { d n } is a decreasing sequence and bounded below. Thus, { d n } is convergent. Let
d n d as  n .
(2.4)

Next, we want to show that d = 0 . We have two cases.

Case 1. When u ( x n , x n 1 ) { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } , as ψ is continuous and φ is lower semi-continuous and from (2.4) we get
ψ ( d ) ψ ( λ d ) φ ( λ d ) .

If λ = 0 , then we have ψ ( d ) 0 d = 0 . If λ 0 , then we have φ ( λ d ) ψ ( λ d ) ψ ( d ) 0 . Thus φ ( λ d ) = 0 , which implies d = 0 .

Case 2. When u ( x n , x n 1 ) = 1 2 d ( x n 1 , x n + 1 ) , we suppose that d 0 ; then
ψ ( d n ) ψ ( λ 2 d ( x n 1 , x n + 1 ) ) φ ( λ 2 d ( x n 1 , x n + 1 ) ) ψ ( λ 2 d ( x n 1 , x n + 1 ) ) ψ ( λ 2 ( d ( x n 1 , x n ) + d ( x n , x n + 1 ) ) ) .

Now, we have two subcases.

Subcase 1. λ < 1 . Then as n we obtain
ψ ( d ) ψ ( λ d ) ,

which leads to a contradiction if d 0 .

Subcase 2. λ = 1 . Then
ψ ( d n ) ψ ( 1 2 d ( x n 1 , x n + 1 ) ) ψ ( 1 2 ( d ( x n 1 , x n ) + d ( x n , x n + 1 ) ) ) .
As n , we have
ψ ( d ) ψ ( 1 2 lim n d ( x n 1 , x n + 1 ) ) ψ ( d ) .
Since ψ is an increasing function, we get
lim n d ( x n 1 , x n + 1 ) = 2 d .
(2.5)
By taking n in ψ ( d n ) ψ ( λ 2 d ( x n 1 , x n + 1 ) ) φ ( λ 2 d ( x n 1 , x n + 1 ) ) and using (2.5), we have
ψ ( d ) ψ ( 1 2 ( 2 d ) ) φ ( 1 2 ( 2 d ) ) ,
i.e.
φ ( d ) 0 .

Hence, φ ( d ) = 0 and then d = 0 .

From the above we obtain d = 0 , i.e.,
d n = d ( x n , x n + 1 ) 0 as  n .
(2.6)
Step 2. We prove that { x n } is a Cauchy sequence. Suppose that { x n } is not a Cauchy sequence, then there exists ε > 0 for which we can find subsequences { x n ( k ) } and { x m ( k ) } of { x n } with n ( k ) > m ( k ) k such that
d ( x n ( k ) , x m ( k ) ) ε / k .
(2.7)
Furthermore, corresponding to m ( k ) , we can choose n ( k ) in such a way that it is the smallest integer with n ( k ) > m ( k ) and satisfying (2.7). Then
d ( x n ( k ) 1 , x m ( k ) ) < ε / k .
Then we have
ε / k d ( x n ( k ) , x m ( k ) ) d ( x n ( k ) , x n ( k ) 1 ) + d ( x n ( k ) 1 , x m ( k ) ) d ( x n ( k ) , x n ( k ) 1 ) + ε / k ,
by taking k , we obtain
lim k d ( x n ( k ) , x m ( k ) ) = 0 ,
which is a contradiction. So, { x n } is a Cauchy sequence in a complete metric space X and hence it is convergent in X. Let
lim n x n = x .
Step 3. Let us now prove that x is a common fixed point of f and g. Put x = x and y = x 2 n + 1 in (2.1) for all n, and we obtain
ψ ( d ( f x , g x 2 n + 1 ) ) ψ ( λ u ( x , x 2 n + 1 ) ) φ ( λ u ( x , x 2 n + 1 ) ) + L N ( x , x 2 n + 1 ) ,
where
u ( x , x 2 n + 1 ) { d ( x , x 2 n + 1 ) , d ( x , f x ) , d ( x 2 n + 1 , g x 2 n + 1 ) , 1 2 ( d ( f x , x 2 n + 1 ) + d ( x , g x 2 n + 1 ) ) }
and
N ( x , x 2 n + 1 ) = min { d ( x , x 2 n + 1 ) , d ( x , f x ) , d ( x 2 n + 1 , g x 2 n + 1 ) , d ( f x , x 2 n + 1 ) , d ( x , g x 2 n + 1 ) } .
Let n , we get
ψ ( d ( f x , x ) ) ψ ( λ lim n u ( x , x 2 n + 1 ) ) φ ( λ lim n u ( x , x 2 n + 1 ) ) ,
where
lim n u ( x , x 2 n + 1 ) { 0 , d ( x , f x ) , 1 2 d ( x , f x ) } .
If d ( x , f x ) 0 , then
ψ ( d ( x , f x ) ) < ψ ( λ d ( x , f x ) ) or ψ ( d ( x , f x ) ) < ψ ( λ 2 d ( x , f x ) ) ,
which is a contradiction. Hence, we obtain
d ( x , f x ) = 0 or x = f x .
(2.8)
Similarly, when we take x = x 2 n and y = x in (2.1) for all n we get
x = g x .
(2.9)

Equations (2.8) and (2.9) show that x is a common fixed point of f and g.

Step 4. Let us now show the uniqueness. Let y be another common fixed point of f and g. Then from (2.1) we have
ψ ( d ( f x , g y ) ) = ψ ( d ( x , y ) ) ψ ( λ u ( x , y ) ) φ ( λ u ( x , y ) ) + L N ( x , y ) ,
where
u ( x , y ) { 0 , d ( x , y ) } and N ( x , y ) = 0 .

Then we obtain x = y . □

Corollary 2.2 Let ( X , d ) be a complete metric space. Suppose ψ : [ 0 , ) [ 0 , ) is an altering distance function and φ : [ 0 , ) [ 0 , ) is a lower semi-continuous function with φ ( t ) = 0 if and only if t = 0 . If f and g are self-maps satisfying the inequality
ψ ( d ( f x , g y ) ) ψ ( λ M ( x , y ) ) φ ( λ M ( x , y ) ) + L N ( x , y ) ,
where
M ( x , y ) = max { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , 1 2 ( d ( f x , y ) + d ( x , g y ) ) }
and
N ( x , y ) = min { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , d ( f x , y ) , d ( x , g y ) } ,

with L 0 and 0 λ 1 , then f has a unique fixed point.

Proof Since M ( x , y ) { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , 1 2 ( d ( f x , y ) + d ( x , g y ) ) } , the result follows from Theorem 2.1. □

Remark 2.3 In Corollary 2.2:
  1. (i)

    If g = f and λ = 1 , we obtain a metric version of Theorem 12 of Aydi et al. [10].

     
  2. (ii)

    If L = 0 and λ = 1 , we get Theorem 2.1 of Doric [18].

     
  3. (iii)

    If ψ ( t ) = t , L = 0 and λ = 1 , we get Theorem 2.1 of Zhang and Song [19].

     

By taking f = g and L = 0 in Corollary 2.2, we obtain the following result.

Corollary 2.4 Let ( X , d ) be a complete metric space. Suppose ψ : [ 0 , ) [ 0 , ) is an altering distance function and φ : [ 0 , ) [ 0 , ) is a lower semi-continuous function with φ ( t ) = 0 if and only if t = 0 . If f is a self-map satisfying the inequality
ψ ( d ( f x , f y ) ) ψ ( λ u ( x , y ) ) φ ( λ u ( x , y ) ) ,
(2.10)

where u ( x , y ) { d ( x , y ) , d ( x , f x ) , d ( y , f y ) , 1 2 ( d ( f x , y ) + d ( x , f y ) ) } , 0 λ 1 , then f has a unique fixed point.

Remark 2.5 Corollary 2.4 extends the main fixed-point result of Dutta and Choudhury [[20], Theorem 2.1] and Theorem 2.2 of Doric [18].

If we take ψ ( t ) = t and φ ( t ) = ( 1 k ) t with k < 1 in Theorem 2.1, we have the following corollary.

Corollary 2.6 Let ( X , d ) be a complete metric space. If f and g are self-maps satisfying the inequality
d ( f x , g y ) α u ( x , y ) + L N ( x , y ) ,
(2.11)
where
u ( x , y ) { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , 1 2 ( d ( f x , y ) + d ( x , g y ) ) }
and
N ( x , y ) = min { d ( x , y ) , d ( x , f x ) , d ( y , g y ) , d ( f x , y ) , d ( x , g y ) } ,

with L 0 and 0 α < 1 , then f and g have a unique common fixed point.

If we take f = g in Corollary 2.6 we obtain the following result.

Corollary 2.7 Let ( X , d ) be a complete metric space. If f is a self-map satisfying the inequality
d ( f x , f y ) α u ( x , y ) + L N ( x , y ) ,
where
u ( x , y ) { d ( x , y ) , d ( x , f x ) , d ( y , f y ) , 1 2 ( d ( f x , y ) + d ( x , f y ) ) }
and
N ( x , y ) = min { d ( x , y ) , d ( x , f x ) , d ( y , f y ) , d ( f x , y ) , d ( x , f y ) } ,

with L 0 and 0 α < 1 , then f has a unique fixed point.

By the aid of Lemma 2.1 of [21], we have the following result as a consequence of Corollary 2.7.

Corollary 2.8 Let ( X , d ) be a complete metric space. If f and g are self-maps satisfying the inequality
d ( f x , f y ) α u ( g x , g y ) + L N ( g x , g y ) ,
where
u ( g x , g y ) { d ( g x , g y ) , d ( g x , f x ) , d ( g y , f y ) , 1 2 ( d ( f x , g y ) + d ( g x , f y ) ) }
and
N ( g x , g y ) = min { d ( g x , g y ) , d ( g x , f x ) , d ( g y , f y ) , d ( f x , g y ) , d ( g x , f y ) } ,

with L 0 and 0 α < 1 , then f and g have a unique common fixed point.

Remark 2.9 Corollary 2.8 extends the results of Abbas et al. [[22], Theorem 2.1] and Jleli et al. [[23], Corollary 3.2].

3 Applications: existence of a common solution to Urysohn integral equations

Throughout this section we take X = C ( [ a , b ] , R ) (the set of continuous functions defined in I = [ a , b ] ). We define the metric d : X × X R by d ( x , y ) = x y for every x , y X . Let φ : R + R + be a function such that

  • φ is lower semi-continuous;

  • φ is increasing;

  • φ ( t ) = 0 t = 0 .

Theorem 3.1 Consider the Urysohn integral equations
x ( t ) = a b K 1 ( t , s , x ( s ) ) d s + h ( t ) , x ( t ) = a b K 2 ( t , s , x ( s ) ) d s + q ( t ) ,
(3.1)
where t I R and x , h , q X . Assume that, for K 1 , K 2 : I × I × R R , we have
| a b K 1 ( t , s , x ( s ) ) d s a b K 2 ( t , s , x ( s ) ) d s + h ( t ) q ( t ) | | x ( t ) y ( t ) | φ ( sup t I | x ( t ) y ( t ) | ) .

Then there exists a solution to (3.1).

Proof Define f , g : X X by f ( x ) = a b K 1 ( t , s , x ( s ) ) d s + h ( t ) and g ( x ) = a b K 2 ( t , s , x ( s ) ) d s + q ( t ) . It is obvious that
f g x y φ ( x y ) .
Thus
d ( f x , g y ) d ( x , y ) φ ( d ( x , y ) )

for all x , y X . Now, all the assumptions of Theorem 2.1 are satisfied with ψ ( t ) = t , for all t R + , u ( x , y ) = d ( x , y ) , and L = 0 . Therefore, f and g have a common fixed point, that is, a solution to the integral equation (3.1). □

Declarations

Acknowledgements

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-DIP-2012-31.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia
(2)
Department of Mathematics, King Abdulaziz University

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© Shaddad et al.; licensee Springer. 2014

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