Open Access

On locally contractive fuzzy set-valued mappings

Journal of Inequalities and Applications20142014:74

https://doi.org/10.1186/1029-242X-2014-74

Received: 28 October 2013

Accepted: 20 January 2014

Published: 13 February 2014

Abstract

We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.

MSC:46S40, 47H10, 54H25.

Keywords

fixed pointfuzzy mappingcontractive mappinglocally contractive

1 Introduction

The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in [1] the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [210] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.

2 Preliminaries

Given a metric space ( X , d ) , denote by C B ( X ) the family of all nonempty closed bounded subsets of ( X , d ) . As usual, for ζ X and A C B ( X ) , we define
d ( ζ , A ) = inf a A d ( ζ , a ) .
Then the Hausdorff metric H on C B ( X ) induced by d is defined as
H ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( A , b ) } ,

for all A , B C B ( X ) .

A fuzzy set in ( X , d ) is a function with domain X and values in I = [ 0 , 1 ] . I X denotes the collection of all fuzzy sets in X. If A is a fuzzy set and ζ X , then the function value A ( ζ ) is called the grade of membership of ζ in A. The α-level set of a fuzzy set A is denoted by A α , and it is defined as follows:
A α = { ζ : A ( ζ ) α } if  α ( 0 , 1 ] , A 0 = closure of  { ζ : A ( ζ ) > 0 } .

According to Heilpern [1], a fuzzy set A in a metric linear space ( X , d ) is said to be an approximate quantity if A α is compact and convex in X, for each α ( 0 , 1 ] , and sup ζ X A ( ζ ) = 1 . The family of all approximate quantities of the metric linear space ( X , d ) is denoted by W ( X ) .

Now, for A , B W ( X ) and α [ 0 , 1 ] , define
D α ( A , B ) = H ( A α , B α ) ,
and
d ( A , B ) = sup α [ 0 , 1 ] D α ( A α , B α ) .

It is well known that d is a metric on W ( X ) .

In case that ( X , d ) is a (non-necessarily linear) metric space, we also define
D α ( A , B ) = H ( A α , B α ) ,

whenever A , B I X and A α , B α C B ( X ) , α [ 0 , 1 ] .

In the sequel the letter will denote the set of positive integer numbers.

The following well-known properties on the Hausdorff metric (see e.g. [11]) will be useful in the next section.

Lemma 2.1 Let ( X , d ) be a metric space and let A , B C B ( X ) with H ( A , B ) < r , r > 0 . If a A , then there exists b B such that d ( a , b ) < r .

Lemma 2.2 Let ( X , d ) be a metric space and let { A n } n = 1 be a sequence in C B ( X ) such that lim n H ( A n , A ) = 0 , for some A C B ( X ) . If ξ n A n , for all n N , and d ( ξ n , ξ ) 0 , then ξ A .

Now, let X be an arbitrary set and let Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into I Y . In fact, a fuzzy mapping T is a fuzzy subset on X × Y with membership function T ( ζ ) . The value T ( ζ ) ( ξ ) is the grade of membership of ξ in T ( ζ ) .

If ( X , d ) is a metric space and T is a (fuzzy) mapping from X into I X , we say that ξ X is a fixed point of T if ξ T ( ξ ) 1 .

We conclude this section with the notion of contractiveness that will be used in our main result.

Definition 2.3 (compare [12])

Let ε ( 0 , ] . A function ψ : [ 0 , ε ) [ 0 , 1 ) is said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition (i.e., lim sup r t + ψ ( r ) < 1 , for all t [ 0 , ε ) ).

Clearly, if ψ : [ 0 , ε ) [ 0 , 1 ) is a nondecreasing function or a nonincreasing function, then it is a MT-function. So the set of MT-functions is a rich class.

3 Fixed points of fuzzy mappings

Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein [13], Beg and Azam [14], Holmes [15], Hu [11], Hu and Rosen [16], Ko and Tasi [17], Kuhfitting [18] and Nadler [19].

Heilpern [1] established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam et al. [4, 5], and Lee and Cho [10] further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem 3.1 below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature.

Let ε ( 0 , ] , and λ ( 0 , 1 ) . A metric space ( X , d ) is said to be ε-chainable if given ζ , ξ X , there exists an ε-chain from ζ to ξ (i.e., a finite set of points ζ = ζ 0 , ζ 1 , ζ 2 , , ζ m = ξ such that d ( ζ j 1 , ζ j ) < ε , for all j = 1 , 2 , , m ). A mapping T : X X is called an ( ε , λ ) uniformly locally contractive mapping if ζ , ζ X and 0 < d ( ζ , ζ ) < ε , implies d ( T ζ , T ξ ) λ d ( ζ , ξ ) . A mapping T : X W ( X ) is called an ( ε , λ ) uniformly locally contractive fuzzy mapping if ζ , ξ X and 0 < d ( ζ , ξ ) < ε , imply d ( T ( ζ ) , T ( ξ ) ) λ d ( ζ , ξ ) . We remark that a globally contractive mapping can be regarded as an ( , λ ) uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.

Theorem 3.1 Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric space and { T i } i = 1 a sequence of fuzzy mappings from X into I X such that, for each ζ X and i N , T i ( ζ ) 1 C B ( X ) . If
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies D 1 ( T i ( ζ ) , T j ( ξ ) ) ψ ( d ( ζ , ξ ) ) d ( ζ , ξ ) ,
(1)

for all i , j N , where ψ : [ 0 , ε ) [ 0 , 1 ) is a MT-function, then the sequence { T i } i = 1 has a common fixed point, i.e., there is ξ X such that ξ T i ( ξ ) 1 , for all i N .

Proof Let ξ 0 be an arbitrary, but fixed element of X. Find ξ 1 X such that ξ 1 T 1 ( ξ 0 ) 1 . Let
ξ 0 = ζ ( 1 , 0 ) , ζ ( 1 , 1 ) , ζ ( 1 , 2 ) , , ζ ( 1 , m ) = ξ 1 T 1 ( ξ 0 ) 1

be an arbitrary ε-chain from ξ 0 to ξ 1 . (We suppose, without loss of generality, that ζ ( 1 , i ) ζ ( 1 , j ) , for each i , j { 0 , 1 , 2 , , m } with i j .)

Since 0 < d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < ε , we deduce that
D 1 ( T 1 ( ζ ( 1 , 0 ) ) , T 2 ( ζ ( 1 , 1 ) ) ) ψ ( d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) ) d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < ψ ( d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) ) d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < ε .
Rename ξ 1 as ζ ( 2 , 0 ) . Since ζ ( 2 , 0 ) T 1 ( ζ ( 1 , 0 ) ) 1 , using Lemma 2.1 we find ζ ( 2 , 1 ) T 2 ( ζ ( 1 , 1 ) ) 1 such that
d ( ζ ( 2 , 0 ) , ζ ( 2 , 1 ) ) < ψ ( d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) ) d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < d ( ζ ( 1 , 0 ) , ζ ( 1 , 1 ) ) < ε .
Similarly we may choose an element ζ ( 2 , 2 ) T 2 ( ζ ( 1 , 2 ) ) 1 such that
d ( ζ ( 2 , 1 ) , ζ ( 2 , 2 ) ) < ψ ( d ( ζ ( 1 , 1 ) , ζ ( 1 , 2 ) ) ) d ( ζ ( 1 , 1 ) , ζ ( 1 , 2 ) ) < d ( ζ ( 1 , 1 ) , ζ ( 1 , 2 ) ) < ε .
Thus we obtain a set { ζ ( 2 , 0 ) , ζ ( 2 , 1 ) , ζ ( 2 , 2 ) , , ζ ( 2 , m ) } of m + 1 points of X such that ζ ( 2 , 0 ) T 1 ( ζ ( 1 , 0 ) ) 1 and ζ ( 2 , j ) T 2 ( ζ ( 1 , j ) ) 1 , for j = 1 , 2 , , m , with
d ( ζ ( 2 , j ) , ζ ( 2 , j + 1 ) ) < ψ ( d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) ) d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) < d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) < ε ,

for j = 0 , 1 , 2 , , m 1 .

Let ζ ( 2 , m ) = ξ 2 . Thus the set of points ξ 1 = ζ ( 2 , 0 ) , ζ ( 2 , 1 ) , ζ ( 2 , 2 ) , , ζ ( 2 , m ) = ξ 2 T 2 ( ξ 1 ) 1 is an ε-chain from ξ 0 to ξ 1 . Rename ξ 2 as ζ ( 3 , 0 ) . Then by the same procedure we obtain an ε-chain
ξ 2 = ζ ( 3 , 0 ) , ζ ( 3 , 1 ) , ζ ( 3 , 2 ) , , ζ ( 3 , m ) = ξ 3 T 3 ( ξ 2 ) 1
from ξ 2 to ξ 3 . Inductively, we obtain
ξ n = ζ ( n + 1 , 0 ) , ζ ( n + 1 , 1 ) , ζ ( n + 1 , 2 ) , , ζ ( n + 1 , m ) = ξ n + 1 T n + 1 ( ξ n ) 1
with
d ( ζ ( n + 1 , j ) , ζ ( n + 1 , j + 1 ) ) < ψ ( d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) ) d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) < d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) < ε ,
(2)

for j = 0 , 1 , 2 , , m 1 .

Consequently, we construct a sequence { ξ n } n = 1 of points of X with
ξ 1 = ζ ( 1 , m ) = ζ ( 2 , 0 ) T 1 ( ξ 0 ) 1 , ξ 2 = ζ ( 2 , m ) = ζ ( 3 , 0 ) T 2 ( ξ 1 ) 1 , ξ 3 = ζ ( 3 , m ) = ζ ( 4 , 0 ) T 3 ( ξ 2 ) 1 , ξ n + 1 = ζ ( n + 1 , m ) = ζ ( n + 2 , 0 ) T n + 1 ( ξ n ) 1 ,

for all n N .

For each j { 0 , 1 , 2 , , m 1 } , we deduce from (2) that { d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) } n = 1 is a decreasing sequence of non-negative real numbers and therefore there exists l j 0 such that
lim n d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) = l j .

By assumption, lim sup t l j + ψ ( t ) < 1 , so there exists n j N such that ψ ( d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) ) < s ( l j ) , for all n n j where lim sup t l j + ψ ( t ) < s ( l j ) < 1 .

Now put
M j = max { max i = 1 , , n j ψ ( d ( ζ ( i , j ) , ζ ( i , j + 1 ) ) ) , s ( l j ) } .
Then, for every n > n j , we obtain
d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) < ψ ( d ( ζ ( n 1 , j ) , ζ ( n 1 , j + 1 ) ) ) d ( ζ ( n 1 , j ) , ζ ( n 1 , j + 1 ) ) < s ( l j ) d ( ζ ( n 1 , j ) , ζ ( n 1 , j + 1 ) ) M j d ( ζ ( n 1 , j ) , ζ ( n 1 , j + 1 ) ) ( M j ) 2 d ( ζ ( n 2 , j ) , ζ ( n 2 , j + 1 ) ) ( M j ) n 1 d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) .
Putting N = max { n j : j = 0 , 1 , 2 , , m 1 } , we have
d ( ξ n 1 , ξ n ) = d ( ζ ( n , 0 ) , ζ ( n , m ) ) j = 0 m 1 d ( ζ ( n , j ) , ζ ( n , j + 1 ) ) < j = 0 m 1 ( M j ) n 1 d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) ,
for all n > N + 1 . Hence
d ( ξ n , ξ p ) d ( ξ n , ξ n + 1 ) + d ( ξ n + 1 , ξ n + 2 ) + + d ( ξ p 1 , ξ p ) < j = 0 m 1 ( M j ) n d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) + + j = 0 m 1 ( M j ) p 1 d ( ζ ( 1 , j ) , ζ ( 1 , j + 1 ) ) ,

whenever p > n > N + 1 .

Since M j < 1 , for all j { 0 , 1 , 2 , , m 1 } , it follows that { ξ n } n = 1 is a Cauchy sequence. Since ( X , d ) is complete, there is ξ X such that ξ n ξ . So for each δ ( 0 , ε ] there is M δ N such that n > M δ implies d ( ξ n , ξ ) < δ . This in view of inequality (1) implies D 1 ( T n + 1 ( ξ n ) , T i ( ξ ) ) < δ , for all i N . Consequently, H ( T n + 1 ( ξ n ) 1 , T i ( ξ ) 1 ) 0 . Since ξ n + 1 T n + 1 ( ξ n ) 1 with d ( ξ n + 1 , ξ ) 0 , we deduce from Lemma 2.2 that ξ T i ( ξ ) 1 , for all i N . This completes the proof. □

Corollary 3.2 Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric space and { T i } i = 1 a sequence of fuzzy mappings from X into I X such that, for each ζ X and i N , T i ( ζ ) 1 C B ( X ) . If
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies D 1 ( T i ( ζ ) , T j ( ξ ) ) λ d ( ζ , ξ ) ,

for all i , j N , where λ ( 0 , 1 ) , then the sequence { T i } i = 1 has a common fixed point.

Proof Apply Theorem 3.1 when ψ is the MT-function defined as ψ ( t ) = λ , for all t [ 0 , ε ) . □

Corollary 3.3 Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric linear space and { T i } i = 1 a sequence of fuzzy mappings from X into W ( X ) satisfying the following condition:
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies d ( T i ( ζ ) , T j ( ξ ) ) ψ ( d ( ζ , ξ ) ) d ( ζ , ξ ) ,

for all i , j N , where ψ : [ 0 , ε ) [ 0 , 1 ) is a MT-function. Then the sequence { T i } i = 1 has a common fixed point.

Proof Since W ( X ) C B ( X ) and D 1 ( T i ( ζ ) , T j ( ξ ) ) d ( T i ( ζ ) , T j ( ξ ) ) , for all i , j N , the result follows immediately from Theorem 3.1. □

Corollary 3.4 Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric linear space and { T i } i = 1 a sequence of fuzzy mappings from X into W ( X ) satisfying the following condition:
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies d ( T i ( ζ ) , T j ( ξ ) ) λ d ( ζ , ξ ) ,

for all i , j N , where λ ( 0 , 1 ) . Then the sequence { T i } i = 1 has a common fixed point.

Corollary 3.5 [4]

Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric linear space and T 1 , T 2 , two fuzzy mappings from X into W ( X ) satisfying the following condition:
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies d ( T i ( ζ ) , T j ( ξ ) ) ψ ( d ( ζ , ξ ) ) d ( ζ , ξ ) ,

for i , j = 1 , 2 , where ψ : [ 0 , ε ) [ 0 , 1 ) is a MT-function. Then T 1 and T 2 have a common fixed point.

Corollary 3.6 [4, 11]

Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric linear space and T: X W ( X ) an ( ε , λ ) uniformly locally contractive fuzzy mapping. Then T has a fixed point.

Corollary 3.7 Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric space and S be a multivalued mapping from X into C B ( X ) satisfying the following condition:
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies H ( S ( ζ ) , S ( ξ ) ) ψ ( d ( ζ , ξ ) ) d ( ζ , ξ ) ,

where ψ : [ 0 , ε ) [ 0 , 1 ) is a MT-function. Then S has a fixed point.

Proof Define a fuzzy mapping T from X into I X as T ( ξ ) ( t ) = 1 if t S ( ξ ) and T ( ξ ) ( t ) = 0 , otherwise. Then T ( ξ ) 1 = S ( ξ ) , for all ξ X , so T ( ξ ) 1 C B ( X ) , for all ξ X . Since
D 1 ( T ( ζ ) , T ( ξ ) ) = H ( T ( ζ ) 1 , T ( ξ ) 1 ) = H ( S ( ζ ) , S ( ξ ) ) ,

for all ζ , ξ X , we deduce that condition (1) of Theorem 3.1 is satisfied for T. Hence T has a fixed point ξ , i.e., ξ T ( ξ ) 1 . We conclude that ξ S ( ξ ) . The proof is complete. □

Corollary 3.8 [13]

Let ε ( 0 , ] , ( X , d ) a complete ε-chainable metric space and S be a multivalued mapping from X into C B ( X ) satisfying the following condition:
ζ , ξ X , 0 < d ( ζ , ξ ) < ε implies H ( S ( ζ ) , S ( ξ ) ) λ d ( ζ , ξ ) ,

where λ ( 0 , 1 ) . Then S has a fixed point.

Corollary 3.9 ([20, 21], see also [9, 13])

Let ( X , d ) be a complete metric space, S a multivalued mapping from X into C B ( X ) and ψ : [ 0 , ) [ 0 , 1 ) a MT-function such that
H ( S ζ , S ξ ) ψ ( d ( ζ , ξ ) ) d ( ζ , ξ ) ,

for all ζ , ξ X . Then S has a fixed point in X.

Proof Apply Corollary 3.8 with ε = . □

We conclude the paper with two examples to support Theorem 3.1 and Corollary 3.2.

Example 3.10 Let ( X , d ) be the compact, and thus complete, metric space such that X = [ 0 , 1 ] , and d ( x , y ) = | x y | , for all x , y X . Let λ be a constant such that λ [ 1 / 14 , 1 ) and let { T k } k = 1 be the sequence of fuzzy mappings defined from X into I X as follows:
if  x = 0 , T k ( x ) ( y ) = { 1 if  y = 0 , 1 / 3 k if  0 < y 1 / 100 , 0 if  1 / 100 < y 1 , k N , if  x 0 , T k ( x ) ( y ) = { 1 if  0 y x / 14 , λ / 2 k if  x / 14 < y x / 12 , λ / 3 k if  x / 12 < y < x , 0 if  x y 1 , k N .
For each x , y X with x y , and i , j N we have
D 1 ( T i ( x ) , T j ( y ) ) = H ( T i ( x ) 1 , T j ( y ) 1 ) = H ( [ 0 , x / 14 ] , [ 0 , y / 14 ] ) = 1 14 | x y | .

Hence, for ψ ( t ) = λ , the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any ε ( 0 , ] , whereas X is not linear. Therefore all previous relevant fixed point results Corollaries 3.3-3.6 on metric linear spaces are not applicable.

Example 3.11 Let ( X , d ) be the complete metric space such that X = [ 0 , ) , d ( x , x ) = 0 , for all x X , and d ( x , y ) = max { x , y } whenever x y (in the sequel we shall write x y instead of max { x , y } ).

Note that a sequence { x n } n = 1 is a Cauchy sequence in ( X , d ) if and only if d ( x n , 0 ) 0 . Moreover, x = 0 is the only non-isolated point of X for the topology induced by d.

Let ψ : [ 0 , ) [ 0 , 1 ) be the MT-function defined as
ψ ( t ) = { 1 / 2 if  0 t 1 , t / ( t + 1 ) if  t > 1 ,
and let { T k } k = 1 be the sequence of fuzzy mappings defined from X into I X as follows:
if  0 x 1 , T k ( x ) ( y ) = { 1 if  x / 4 k y x / 2 k , 0 otherwise, k N , if  x > 1 , T k ( x ) ( y ) = { 1 if  x / 2 k y < x 2 / k ( 1 + x ) , 0 otherwise , k N .
Observe that, for 0 x 1 ,
T k ( x ) 1 = [ x 4 k , x 2 k ] ,
and, for x > 1 ,
T k ( x ) 1 = [ x 2 k , x 2 k ( 1 + x ) ) .

Therefore T k ( x ) 1 C B ( X ) , for all x X and k N (recall that each x 0 is an isolated point for the induced topology, so every bounded interval belongs to C B ( X ) ).

We show that condition (1) of Theorem 3.1 is satisfied for ε = and ψ as defined above. Indeed, let x , y X with x y and j , k N . Assume without loss of generality that x > y .

If x , y > 1 , for each b T j ( y ) 1 , we obtain
d ( T k ( x ) 1 , b ) = inf a T k ( x ) 1 ( a b ) x 2 k ( 1 + x ) b x 2 k ( 1 + x ) y 2 j ( 1 + y ) .
Similarly, for each a T k ( x ) 1 , we obtain
d ( a , T j ( y ) 1 ) x 2 k ( 1 + x ) y 2 j ( 1 + y ) .
Consequently
D 1 ( T k ( x ) , T j ( y ) ) = H ( T k ( x ) 1 , T j ( y ) 1 ) x 2 k ( 1 + x ) y 2 j ( 1 + y ) ( x y ) 2 1 + ( x y ) = d ( x , y ) 1 + d ( x , y ) d ( x , y ) = ψ ( d ( x , y ) ) d ( x , y ) .
If x > 1 and y 1 , we deduce, in a similar way, that
D 1 ( T k ( x ) , T j ( y ) ) = H ( T k ( x ) 1 , T j ( y ) 1 ) x 2 k ( 1 + x ) y 2 j x 2 1 + x y 2 x 2 1 + x x 2 = x 2 1 + x = ( x y ) 2 1 + ( x y ) = d ( x , y ) 1 + d ( x , y ) d ( x , y ) = ψ ( d ( x , y ) ) d ( x , y ) .
Finally, if x , y 1 , we deduce
D 1 ( T k ( x ) , T j ( y ) ) = H ( T k ( x ) 1 , T j ( y ) 1 ) x 2 k y 2 j x y 2 = ψ ( d ( x , y ) ) d ( x , y ) .

We have shown that all conditions of Theorem 3.1 are satisfied (in fact x = 0 is the only fixed point of T).

Declarations

Acknowledgements

The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

Authors’ Affiliations

(1)
Department of Mathematics, COMSATS Institute of Information Technology
(2)
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València

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© Ahmad et al.; licensee Springer. 2014

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