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On locally contractive fuzzy set-valued mappings

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.

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 CB(X) the family of all nonempty closed bounded subsets of (X,d). As usual, for ζX and ACB(X), we define

d(ζ,A)= inf a A d(ζ,a).

Then the Hausdorff metric H on CB(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,BCB(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 ζX, 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,BW(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 CB(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,BCB(X) with H(A,B)<r, r>0. If aA, then there exists bB 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 CB(X) such that lim n H( A n ,A)=0, for some ACB(X). If ξ n A n , for all nN, and d( ξ n ,ξ)0, then ξA.

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 ξX is a fixed point of T if ξT ( 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,ξX, 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:XX is called an (ε,λ) uniformly locally contractive mapping if ζ,ζX and 0<d(ζ,ζ)<ε, implies d(Tζ,Tξ)λd(ζ,ξ). A mapping T:XW(X) is called an (ε,λ) uniformly locally contractive fuzzy mapping if ζ,ξX and 0<d(ζ,ξ)<ε, imply d (T(ζ),T(ξ))λd(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 ζX and iN, T i ( ζ ) 1 CB(X). If

ζ,ξX,0<d(ζ,ξ)<εimplies D 1 ( T i ( ζ ) , T j ( ξ ) ) ψ ( d ( ζ , ξ ) ) d(ζ,ξ),
(1)

for all i,jN, 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 iN.

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 ij.)

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,,m1.

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,,m1.

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 nN.

For each j{0,1,2,,m1}, 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,,m1}, 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,,m1}, 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 iN. 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 iN. 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 ζX and iN, T i ( ζ ) 1 CB(X). If

ζ,ξX,0<d(ζ,ξ)<εimplies D 1 ( T i ( ζ ) , T j ( ξ ) ) λd(ζ,ξ),

for all i,jN, 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,ξX,0<d(ζ,ξ)<εimplies d ( T i ( ζ ) , T j ( ξ ) ) ψ ( d ( ζ , ξ ) ) d(ζ,ξ),

for all i,jN, 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)CB(X) and D 1 ( T i (ζ), T j (ξ)) d ( T i (ζ), T j (ξ)), for all i,jN, 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,ξX,0<d(ζ,ξ)<εimplies d ( T i ( ζ ) , T j ( ξ ) ) λd(ζ,ξ),

for all i,jN, 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,ξX,0<d(ζ,ξ)<εimplies 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: XW(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 CB(X) satisfying the following condition:

ζ,ξX,0<d(ζ,ξ)<εimpliesH ( 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 tS(ξ) and T(ξ)(t)=0, otherwise. Then T ( ξ ) 1 =S(ξ), for all ξX, so T ( ξ ) 1 CB(X), for all ξX. Since

D 1 ( T ( ζ ) , T ( ξ ) ) =H ( T ( ζ ) 1 , T ( ξ ) 1 ) =H ( S ( ζ ) , S ( ξ ) ) ,

for all ζ,ξX, 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 CB(X) satisfying the following condition:

ζ,ξX,0<d(ζ,ξ)<εimpliesH ( S ( ζ ) , S ( ξ ) ) λd(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 CB(X) and ψ:[0,)[0,1) a MT-function such that

H(Sζ,Sξ)ψ ( d ( ζ , ξ ) ) d(ζ,ξ),

for all ζ,ξX. 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)=|xy|, for all x,yX. 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,yX with xy, and i,jN 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 |xy|.

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 xX, and d(x,y)=max{x,y} whenever xy (in the sequel we shall write xy 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 0x1,

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 CB(X), for all xX and kN (recall that each x0 is an isolated point for the induced topology, so every bounded interval belongs to CB(X)).

We show that condition (1) of Theorem 3.1 is satisfied for ε= and ψ as defined above. Indeed, let x,yX with xy and j,kN. 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 (ab) 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 y1, 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,y1, 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).

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Acknowledgements

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

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Ahmad, J., Azam, A. & Romaguera, S. On locally contractive fuzzy set-valued mappings. J Inequal Appl 2014, 74 (2014). https://doi.org/10.1186/1029-242X-2014-74

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Keywords

  • fixed point
  • fuzzy mapping
  • contractive mapping
  • locally contractive