^{1}

^{2}

^{*}

We obtain two generalizations of a known theorem of A. Alam and M. Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702) showing that some standard proofs can be obtained involving only Cauchy sequences of the successive approximations. Suitable examples prove the effective generalization of our results in metric spaces not necessarily complete.

Fixed point theorems involving contraction conditions under preserving relations are known in literature (cf. [

We start with some known definitions [

Definition 1. Let X be a nonempty set and ℜ be a binary relation (eventually partial) defined on X. A sequence { x n } of X is called ℜ -preserving if ( x n , x n + 1 ) ∈ ℜ for every n = 0 , 1 , 2 , ⋯ .

From now on we consider such binary relations and we write simply ℜ ⊆ X 2 .

Definition 2. Let ( X , d ) be a nonempty metric space and ℜ ⊆ X 2 is called d-self-closed if whenever { x n } is ℜ -preserving and converging to a point x ∈ X , then there exists a subsequence { x n ( k ) } of { x n } such that either ( x n ( k ) , x ) ∈ ℜ or ( x , x n ( k ) ) ∈ ℜ for every k = 0 , 1 , 2 , ⋯ .

Definition 3. (cf. [

1) z 0 = x and z k = y ;

2) ( z i , z i + 1 ) ∈ ℜ for each i = 0 , ⋯ , k − 1 .

Notice that a path of length k involves k + 1 elements of X, although they are not necessarily distinct. In [

Theorem 1. Let ( X , ≤ ) be a partially ordered set and there exists a metric d : X × X → [ 0, + ∞ ) . Let T be a selfmap of X such that

1) T is monotone non-decreasing;

2) there exists a point x 0 ∈ X such that x 0 ≤ T ( x 0 ) ;

3) if { x n } is a non-decreasing Cauchy sequence in X, then { x n } converges to z ∈ X and T ( x n ) ≤ z for every n;

4) there exists c ∈ [ 0,1 ) such that d ( T ( x ) , T ( y ) ) ≤ c ⋅ d ( x , y ) for all ( x , y ) ∈ X 2 with x ≥ y ;

then T has a fixed point w ∈ X such that x 0 ≤ w .

In [

Theorem 2. Let ( X , d ) be a complete metric space, ℜ ⊆ X 2 and T be a selfmap of X such that

1) there exists in X a point x 0 such that ( x 0 , T x 0 ) ∈ ℜ ;

2) ℜ is T-closed, that is ( x , y ) ∈ ℜ implies ( T x , T y ) ∈ ℜ ;

3) ℜ is d-self-closed;

4) there exists c ∈ [ 0,1 ) such that d ( T ( x ) , T ( y ) ) ≤ c ⋅ d ( x , y ) for all pair ( x , y ) ∈ ℜ .

Then T has a fixed point. Moreover, if there exists a ℜ -path from x to y for all x , y ∈ X , then this fixed point is unique.

Now we unify Theorem 1 and 2 with the following:

Theorem 3. Let ( X , d ) be a metric space, ℜ ⊆ X 2 and T be a selfmap of X. Suppose that

1) there exists in X a point x 0 such that ( x 0 , T x 0 ) ∈ ℜ ;

2) ℜ is T-closed;

3) for any sequence { y n } ℜ -preserving, n = 0 , 1 , 2 , ⋯ , which is Cauchy and converging to a point y ∈ X , there exists a subsequence { y n ( k ) } of { y n } such that either ( y n ( k ) , y ) ∈ ℜ or ( y , y n ( k ) ) ∈ ℜ for every k = 0 , 1 , 2 , ⋯ ;

4) there exists c ∈ [ 0,1 ] such that d ( T x , T y ) ≤ c ⋅ d ( x , y ) for all ( x , y ) ∈ ℜ .

Then T has a fixed point z in X and there exists a sequence { z k } such that either ( T z k , z ) ∈ ℜ or ( z , T z k ) ∈ ℜ for every k = 0 , 1 , 2 , ⋯ . Moreover,

5) if there exists a ℜ -path from x to y for all x , y ∈ X , then this fixed point is unique.

Proof. Let x 0 ≠ T x 0 otherwise the thesis is trivial. Put h = d ( x 0 , T x 0 ) > 0 and y n = T n ( x 0 ) for every n = 0 , 1 , 2 , ⋯ , so we have

T 0 x 0 = x 0 = y 0 , T 1 x 0 = T x 0 = T y 0 = y 1 , T 2 x 0 = T y 1 = y 2 , ⋯ , T y n = y n + 1 for n = 0 , 1 , ⋯

Because of properties 1) and 2), the sequence { y n } is ℜ -preserving. In virtue of property 4), we have that

d ( y n + 1 , y n ) = d ( T n + 1 x 0 , T n x 0 ) ≤ c n ⋅ h for n = 1 , 2 , ⋯

and hence { y n } is a Cauchy sequence. By property 3), { y n } converges to a point z in X and there exists a subsequence { y n ( k ) } of { y n } such that either ( y n ( k ) , z ) ∈ ℜ or ( z , y n ( k ) ) ∈ ℜ for all k = 0 , 1 , 2 , ⋯ . This implies that

d ( y n ( k ) + 1 , T z ) = d ( T y n ( k ) , T z ) ≤ c ⋅ d ( y n ( k ) , z )

by property 4) and passing k → ∞ , we have d ( z , T z ) = 0 and therefore z is a fixed point of T. By setting z k = y n ( k ) for every k = 0 , 1 , 2 , ⋯ , we have ( T z k , z ) ∈ ℜ if ( z k , z ) ∈ ℜ or ( z , T z k ) ∈ ℜ if ( z , z k ) ∈ ℜ for every k = 0 , 1 , 2 , ⋯ because of property 2). If property 5) holds, then it is a routine to prove that the fixed point is unique (cf., e.g. [

Remark 1. Theorem 1 is generalized from Theorem 3 by defining the non-decreasing order “ ≤ ” as relation R ⊆ X 2 . Theorem 2 is generalized from Theorem 3 because the condition 3) of Theorem 2, i.e. Definition 2, is restricted only to Cauchy sequences and moreover the hypothesis that X is complete does not appear in Theorem 3 as well.

In the following example Theorem 3, inspired to Example 1 of [

Example 1. Let X = { ( x , x ) : x ∈ ( − 1 , 1 ] } ⊆ R 2 endowed with the Euclidean metric d. Then ( X , d ) is a non-complete metric space and define ℜ as ( ( x , x ) , ( y , y ) ) ∈ ℜ if x ≤ y and x ≠ 0 , y ≠ 0 , ( ( x , x ) , ( 0,0 ) ) ∈ ℜ and ( ( 0,0 ) , ( x , x ) ) ∈ ℜ if x ∈ ( − 1,1 ] . Let c ∈ [ 0,1 ) and define T : X → X as T ( ( x , x ) ) = ( c x , c x ) if x < 0 and f ( ( x , x ) ) = ( 0 , 0 ) if x ≥ 0 . It is immediate to verify that property 1) holds since ( ( 0,0 ) , ( 0,0 ) ) = ( ( 0,0 ) , ( T 0, T 0 ) ) ∈ ℜ , moreover properties 2) and 3) hold trivially. Additionally we have that

d ( T ( x , x ) , T ( y , y ) ) = d ( ( k x , k x ) , ( k y , k y ) ) = 2 1 / 2 ⋅ c ⋅ | x − y | = c ⋅ d ( ( x , x ) , ( y , y ) ) if x < y < 0,

d ( T ( x , x ) , T ( y , y ) ) = d ( ( k x , k x ) , ( 0,0 ) ) = 2 1 / 2 ⋅ c ⋅ x = k ⋅ d ( ( x , x ) , ( y , y ) ) if x ≤ 0, y > 0,

d ( T ( x , x ) , T ( y , y ) ) = d ( ( 0,0 ) , ( 0,0 ) ) = 0 < c ⋅ d ( ( x , x ) , ( y , y ) ) if 0 < x ≤ y ,

thus property 4) holds. Also property 5) holds because there exists at least an ℜ -path of length 2, i.e. ( ( x , x ) , ( 0,0 ) ) ∈ ℜ and ( ( 0,0 ) , ( y , y ) ) ∈ ℜ , joining two any points ( x , x ) , ( y , y ) of X. Indeed ( 0,0 ) is the unique fixed point of T but Theorem 2 is not applicable because X is not complete.

Remark 2. If 5) does not hold, Theorem 3 does not guarantee the uniqueness of the fixed point as proved in the following example:

Example 2. Let X = [ 0 , 1 ] − { 1 / 2 } be endowed with metric d ( x , y ) = | x − y | for all x , y ∈ X . Define ℜ ⊆ X 2 as follows: ( x , y ) ∈ ℜ if for all x , y ∈ X such that 0 ≤ x ≤ y < 1 / 2 or 1 / 2 < x ≤ y ≤ 1 . Then X is a metric space with the partially defined binary relation ℜ . Define T : X → X as T x = x / 2 if 0 ≤ x < 1 / 2 and T x = ( x + 1 ) / 2 if 1 / 2 < x ≤ 1 . Then property 1) holds because 1 / 2 < x 0 ≤ T x 0 if x 0 ∈ ( 1 / 2 ,1 ] . The property 2) holds because T is strictly increasing in both intervals [ 0, 1 / 2 ) and ( 1 / 2 ,1 ] . The property 3) holds because it is enough to take strictly increasing sequences in ( 1 / 2 ,1 ] . Property 4) holds also for c = 1 / 2 . Property 5) fails because if x ∈ [ 0, 1 / 2 ] and y ∈ ( 1 / 2 ,1 ] , for any finite ℜ -path of length k, { z 0 , z 1 , z 2 , ⋯ , z k } , there exists at least certainly some m ∈ { 0,1, ⋯ , k − 1 } such that z m ∈ [ 0, 1 / 2 ) and z m + 1 ∈ ( 1 / 2 ,1 ] , hence ( z m , z m + 1 ) ∈ ℜ . Note that T has two fixed points which are 0 and 1.

Remark 3. Theorem 2 is not applicable to Example 2 because X is not complete.

In [

Theorem 4. Let ( X , ≤ ) be a nonempty partially ordered set and there exists a metric d : X × X → [ 0, + ∞ ) . Let T be a selfmap of X such that

1) there exists a point x 0 ∈ X such that x 0 ≤ T ( x 0 ) ;

2) T is continuous and non-decreasing;

3) if { x n } is a non-decreasing Cauchy sequence in X, then { T x n } converges to a point z ∈ X ;

4) there exists c ∈ [ 0,1 ] such that d ( T ( x ) , T ( y ) ) ≤ c ⋅ d ( x , y ) for all ( x , y ) ∈ X 2 with x ≥ y . Then T has a fixed point.

In the case T is assumed continuous, Theorem 2 becomes [

Theorem 5. Let ( X , d ) be a complete metric space, ℜ ⊆ X 2 and T be a selfmap of X such that

1) there exists at least a point ( x 0 , T x 0 ) ∈ ℜ ;

2) ℜ is T-closed;

3) T is continuous;

4) there exists c ∈ [ 0,1 ] such that d ( T ( x ) , T ( y ) ) ≤ c ⋅ d ( x , y ) for all ( x , y ) ∈ ℜ .

Then T has a fixed point.

Now we unify Theorems 4 and 5 with the following:

Theorem 6. Let ( X , d ) be a metric space, ℜ ⊆ X 2 and T be a selfmap of X. Suppose that

1) there exists in X a point x 0 such that ( x 0 , T x 0 ) ∈ ℜ ;

2a) ℜ is T-closed;

2b) T continuous;

3) if x_{n} is a ℜ -preserving Cauchy sequence in X, then Tx_{n} converges to a point z ∈ X ;

4) there exists c ∈ [ 0,1 ] such that d ( T x , T y ) ≤ c ⋅ d ( x , y ) for all ( x , y ) ∈ ℜ .

Then T has a fixed point in X.

Proof. As in the proof of Theorem 3, let x 0 ≠ T x 0 , h = d ( x 0 , T x 0 ) > 0 and y n = T n ( x 0 ) for every n = 0 , 1 , 2 , ⋯ . Because of properties (1) and (2.1), the sequence { y n } is ℜ -preserving. In virtue of property 4), we have that

d ( y n + 1 , y n ) = d ( T n + 1 x 0 , T n x 0 ) ≤ c n ⋅ h for n = 1,2, ⋯

and hence { y n } is a Cauchy sequence. By property 3), { T y n } converges to a point z and therefore { T T y n } = { T y n + 1 } converges to T z because of property 2.2), thus z = T z because of the uniqueness of the limit.

Remark 4. Theorem 4 is generalized from Theorem 6 by defining the non-decreasing order “ ≤ ” as relation ℜ ⊆ X 2 . Theorem 5 is generalized from Theorem 6 because if { x n } is a ℜ -preserving Cauchy sequence in X, the completeness of X and the continuity of T assure that T x n converges to a point of X, i.e. the property 3) of Theorem 6 holds. The following example shows Theorem 5 is not applicable but Theorem 6 holds.

Example 3. Let X = [ 0 , 1 ] − 1 / 5 with the metric d ( x , y ) = | x − y | for all x , y ∈ X . Define ℜ ⊆ X 2 as follows: ( x , y ) ∈ ℜ if x ≤ y for all x , y ∈ X . Define T : X → X as T x = ( x + 4 ) / 5 for any x ∈ X . Obviously T is continuous in X and ℜ is T-closed. If { x n } is a ℜ -preserving (that is monotone non-decreasing) Cauchy sequence in X, then T x n is a monotone non-decreasing bounded sequence and hence converging to a point z ∈ X , thus properties 1), 2), 3) hold. Too property 4) holds because it is enough to assume k = 1 / 5 . ( X , d ) is a metric space not complete, so Theorem 5 is not applicable while all the assumptions of Theorem 6 (or Theorem 4) are satisfied and 1 is the (unique) fixed point of T.

Remark 5. The uniqueness of the fixed point can be guaranteed from several additional properties of the relation ℜ (cf. [

Example 4. Consider X = [ 0 , 2 ] equipped with usual metric d ( x , y ) = | x − y | for all x , y ∈ X . ( X , d ) is a complete metric space. Define ℜ ⊆ X 2 as

ℜ = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) , ( 0 , 2 ) }

and T : X → X as T 0 = 1 / 4 , T ( x ) = 0 if 0 < x ≤ 1 , T x = 1 if 1 < x ≤ 2 . ℜ is T-closed but T is not continuous. Consider any ℜ -preserving sequence { x n } , then ( x n , x n + 1 ) ∈ ℜ − { ( 0,2 ) } for all n = 0 , 1 , 2 , ⋯ . Hence x n = 0 or x n = 1 for all n = 0 , 1 , 2 , ⋯ . If { x n } is a ℜ -preserving Cauchy sequence in X, then we have definitively x n = 0 (resp., x n = 1 ), i.e. there exists some suitable integer m such that x n = 0 (resp., x n = 1 ) for every integer n > m , which implies that T x n = 1 / 4 (resp., T x n = 0 ) for all n > m . Further

d ( T 0 , T 0 ) = d ( T 1 , T 1 ) = 0 ,

d ( T 1 , T 0 ) = d ( T 0 , T 1 ) = 1 / 4 ≤ c ⋅ 1 = c d ( 1 , 0 ) ,

d ( T 0 , T 2 ) = 3 / 4 ≤ c ⋅ 2 = c ⋅ d ( 0 , 2 ) ,

where c = 1 / 2 . Thus all the hypothesis of Theorem 6 hold except property 2.2 but T has no fixed points.

We have generalized fixed point theorems for theoretic-relation contractions about continuous selfmaps of metric spaces. Suitable examples prove the effective generalization of our results in metric spaces not necessarily complete.

Future studies shall be necessary for establishing extensions of the results here presented, essentially common fixed point theorems involving Cauchy sequences of Jungck type [

Funds of the “Dipartimento di Architettura” (Università degli Studi di Napoli Federico II, Italy) of the second author cover this research.

The authors declare that they have no conflict of interest.

Diviccaro, M.L. and Sessa, S. (2019) Relation Contractive Selfmaps Involving Cauchy Sequences. Advances in Pure Mathematics, 9, 421-428. https://doi.org/10.4236/apm.2019.95020