Open Access

Erratum to: Fixed point theorems of contractive mappings in cone b-metric spacesand applications

Fixed Point Theory and Applications20142014:55

https://doi.org/10.1186/1687-1812-2014-55

Received: 26 November 2013

Accepted: 26 November 2013

Published: 5 March 2014

The original article was published in Fixed Point Theory and Applications 2013 2013:112

Correction

In this note we correct some errors that appeared in the article (Huang and Xu in FixedPoint Theory Appl. 2013:112, 2013) by modifying some conditions in the main theorems andexamples.

After examining the proofs of the main results in [1], we can find that there is something wrong with the proof of the Cauchy sequencein [[1], Theorem 2.1]. This leads to subsequent errors in Theorem 2.3 andrelated examples in [1]. We also find that it is not rigorous to use the corresponding lemmas, and so theproof is inaccurate. The detailed reasons are given in the following.

On p.5 in [1], we conclude that
s p λ m + 1 s λ d ( x 1 , x 0 ) + s p 1 λ m d ( x 1 , x 0 ) θ
as m for any p 1 . This is incorrect. Indeed, note that taking λ = 1 s > 1 s and p = m + 1 leads to
s p λ m + 1 s λ d ( x 1 , x 0 ) + s p 1 λ m d ( x 1 , x 0 ) = s m + 2 2 s 3 2 1 d ( x 1 , x 0 ) + s m 2 d ( x 1 , x 0 ) θ

as m . Therefore, it is impossible to utilize [[1], Lemma 1.8, 1.9] and demonstrate that { x n } is a Cauchy sequence.

In this note, we would like to slightly modify only one of the used conditions to achieveour claim.

The following theorem is a modification to [[1], Theorem 2.1]. The proof is the same as that in [1] except the proof of the Cauchy sequence. We will attain the desired goal by usingthe new modified condition λ [ 0 , 1 s ) instead of λ [ 0 , 1 ) .

Theorem 2.1 Let ( X , d ) be a complete cone b-metric space with the coefficient s 1 . Suppose that the mapping T : X X satisfies the contractive condition
d ( T x , T y ) λ d ( x , y ) for  x , y X ,

where λ [ 0 , 1 s ) is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence { T n x } converges to the fixed point.

Proof In order to show that { x n } is a Cauchy sequence, we only need the following calculations.For any m 1 , p 1 , it follows that
d ( x m , x m + p ) s [ d ( x m , x m + 1 ) + d ( x m + 1 , x m + p ) ] s d ( x m , x m + 1 ) + s 2 [ d ( x m + 1 , x m + 2 ) + d ( x m + 2 , x m + p ) ] s d ( x m , x m + 1 ) + s 2 d ( x m + 1 , x m + 2 ) + s 3 d ( x m + 2 , x m + 3 ) + + s p 1 d ( x m + p 2 , x m + p 1 ) + s p 1 d ( x m + p 1 , x m + p ) s λ m d ( x 1 , x 0 ) + s 2 λ m + 1 d ( x 1 , x 0 ) + s 3 λ m + 2 d ( x 1 , x 0 ) + + s p 1 λ m + p 2 d ( x 1 , x 0 ) + s p λ m + p 1 d ( x 1 , x 0 ) = s λ m [ 1 + s λ + s 2 λ 2 + + ( s λ ) p 1 ] d ( x 1 , x 0 ) s λ m 1 s λ d ( x 1 , x 0 ) .
Let θ c be given. Notice that s λ m 1 s λ d ( x 1 , x 0 ) θ as m for any p. Making full use of [[1], Lemma 1.8], we find m 0 N such that
s λ m 1 s λ d ( x 1 , x 0 ) c
for each m > m 0 . Thus,
d ( x m , x m + p ) s λ m 1 s λ d ( x 1 , x 0 ) c

for all m 1 , p 1 . So, by [[1], Lemma 1.9], { x n } is a Cauchy sequence in ( X , d ) . The proof is completed. □

As is indicated in the reviewer’s comments, [[1], Example 2.2] is too trivial. Therefore, [[1], Example 2.2] is withdrawn. Now we give another example as follows.

Example 2.2 Let X = [ 0 , 0.48 ] , E = R 2 and let 1 p 6 be a constant. Take P = { ( x , y ) E : x , y 0 } . We define d : X × X E as
d ( x , y ) = ( | x y | p , | x y | p ) for all  x , y X .
Then ( X , d ) is a complete cone b-metric space with s = 2 p 1 . Let us define T : X X as
T x = 1 2 ( cos x 2 | x 1 2 | ) for all  x X .
Thus, for all x , y X , we have
d ( T x , T y ) = ( | T x T y | p , | T x T y | p ) = 1 2 p ( | ( cos x 2 cos y 2 ) ( | x 1 2 | | y 1 2 | ) | p , | ( cos x 2 cos y 2 ) ( | x 1 2 | | y 1 2 | ) | p ) 1 2 p ( ( | cos x 2 cos y 2 | + | x y | ) p , ( | cos x 2 cos y 2 | + | x y | ) p ) 1 2 p ( ( | x + y | 8 | x y | + | x y | ) p , ( | x + y | 8 | x y | + | x y | ) p ) 0.56 p ( | x y | p , | x y | p ) < 1 2 p 1 ( | x y | p , | x y | p ) .

Hence, by Theorem 2.1, there exists x 0 X (in fact, it satisfies 0.472251591454 < x 0 < 0.472251591479 ) such that x 0 is the unique fixed point of T.

For the same reason, we need to use the new condition λ 1 + λ 2 + s ( λ 3 + λ 4 ) < 2 1 + s instead of the original condition λ 1 + λ 2 + s ( λ 3 + λ 4 ) < min { 1 , 2 s } in [[1], Theorem 2.3]. The correct statement is as follows.

Theorem 2.3 Let ( X , d ) be a complete cone b-metric space with the coefficient s 1 . Suppose that the mapping T : X X satisfies the contractive condition
d ( T x , T y ) λ 1 d ( x , T x ) + λ 2 d ( y , T y ) + λ 3 d ( x , T y ) + λ 4 d ( y , T x ) for  x , y X ,

where the constant λ i [ 0 , 1 ) and λ 1 + λ 2 + s ( λ 3 + λ 4 ) < 2 1 + s , i = 1 , 2 , 3 , 4 . Then T has a unique fixed point in X. Moreover, the iterative sequence { T n x } converges to the fixed point.

Proof Following an identical argument that is given in [[1], Theorem 2.3] except substituting 0 λ 1 for 0 λ 1 s in line 26 of p.6 in [1], we obtain the proof of Theorem 2.3. □

In addition, based on the changes of Theorem 2.1, we need to change the condition h 2 < min { δ M 2 , 1 L 2 } into h 2 < min { δ M 2 , 1 2 L 2 } for [[1], Example 3.1]. Let us give the corrected example.

We now apply Theorem 2.1 to the first-order periodic boundary problem
{ d x d t = F ( t , x ( t ) ) , x ( 0 ) = ξ ,
(2.1)

where F : [ h , h ] × [ ξ δ , ξ + δ ] is a continuous function.

Example 2.4 Consider boundary problem (2.1) with the continuous function F,and suppose that F ( x , y ) satisfies the local Lipschitz condition, i.e., if | x | h , y 1 , y 2 [ ξ δ , ξ + δ ] , it induces
| F ( x , y 1 ) F ( x , y 2 ) | L | y 1 y 2 | .

Set M = max [ h , h ] × [ ξ δ , ξ + δ ] | F ( x , y ) | such that h 2 < min { δ M 2 , 1 2 L 2 } , then there exists a unique solution of (2.1).

Proof Let X = E = C ( [ h , h ] ) and P = { u E : u 0 } . Put d : X × X E as d ( x , y ) = f ( t ) max h t h | x ( t ) y ( t ) | 2 with f : [ h , h ] R such that f ( t ) = e t . It is clear that ( X , d ) is a complete cone b-metric space with s = 2 .

Note that (2.1) is equivalent to the integral equation
x ( t ) = ξ + 0 t F ( τ , x ( τ ) ) d τ .
Define a mapping T : C ( [ h , h ] ) R by T x ( t ) = ξ + 0 t F ( τ , x ( τ ) ) d τ . If
x ( t ) , y ( t ) B ( ξ , δ f ) { φ ( t ) C ( [ h , h ] ) : d ( ξ , φ ) δ f } ,
then from
d ( T x , T y ) = f ( t ) max h t h | 0 t F ( τ , x ( τ ) ) d τ 0 t F ( τ , y ( τ ) ) d τ | 2 = f ( t ) max h t h | 0 t [ F ( τ , x ( τ ) ) F ( τ , y ( τ ) ) ] d τ | 2 h 2 f ( t ) max h τ h | F ( τ , x ( τ ) ) F ( τ , y ( τ ) ) | 2 h 2 L 2 f ( t ) max h τ h | x ( τ ) y ( τ ) | 2 = h 2 L 2 d ( x , y ) ,
and
d ( T x , ξ ) = f ( t ) max h t h | 0 t F ( τ , x ( τ ) ) d τ | 2 h 2 f max h τ h | F ( τ , x ( τ ) ) | 2 h 2 M 2 f δ f ,

we speculate that T : B ( ξ , δ f ) B ( ξ , δ f ) is a contractive mapping.

Finally, we prove that ( B ( ξ , δ f ) , d ) is complete. In fact, suppose that { x n } is a Cauchy sequence in B ( ξ , δ f ) . Then { x n } is also a Cauchy sequence in X. Since ( X , d ) is complete, there is x X such that x n x ( n ). So, for each c int P , there exists N, whenever n > N , we obtain d ( x n , x ) c . Thus, it follows from
d ( ξ , x ) d ( x n , ξ ) + d ( x n , x ) δ f + c

and Lemma 1.12 in [1] that d ( ξ , x ) δ f , which means x B ( ξ , δ f ) , that is, ( B ( ξ , δ f ) , d ) is complete. □

Owing to the above statement, all conditions of Theorem 2.1 are satisfied. HenceT has a unique fixed point x ( t ) B ( ξ , δ f ) . That is to say, there exists a unique solution of (2.1).

Remark 2.5 Theorem 2.1 and Theorem 2.3 generalize and improve thecorresponding results in [24].

Notes

Declarations

Acknowledgements

The authors thank the referees, the editors and the readers including Prof. SriramBalasubramanian and Prof. Reny George. Special thanks are due to Prof. Ravi P. Agarwal andProf. Ljubomir Ciric, who have made a number of valuable comments and suggestions, whichhave improved [1] greatly. The research is partially supported by the PhD Start-up Fund ofHanshan Normal University, Guangdong Province, China (No. QD20110920).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hubei Normal University
(2)
Department of Mathematics and Statistics, Hanshan Normal University

References

  1. Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces andapplications. Fixed Point Theory Appl. 2013., 2013: Article ID 112Google Scholar
  2. Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121 10.1155/2010/978121Google Scholar
  3. Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398 10.1155/2010/315398Google Scholar
  4. Shah MH, Smić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metrictype spaces for b -metric spaces. J. Comput. Anal. Appl. 2012, 14(2):290–297.MathSciNetMATHGoogle Scholar

Copyright

© Xu and Huang; licensee Springer. 2014

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