Received: 31 May 2017
DOI: 10.1002/mma.4601
RESEARCH ARTICLE
A note on “Convergence and best proximity points
for Berinde's cyclic contraction with proximally
complete property”
Abdelbasset Felhi
Department of Mathematics, Bizerte
Preparatory Engineering Institute,
Carthage University, Tunisia
Correspondence
Abdelbasset Felhi, Department of
Mathematics, Bizerte Preparatory
Engineering Institute, Carthage
University, Tunisia.
Email: abdelbassetfelhi@gmail.com
Communicated by: W. Sprößig
MOS Classification: 47H09; 47H10; 54H25
Recently, Sanhan and Mongkolkeha introduced the concept of Berinde's cyclic
contraction, and they established some results. Unfortunately, these results
seem to be incorrect. In this paper, some counterexamples are given.
KEYWORDS
Berinde's contraction, best proximity point, cyclic contraction mapping, proximally complete
property
1 ON THE RESULTS IN SANHAN AND MONGKOLKEHA
1
In Sanhan and Mongkolkeha,
1
the authors have introduced the notion of Berinde's cyclic contraction, and they proved
some best proximity points for such mapping by using the proximally complete property. We recall first some definitions
and notations.
Definition 1.1.
1
Let A and B be nonempty subsets of a metric space (X, d). A mapping T ∶ A ∪ B → A ∪ B is called a
Berinde's cyclic contraction if T(A) ⊂ B, T(B) ⊂ A and there exist ∈[0, 1) and some L ≥ 0 such that
d(Tx, Ty) ≤ d(x, y)+(1 - )d(A, B)+ Ld(y, Tx) ∀(x, y)∈ A × B,
where d(A, B)= inf {d(x, y)∶ x ∈ A, y ∈ B}.
Definition 1.2.
2
Let A and B be nonempty subsets of a metric space (X, d). A sequence (x
n
) in A ∪ B, with x
2n
∈ A
and x
2n+1
∈ B, is said to be a cyclically Cauchy sequence if and only if for any > 0 there exists N ∈ N such that
d(x
n
, x
m
) < d(A, B)+ , when n is even, m is odd and n, m ≤ N.
Definition 1.3.
2
A pair (A, B) of nonempty subsets of a metric space (X, d) is called semi-sharp proximally if, for each
(x, y)∈ A × B, there exists at most (y
′
, x
′
)∈ A × B such that d(x, x
′
)= d(y, y
′
)= d(A, B).
Definition 1.4.
2
A pair (A, B) of nonempty subsets of a metric space (X, d) is proximally complete if any cyclically
Cauchy sequence (x
n
) in A ∪ B the sequences (x
2n
) and (x
2n+1
) have convergent subsequences in A and B, respectively.
The following results are due in Sanhan and Mongkolkeha.
1
Theorem 1.5. (Theorem 3.5 in Sanhan and Mongkolkeha
1
) Let (A, B) be proximally complete pair in a metric space
(X, d). If T ∶ A ∪ B → A ∪ B is a Berinde's cyclic contraction, then there exists (x, y)∈ A × B such that d(x, Tx)= d(y, Ty)=
d(x, y)= d(A, B).
Math Meth Appl Sci. 2017;1–4. wileyonlinelibrary.com/journal/mma Copyright © 2017 John Wiley & Sons, Ltd. 1