Correction: Singh, Y. Mahendra, et al. F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application. Mathematics 2018, 6, 105

Y. Mahendra Singh; Mohammad Saeed Khan; Shin Min Kang

doi:10.3390/math7040337

,

and

¹

Department of Humanities and Basic Sciences, Manipur Institute of Technology, Takyelpat 795004, India

²

Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, 123 Al-Khod, Muscat, Oman

³

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea

⁴

Center for General Education, China Medical University, Taichung 40402, Taiwan

Mathematics2019, 7(4), 337;https://doi.org/10.3390/math7040337

Version Notes

Order Reprints

We found some errors in Lemma 1 of our paper [1], thus, we would like to make the following corrections:

Instead of the following Lemma 1 [1]:

Lemma 1.

Let

(X, d)

be a metric space and

T : X \to X

be an α-F-convex contraction satisfying the following conditions:

(i): T is α-admissible;
(ii): there exists $x_{0} \in X$ such that $α (x_{0}, T x_{0}) \geq 1$ .

Define a sequence

{x_{n}}

in X by

x_{n + 1} = T x_{n} = T^{n + 1} x_{0}

for all

n \geq 0

. Then

{d^{p} (x_{n}, x_{n + 1})}

is strictly non-increasing sequence in X.

It should read:

Lemma 2.

Let

(X, d)

be a metric space and

T : X \to X

be an

α - F

-convex contraction satisfying the conditions:

(i): T is α-admissible;
(ii): there exists $x_{0} \in X$ such that $α (x_{0}, T x_{0}) \geq 1$ .

Define a sequence

{x_{n}}

in X by

x_{n + 1} = T x_{n} = T^{n + 1} x_{0}

for all

n \geq 0

, then

F (d^{p} (x_{n}, x_{n + 1})) \leq F (v) - l τ

, whenever

n = 2 l

or

n = 2 l + 1

for

l \geq 1

.

Proof.

Following the same steps as in Lemma 1, the last paragraph was replaced with the following: Therefore,

v > d^{p} (x_{2}, x_{3})

and hence

F (d^{p} (x_{2}, x_{3})) \leq F (v) - τ

. By a similar argument, we obtain

F (d^{p} (x_{3}, x_{4})) \leq F (v) - τ;

continuing in these way, we arrive at

F (d^{p} (x_{n}, x_{n + 1})) \leq F (v) - l τ,

whenever

n = 2 l

or

n = 2 l + 1

for

l \geq 1

. ☐

In the proof of the Theorem 2 [1], instead of the following:

“By Lemma 1,

{d^{p} (x_{n}, x_{n + 1})}

is strictly non-increasing sequence. Therefore,

\begin{matrix} F (d^{p} (x_{n}, x_{n + 1})) \leq F (d^{p} (x_{n - 2}, x_{n - 1})) - τ \leq \dots \leq F (v) - l τ \end{matrix}

whenever

n = 2 l

or

n = 2 l + 1

for

l \geq 1

”.

It should be: By Lemma 1, we obtain:

\begin{matrix} F (d^{p} (x_{n}, x_{n + 1})) \leq F (v) - l τ, \end{matrix}

whenever

n = 2 l

or

n = 2 l + 1

for

l \geq 1

. The rest of the proof is unaltered.

The authors apologize to all the readers for any inconvenience this may have caused.

Reference

Singh, Y.M.; Khan, M.S.; Kang, S.M. On interpolative F-convex contraction and fixed point theorems with and application. Mathematics 2018, 6, 105. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Correction: Singh, Y. Mahendra, et al. F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application. Mathematics 2018, 6, 105

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