A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

Gwang-Myong Kim; Gum-Sik Kang; Chol Jin Kil; Jin-Sim Kim

doi:doi:10.11648/j.engmath.20250902.11

Research Article |

| Peer-Reviewed

A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

Gwang-Myong Kim

, Gum-Sik Kang

, Chol Jin Kil

, Jin-Sim Kim^*

Published in Engineering Mathematics (Volume 9, Issue 2)

Received: 4 September 2025 Accepted: 30 September 2025 Published: 30 October 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

Published in	Engineering Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.engmath.20250902.11
Page(s)	26-30
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Fixed Point, Banach Contraction, Double-Composed Metric Space, Fredholm Integral Equation

1. Introduction and Preliminaries

Fixed point theory has wide applications in various fields including nonlinear analysis, engineering, computer science, mathematical modeling, etc.,

[1]

. Banach’s contraction principle

[2]

plays a key role in fixed point theory. Many researchers have been engaged in research to extend and improve Banach’s contraction principle, where recent attention is focused on generalization of metric spaces. Several abstract spaces that generalize triangular inequalities such as b-metric space

[3]

, extended b-metric space

[4]

, controlled metric space

[5]

, and double controlled metric space

[6]

have been introduced in the field of fixed point theory. In 2023, Ayoob et al.

[7]

defined a new extension of metric space by using the composition of functions, namely, double-composed metric space, as follows.

Definition 1.1 (

[7]

). Let

F, G : [0, \infty) \to [0, \infty)

be two non-constant functions. A double-composed metric on

Κ \neq \emptyset

is a function

D_{c} : Κ \times Κ \to [0, \infty)

such that

\forall κ, ι, ν \in Κ

κ = ι ⟺ D_{c} (κ, ι) = 0;

D_{c} (κ, ι) = D_{c} (ι, κ);

D_{c} (κ, ι) \leq F (D_{c} (κ, ν)) + G (D_{c} (ν, ι))

The pair

(Κ, D_{c})

is called a double-composed metric space. Obviously, each metric space is a double-composed metric space via

F (x) = G (x) = x

for all

x \geq 0

More recently, several papers have been published concerning the double-composed metric space and its generalization, see

[8-15]

Example 1.1 (

[8]

). Let

Κ = R

. Define

D_{c} : Κ \times Κ \to [0, \infty)

D_{c} (κ, ι) = e^{|κ - ι|} - 1

, for all

κ, ι \in Κ

. Then

(Κ, D_{c})

is a double-composed metric space via

F, G : [0, \infty) \to [0, \infty)

with

F (x) = G (x) = \frac{x^{2} + 2 x}{2}

for all

x \geq 0

, which is not a standard metric space.

The authors of

[7]

presented some concepts concerning double-composed metric spaces.

Definition 1.2 (

[7]

). Let

(Κ, D_{c})

be a double-composed metric space.

{κ_{i}} \subset Κ

converges to

κ

\lim_{i \to \infty} D_{c} (κ_{i}, κ) = 0

for some

κ \in Κ

{κ_{i}} \subset Κ

is said to Cauchy if

\lim_{i, j \to \infty} D_{c} (κ_{i}, κ_{j}) = 0

;

(Κ, D_{c})

is said to complete if every Cauchy sequence on

Κ

converges to some point of

Κ

Proposition 1.1 (

[7]

). Let

(Κ, D_{c})

be a double-composed metric space via two non-constant functions

F, G : [0, \infty) \to [0, \infty)

. If

F

and

G

are continuous and

F (0) = G (0) = 0

, then every convergent sequence has a unique limit.

The following theorem is Banach-type fixed point result in the framework of double-composed metric space proposed in

[7]

Theorem 1.1 (

[7]

). Let

(Κ, D_{c})

be a complete double-composed metric space via continuous and non-decreasing functions

F, G : [0, \infty) \to [0, \infty)

with

F (0) = G (0) = 0

, and let

T : Κ \to Κ

be a mapping such that

D_{c} (T κ, T ι) \leq λ D_{c} (κ, ι)

\forall κ, ι \in Κ

where

λ \in (0, 1)

. Define a sequence

{κ_{i}}

κ_{i} = T^{i} κ_{0}

for

κ_{0}

. Assume that:

\lim_{i, j \to \infty} (\sum_{n = j}^{i - 2} G^{n - j} F (λ^{n} D_{c} (κ_{0}, κ_{1})) + G^{i - j - 1} (λ^{i - 1} D_{c} (κ_{0}, κ_{1}))) = 0

where

G^{n - j} F (λ^{n} D_{c} (κ_{0}, κ_{1}))

and

G^{i - j - 1} (λ^{i - 1} D_{c} (κ_{0}, κ_{1}))

denotes the composite function.

G

is sub-additive.

Then

T

has a unique fixed point.

In this paper, we prove Banach-type fixed point result in double-composed metric space without the assumptions 1) and 2) of Theorem 1.1. We also present an example showing the significance of the obtained result. Moreover, we show the existence of a solution of Fredholm integral equations by applying obtained fixed point result.

2. Main Results

We reconsider the Banach-type fixed point theorem in double-composed metric spaces without the assumptions 1) and 2) of Theorem 1.1, while we use the assumption that

F

and

G

are strictly increasing instead of the assumption that they are non-decreasing.

Theorem 2.1. Let

(Κ, D_{c})

be a complete double composed metric space via two continuous and strictly increasing functions

F, G : [0, \infty) \to [0, \infty)

with

F (0) = G (0) = 0

, and let

T : Κ \to Κ

be a mapping such that

D_{c} (T κ, T ι) \leq λ D_{c} (κ, ι)

\forall κ, ι \in Κ

,(1)

where

λ \in [0, 1)

. Then

T

has a unique fixed point.

Proof. Since

λ \in [0, 1)

, for given

0 < η < 1

, we can take

n \in N

such that

λ^{n} < \min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\}

.(2)

Let us take

κ_{0} \in Κ

arbitrarily. We define

Ψ ≔ T^{n}

and construct a sequence

\{κ_{i}\}

κ_{i} = Ψ^{i} κ_{0}

Firstly, we will see that

\{κ_{i}\}

is a Cauchy sequence.

From (1), we obtain

D_{c} (Ψκ, Ψι) = D_{c} (T^{n} κ, T^{n} ι) \leq λ D_{c} (T^{n - 1} κ, T^{n - 1} ι) \dots \leq λ^{n} D_{c} (κ, ι)

(3)

By (3), we get for all

i \in N

D_{c} (κ_{i}, κ_{i + 1}) = D_{c} (Ψ κ_{i - 1}, {Ψκ}_{i}) \leq λ^{n} D_{c} (κ_{i - 1}, κ_{i}) \leq λ^{ni} D_{c} (κ_{0}, κ_{1})

Thus,

\lim_{i \to \infty} D_{c} (κ_{i}, κ_{i + 1}) = 0

.(4)

So we can take

h \in N

satisfying

D_{c} (κ_{h}, κ_{h + 1}) < \min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\}

.(5)

Denote by

B_{dc} [κ_{h}, η / 2] : = \{ι \in Κ : D_{c} (κ_{h}, ι) \leq η / 2\}

. Since it is clear that

κ_{h} \in B_{dc} [κ_{h}, η / 2]

, it holds that

B_{dc} [κ_{h}, η / 2] \neq \emptyset

. Let us take an arbitrary

ν \in B_{dc} [κ_{h}, η / 2]

. Then,

D_{c} (κ_{h}, ν) \leq η / 2

By (2) and (3), we get

D_{c} (Ψ κ_{h}, Ψν) \leq λ^{n} D_{c} (κ_{h}, ν) \leq \min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\} ∙ \frac{η}{2} \leq \min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\}

.(6)

By triangle inequality (iii), (5), (6) and the fact that

F

and

G

are strictly increasing, we get

D_{c} (κ_{h}, Ψν) \leq F (D_{c} (κ_{h}, κ_{h + 1})) + G (D_{c} (κ_{h + 1}, Ψν))

= F (D_{c} (κ_{h}, κ_{h + 1})) + G (D_{c} (Ψ κ_{h}, Ψν))

\leq F (\min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\}) + G (\min \{F^{- 1} (\frac{η}{4}), G^{- 1} (\frac{η}{4})\})

\leq F (F^{- 1} (\frac{η}{4})) + G (G^{- 1} (\frac{η}{4}))

\leq \frac{η}{2}

Hence,

Ψν \in B_{dc} [κ_{h}, η / 2]

. This means that

Ψ

maps

B_{dc} [κ_{h}, η / 2]

into itself. Thus we have

Ψ κ_{h} \in B_{dc} [κ_{h}, η / 2]

for

κ_{h} \in B_{dc} [κ_{h}, η / 2]

. Repeating this process, we get

κ_{j} \in B_{dc} [κ_{h}, η / 2]

for all

j > h

For all

j > i \geq h

(where

i = h + l

for some

l \geq 0

), by (3) and

κ_{j - l} \in B_{dc} [κ_{h}, η / 2]

, we have

D_{c} (κ_{i}, κ_{j}) = D_{c} (Ψ κ_{i - 1}, Ψ κ_{j - 1}) \leq λ^{n} D_{c} (κ_{i - 1}, κ_{j - 1}) \leq λ D_{c} (κ_{i - 1}, κ_{j - 1})

\leq λ^{2} D_{c} (κ_{i - 2}, κ_{j - 2})

\dots

\leq λ^{l} ∙ D_{c} (κ_{h}, κ_{j - l})

\leq D_{c} (κ_{h}, κ_{j - l})

\leq \frac{η}{2} < η

Thus, we have

\lim_{i, j \to \infty} D_{c} (κ_{i}, κ_{j}) = 0

and

{κ_{i}}

is Cauchy. By the completeness of

Κ

, there exists

κ^{*} \in Κ

such that

\lim_{i \to \infty} D_{c} (κ_{i}, κ^{*}) = 0

Secondly, we will prove that

κ^{*}

is a unique fixed point of

Ψ

. By triangle inequality (iii) and (3),

D_{c} (κ^{*}, Ψ κ^{*}) \leq F (D_{c} (κ^{*}, κ_{i + 1})) + G (D_{c} (κ_{i + 1}, Ψ κ^{*}))

= F (D_{c} (κ^{*}, κ_{i + 1})) + G (D_{c} (Ψ κ_{i}, Ψ κ^{*}))

\leq F (D_{c} (κ^{*}, κ_{i + 1})) + G (λ^{n} D_{c} (κ_{i}, κ^{*}))

for all

i \in N

Taking the limit as

i \to \infty

, by the continuity of

F

and

G

we obtain that

D_{c} (κ^{*}, Ψ κ^{*}) \leq F (\lim_{i \to \infty} D_{c} (κ^{*}, κ_{i + 1})) + G (λ^{n} \lim_{i \to \infty} D_{c} (κ_{i}, κ^{*})) = F (0) + G (0) = 0

κ^{*}

is a fixed point of

Ψ

Now, let

Ψ

has two fixed points

κ^{*}

and

ι^{*}

, i.e.,

κ^{*} = Ψ κ^{*}

and

ι^{*} = Ψ ι^{*}

By (3)

D_{c} (κ^{*}, ι^{*}) = D_{c} (Ψ κ^{*}, Ψ ι^{*}) \leq λ^{n} D_{c} (κ^{*}, ι^{*})

That is

D_{c} (κ^{*}, ι^{*}) = 0

, and the fixed point of

Ψ

is unique.

Finally, since

κ^{*} = Ψ κ^{*} = T^{n} κ^{*}

, we get

T κ^{*} = T^{n + 1} κ^{*} = Ψ (T κ^{*})

. That is,

T κ^{*}

is also a fixed point of

Ψ

. Since the fixed point of

Ψ

is unique,

κ^{*} = T κ^{*}

. Consequently,

κ^{*}

is a fixed point of

T

. The uniqueness can be easily showed.

Example 2.1. We discuss the double composed metric space

(Κ, D_{c})

defined in Example 1.1. The completeness of

Κ

can be easily checked, and

F

and

G

are continuous and strictly increasing with

F (0) = G (0) = 0

Take

T : Κ \to Κ

T κ = \frac{κ}{3}

Since

e^{x / 3} - 1 \leq \frac{1}{3} (e^{x / 3} - 1)

\forall x \in R

, we have

D_{c} (T κ, T ι) = e^{|T κ - T ι|} - 1 = e^{\frac{|κ - ι|}{3}} - 1 \leq \frac{1}{3} (e^{|κ - ι|} - 1) \leq λ D_{c} (κ, ι)

for all

λ \in [\frac{1}{3}, 1)

. Thus all the conditions of Theorem 2.1 are satisfied and

T

has a unique fixed point

κ^{*} = 0

. On the other hand, since

G

is not sub-additive, so we cannot apply Theorem 1.1 (Theorem 3 in

[7]

3. Application to Nonlinear Integral Equations

We apply Theorem 2.1 to discuss the existence of a solution of Fredholm integral equation as follows:

κ (t) = \int_{0}^{1} H (t, s, κ (s)) ds

\forall t, s \in [0, 1]

,(7)

where

H : {[0, 1]}^{2} \times R \to R

is given continuous function.

Let

Κ = C ([0, 1], R)

. Define

D_{c} : Κ \times Κ \to [0, \infty)

D_{c} (κ, ι) = e^{\max_{t \in [0, 1]} (|κ (t) - ι (t)|)} - 1

\forall κ, ι \in Κ

From Example 1.1, we can easily check that

(Κ, D_{c})

is a complete double-composed metric space via

F, G : [0, \infty) \to [0, \infty)

with

F (x) = G (x) = (x^{2} + 2 x) / 2

for all

x \geq 0

Theorem 3.1. Suppose that

e^{|H (t, s, κ (s)) - H (t, s, ι (s))|} - 1 \leq λ (e^{|κ (s) - ι (s)|} - 1)

(8)

for all

κ, ι \in Κ

and

t, s \in [0, 1]

, where

λ \in [0, 1)

. Then the integral equation (7) has a unique solution in

Κ

Proof. The inequality (8) is equivalent to the following inequality.

|H (t, s, κ (s)) - H (t, s, ι (s))| \leq \ln (λ (e^{|κ (s) - ι (s)|} - 1) + 1)

(9)

for all

κ, ι \in Κ

and

t, s \in [0, 1]

We define the integral operator

T : Κ \to Κ

T κ (t) = \int_{0}^{1} H (t, s, κ (s)) ds

\forall t, s \in [0, 1]

By (9), we get

e^{|T κ (t) - T ι (t)|} - 1 = e^{|\int_{0}^{1} H (t, s, κ (s)) ds - \int_{0}^{1} H (t, s, ι (s)) ds|} - 1 \leq e^{\int_{0}^{1} |H (t, s, κ (s)) - H (t, s, ι (s))| ds} - 1

\leq e^{\int_{0}^{1} \ln (λ (e^{|κ (s) - ι (s)|} - 1) + 1) ds} - 1

\leq e^{\ln (λ (e^{\max_{s \in [0, 1]} |κ (s) - ι (s)|} - 1) + 1) \int_{0}^{1} ds} - 1

= λ (e^{\max_{s \in [0, 1]} |κ (s) - ι (s)|} - 1)

\leq λ (e^{\max_{t \in [0, 1]} |κ (t) - ι (t)|} - 1)

for all

t \in [0, 1]

and

κ, ι \in Κ

So,

e^{\max_{t \in [0, 1]} |T κ (t) - T ι (t)|} - 1 \leq λ (e^{\max_{t \in [0, 1]} |κ (t) - ι (t)|} - 1)

that is,

D_{c} (T κ, T ι) \leq λ D_{c} (κ, ι)

By Theorem 2.1,

T

has a unique fixed point, which means that (7) has a unique solution in

Κ

4. Conclusions

In this paper, we newly discussed the Banach-type fixed point result in the framework of double-composed metric spaces. We have established the Banach-type fixed point result without the assumptions (1) and (2) of Theorem 1.1, instead replacing the assumption that

F

and

G

are non-decreasing with the assumption of strictly increasing. And we presented an example supporting our fixed point result. Finally, we show the existence of a solution of Fredholm integral equations by applying obtained fixed point result.

Acknowledgments

The authors would like to express their gratitude to the handling editor and reviewers for their constructive comments.

Author Contributions

Gwang-Myong Kim: Writing - original draft, Methodology

Gum-Sik Kang: Writing - original draft, Conceptualization

Chol Jin Kil: Writing - review & editing, Supervision

Jin-Sim Kim: Formal Analysis, Data curation

Funding

This work is not supported by any external funding.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Bucur, About applications of the fixed point theory, Scientific Bulletin, 2017, XXII(1), 43.
[2]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 1922, 3(1), 133-181. https://doi.org/10.4064/fm-3-1-133-181
[3]	I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 1989, 30, 26-37.
[4]	T. Kamran, M. Samreen, and Q. Ul Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 2017, 5(2), 19. https://doi.org/10.3390/math5020019
[5]	N. Mlaiki, H. Aydi, N. Souayah, and T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 2018, 6(10), 194. https://doi.org/10.3390/math6100194
[6]	T. Abdeljawad, N. Mlaiki, H. Aydi, and N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 2018, 6(12), 320. https://doi.org/10.3390/math612320
[7]	I. Ayoob, N. Z. Chuan, and N. Mlaiki, Double-Composed Metric Spaces, Mathematics, 2023, 11(8), 1866. https://doi.org/10.3390/math11081866
[8]	C. J. Kil, C. S. Yu, and U. C. Han, Fixed point results for some rational type contractions in double-composed metric spaces and applications, Informatica, 2023, 34(12), 105-130.
[9]	F. M. Azmi, I. Ayoob, N. Mlaiki, Exploring Double Composed Partial Metric Spaces: A Novel Approach to Fixed Point Theorems, Int. J. Anal. Appl., 2024, 22, 192. https://doi.org/10.28924/2291-8639-22-2024-192
[10]	A. A. Hijab, L. K. Shaakir, S. Aljohani, N. Mlaiki, Double composed metric-like spaces via some fixed point theorems, AIMS Math., 2024, 9(10), 27205-27219. https://dx.doi.org/10.3934/math.20241322
[11]	C. J. Kil, K. Ho, W. Yang, U. Kim, Triple-Composed Metric Spaces and Related Fixed Point Results With Application, Journal of Function Spaces, 2024, Article ID 6466538, 9 pages. https://doi.org/10.1155/2024/6466538
[12]	C. J. Kil, B. Kim, S. Jon, H. Rim, Discussion on fixed points of nonlinear F-contractions in partially ordered double-composed metric-like spaces, Asian-European Journal of Mathematics, 2025, 18(11), 1-19. https://dx.doi.org/10.1142/S1793557125500469
[13]	A. A. Hijab, L. Shaakir, S. Aljohani, N. M. Mlaiki, Common Fixed Point Results in Double-Composed Cone Metric-Like Spaces for Some Generalized Rational Contractions with Applications, European Journal of Pure and Applied Mathematics, 2025, 18(2), 6029.
[14]	A. A. Hijab, L. K. Shaakir, S. Aljohani, N. Mlaiki, Fredholm integral equation in composed-cone metric spaces, Bound. Value Probl., 2024, 2024, 64. https://doi.org/10.1186/s13661-024-01876-w
[15]	C. J. Kil, B. Kim, Un-Ryong Rim, Discussion on Fixed Points for (α, F, Φ)-Contractions in Double-Composed Metric Spaces, Boletin de la Sociedad Mathematica Mexicana, 2025, 31(118), 1-20. https://doi.org/10.1007/s40590-025-00802-z

Cite This Article

Plain Text BibTeX RIS

APA Style

Kim, G., Kang, G., Kil, C. J., Kim, J. (2025). A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Engineering Mathematics, 9(2), 26-30. https://doi.org/10.11648/j.engmath.20250902.11

Copy | Download

ACS Style

Kim, G.; Kang, G.; Kil, C. J.; Kim, J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng. Math. 2025, 9(2), 26-30. doi: 10.11648/j.engmath.20250902.11

Copy | Download

AMA Style

Kim G, Kang G, Kil CJ, Kim J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng Math. 2025;9(2):26-30. doi: 10.11648/j.engmath.20250902.11

Copy | Download

@article{10.11648/j.engmath.20250902.11,
  author = {Gwang-Myong Kim and Gum-Sik Kang and Chol Jin Kil and Jin-Sim Kim},
  title = {A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {26-30},
  doi = {10.11648/j.engmath.20250902.11},
  url = {https://doi.org/10.11648/j.engmath.20250902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.11},
  abstract = {One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

AU  - Gwang-Myong Kim
AU  - Gum-Sik Kang
AU  - Chol Jin Kil
AU  - Jin-Sim Kim
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.11
DO  - 10.11648/j.engmath.20250902.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 26
EP  - 30
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.11
AB  - One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

VL  - 9
IS  - 2
ER  -

Copy | Download

Author Information

Gwang-Myong Kim

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0006-5810-0487
Gum-Sik Kang

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0007-3525-822X
Chol Jin Kil

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0008-9236-7139
Jin-Sim Kim

International Technology Cooperation Center, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0000-0002-7937-9185

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Kim, G., Kang, G., Kil, C. J., Kim, J. (2025). A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Engineering Mathematics, 9(2), 26-30. https://doi.org/10.11648/j.engmath.20250902.11

Copy | Download

ACS Style

Kim, G.; Kang, G.; Kil, C. J.; Kim, J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng. Math. 2025, 9(2), 26-30. doi: 10.11648/j.engmath.20250902.11

Copy | Download

AMA Style

Kim G, Kang G, Kil CJ, Kim J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng Math. 2025;9(2):26-30. doi: 10.11648/j.engmath.20250902.11

Copy | Download

@article{10.11648/j.engmath.20250902.11,
  author = {Gwang-Myong Kim and Gum-Sik Kang and Chol Jin Kil and Jin-Sim Kim},
  title = {A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {26-30},
  doi = {10.11648/j.engmath.20250902.11},
  url = {https://doi.org/10.11648/j.engmath.20250902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.11},
  abstract = {One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

AU  - Gwang-Myong Kim
AU  - Gum-Sik Kang
AU  - Chol Jin Kil
AU  - Jin-Sim Kim
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.11
DO  - 10.11648/j.engmath.20250902.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 26
EP  - 30
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.11
AB  - One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

VL  - 9
IS  - 2
ER  -

Copy | Download