Applications of fixed point theorems in extended cone $ b $-metric spaces over Banach algebras to integral equations arising in signal processing

Nura Alotaibi; Jamshaid Ahmad; Nura Alotaibi; Jamshaid Ahmad

doi:10.3934/math.20251317

AIMS Mathematics

2025, Volume 10, Issue 12: 29958-29988. doi: 10.3934/math.20251317

Previous Article Next Article

Research article

Applications of fixed point theorems in extended cone $ b $-metric spaces over Banach algebras to integral equations arising in signal processing

Nura Alotaibi ¹,
Jamshaid Ahmad ^{2,3
,
,}

1.
Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia; nthmalabubi@ju.edu.sa
2.
Department of Mathematics, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South Africa; jkhan@uj.edu.sa

Received: 21 August 2025 Revised: 19 November 2025 Accepted: 01 December 2025 Published: 19 December 2025
MSC : 46S40, 47H10, 54H25

The aim of this research article is to explore the notion of extended cone $ b $-metric spaces over Banach algebras and to establish new fixed point theorems for both single-valued mappings and multi-valued mappings. These results extend and generalize several well-known findings in metric fixed point theory. To demonstrate the applicability of our theoretical results, we provide illustrative examples that highlight the distinct features of our approach. Additionally, we showcase the practical relevance of our primary theorem by solving a Fredholm integral equation of the second kind, which arises in the context of radar and sonar signal processing.
- fixed point,
- extended cone $ b $-metric spaces,
- single-valued mappings,
- Banach algebras,
- multivalued mappings,
- Fredholm integral equations
Citation: Nura Alotaibi, Jamshaid Ahmad. Applications of fixed point theorems in extended cone $ b $-metric spaces over Banach algebras to integral equations arising in signal processing[J]. AIMS Mathematics, 2025, 10(12): 29958-29988. doi: 10.3934/math.20251317

Related Papers:

Abstract

The aim of this research article is to explore the notion of extended cone $ b $-metric spaces over Banach algebras and to establish new fixed point theorems for both single-valued mappings and multi-valued mappings. These results extend and generalize several well-known findings in metric fixed point theory. To demonstrate the applicability of our theoretical results, we provide illustrative examples that highlight the distinct features of our approach. Additionally, we showcase the practical relevance of our primary theorem by solving a Fredholm integral equation of the second kind, which arises in the context of radar and sonar signal processing.

References

[1]	M. M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Matem. Palermo, 22 (1906), 1–72. https://doi.org/10.1007/BF03018603 doi: 10.1007/BF03018603
[2]	S. G. Matthews, Partial metric topology, Ann. NY. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x doi: 10.1111/j.1749-6632.1994.tb44144.x
[3]	I. A. Bakhtin, The contraction mapping principle in almost metric space, Funct. Anal., Gos. Ped. Inst. Unianowsk, 30 (1989), 26–37.
[4]	T. Kamran, M. Samreen, Q. Ul Ain, A generalization of $b$-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. https://doi.org/10.3390/math5020019 doi: 10.3390/math5020019
[5]	A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. https://doi.org/10.5486/PMD.2000.2133 doi: 10.5486/PMD.2000.2133
[6]	L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 doi: 10.1016/j.jmaa.2005.03.087
[7]	N. Faried, S. M. A. Abou Bakr, H. A. El-Ghaffar, S. S. S. Almassri, Towards coupled coincidence theorems of generalized admissible types of mappings on partial satisfactory cone metric spaces and some applications, AIMS Mathematics, 8 (2023), 8431–8459. https://doi.org/10.3934/math.2023425 doi: 10.3934/math.2023425
[8]	H. Liu, S. Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013), 320. https://doi.org/10.1186/1687-1812-2013-320 doi: 10.1186/1687-1812-2013-320
[9]	H. Liu, S. Y. Xu, Fixed point theorems of quasi-contractions on cone metric spaces with Banach algebras, Abstr. Appl. Anal., 2013. https://doi.org/10.1155/2013/187348 doi: 10.1155/2013/187348
[10]	S. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014 (2014), 102. https://doi.org/10.1186/1687-1812-2014-102 doi: 10.1186/1687-1812-2014-102
[11]	S. M. A. Abou Bakr, H. K. Hussein, Some results on coupled coincidence points in vector quasi cone metric spaces over Banach algebras with satisfactory cones, J. Math. Comput. Sci., 31 (2023), 305–317. http://dx.doi.org/10.22436/jmcs.031.03.06 doi: 10.22436/jmcs.031.03.06
[12]	H. Huang, S. Radenović, Some fixed point results of generalized lipschitz mappings on cone $b$-metric spaces over Banach algebras, J. Comput. Anal. Appl., 20 (2016), 566–583.
[13]	H. Huang, S. Radenović, G. Deng, A sharp generalization on cone $b$-metric space over Banach algebra, J. Nonlinear Sci. Appl., 10 (2017), 429–435. https://doi.org/0.22436/jnsa.010.02.09
[14]	W. Shatanawi, Z. D. Mitrovic, N. Hussain, S. Radenović, On generalized Hardy–Rogers type $\alpha$-admissible mappings in cone $b$-metric spaces over Banach algebras, Symmetry, 12 (2020), 81. https://doi.org/10.3390/sym12010081 doi: 10.3390/sym12010081
[15]	K. Roy, S. Panja, M. Saha, A generalized fixed point theorem in an extended cone $b$-metric space over Banach algebra with its application to dynamical programming, Appl. Math. E-Notes, 21 (2021), 209–219.
[16]	S. Banach, Sur les operations dans les ensembles abstraits etleur application aux equations integrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[17]	V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–53.
[18]	B. Samet, C. Vetro, P. Vetro, Fixed point theorem for $\alpha $-$\psi $ contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165. https://doi.org/10.1016/j.na.2011.10.014. doi: 10.1016/j.na.2011.10.014
[19]	S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488.
[20]	S. H. Cho, J. S. Bae, Fixed point theorems for multi-valued maps in cone metric spaces, Fixed Point Theory Appl., 2011 (2011), 87. https://doi.org/10.1186/1687-1812-2011-87. doi: 10.1186/1687-1812-2011-87
[21]	A. Azam, N. Mehmood, J. Ahmad, S. Radenović, Multivalued fixed point theorems in cone $b$-metric spaces, J. Inequal. Appl., 2013 (2013), 582. https://doi.org/10.1186/1029-242X-2013-582 doi: 10.1186/1029-242X-2013-582
[22]	M. A. Kutbi, J. Ahmad, A. E. Al-Mazrooei, N. Hussain, Multivalued fixed point theorems in cone $b$-metric spaces over Banach algebra with applications, J. Math. Anal., 9 (2018), 52–64.
[23]	S. Janković, Z. Kadelburg, S. Radenović, On cone metric spaces: A survey, Nonlinear Anal., 74 (2011), 2591–2601. https://doi.org/10.1016/j.na.2010.12.014 doi: 10.1016/j.na.2010.12.014
[24]	M. A. Kutbi, J. Ahmad, A. Azam, On fixed points of $\alpha$-$\psi$-contractive multi-valued mappings in cone metric spaces, Abstr. Appl. Anal., 2013. https://doi.org/10.1155/2013/313782 doi: 10.1155/2013/313782
[25]	S. Radenović, B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl., 57 (2009), 1701–1707. https://doi.org/10.1016/j.camwa.2009.03.058 doi: 10.1016/j.camwa.2009.03.058
[26]	W. Ullah, M. Samreen, T. Kamran, Fixed points of mappings on extended cone $b$-metric space over real Banach algebra, Filomat, 36 (2022), 853–868. https://doi.org/10.2298/FIL2203853U doi: 10.2298/FIL2203853U
[27]	S. M. A. Abou Bakr, Coupled fixed point theorems for some type of contraction mappings in $b$-cone and $b$-theta cone metric spaces, J. Math., 2021. https://doi.org/10.1155/2021/5569674 doi: 10.1155/2021/5569674
[28]	J. H. Wilson, Solution of integral equation important in signal processing of radar and sonar hydrophone array outputs, SIAM J. Appl. Math., 39 (1980), 1–7. https://doi.org/10.1137/0139001 doi: 10.1137/0139001
[29]	R. Kalaba, On the integral equations of optimal signal detection and estimation theory, J. Optim. Theory Appl., 8 (1971), 110–114. https://doi.org/10.1007/BF00928471 doi: 10.1007/BF00928471
[30]	W. Li, G. R. Liu, V. K. Varadan, Fredholm integral equation for shape reconstruction of an underwater object using backscattered ramp response signatures, J. Acoust. Soc. Am., 116 (2004), 1436–1445. https://doi.org/10.1121/1.1776190 doi: 10.1121/1.1776190
[31]	W. Rudin, Functional analysis, 2 Eds., NewYork: McGraw-Hill, 1991.
[32]	M. Abbas, V. Rakocević, B. T. Leyew, Common fixed points of ($\alpha $-$\psi $)-generalized rational multivalued contractions in dislocated quasi $b$-metric spaces and applications, Filomat, 31 (2017), 3263–3284. https://doi.org/10.2298/FIL1711263A doi: 10.2298/FIL1711263A
[33]	X. Shi, W. Wang, P. Long, H. Huang, Z. Jiang, Fixed point theorems in extended cone $b$-metric-like spaces over Banach algebras, J. Math. Anal. Appl., 548 (2025), 129427. https://doi.org/10.1016/j.jmaa.2025.129427 doi: 10.1016/j.jmaa.2025.129427
[34]	I. Shereen, Q. Kiran, A. Aloqaily, H. Aydi, N. Mlaiki, On new extended cone $b$-metric-like spaces over a real Banach algebra, J. Inequal. Appl., 2024 (2024), 129. https://doi.org/10.1186/s13660-024-03205-2 doi: 10.1186/s13660-024-03205-2
[35]	H. A. Hammad, M-F. Bota, L. Guran, Wardowski's contraction and fixed point technique for solving systems of functional and integral equations, J. Funct. Spaces, 2021. https://doi.org/10.1155/2021/7017046 doi: 10.1155/2021/7017046
[36]	H. A. Hammad, M. De la Sen, H. Aydi, Generalized dynamic process for an extended multi-valued $F$-contraction in metric-like spaces with applications, Alex. Eng. J., 59 (2020), 3817–3825. https://doi.org/10.1016/j.aej.2020.06.037 doi: 10.1016/j.aej.2020.06.037
[37]	Y-K. Ma, M. M. Raja, A. Shukla, V. Vijayakumar, K. S. Nisar, K. Thilagavathi, New results on approximate controllability of fractional delay integrodifferential systems of order $1 < r < 2$ with Sobolev-type, Alex. Eng. J., 81 (2023), 501–518. https://doi.org/10.1016/j.aej.2023.09.043 doi: 10.1016/j.aej.2023.09.043
[38]	M. M. Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, W. Albalawi, A-H. Abdel-Aty, A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $r\in (1, 2)$ with impulses, AIMS Mathematics, 8 (2023), 10802–10821. https://doi.org/10.3934/math.2023548 doi: 10.3934/math.2023548

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)