On a class of bivalued hybrid cyclic and non-cyclic contractive self-mappings on unions of possibly disjoint subsets in metric spaces

Manuel De la Sen
Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (EHU), 48940 Leioa, Spain
Asier Ibeas
Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Barcelona, Spain
Hasanen A. Hammad
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia; Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

Article ID: 4549

Keywords: cyclic/non-cyclic hybrid bivalued contractive mappings; cyclic mappings; best proximity points

Abstract

This paper presents and analyzes a class of self-mappings in metric spaces which are endowed of mixed characteristics between that of contractive cyclic self-mappings on union of subsets and that of contractive mappings within the individual subsets. Stability analysis of dynamic systems can be focused on by formalizing the relationship between cyclic contractive mappings over non-necessarily intersecting subsets and the stability criteria for switched systems with cyclic dynamics. The case of disjoint set structures necessitates the application of best proximity theory. Therefore, this manuscript introduces the mentioned class of hybrid cyclic/non-cyclic bivalued contractive self-mappings which can operate iteration-by-iteration in either a cyclic operation mode (the selected image lies in the next adjacent subset) or in an intra mode operation (the selected image lies in the current subset). Unlike cyclic operators, these proposed self-mappings allow dynamic switches to either the current subset or the next adjacent one in a cyclic disposal, governed by an iteration-dependent image selection sequence. This “modus-operandi” is possible since one of the images of each point at each iteration is activated for the cyclic operation mode while the other one is activated for an intra mode mode within some subset. Under the key assumptions that the subsets are closed, at least one best proximity set is a singleton, and the subset itself is boundedly compact, new boundedness and convergence theorems are established. It is emphasized how cyclic contractive properties are useful in the context of asymptotic stability of hybrid systems, validating the theoretical framework through illustrative examples.

Published

2026-07-30

How to Cite

De la Sen, M., Ibeas, A., & Hammad, H. A. (2026). On a class of bivalued hybrid cyclic and non-cyclic contractive self-mappings on unions of possibly disjoint subsets in metric spaces. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4549

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Vol. 33 No. 3 (2026): In Progress

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Eldred AA, Veeramani P. Existence and convergence of best proximity points. Journal of Mathematical Analysis and Applications. 2006; 323(2): 1001–1006. doi: 10.1016/j.jmaa.2005.10.081

[2]Karpagam S, Agrawal S. Best Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions. Fixed Point Theory and Applications. 2009; 2009(1): 197308. doi: 10.1155/2009/197308

[3]Karpagam S, Agrawal S. Existence of best proximity points of p-cyclic contractions. Fixed Point Theory. 2012; 13(1): 99–105. Available online: https://www.math.ubbcluj.ro/~nodeacj/download.php?f=121Karpagam-final.pdf

[4]De La Sen M, Singh SL, Gordji ME, et al. Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition. Fixed Point Theory and Applications. 2013; 2013(1): 324. doi: 10.1186/1687-1812-2013-324

[5]Weng S, Xiao H. Related Fixed Point Results for LB3S2‐Type Cyclic Mapping in Extended b‐Metric Spaces With an Application. Journal of Mathematics. 2025; 2025(1): 8867702. doi: 10.1155/jom/8867702

[6]Ali WF, Ahmad BAA. Best Proximity Point for Some Type of Cyclic Mappingin Strong ḇ-Metric Space. Iraqi Journal of Science. 2025; 1996–2002. doi: 10.24996/ijs.2025.66.5.18

[7]Wu D, Weng S. Results for ϕδ-type cyclic mapping on extended b-metric space. Journal of Physics: Conference Series. 2025; 2964(1): 012082. doi: 10.1088/1742-6596/2964/1/012082

[8]Gabeleh M, Uyanık Ekici E, Aphane M. Best Proximity Theory in Metrically Convex Menger PM-Spaces via Cyclic Kannan Maps. Symmetry. 2025; 17(9): 1549. doi: 10.3390/sym17091549

[9]Hafshejani AS. On the error estimates for the sequence of successive approximations for cyclic φ–contractions in metric spaces. Carpathian Journal of Mathematics. 2024; 41(1): 193–204. doi: 10.37193/CJM.2025.01.13

[10]Ghosh S, Bhardwaj R, Narayan S, et al. A result concerning best proximity point from the perspective of G-metric spaces using control functions with an application. Advances in Mathematical Sciences and Application. 2025; 34(1): 33–46. Available online: https://www.researchgate.net/publication/391346551_A_Result_Concerning_Best_Proximity_Point_from_the_perspective_of_G-Metric_Spaces_using_Control_Functions_with_an_Application_Adv_Math_Sci_Appl34133-46ISSN_1343-4373

[11]Unni AS, Pragadeeswarar V. Proximal iterated function systems using cyclic Meir-Keeler contractions and an application to fractal theory. Fixed Point Theory and Algorithms for Sciences and Engineering. 2025; 2025(1): 3. doi: 10.1186/s13663-025-00784-7

[12]Di Bari C, Suzuki T, Vetro C. Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications. 2008; 69(11): 3790–3794. doi: 10.1016/j.na.2007.10.014

[13]Pasupathi R, Chand AKB, Navascués MA. Cyclic Meir-Keeler Contraction and Its Fractals. Numerical Functional Analysis and Optimization. 2021; 42(9): 1053–1072. doi: 10.1080/01630563.2021.1937215

[14]Pasupathi R, Chand AKB, Navascués MA. Cyclic iterated function systems. Journal of Fixed Point Theory and Applications. 2020; 22(3): 58. doi: 10.1007/s11784-020-00790-9

[15]Ishtiaq T, Batool A. Approximating Fixed Points of Generalized Cyclic Enriched Contraction Mapping Using Ishi Iteration Scheme with Application. Global Integrated Mathematics. 2026; 2(1): 28–40. doi: 10.64229/e7pt2x51

[16]Saluja GS. Some fixed point theorems via cyclic contractive conditions in s-metric spaces. Facta Universitatis, Series: Mathematics and Informatics. 2021; 377. doi: 10.22190/FUMI200811028S

[17]Gupta A. Cyclic contraction on an S-metric space. International Journal of Analysis and Applications. 2013; 3(2): 119–130. Available online: https://www.etamaths.com/index.php/ijaa/article/view/84

[18]Chaipornjareansri S. Some best proximity point results for MT-rational cyclic contractions in S- metric space. Communications in Mathematics and Applications. 2020; 11(4): 587–600. Available online: https://rgnpublications.com/journals/index.php/cma/article/download/1413/1123

[19]Rada E. Novel Fixed Point Theorems and Stability Analysis for Cyclic Contractions in b-G-metric Spaces via λ-Iteration. European Journal of Pure and Applied Mathematics. 2026; 19(1): 6907. doi: 10.29020/nybg.ejpam.v19i1.6907

[20]De La Sen M, Ibeas A. Stability Results for Switched Linear Systems with Constant Discrete Delays. Mathematical Problems in Engineering. 2008; 2008(1): 543145. doi: 10.1155/2008/543145

[21]Ibeas A, De La Sen M. Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function. Applied Mathematics Letters. 2009; 22(10): 1549–1555. doi: 10.1016/j.aml.2009.03.023

[22]Kulenović MRS, Sullivan R. Basins of Attraction for Third-Order Sigmoid Beverton Holt Difference Equation. Mathematics. 2025; 13(18): 2990. doi: 10.3390/math13182990

[23]Kessy JA, Wangwe L. On a fixed point theorem of Karapinar for interpolative contractions. Research in Mathematics. 2025; 12(1): 2526218. doi: 10.1080/27684830.2025.2526218

[24]Abbas A, Ali A, Al Sulami H, et al. Recent Advancements in KRH-Interpolative-Type Contractions. Symmetry. 2023; 15(8): 1515. doi: 10.3390/sym15081515

[25]Devi D, Debnath P. Fixed points of two interpolative cyclic contractions in b-metric spaces. Heliyon. 2025; 11(1): e41667. doi: 10.1016/j.heliyon.2025.e41667

[26]Nazam M, Aydi H, Hussain A. Existence theorems for (Ψ,Φ)-orthogonal interpolative contractions and an application to fractional differential equations. Optimization. 2023; 72(7): 1899–1929. doi: 10.1080/02331934.2022.2043858

[27]Tomar A, Rana US, Kumar V. Fixed point, its geometry, and application via ω‐interpolative contraction of Suzuki type mapping. Mathematical Methods in the Applied Sciences. 2024; 47(5): 3507–3528. doi: 10.1002/mma.8871

[28]Zeng HB, Chen YJ, He Y, et al. A delay-derivative-dependent switched system model method for stability analysis of linear systems with time-varying delay. Automatica. 2025; 175: 112183. doi: 10.1016/j.automatica.2025.112183

[29]Xu L, Bao B, Hu H. Stability of impulsive delayed switched systems with conformable fractional-order derivatives. International Journal of Systems Science. 2025; 56(6): 1271–1288. doi: 10.1080/00207721.2024.2421454

[30]Liu S, Zhao N, Zhang L, et al. Adaptive neural hierarchical sliding mode control for uncertain switched underactuated nonlinear systems against unmodeled dynamics and input delay. Asian Journal of Control. 2025; 27(3): 1552–1569. doi: 10.1002/asjc.3528

[31]Chen H, Zong G, Shen M, et al. Finite-Time Resilient Control of Networked Markov Switched Nonlinear Systems: A Relaxed Design. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 2025; 55(4): 2569–2579. doi: 10.1109/TSMC.2025.3526321

[32]Zhao ZL, Zhang RF. Artificial Intelligence-Enhanced Mathematical Derivation method: Exact solutions of the Benjamin–Bona–Mahony equation. Engineering Applications of Artificial Intelligence. 2026; 173: 114464. doi: 10.1016/j.engappai.2026.114464

[33]Wang J, Liu Y, Yan L, et al. Fractional sub-equation neural networks (fSENNs) method for exact solutions of space–time fractional partial differential equations. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2025; 35(4): 043110. doi: 10.1063/5.0259937

[34]Zhang H, Zhang R, Liu Q. A Novel Multi-Modal Neurosymbolic Reasoning Intelligent Algorithm for BLMP Equation. Chinese Physics Letters. 2025; 42(10): 100002. doi: 10.1088/0256-307X/42/10/100002

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