10.57647/cna.2025.h13d

An Application of Fixed Point Theorems to Dynamic Programming in a Generalized Distance Space

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  • Atefeh Sojoudi1,
  • Hasan Hosseinzadeh*1,
  • Kushal Roy2,
  • Vahid Parvaneh3,
  1. Department of Mathematics, Ard.C., Islamic Azad University, Ardabil, Iran
  2. Department of Mathematics, Dr. B.C. Roy Engineering College, Durgapur-713206, West Bengal, India
  3. Department of Mathematics, Ker.C., Islamic Azad University, Kermanshah, Iran

Received: 2025-02-17

Revised: 2025-05-08

Accepted: 2025-07-14

Published in Issue 2025-12-30

Copyright (c) 2025 Atefeh Sojoudi, Hasan Hosseinzadeh, Kushal Roy, Vahid Parvaneh (Author)

Creative Commons License

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

How to Cite

An Application of Fixed Point Theorems to Dynamic Programming in a Generalized Distance Space. (2025). Communications in Nonlinear Analysis, 13(2). https://doi.org/10.57647/cna.2025.h13d

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Abstract

This study introduces a novel concept of a three-variable distance space, which extends several existing metric structures, including S-metric space, Sb-metric, Sθ -metric space, and Sp-metric space. Various topological properties of these spaces are examined. A fixed point theorem for generalized weak contraction mappings in this framework is established, accompanied by several related corollaries. To validate the space and the derived fixed-point theorem, illustrative examples are provided. Finally, the fixed point theorem is applied to analyze the solution of a functionalequation arising in dynamic programming.

Keywords

  • Fixed point,
  • S-metric space,
  • S∆-metric space,
  • Weak contraction mapping,
  • Dynamic programming,
  • MSC 2020: 47H10, 54H25
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References

  1. Sedghi S, Shobe N, Aliouche A. A generalization of fixed point theorems in S-metric spaces. Matematicˇki Vesnik. 2012;64:258-66.
  2. Roy K, Saha M, Parvaneh V, Khorshidi M. A new variant of symmetric distance spaces and an extension of the Banach fixed-point theorem. Journal of Mathematics. 2022:9282762. doi:10.1155/2022/9282762.
  3. Bakhtin IA. The contraction mapping principle in quasi-metric spaces. Functional Analysis. 1989;30. Ulyanovsk Gos. Pedagogical Institute, Ulyanovsk.
  4. Kamran T, Samreen M, Ain QU. A generalization of b-metric space and some fixed point theorems. Mathematics. 2017;5(2):19. doi:10.3390/math5020019.
  5. Parvaneh V, Ghoncheh SJH. Fixed points of (ψ, ϕ)Ω-contractive mappings in ordered p-metric spaces. Global Analysis and Discrete Mathematics. 2020;4(1):15-29. doi:10.22128/gadm.2019.290.1019.
  6. Jleli M, Samet B. On Banach’s fixed point theorem in perturbed metric spaces. Journal of Applied Analysis and Computation. 2025;15(2):993-1001. doi:10.11948/20240242.
  7. Souayah N, Mlaiki N. A fixed point theorem in Sb-metric spaces. Journal of Mathematics and Computer Science. 2016;16:131-9.
  8. Rohen Y, Dosenovic´ T, Radenovic´ S. A note on the paper “A fixed point theorem in Sb-metric spaces”. Filomat. 2017;31(11):3335-46. doi:10.2298/FIL1711335R.
  9. Mlaiki N. Extended Sb-metric spaces. Journal of Mathematical Analysis. 2018;9:124-35.
  10. Mustafa Z, Shahkoohi RJ, Parvaneh V, Kadelburg Z, Jaradat MMM. Ordered Sp-metric spaces and some fixed point theorems for contractive map-pings with application to periodic boundary value problems. Fixed Point Theory and Applications. 2019;2019(1):16. doi:10.1186/s13663-019-0651-4.
  11. Mukheimer A. Extended partial Sb-metric spaces. Axioms. 2018;7(4):87. doi:10.3390/axioms7040087.
  12. Bellman R. Dynamic Programming. Princeton, NJ: Princeton University Press; 1957.
  13. Liu Z, Ume JS, Kang SM. On properties of solutions for two functional equations arising in dynamic programming. Fixed Point Theory and Applications. 2010:905858. doi:10.1155/2010/905858.
  14. Panja S, Roy K, Saha M. Fixed points for a class of extended interpolative ΨF -contraction maps over a b-metric space and its application to dynamical programming. Scientific Bulletin of the Politehnica University of Bucharest, Series A: Applied Mathematics and Physics. 2021;83:59-70.
  15. Roy K, Panja S, Saha M. A generalized fixed point theorem in an extended cone b-metric space over Banach algebra with its application to dynamical programming. Applied Mathematics E Notes. 2021;21:209-19.