Tripled Fixed Point Approaches and Hyers-Ulam Stability With Applications

Main Article Content

Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen


In this paper, we present tripling fixed point results for extended contractive mappings in the context of a generalized metric space. Many publications in the literature are improved, unified, and generalized by our theoretical results. Furthermore, the Ulam-Hyers stability problem for the tripled fixed point problem in vector-valued metric spaces has been examined as a stability analysis for fixed point approaches. Finally, as a type of application to support our research, the theoretical conclusions are used to explore the existence and uniqueness of solutions to a periodic boundary value problem.

Article Details


  1. A. Bica, S. Mure¸san, H. Oros, Existence Result and Numerical Method for a Delay Integro-Differential Equation Arising in Infectious Diseases, in: Proceedings of the 11-th Conference on Applied and Industrial Mathematics, Oradea Romania, 34-41, (2003).
  2. A. Canada, A. Zertiti, Systems of Nonlinear Delay Integral Equations Modeling Population Growth in a Periodic Environment, Commentat. Math. Univ. Carol. 35 (1994), 633–644.
  3. D. Guo, V. Lakshmikantham, Positive Solutions of Nonlinear Integral Equations Arising in Infectious Diseases, J. Math. Anal. Appl. 134 (1988) 1–8.
  4. A.I. Perov, On the Cauchy Problem for a System of Ordinary Differential Equations, Priblizhen. Met. Reshen. Differ. Uravn. 2 (1964), 115–134.
  5. A. Petru¸sel, Multivalued Weakly Picard Operators and Applications, Sci. Math. Jpn. 59 (2004), 169–202.
  6. S. Banach, Sur les Opérations dans les Ensembles Abstraits et Leur Application aux Équations Intégrales, Fund. Math. 3 (1922), 133–181.
  7. V. Berinde, M. Borcut, Tripled Fixed Point Theorems for Contractive Type Mappings in Partially Ordered Metric Spaces, Nonlinear Anal.: Theory Methods Appl. 74 (2011), 4889–4897.
  8. M. Borcut, Tripled Coincidence Theorems for Contractive Type Mappings in Partially Ordered Metric Spaces, Appl. Math. Comput. 218 (2012), 7339–7346.
  9. A. Amini-Harandi, Coupled and Tripled Fixed Point Theory in Partially Ordered Metric Spaces With Application to Initial Value Problem, Math. Comput. Model. 57 (2013), 2343–2348.
  10. Z. Kadelburg, S. Radenovi´c, Fixed point and tripled fixed point theorems under Pata-type conditions in ordered metric spaces, Int. J. Anal. Appl. 6 (2014), 113–122.
  11. R. Vats, K. Tas, V. Sihag, A. Kumar, Triple Fixed Point Theorems via α−Series in Partially Ordered Metric Spaces, J. Ineq. Appl. 2014 (2014), 176.
  12. H.A. Hammad, H. Aydi, M. De la Sen, New Contributions for Tripled Fixed Point Methodologies via a Generalized Variational Principle With Applications, Alexandria Eng. J. 61 (2022), 2687–2696.
  13. H.A. Hammad, M.D. La Sen, A Technique of Tripled Coincidence Points for Solving a System of Nonlinear Integral Equations in POCML Spaces, J. Inequal. Appl. 2020 (2020), 211.
  14. H.A. Hammad, M. Zayed, Solving Systems of Coupled Nonlinear Atangana–baleanu-Type Fractional Differential Equations, Bound. Value Probl. 2022 (2022), 101.
  15. Humaira, H.A. Hammad, M. Sarwar, M. De la Sen, Existence Theorem for a Unique Solution to a Coupled System of Impulsive Fractional Differential Equations in Complex-Valued Fuzzy Metric Spaces, Adv. Differ. Equ. 2021 (2021), 242.
  16. H.A. Hammad, M. De la Sen, Analytical Solution of Urysohn Integral Equations by Fixed Point Technique in Complex Valued Metric Spaces, Mathematics. 7 (2019), 852.
  17. H.A. Hammad, M. De la Sen, Stability and Controllability Study for Mixed Integral Fractional Delay Dynamic Systems Endowed With Impulsive Effects on Time Scales, Fractal Fract. 7 (2023), 92.
  18. H.A. Hammad, H. Aydi, H. I¸sık, M. De la Sen, Existence and Stability Results for a Coupled System of Impulsive Fractional Differential Equations With Hadamard Fractional Derivatives, AIMS Math. 8 (2023), 6913–6941.
  19. H.A. Hammad, M. De la Sen and H. Aydi, Generalized Dynamic Process for an Extended Multi-Valued F−Contraction in Metric-Like Spaces With Applications, Alexandria Eng. J. 59 (2020), 3817–3825.
  20. H.A. Hammad, H. Aydi, M. Zayed, Involvement of the Topological Degree Theory for Solving a Tripled System of Multi-Point Boundary Value Problems, AIMS Math. 8 (2022), 2257–2271.
  21. 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 (2021), 7017046.
  22. S.M. Ulam, A Collection of Mathematical Problems, Inter-Science, New York, (1968).
  23. D.H. Hyers, On the Stability of the Linear Functional Equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.
  24. R. Rizwan, Existence Theory and Stability Analysis of Fractional Langevin Equation, Int. J. Nonlinear Sci. Numer. Simul. 20 (2019), 833–848.
  25. J. Wang, K. Shah, A. Ali, Existence and Hyers–Ulam Stability of Fractional Nonlinear Impulsive Switched Coupled Evolution Equations, Math. Methods Appl. Sci. 41 (2018), 2392–2402.
  26. M.A. Almalahi, M.S. Abdo, S.K. Panchal, Existence and Ulam-Hyers Stability Results of a Coupled System of ψ−Hilfer Sequential Fractional Differential Equations, Results Appl. Math. 10 (2021), 100142.
  27. B. Ahmad, S.K. Ntouyas, A. Alsaedi, Sequential Fractional Differential Equations and Inclusions With SemiPeriodic and Nonlocal Integro-Multipoint Boundary Conditions, J. King Saud Univ. - Sci. 31 (2019), 184–193.
  28. R.S. Varga, Matrix Iterative Analysis, Springer, Berlin, Heidelberg, 2000.
  29. H. Hosseinzadeh, Some fixed point theorems in generalized metric spaces endowed with vector-valued metrics and application in nonlinear matrix equations, Sahand Commun. Math. Anal. 17 (2020), 37–53.
  30. H. Hosseinzadeh, A. Jabbari, A. Razani, Fixed point theorems and common fixed point theorems on spaces equipped with vector-valued metrics, Ukr. Math. J. 65 (2013), 814–822.
  31. G. Allaire, S.M. Kaber, K. Trabelsi, Numerical Linear Algebra, Texts in Applied Mathematics, 55, Springer, New York, 2008.
  32. R. Precup, The Role of Matrices That Are Convergent to Zero in the Study of Semilinear Operator Systems, Math. Comput. Model. 49 (2009), 703–708.
  33. T.G. Bhaskar, V. Lakshmikantham, Fixed Point Theorems in Partially Ordered Metric Spaces and Applications, Nonlinear Anal.: Theory Methods Appl. 65 (2006), 1379–1393.
  34. D. Guo, V. Lakshmikantham, Coupled Fixed Points of Nonlinear Operators With Applications, Nonlinear Anal.: Theory Methods Appl. 11 (1987), 623–632.
  35. D. Guo, Y.J. Cho, J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Sci. Publishers Inc., Hauppauge, NY, (2004).
  36. S. Hong, Fixed Points for Mixed Monotone Multivalued Operators in Banach Spaces With Applications, J. Math. Anal. Appl. 337 (2008), 333–342.
  37. M.D. Rus, The Method of Monotone Iterations for Mixed Monotone Operators, Ph.D. Thesis, Universitatea BabesBolyai, Cluj-Napoca, (2010).