Generalized \(\theta\)-\(\Phi_{\aleph}\)-Contractions and \(\theta\)-\(\mathscr{F}_{\aleph}\)-Expansion of Darbo-Type and Their Applications

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DOI: https://doi.org/10.28924/2291-8639-23-2025-220
Published: Sep 8, 2025
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Muhammad Sarwar, Mian Bahadur Zada, Syed Khayyam Shah, Haroon Rashid, Kamaleldin Abodayeh, Chanon Promsakon, Thanin Sitthiwirattham, Generalized \(\theta\)-\(\Phi_{\aleph}\)-Contractions and \(\theta\)-\(\mathscr{F}_{\aleph}\)-Expansion of Darbo-Type and Their Applications, Int. J. Anal. Appl., 23 (2025), 220.

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Muhammad Sarwar, Mian Bahadur Zada, Syed Khayyam Shah, Haroon Rashid, Kamaleldin Abodayeh, Chanon Promsakon, Thanin Sitthiwirattham

Abstract

In this work, we present the concept of \(\theta\)-\(\Phi_{\aleph}\)-contraction, \(\theta\)-\(\Phi_{\aleph}\)-Suzuki contraction, \(\theta\)-\(\Phi_{\aleph}\)-Kannan type contraction, and \(\theta\)-\(\mathscr{F}_{\aleph}\)-expansion, and establish some novel fixed point theorems in the light of Banach space. In order to verify our results, we construct some examples. Furthermore, we use our results to check the existence of a solution to differential equations.

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References

  1. J. Banaś, On Measures of Noncompactness in Banach Spaces, Comment. Math. Univ. Carol. 21 (1980), 131–143. https://eudml.org/doc/17021.
  2. G. Darbo, Punti Uniti in Trasformazioni a Codominio Non Compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92.
  3. V. Parvaneh, N. Hussain, M. Khorshidi, N. Mlaiki, H. Aydi, Fixed Point Results for Generalized $mathcal{F}$-Contractions in Modular B-Metric Spaces with Applications, Mathematics 7 (2019), 887. https://doi.org/10.3390/math7100887.
  4. J. Garcia-Falset, K. Latrach, On Darbo-Sadovskii's Fixed Point Theorems Type for Abstract Measures of (weak) Noncompactness, Bull. Belg. Math. Soc. - Simon Stevin 22 (2015), 797–812. https://doi.org/10.36045/bbms/1450389249.
  5. S. Banach, Sur les Opérations dans les Ensembles Abstraits et Leur Application aux Équations Intégrales, Fundam. Math. 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181.
  6. L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1911), 97–115. https://eudml.org/doc/158520.
  7. J. Chen, X. Tang, Generalizations of Darbo’s Fixed Point Theorem via Simulation Functions with Application to Functional Integral Equations, J. Comput. Appl. Math. 296 (2016), 564–575. https://doi.org/10.1016/j.cam.2015.10.012.
  8. G. Darbo, Punti Uniti in Trasformazioni a Codominio Non Compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92. http://www.numdam.org/item?id=RSMUP_1955__24__84_0.
  9. A.A. Gillespie, B.B. Williams, V. Lakshmikantham, Fixed Point Theorems for Expanding Maps, Appl. Anal. 14 (1983), 161–165. https://doi.org/10.1080/00036818308839419.
  10. J. Górnicki, Fixed Point Theorems for $mathcal{F}$-Expanding Mappings, Fixed Point Theory Appl. 2017 (2016), 9. https://doi.org/10.1186/s13663-017-0602-3.
  11. M. Jleli, B. Samet, A New Generalization of the Banach Contraction Principle, J. Inequal. Appl. 2014 (2014), 38. https://doi.org/10.1186/1029-242x-2014-38.
  12. M. Jleli, E. Karapinar, D. O’Regan, B. Samet, Some Generalizations of Darbo’s Theorem and Applications to Fractional Integral Equations, Fixed Point Theory Appl. 2016 (2016), 11. https://doi.org/10.1186/s13663-016-0497-4.
  13. S.M. Kang, Fixed Point for Expansion Mappings, Math. Japonica 38 (1993), 713–717. https://cir.nii.ac.jp/crid/1573668924494060288.
  14. C. Kuratowski, Sur les Espaces Complets, Fundam. Math. 15 (1930), 301–309. https://doi.org/10.4064/fm-15-1-301-309.
  15. S. Park, B.E. Rhoades, Some Fixed Point Theorems for Expansion Mappings, Math. Japon. 33 (1988), 129–132.
  16. V. Parvaneh, N. Hussain, M. Khorshidi, N. Mlaiki, H. Aydi, Fixed Point Results for Generalized $mathcal{F}$-Contractions in Modular $b$-Metric Spaces with Applications, Mathematics 7 (2019), 887. https://doi.org/10.3390/math7100887.
  17. J. Schauder, Der Fixpunktsatz in Funktionalraümen, Stud. Math. 2 (1930), 171–180. http://eudml.org/doc/217247.
  18. D. Wardowski, Fixed Points of a New Type of Contractive Mappings in Complete Metric Spaces, Fixed Point Theory Appl. 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94.
  19. D. Wardowski, Solving Existence Problems via $mathcal{F}$-Contractions, Proc. Am. Math. Soc. 146 (2017), 1585–1598. https://doi.org/10.1090/proc/13808.
  20. S.Z. Wang, B.Y. Li, Z.M. Gao, K.K. Iséki, Some Fixed Point Theorems on Expansion Mappings, Math. Japon. 29 (1984), 631–636.
  21. M. Al-Refai, K. Pal, New Aspects of Caputo-Fabrizio Fractional Derivative, Prog. Fract. Differ. Appl. 5 (2019), 157–166. https://doi.org/10.18576/pfda/050206.
  22. M. Younis, D. Singh, D. Gopal, A. Goyal, M.S. Rathore, On Applications of Generalized $mathcal{F}$-Contraction to Differential Equations, Nonlinear Funct. Anal. Appl. 24 (2019), 155–174.
  23. M.B. Zada, M. Sarwar, T. Abdeljawad, F. Jarad, Generalized Darbo-Type $mathcal{F}$-Contraction and $mathcal{F}$-Expansion and Its Applications to a Nonlinear Fractional-Order Differential Equation, J. Funct. Spaces 2020 (2020), 4581035. https://doi.org/10.1155/2020/4581035.