Fixed-Point Approaches for Multi-Valued Contraction Mappings in a Novel Space With an Application

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-80
Published: May 20, 2024
Cite as:
Hasanen A. Hammad, Manuel De la Sen, Fixed-Point Approaches for Multi-Valued Contraction Mappings in a Novel Space With an Application, Int. J. Anal. Appl., 22 (2024), 80.

Main Article Content

Hasanen A. Hammad, Manuel De la Sen

Abstract

The vital goals of this manuscript are to combine metric-like spaces with S-metric spaces under a control function to obtain a new space called the controlled S-metric-like spaces (CSMLSs, for short). Under this name, many fixed-point (FP) results have been obtained for multi-valued mappings (MVMs). In addition, we present several non-trivial examples to back up our statements. The results obtained generalize and unify many results in the same direction. Finally, to support and test the adequacy of the theoretical results, the existence of the solution to the differential inclusion problem (DIP) was studied as a type of application.

Article Details

References

  1. P. Saipara, K. Khammahawong, P. Kumam, Fixed-Point Theorem for a Generalized Almost Hardy-Rogers-Type F Contraction on Metric-Like Spaces, Math. Meth. Appl. Sci. 42 (2019), 5898–5919. https://doi.org/10.1002/mma.5793.
  2. A. Tomar, R. Sharma, Some Coincidence and Common Fixed Point Theorems Concerning F−Contraction and Applications, J. Int. Math. Virtual Inst. 8 (2018), 181–198.
  3. H.A. Hammad, M. De la Sen, Fixed-Point Results for a Generalized Almost (s, q)-Jaggi F-Contraction-Type on b-Metric-Like Spaces, Mathematics. 8 (2020), 63. https://doi.org/10.3390/math8010063.
  4. H.A. Hammad, M. De la Sen, H. Aydi, Analytical Solution for Differential and Nonlinear Integral Equations via F$e−Suzuki Contractions in Modified $e−Metric-Like Spaces, J. Function Spaces, 2021 (2021), 6128586. https://doi.org/10.1155/2021/6128586.
  5. 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. https://doi.org/10.3390/math7090852.
  6. M. Berzig, S. Chandok, M.S. Khan, Generalized Krasnoselskii Fixed Point Theorem Involving Auxiliary Functions in Bimetric Spaces and Application to Two-Point Boundary Value Problem, Appl. Math. Comput. 248 (2014), 323–327. https://doi.org/10.1016/j.amc.2014.09.096.
  7. D. Gopal, M. Abbas, C. Vetro, Some New Fixed Point Theorems in Menger Pm-Spaces With Application to Volterra Type Integral Equation, Appl. Math. Comput. 232 (2014), 955–967. https://doi.org/10.1016/j.amc.2014.01.135.
  8. Z. Liu, X. Li, S.M. Kang, S.Y. Cho, Fixed Point Theorems for Mappings Satisfying Contractive Conditions of Integral Type and Applications, Fixed Point Theory Appl 2011 (2011), 64. https://doi.org/10.1186/1687-1812-2011-64.
  9. M.M.A. Taleb, V.C. Borkar, Some Rational Contraction and Applications of Fixed Point Theorems to F-Metric Space in Differential Equations, J. Math. Comput. Sci. 12 (2022), 133. https://doi.org/10.28919/jmcs/7266.
  10. M. Fréchet, Sur Quelques Points du Calcul Fonctionnel, Rend. Circ. Matem. Palermo. 22 (1906), 1–72. https://doi.org/10.1007/bf03018603.
  11. I.A. Bakhtin, The Contraction Mapping Principle in Almost Metric Space. Funct. Anal. 30 (1989), 26–37.
  12. A. Branciari, A Fixed Point Theorem of Banach-Caccioppoli Type on a Class of Generalized Metric Spaces, Publ. Math. Debrecen. 57 (2000), 31–37.
  13. P. Debnath, N. Konwar, S. Radenovi´c, eds., Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences, Springer, Singapore, 2021. https://doi.org/10.1007/978-981-16-4896-0.
  14. R.A. Rashwan, H.A. Hammad, M.G. Mahmoud, Common Fixed Point Results for Weakly Compatible Mappings Under Implicit Relations in Complex Valued G-Metric Spaces, Inf. Sci. Lett. 8 (2019), 111–119. https://doi.org/10.18576/isl/080305.
  15. 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. https://doi.org/10.3934/math.2023117.
  16. 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. https://doi.org/10.1155/2021/7017046.
  17. S. Czerwik, Contraction Mappings in b-Metric Spaces, Acta Math. Inf. Univ. Ostrav. 1 (1993), 5–11. http://dml.cz/dmlcz/120469.
  18. 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.
  19. N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled Metric Type Spaces and the Related Contraction Principle, Mathematics. 6 (2018), 194. https://doi.org/10.3390/math6100194.
  20. D. Lateef, Fisher Type Fixed Point Results in Controlled Metric Spaces, J. Math. Comput. Sci. 20 (2020), 234–240. https://doi.org/10.22436/jmcs.020.03.06.
  21. M.S. Sezen, Controlled Fuzzy Metric Spaces and Some Related Fixed Point Results, Numer. Meth. Part. Diff. Equ. 37 (2020), 583–593. https://doi.org/10.1002/num.22541.
  22. S. Tasneem, K. Gopalani, T. Abdeljawad, A Different Approach to Fixed Point Theorems on Triple Controlled Metric Type Spaces with a Numerical Experiment, Dynamic Systems Appl., 30 (2021), 111–130. https://doi.org/10.46719/dsa20213018.
  23. H.A. Hammad, H. Aydi, Y.U. Gaba, Exciting Fixed Point Results on a Novel Space with Supportive Applications, J. Function Spaces. 2021 (2021), 6613774. https://doi.org/10.1155/2021/6613774.
  24. 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, Alexandria Eng. J. 59 (2020), 3817–3825. https://doi.org/10.1016/j.aej.2020.06.037.
  25. 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. https://doi.org/10.1016/j.aej.2021.07.042.
  26. S. Sedghi, N. Shobe, A. Aliouche, A Generalization of Fixed Point Theorem in S−Metric Spaces, Mat. Vesnik. 64 (2012), 258–266.
  27. B.C. Dhage, Generalized Metric Spaces Mappings With Fixed Point, Bull. Cal. Math. Soc. 84 (1992), 329–336.
  28. S. Sedghi, N. Shobe, H. Zhou, A Common Fixed Point Theorem in D−Metric Space, Fixed Point Theory Appl. 2007 (2007), 27906. https://doi.org/10.1155/2007/27906.
  29. S.B. Nadler, Jr., Multi-Valued Contraction Mappings, Pac. J. Math. 30 (1969), 475–488. https://doi.org/10.2140/pjm.1969.30.475.
  30. M. Berinde, V. Berinde, On a General Class of Multi-Valued Weakly Picard Mappings, J. Math. Anal. Appl. 326 (2007), 772–782. https://doi.org/10.1016/j.jmaa.2006.03.016.
  31. L.B. Ciric, Fixed Point Theory. Contraction Mapping Principle, FME Press, Beograd, Serbia, (2003).
  32. S. Itoh, Multi-Valued Generalized Contractions and Fixed Point Theorems, Comment. Math. Univ. Carol. 18 (1977), 247–258. http://dml.cz/dmlcz/105770.
  33. H. Kaneko, A General Principle for Fixed Points of Contractive Multi-Valued Mappings, Math. Japon. 31 (1986), 407–411.
  34. S. Reich, Approximate Selections, Best Approximations, Fixed Points, and Invariant Sets, J. Math. Anal. Appl. 62 (1978), 104–113. https://doi.org/10.1016/0022-247x(78)90222-6.
  35. H.A. Hammad, M. De la Sen, PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application, Mathematics. 7 (2019), 52. https://doi.org/10.3390/math7010052.
  36. H.A. Hammad, M. De la Sen, A Fixed Point Technique for Set-Valued Contractions with Supportive Applications, Adv. Math. Phys. 2021 (2021), 6880478. https://doi.org/10.1155/2021/6880478.
  37. M.A. Alghamdi, N. Hussain, P. Salimi, Fixed Point and Coupled Fixed Point Theorems on b-Metric-Like Spaces, J. Inequal. Appl. 2013 (2013), 402. https://doi.org/10.1186/1029-242x-2013-402.
  38. M. Abuloha, D. Rizk, K. Abodayeh, N. Mlaiki, T. Abdeljawad, New Results in Controlled Metric Type Spaces, J. Math. 2021 (2021), 5575512. https://doi.org/10.1155/2021/5575512.
  39. A. Pourgholam, M. Sabbaghan, F. Taleghani, Common Fixed Points of Single-Valued and Multi-Valued Mappings in S-Metric Spaces, J. Indones. Math. Soc. 28 (2022), 19–30. https://doi.org/10.22342/jims.28.1.1106.19-30.
  40. N. Souayaha, N. Mlaikib, A Fixed Point Theorem in Sb -Metric Spaces, J. Math. Comput. Sci. 16 (2016), 131–139. https://doi.org/10.22436/jmcs.016.02.01.