10.1007/s40096-023-00521-4

Existence of the solution of nonlinear fractional differential equations via new best proximity point results

HTML PDF
  • Mustafa Aslantaş*1,
  1. Department of Mathematics, Çankırı Karatekin University, Çankırı, TR

Published 2024-01-16

How to Cite

Aslantaş, M. (2024). Existence of the solution of nonlinear fractional differential equations via new best proximity point results. Mathematical Sciences, 18(4 (December 2024). https://doi.org/10.1007/s40096-023-00521-4
Crossref
Scopus
Google Scholar
Europe PMC

HTML views: 12

PDF views: 11

Abstract

Abstract In this paper, we obtain some best proximity point results by introducing the concepts of proximal p -contractions of the first type and proximal p -contractions of the second type on partial metric spaces. Thus, some famous results in the literature such as the main result of Altun et al. (Acta Math Hung 162:393–402, 2020) and Basha (J Approx Theory 163(11):1772–1781, 2011) have been extended. Also, we provide some examples where our results are applicable and the results in Haghi et al. (Topol Appl 160:450–454, 2013) are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations via our results.

Keywords

  • Best proximity point,
  • Partial metric space,
  • Nonlinear fractional differential equations
HTML PDF

References

  1. Altun and Simsek (2008) Some fixed point theorems on dualistic partial metric spaces (pp. 1-9)
  2. Altun et al. (2010) Generalized contractions on partial metric spaces 157(18) (pp. 2778-2785) https://doi.org/10.1016/j.topol.2010.08.017
  3. Altun et al. (2020) Best proximity point results for p-proximal contractions (pp. 393-402) https://doi.org/10.1007/s10474-020-01036-3
  4. Aslantas (2022) Finding a solution to an optimization problem and an application 194(1) (pp. 121-141) https://doi.org/10.1007/s10957-022-02011-4
  5. Aslantas, M.: A new contribution to best proximity point theory on quasi metric spaces and an application to nonlinear integral equations. Optimization 1–14 (2022)
  6. Aydi et al. (2012) Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces (pp. 3234-3242) https://doi.org/10.1016/j.topol.2012.06.012
  7. Banach (1922) Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales (pp. 133-181) https://doi.org/10.4064/fm-3-1-133-181
  8. Basha (2011) Best proximity point theorems 163(11) (pp. 1772-1781) https://doi.org/10.1016/j.jat.2011.06.012
  9. Basha (2010) Extensions of Banach’s contraction principle 31(5) (pp. 569-576) https://doi.org/10.1080/01630563.2010.485713
  10. Basha and Veeramani (1977) Best approximations and best proximity pairs (pp. 289-300)
  11. Ćirić et al. (2011) Common fixed points of generalized contractions on partial metric spaces and an application 218(6) (pp. 2398-2406)
  12. El-Hady and Ogrekci (2021) On Hyers–Ulam–Rassias stability of fractional differential equations with Caputo derivative (pp. 325-332) https://doi.org/10.22436/jmcs.022.04.02
  13. Haghi et al. (2013) Be careful on partial metric fixed point results (pp. 450-454) https://doi.org/10.1016/j.topol.2012.11.004
  14. Hattaf et al. (2022) Gronwall inequality and existence of solutions for differential equations with generalized Hattaf fractional derivative (pp. 18-27) https://doi.org/10.22436/jmcs.027.01.02
  15. Kadelburg and Radenovic (2014) Fixed point and tripled fixed point theorems under Pata-Type conditions in ordered metric spaces (pp. 113-122)
  16. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam (2006)
  17. Long (2022) Cauchy problem for inhomogeneous fractional nonclassical diffusion equation on the sphere 25(4) (pp. 303-311) https://doi.org/10.22436/jmcs.025.04.01
  18. Matthews (1994) Partial metric topology (pp. 183-197) https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
  19. Popescu (2008) A new type of contractive mappings in complete metric spaces (pp. 479-482)
  20. Raj (2013) Best proximity point theorems for non-self mappings 14(2) (pp. 447-454)
  21. Reich (1972) Fixed points of contractive functions (pp. 26-42)
  22. Romaguera (2012) Fixed point theorems for generalized contractions on partial metric spaces (pp. 194-199) https://doi.org/10.1016/j.topol.2011.08.026
  23. Sahin (2022) Existence and uniqueness results for nonlinear fractional differential equations via new Q-function (pp. 1-21) https://doi.org/10.1007/s43036-021-00168-9
  24. Sahin (2022) A new best proximity point result with an application to nonlinear Fredholm integral equations 10(4) (pp. 1-14) https://doi.org/10.3390/math10040665
  25. Sahin et al. (2023) Some extended results for multivalued F-contraction mappings 12(2) https://doi.org/10.3390/axioms12020116
  26. Samko et al. (1993) Gordon and Breach
  27. Youssef (2022) Generalized fractional delay functional equations with Riemann–Stieltjes and infinite point nonlocal conditions (pp. 33-48) https://doi.org/10.22436/jmcs.024.01.04