DOI QR코드

DOI QR Code

FIXED POINT THEOREMS FOR THE MODIFIED SIMULATION FUNCTION AND APPLICATIONS TO FRACTIONAL ECONOMICS SYSTEMS

  • Nashine, Hemant Kumar (Applied Analysis Research Group, Faculty of Mathematics and Statistics Ton Duc Thang University) ;
  • Ibrahim, Rabha W. (Nonlinear Dynamics Research Center (NDRC), Ajman University) ;
  • Cho, Yeol Je (Center for General Education, China Medical University) ;
  • Kim, Jong Kyu (Department of Mathematics Education, Kyungnam University)
  • Received : 2020.08.06
  • Accepted : 2020.10.10
  • Published : 2021.03.15

Abstract

In this paper, first, we prove some common fixed point theorems for the generalized contraction condition under newly defined modified simulation function which generalize and include many results in the literature. Second, we give two numerical examples with graphical representations for verifying the proposed results. Third, we discuss and study a set of common fixed point theorems for two pairs (finite families) of self-mappings. Finally, we give some applications of our results in discrete and functional fractional economic systems.

Keywords

References

  1. R.P. Agarwal, A.A. Lupulescu and D. O'Regan, Lp-solutions for a class of fractional integral equations, J. Integral Equat. Appl., 29 (2017), 251-270. https://doi.org/10.1216/JIE-2017-29-2-251
  2. R. Arab, R. Allahyari and A. Haghighi, Existence of solutions of infinite systems of integral equations in Frechet spaces, Inter. J. Nonlinear Anal. Appl., 7 (2016), 205-216.
  3. D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969) 458-465. https://doi.org/10.1090/S0002-9939-1969-0239559-9
  4. M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015), 73-85.
  5. C. Cattani, H.M. Srivastava and X.J. Yang, Fractional Dynamics, Walter de Gruyter GmbH, Berlin/Boston (2015).
  6. L.B. Ciric, A generalization of Banach contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273. https://doi.org/10.1090/S0002-9939-1974-0356011-2
  7. D. Dukic, Z. Kadelburg and S. Radenovic, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abst. Appl. Anal., 2011, Art. ID 561245, 13 pp.
  8. M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608. https://doi.org/10.1090/S0002-9939-1973-0334176-5
  9. B. Hazarika, R. Arab and H.K. Nashine, Applications of measure of non-compactness and modified simulation function for solvability of nonlinear functional integral equations, Filomat, 33:17 (2019), 5427-5439. https://doi.org/10.2298/fil1917427h
  10. R.W. Ibrahim and M. Darus, Weakly solutions for fractional integral equation: Volterra type, Inter. J. Modern Theoretical Physics, 2 (2013), 42-52.
  11. G. Jungck, Compatible mappings and common fixed points, Inter. J. Math. Math. Sci., (1986), 771-779.
  12. F. Khojasteh, S. Shukla and S. Radenovic, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-1194. https://doi.org/10.2298/FIL1506189K
  13. S. Radenovic, Z. Kadelburg, D. Jandrlic and A. Jandrlic, Some results on weak contraction maps, Bull. Iranian Math. Soc., 38 (2012), 625-645.
  14. A. Gasull and A. Geyer, Traveling surface waves of moderate amplitude in shallow water, Nonlinear Anal. 102 (2014), 105-119. https://doi.org/10.1016/j.na.2014.02.005
  15. M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces, Chaos Solit. Fract., 42 (2009), 3121-3129. https://doi.org/10.1016/j.chaos.2009.04.017
  16. N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177-188. https://doi.org/10.1016/0022-247x(89)90214-x
  17. B.E. Rhoades, A comparison of various definations of contractive mappings, Proc. Amer. Math. Soc., 226 (1977), 257-290. https://doi.org/10.1090/S0002-9947-1977-0433430-4
  18. B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis, 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
  19. A.F. Roldan Lopez-de-Hierro, E. Karapnar, C. Roldan-Lopez-de-Hierro and J. Martnez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345-355. https://doi.org/10.1016/j.cam.2014.07.011
  20. X.J. Yang, D. Baleanu and H.M. Srivastava, Local Fractional Integral Transforms and Their Applications, Published by Elsevier Ltd. (2016).
  21. Q. Zhang and Y. Song, Fixed point theory for generalized φ-weakly contraction, Appl. Math. Letts., 22 (2009), 75-78. https://doi.org/10.1016/j.aml.2008.02.007