Nonlinear Functional Analysis and Applications

  • 제26권2호
  • /
  • Pages.303-322
  • /
  • 2021
  • /
  • 1229-1595(pISSN)
  • /
  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF SOLUTION OF DIFFERENTIAL EQUATION VIA FIXED POINT IN COMPLEX VALUED b-METRIC SPACES

  • Mebawondu, A.A. (School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)) ;
  • Abass, H.A. (School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)) ;
  • Aibinu, M.O. (Institute for Systems Science & KZN e-Skills CoLab Durban University of Technology, DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)) ;
  • Narain, O.K. (School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal)
  • 투고 : 2020.09.14
  • 심사 : 2021.02.05
  • 발행 : 2021.06.15
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The concepts of new classes of mappings are introduced in the spaces which are more general space than the usual metric spaces. The existence and uniqueness of common fixed points and fixed point results are established in the setting of complete complex valued b-metric spaces. An illustration is given by establishing the existence of solution of periodic differential equations in the framework of a complete complex valued b-metric spaces.

키워드

과제정보

The financial support from Department of Science and Innovation and National Research Foundation, Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF CoE-MaSS), Republic of South Africa is acknowledged with thanks by the first and second authors for the Doctoral Bursary and by the third author for the Postdoctoral Fellowship. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

참고문헌

  1. C.T. Aage and J.N .Salunke, Fixed points for weak contractions in G-metric spaces, Appl. Math. E-Notes, 12 (2012), 23-28.
  2. M.O. Aibinu and C.E. Chidume, Existence and uniqueness of solutions of integral equations of Hammerstein type, M.Sc Thesis, African University of Science and Technology, Abuja, Nigeria, 2013.
  3. M.O. Aibinu and P. Pillay, Iterative algorithms for solutions of nonlinear equations in Banach spaces, Ph.D Thesis, University of KwaZulu-Natal, South Africa, 2019.
  4. A.H. Ansari, Note on ϕ-ψ-contractive type mappings and related fixed point, in Proceedings of the 2nd Regional Conference on Mathematics and Applications, Payame Noor University, Tonekabon, Iran, (2014), 377-380.
  5. H. Ayadi, T. Wongyat and W. Sintunavaret, On new evolutions of Ris result via wdistance and the study on the solution for nonlinear integral equations and fractional differential equations, Advances in Differences Equations, 2018:132. (2018). https://doi.org/10.1186/s13662-018-1590-2
  6. H. Ayadi, E. Karapinar and W. Shatanawi, Coupled fixed point results for (ψ, φ)-weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., 62 (2011), 4449-4460. https://doi.org/10.1016/j.camwa.2011.10.021
  7. A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32(3) (2011), 243-253. https://doi.org/10.1080/01630563.2011.533046
  8. S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  9. M. Boriceanu, M. Bota and A. Petrusel, Mutivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367-377. https://doi.org/10.2478/s11533-010-0009-4
  10. Y.J. Cho, B.E. Rhoades, R. Saadait, B. Samet and W. Shatanawi, Nonlinear coupled fixed point theorems in ordered generalized metric space with integral type, Fixed point theory Appl., 2012:8 (2012).
  11. S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11.
  12. A.K. Dubey and M. Tripathi, Common Fixed Point Theorem in Complex Valued bmetric Space for Rational Contractions, J. Inf. Math. Sci., 7 (2015), 149-161.
  13. A.K. Dubey, U. Mishra and W.H. Lim, Generalized common fixed point results for three self-mappings in complex valued b-metric spaces, Nonlinear Funct. Anal. Appl., 24(2) (2019), 255-267.
  14. A.K. Dubey, U. Mishra and W.H. Lim, Some new fixed point theorems for generalized contractions involving rational expressions in complex valued b-metric spaces, Nonlinear Funct. Anal. Appl., 24(3) (2019), 477-483.
  15. G. Jungck and N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl., 325(2) (2007), 1003-1012. https://doi.org/10.1016/j.jmaa.2006.02.058
  16. F. Khojasteh, S. Shukla and S. Radenovi, A new approach to study fixed point theorems via simulation functions, Filomat, 29(6) (2015), 1189-1194. https://doi.org/10.2298/FIL1506189K
  17. P. Kumam, D. Gopal and L. Budhiyi, A new fixed point theorem under Suzuki type Z-contraction mappings, J. Math. Anal., 8(1) (2017), 113-119.
  18. X.L. Liu, A.H. Ansari, S. Chandok and S. Radenovic On some results in metric spaces using auxillary simulation functions via new functions, J. Comput. Anal. Appl., 24(6) (2018), 1103-1114.
  19. J.J. Nieto and R. Rodriguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation, Acta Math. Sin., 23 (2007), 2205-2212. https://doi.org/10.1007/s10114-005-0769-0
  20. B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47(4) (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
  21. A.F. Roldan Lopex de Hierro, E. Karapinar and D. O'Regan, 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
  22. J.K.P.R. Rao, P.R. Swamy and J.R. Prasad, A common fixed point theorem in complex valued b-metric spaces, Bull. Maths. Stat. res., 1 (2013), 1-8.
  23. J.R. Roshan, V. Parvaneh and Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 229-245. https://doi.org/10.22436/jnsa.007.04.01
  24. P. Salimi and V. Pasquale, A result of Suzuki type in partial G-metric spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 34(2) (2014), 274-284.
  25. B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014
  26. S.L. Singh, R. Kamal, M. De la Sen M. and R. Chugh, A fixed point theorem for generalized weak contractions, Filomat, 9(7) (2015), 1481-1490.
  27. W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, J. Fixed Point Theory Appl., 18 (2016), 397-416. DOI 10.1007/s11784-015-0276-6, 1-20.
  28. W. Sintunavarat, Y.J. Cho and P. Kumam, Urysohn integral equations approach by common fixed point in complex valued metric spaces, J. Adv. Difference Equ., 2013:49 ( 2013), 1-14. https://doi.org/10.1186/1687-1847-2013-1
  29. W. Sintunavarat and P. Kumam, Generalized common fixed point theorems in complex valued metric spaces and applications, J. Inequal. Appl., 2012:84 (2012), 1-12. https://doi.org/10.1186/1029-242X-2012-1
  30. W. Sintunavarat, Z.M. Bahadur and M. Sarwar, Common solution of Urysohn integral equations with the help of common fixed point results in complex valued metric spaces, Rev. R. Acad. Cienc. Exactas F's. Nat. Ser. A Math. RACSAM, 111(2) (2017), 531-545. https://doi.org/10.1007/s13398-016-0309-z
  31. W. Sintunavarat and A. Al-Rawashdeh, Common fixed point of almost generalized (ψ, φ)-contractive mappings in ordered metric space, Fixed point theory Appl., 2012;80 (2012).
  32. W. Sintunavarat, Some fixed point results for a generalized ψ-weak contraction mappings in orbitally metric space, Chaos Solitons Fractals, 45 (2012), 520-526. https://doi.org/10.1016/j.chaos.2012.01.015