Title: An Extended S-Iteration Scheme for G-Contractive Type Mappings in b-Metric Spaces with Graph
Author(s): Nilakshi Goswami, Nehjamang Haokip, Vishnu Narayan Mishra
Pages: 33-49
Cite as:
Nilakshi Goswami, Nehjamang Haokip, Vishnu Narayan Mishra, An Extended S-Iteration Scheme for G-Contractive Type Mappings in b-Metric Spaces with Graph, Int. J. Anal. Appl., 18 (1) (2020), 33-49.


In this paper, we introduce an extended S-iteration scheme for G-contractive type mappings and prove ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. We also give a numerical example in support of our result and compare the convergence rate between the studied iteration and the modified S-iteration.

Full Text: PDF



  1. A. A. Abdelhakim, A convexity of functions on convex metric spaces of Takahashi and applications, J. Egypt. Math. Soc. 24 (2016), 348–354. Google Scholar

  2. R. P. Agarwal, D. ORegan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal. 8 (2007), 61–79. Google Scholar

  3. V. Berinde, Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002. Google Scholar

  4. R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole Cengage Learning, 9th edition, Boston, 2010. Google Scholar

  5. S. Czerwik, Contraction mappings in b-metric spaces, Acta. Math. Inform. Univ. Ostra. 1 (1993), 5–11. Google Scholar

  6. D. Das and N. Goswami, Some fixed point theorems on the sum and product of operators in tensor product spaces, Int. J. Pure Appl. Math. 109 (2016), no.3, 651–663. Google Scholar

  7. D. Das, N. Goswami and V.N. Mishra, Some results on fixed point theorems in Banach algebras, Int. J. Anal. Appl. 13 (2017), no. 1, 32–40. Google Scholar

  8. D. Das, N. Goswami and V.N. Mishra, Some results on the projective cone normed tensor product spaces over Banach algebras, (online available) Bol. Soc. Paran. Mat. (3s.) 38 (2020), no. 1, 197–221. Google Scholar

  9. Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis (2014), Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India. Google Scholar

  10. Deepmala and H. K. Pathak, A study on some problems on existence of solutions for nonlinear functional-integral equations, Acta Math. Sci. 33 B(5) (2013), 1305–1313. Google Scholar

  11. H. Fukhar-ud-din, Existence and approximation of fixed points in convex metric spaces, Carpathian J. Math. 30 (2014), 175–185. Google Scholar

  12. H. Fukhar-ud-din, One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat. 44 (2015), 1023–1031. Google Scholar

  13. N. Goswami, N. Haokip and V. N. Mishra, F-contractive type mappings in b-metric spaces and some related fixed point results, Fixed Point Theory and Applications 2019 (2019), 13. Google Scholar

  14. N. Haokip and N. Goswami, Some fixed point theorems for generalized Kannan type mappings in b-metric spaces, Proyecciones (Antofagasta) 38 (4) (2019), 763-782. Google Scholar

  15. S. Ishikawa, Fixed Point by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147–150. Google Scholar

  16. G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), 261–263. Google Scholar

  17. W. A. Kirk and W. O. Ray, A note on Lipschitzian mappings in convex metric spaces, Canad. Math. Bull. 20 (1977), 463–466. Google Scholar

  18. T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179 182. Google Scholar

  19. X. Liu, M. Zhou, L.N. Mishra, V.N. Mishra and B. Damjanovi´c, Common fixed point theorem of six self-mappings in Menger spaces using (CLRST ) property, Open Math. 16 (2018), 1423–1434. Google Scholar

  20. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 44 (1953), 506–510. Google Scholar

  21. L. N. Mishra, H.M. Srivastava and M. Sen, On existence results for some nonlinear functional-integral equations in Banach algebra with applications, Int. J. Anal. Appl., 11 (2016), 1–10. Google Scholar

  22. L. N. Mishra, K. Jyoti, A. Rani and Vandana, Fixed point theorems with digital contractions image processing, Nonlinear Sci. Lett. A, 9 (2018), 104–115. Google Scholar

  23. L. N. Mishra, S. K. Tiwari, V. N. Mishra and I. A. Khan, Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, J. Funct. Spaces, 2015 (2015), Article ID 960827, 8 pages. Google Scholar

  24. L. N. Mishra, S. K. Tiwari and V. N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, J. Appl. Anal. Comput. 5 (2015), 600–612. Google Scholar

  25. M. O. Olatinwo and C. O. Imoru, Some convergence results for the Jungck-Mann and Jungck-Ishikawa processes in the class of generalized Zamfirescu operators, Acta Math. Univ. Comenianae LXXVII (2008), 299–304. Google Scholar

  26. H. K. Pathak and Deepmala, Common fixed point theorems for PD-operator pairs under Relaxed conditions with applications, J. Comput. Appl. Math. 239 (2013), 103–113. Google Scholar

  27. B. Patir, N. Goswami and L. N. Mishra, Fixed point theorems in fuzzy metric spaces for mappings with some contractive type conditions, Korean J. Math. 26 (2018), 307–326. Google Scholar

  28. B. Patir, N. Goswami and V. N. Mishra, Some results on fixed point theory for a class of generalized nonexpansive mappings, Fixed Point Theory Appl. 2018 (2018), 19. Google Scholar

  29. T. Rasham, A. Shoaib, C. Park and M. Arshad, Fixed Point Results for a Pair of Multi Dominated Mappings on a Smallest Subset in K-Sequentially Dislocated quasi Metric Space with Application, J. Comput. Anal. Appl., 25 (5) (2018), 975–986. Google Scholar

  30. T. Rasham, A. Shoaib, N. Hussain, B. S. Alamri and M. Arshad, Multivalued Fixed Point Results in Dislocated b-Metric Spaces with Application to the System of Nonlinear Integral Equations, Symmetry, 11 (1) (2019), 40. Google Scholar

  31. T. Rasham, A. Shoaib, B. S. Alamri and M. Arshad, Fixed Point Results for Multivalued Contractive Mappings Endowed With Graphic Structure, J. Math. 2018 (2018) Article ID 5816364, 8 pages. Google Scholar

  32. T. Rasham, A. Shoaib, B. S. Alamri and M. Arshad, Multivalued fixed point results for new generalized F-Dominted contractive mappings on dislocated metric space with application, J. Funct. Spaces, 2018 (2018), Article ID 4808764, 12 pages. Google Scholar

  33. T. Rasham, A. Shoaib, B. A. S. Alamri, A. Asif and M. Arshad, Fixed Point Results for α ∗-ψ-Dominated Multivalued Contractive Mappings Endowed with Graphic Structure, Mathematics 7 (2019), 307. Google Scholar

  34. A. Razani and M. Bagherboum, Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces, Fixed Point Theory Appl. 2013 (2013), 331. Google Scholar

  35. B. E. Rhoades, Comments on two fixed point iteration method, J. Math. Anal. Appl. 56 (1976), 741–750. Google Scholar

  36. G. S. Saluja, Strong and ∆-convergence of modified two-step iterations for nearly asymptotically nonexpansive mappings in hyperbolic spaces, Int. J. Anal. Appl. 8 (2015), 39–52. Google Scholar

  37. S. Shahzad and R. Al-Dubiban, Approximating common fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J. 13 (2006), 529–537. Google Scholar

  38. T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8 (1996), 197–203. Google Scholar

  39. S. L. Singh, C. Bhatnagar and S. N. Mishra, Stability of Jungck-Type Iterative Procedures, Int. J. Math. Math. Sci. 19 (2005), 3035–3043. Google Scholar

  40. S. Suparatulatorn, W. Cholamjiak and S. Suantai, A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs, Numer. Algorithms 77 (2017), 479–490. Google Scholar

  41. W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970), 142–149. Google Scholar


Copyright © 2020 IJAA, unless otherwise stated.