##### Title: An Extended S-Iteration Scheme for G-Contractive Type Mappings in b-Metric Spaces with Graph

##### 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.#### Abstract

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.

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#### References

- A. A. Abdelhakim, A convexity of functions on convex metric spaces of Takahashi and applications, J. Egypt. Math. Soc. 24 (2016), 348–354.
- 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.
- V. Berinde, Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002.
- R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole Cengage Learning, 9th edition, Boston, 2010.
- S. Czerwik, Contraction mappings in b-metric spaces, Acta. Math. Inform. Univ. Ostra. 1 (1993), 5–11.
- 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.
- 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.
- 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.
- 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.
- 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.
- H. Fukhar-ud-din, Existence and approximation of fixed points in convex metric spaces, Carpathian J. Math. 30 (2014), 175–185.
- 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.
- 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.
- 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.
- S. Ishikawa, Fixed Point by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
- G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), 261–263.
- W. A. Kirk and W. O. Ray, A note on Lipschitzian mappings in convex metric spaces, Canad. Math. Bull. 20 (1977), 463–466.
- T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179 182.
- 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.
- W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 44 (1953), 506–510.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- B. E. Rhoades, Comments on two fixed point iteration method, J. Math. Anal. Appl. 56 (1976), 741–750.
- 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.
- S. Shahzad and R. Al-Dubiban, Approximating common fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J. 13 (2006), 529–537.
- T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8 (1996), 197–203.
- S. L. Singh, C. Bhatnagar and S. N. Mishra, Stability of Jungck-Type Iterative Procedures, Int. J. Math. Math. Sci. 19 (2005), 3035–3043.
- 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.
- W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970), 142–149.