Some fixed point results for nonexpansive and G-nonexpansive mappings in Hilbert spaces
Abstract
This paper is concerned with the convergence of a three-step Picard iteration scheme introduced by Javid et al., for nonexpansive and G− nonexpansive mappings in Hilbert spaces endowed with binary relation under some assumptions. After that same results are obtained by replacing binary relation with directed graph. Main results are justified with some numerical examples. Here, an application of fixed point theory is discussed to obtain solution of system of equations and fastness of a three-step Picard iteration scheme with other known iteration schemes.
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References
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19. Mann W.R., Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506-510.
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25. Schu J., Weak and strong convergence of fixed point of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), No. 1, 153-159.
26. Phuengrattana W., Suantai S., On the rate of convergence of Mann, Ishikawa, Noor and SP− iterations for continuous functions on an arbitrary interval, J. Comp. Appl.Math., 235 (2011), 3006–3014.
27. Tiammee J,, Kaewkhao A., Suantai S., On Browder’s convergence theorem and Halpern iteration process for G− nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory and Appl., 187 (2015), 1-12.
28. Thakur D., Thakur B.S., Postolache M., New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl., 2014:328 (2014), 1-15.
29. Tripak O., Common fixed points of G− nonexpansive mappings on Banach spaces with a graph, Fixed Point Theory Appl., 2016:87 (2016), 1-8.
30. Vetro F., Fixed point results for nonexpansive mappings on metric spaces, Filomat, 29 (2015), No. 9, 2011–2020
2. Alfuraidan M.R., Khamsi M.A., Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph, Fixed Point Theory and Appl., 44 (2015), 1-10.
3. Faeem A., Javid A., et al., Approximation of fixed points and the solution of a nonlinear integral equation, Nonlinear Functional Analysis and Applications, 26 (2021), No. 5, 869-885.
4. Banach S., Sur les operations dans ensembles abstraits et leur application aux equations integrales, Fundamenta Math ematicae, 3 (1922), 133-181.
5. Browder F.E., Nonexpansive nonlinear operators in Banach space, Proc. Nat. Sci., USA, 54 (1965), 1041-1044.
6. Browder F.E., Petryshyn W.V., Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Math ematical Analysis and Applications, 20 (1967), No. 2, 197–228.
7. Byrne C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), No. 1, 103–120.
8. Chugh R., Kumar V. Kumar S., Strong Convergence of a new three-step iterative scheme in Banach spaces, Amer. J. Comp. Math., 2 (2012), 345–357.
9. Edelstein M., On nonexpansive mappings, Proc. Amer. Math. Soc., 15 (1964), 689-695.
10. Echenique F.A., Short and constructive proof of Tarski’s fixed point theorem, Internat. J. Game Theory , 33 (2005), No. 2, 215-218.
11. Espinola R., Kirk W.A., Fixed point theorems in R− trees with applications to graph theory, Topology Appl., 153 (2006), 1046-1055.
12. Ishikawa S., Fixed point by new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
13. Jachymski J.,The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), No. 4, 1359-1373.
14. Khan S.H., A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013 (2013), 1–10.
15. Kim J.K., Dashputre S., Lim W.H., Approximation of fixed points for multi-valued nonexpansive mappings in Banach space, Global j. pure Appl. Math., 12 (2016), No. 6, 4901-4912.
16. Kirk W.A., Ray W.O., Fixed point theorems for mappings define on unbounded sets in Banach spaces, Studia mathematica, 114 (1979), 127-138.
17. Krasnoselskii M.A., Two remarks on the method of successive approximations, Uspehi Mat. Nauk., 10 (1955), No. 1(63), 123-127.
18. Khamsi M.A, Riech S., Nonexpansive mappings and semigroups in hyperconvex spaces, Mathematica Japonica, 35 (1990), 467-471.
19. Mann W.R., Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506-510.
20. Noor M.A., New approximation schemes for general variation inequality, J. Math. Anal. Appl., 251 (2000), 221-229.
21. Picard E., Memoire sur la theorie des equations aux derives partielles et la methode des approximations successives, J. Math. Pures et Appl., 6 (1890), 145-210.
22. Podilchuk C.I., Mammone R.J., Image recovery by convex projections using a least-squares constraint, Journal of the Optical Society of America, 7 (1990), No. 3, 517–521.
23. Reich S., Shafrir I., The asymptotic behaviour of firmly nonexpansive mappings, Proc. Amer. Math. Soc., 101 (1987), 246-250.
24. Sahu D.R., Applications of the S− iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory, 12 (2011), 187–204.
25. Schu J., Weak and strong convergence of fixed point of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), No. 1, 153-159.
26. Phuengrattana W., Suantai S., On the rate of convergence of Mann, Ishikawa, Noor and SP− iterations for continuous functions on an arbitrary interval, J. Comp. Appl.Math., 235 (2011), 3006–3014.
27. Tiammee J,, Kaewkhao A., Suantai S., On Browder’s convergence theorem and Halpern iteration process for G− nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory and Appl., 187 (2015), 1-12.
28. Thakur D., Thakur B.S., Postolache M., New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl., 2014:328 (2014), 1-15.
29. Tripak O., Common fixed points of G− nonexpansive mappings on Banach spaces with a graph, Fixed Point Theory Appl., 2016:87 (2016), 1-8.
30. Vetro F., Fixed point results for nonexpansive mappings on metric spaces, Filomat, 29 (2015), No. 9, 2011–2020
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2025-09-17
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