A New Iterative Algorithms for Finite Family of Resolvent Operators

Authors

Mustafa Dawood Talak Alobadi Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad , Baghdad, Iraq https://orcid.org/0009-0001-9713-9804
Zena Hussein Maibed Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad , Baghdad, Iraq https://orcid.org/0000-0002-3369-7313

DOI:

https://doi.org/10.30526/37.2.3301

Keywords:

Nonexpansive Mapping, Metric Projection, Strictly Pseudo Contraction Mapping, Strongly Pseudo-Contractive, Fixed Point

Abstract

Depending on the needs and requirements of keeping up with the scientific procession, researchers tend to find new recurrence schemes or develop previous recurrence schemes that will help researchers reach the fixed point and solution of variational inequality. This article aims to provide novel approaches to finding a common fixed point of different types of important mappings and the set of zeros of maximal monotone operators. Also, we studied the weak and strong convergence of the proposed iterative method under some suitable conditions. To achieve this goal, we will introduce a new technical method of resolvent operators and metric projection using different types of function sequences, including sequence of maximal monotone operators, sequence of k-strictly pseudo-contractive mappings, and sequence of non-expansive mappings defined on nonempty convex- closed subset of Hilbert space.

Author Biography

Mustafa Dawood Talak Alobadi, Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad , Baghdad, Iraq

.

References

Browder, F.E. Fixed-Point Theorems for Noncompact Mappings in Hilbert Space. Proc. Natl. Acad. Sci. 1965, 53, 1272–1276. DOI: https://doi.org/10.1073/pnas.53.6.1272

Browder, F.E. Convergence of Approximants to Fixed Points of Nonexpansive Nonlinear Mappings in Banach Spaces. Arch. Ration. Mech. Anal. 1967, 24, 82–90.

Browder, F.E.; Petryshyn, W. V Construction of Fixed Points of Nonlinear Mappings in Hilbert Space. J. Math. Anal. Appl. 1967, 20, 197–228. DOI: https://doi.org/10.1016/0022-247X(67)90085-6.

Mann, W.R. Mean Value Methods in Iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. DOI: http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3

Halpern, B. Fixed Points of Nonexpanding Maps. 1967. DOI: http://dx.doi.org/10.1090/S0002-9904-1967-11864-0

Goebel, K.; Kirk, W.A. Topics in Metric Fixed Point Theory; Cambridge university press, 1990; ISBN 0521382890.

Goebel, K.; Simeon, R. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings; Dekker, 1984; ISBN 0824772237.

Bauschke, H.H. The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Space. J. Math. Anal. Appl. 1996, 202, 150–159. DOI : https://doi.org/10.1006/jmaa.1996.0308.

Xu, H.-K. Iterative Algorithms for Nonlinear Operators. J. London Math. Soc. 2002, 66, 240–256.DOI: http://dx.doi.org/10.1112/S0024610702003332

Xu, H.K. Remarks on an Iterative Method for Nonexpansive Mappings. Comm. Appl. Nonlinear Anal 2003, 10, 67–75. DOI: https://doi.org/10.1016/j.jmaa.2005.11.057

Byrne, C. A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image Reconstruction. Inverse Probl. 2003, 20, 103. DOI: http://dx.doi.org/10.1088/0266-5611/20/1/006

Kim, T.-H.; Xu, H.-K. Strong Convergence of Modified Mann Iterations. Nonlinear Anal. Theory, Methods Appl. 2005, 61, 51–60.DOI: https://doi.org/10.1016/j.na.2004.11.011

Maibed, Z.H. Common Fixed Point Problem for Classes of Nonlinear Maps in Hilbert Space. In Proceedings of the IOP Conference Series: Materials Science and Engineering; IOP Publishing, 2020; Vol. 871, p. 12037. DOI: https://doi.org/10.1088/1757-899X/871/1/012037

Kim,T.-H.;Xu, H.-K. Strong Convergence of Modified Mann Iterations for Asymptotically Nonexpansive Mappings and Semigroups. Nonlinear Anal. Theory, Methods Appl. 2006, 64, 1140–1152. DOI: https://doi.org/10.1016/j.na.2005.05.059 .

Xu, H.-K. Strong Convergence of an Iterative Method for Nonexpansive and Accretive Operators. J. Math. Anal. Appl. 2006, 314, 631–643. DOI: https://doi.org/10.1016/j.jmaa.2005.04.082

Maibed, Z.H.; Thajil, A.Q. Zenali Iteration Method For Approximating Fixed Point of A Î´ ZA-Quasi Contractive Mappings. Ibn AL-Haitham J. Pure Appl. Sci. 2021, 34, 78–92.DOI: https://doi.org/10.30526/34.4.2705

Hamed, M.M.; Jamil, Z.Z. Stability And Data Dependence Results For The Mann Iteration Schemes on N-Banach Space. Iraqi J. Sci. 2020, 1456–1460. DOI: https://doi.org/10.24996/ijs.2020.61.6.25

Abed, S.S.; Ali, R.S.A. Fixed Point for Asymptotically Non-Expansive Mappings in 2-Banach Space. Ibn AL-Haitham J. Pure Appl. Sci. 2017, 27, 343–350. DOI https://doi.org/10.1006/jmaa.2001.7649

Abed, S.S.; Maibed, Z.H. Proximal Schemes by Family of Szl–Widering Mappings. In Proceedings of the IOP Conference Series: Materials Science and Engineering; IOP Publishing, 2019; Vol. 571, p. 12006. DOI: https://doi.org/10.1088/issn.1757-899X .

Jamil, Z.Z.; Abed, M.B. On a Modified SP-Iterative Scheme for Approximating Fixed Point of a Contraction Mapping. Iraqi J. Sci. 2015, 56, 3230–3239. DOI: https://ijs.uobaghdad.edu.iq/index.php/eijs/article/view/9429

Maibed, Z.H.; Thajil, A.Q. Equivalence of Some Iterations for Class of Quasi Contractive Mappings. In Proceedings of the Journal of Physics: Conference Series; IOP Publishing, 2021; Vol. 1879, p. 22115. DOI : http://dx.doi.org/10.1088/1742-6596/1879/2/022115

Reich, S. Weak Convergence Theorems for Nonexpansive Mappings in Banach Spaces. J.Math. Anal. Appl 1979, 67, 274–276. DOI:https://doi.org/10.1016/0022-247X(79)90024-6

A, Z.H.M. and A.M. On the Convergence of New Iteration Schems by Resolvent ZA-Jungck Mapping. J. Interdiscip. Math. to Appear 2022. DOI : https://doi.org/10.30526/36.1.2917 .

T. Suzuki, W.T. Weak and Strong Convergence Theorems for Nonexpansive Mapping in Banach Spaces. Nonlinear Anal. 2001, 47, 2805–2815. http://dx.doi.org/10.1016/S0362-546X(01)00399-6

Nakajo, K.; Takahashi, W. Strong Convergence Theorems for Nonexpansive Mappings and Nonexpansive Semigroups. J. Math. Anal. Appl. 2003, 279, 372–379. DOI: http://dx.doi.org/10.1016/S0022-247X(02)00458-4 .

Takahashi, W.; Toyoda, M. Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings. J. Optim. Theory Appl. 2003, 118, 417–428. DOI : http://dx.doi.org/10.1023/A:1025407607560 .

Chen, R.; He, H. Viscosity Approximation of Common Fixed Points of Nonexpansive Semigroups in Banach Space. Appl. Math. Lett. 2007, 20, 751–757. DOI : DOI:10.1007/s10013-013-0033-3 .

Marino, G.; Xu, H.-K. Weak and Strong Convergence Theorems for Strict Pseudo-Contractions in Hilbert Spaces. J. Math. Anal. Appl. 2007, 329, 336–346.DOI: https://doi.org/10.1016/j.jmaa.2006.06.055 .

Rhoades, B.E. Fixed Point Iterations Using Infinite Matrices. Trans. Am. Math. Soc. 1974, 196, 161–176.DOI: http://dx.doi.org/10.2307/1997020 .

Acedo, G.L.; Xu, H.-K. Iterative Methods for Strict Pseudo-Contractions in Hilbert Spaces. Nonlinear Anal. Theory, Methods Appl. 2007, 67, 2258–2271.DOI : http://dx.doi.org/10.1016/j.na.2006.08.036 .

Genel, A.; Lindenstrauss, J. An Example Concerning Fixed Points. Isr. J. Math. 1975, 22, 81–86. https://doi.org/10.1007/BF02757276 .

Güler, O. On the Convergence of the Proximal Point Algorithm for Convex Minimization. SIAM J. Control Optim. 1991, 29, 403–419. DOI : http://dx.doi.org/10.1137/0329022

Liu, L.-S. Ishikawa and Mann Iterative Process with Errors for Nonlinear Strongly Accretive Mappings in Banach Spaces. J. Math. Anal. Appl. 1995, 194, 114–125. https://doi.org/10.1006/jmaa.1998.5987