Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

Vasile Berinde

doi:10.2478/awutm-2018-0013

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Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 56 (2018): Issue 2 (December 2018)

By: Vasile Berinde

Open Access

|Apr 2020

Abstract

In this paper, we introduce and study the class of enriched strictly pseudocontractive mappings in Hilbert spaces and extend some convergence theorems, i.e., Theorem 12 in [Brow-der, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 3.1 in [Marino, G., Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346], from the class of strictly pseudocontractive mappings to that of enriched strictly pseudocontractive mappings and thus include many other important related results from literature as particular cases.

References

[1] V. Berinde, Approximating fixed points of Lipschitzian generalized pseudo-contractions, in: Mathematics & Mathematics education (Bethlehem, 2000), World Sci. Publ., River Edge, NJ, 2002, 73-81.10.1142/9789812778390_0006
Search in Google Scholar Back to article
[2] V. Berinde, Iterative approximation of fixed points for pseudo-contractive operators, Semin. Fixed Point Theory Cluj-Napoca3 (2002), 209-215.
Search in Google Scholar Back to article
[3] V. Berinde, Iterative approximation of fixed points, Second edition. Lecture Notes in Mathematics. 1912, Springer, 2007.10.1109/SYNASC.2007.49
Search in Google Scholar Back to article
[4] V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math.35 (2019), 293-304.10.37193/CJM.2019.03.04
Search in Google Scholar Back to article
[5] V. Berinde, Iterative approximation of fixed points of enriched quasi-nonexpansive mappings and applications, (submitted).
Search in Google Scholar Back to article
[6] V. Berinde, Approximating fixed points of enriched demicontractive mappings by Krasnoselskij-Mann iteration in Hilbert spaces, (submitted).
Search in Google Scholar Back to article
[7] V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij-Mann iteration in Banach spaces, (submitted).
Search in Google Scholar Back to article
[8] V. Berinde, M. Păcurar, Fixed point theorems of Kannan type mappings with applications to split feasibility and variational inequality problems, (submitted).
Search in Google Scholar Back to article
[9] V. Berinde, M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces, (submitted).
Search in Google Scholar Back to article
[10] V. Berinde, M. Păcurar, Approximating fixed points of enriched Chatterjea type mappings, (submitted).
Search in Google Scholar Back to article
[11] V. Berinde, M. Petric, Fixed point theorems for cyclic non-self single-valued almost contractions, Carpathian J. Math.31 (2015), 289-296.10.37193/CJM.2015.03.04
Search in Google Scholar Back to article
[12] V. Berinde, A. R. Khan, M. Păcurar, Coupled solutions for a bivariate weakly nonexpansive operator by iterations, Fixed Point Theory Appl. (2014), 12 pp.10.1186/1687-1812-2014-149
Search in Google Scholar Back to article
[13] M. Bethke, Approximation of fixed points of strictly pseudocontractive operators (German), Wiss. Z. Pädagog. Hochsch. ”Liselotte Herrmann” Güstrow Math.-Natur. Fak.27 (1989), 263-270.
Search in Google Scholar Back to article
[14] F. E. Browder, W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc.72 (1966), 571-575.10.1090/S0002-9904-1966-11544-6
Search in Google Scholar Back to article
[15] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl.20 (1967), 197-228.10.1016/0022-247X(67)90085-6
Search in Google Scholar Back to article
[16] C. E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, 1965, Springer, 2009.10.1007/978-1-84882-190-3
Search in Google Scholar Back to article
[17] L. Deng, On Chidume’s open questions, J. Math. Anal. Appl.174 (1993), 441-449.10.1006/jmaa.1993.1129
Search in Google Scholar Back to article
[18] H. Fukhar-ud-din, V. Berinde, Fixed point iterations for Prešić-Kannan nonexpansive mappings in product convex metric spaces, Acta Univ. Sapientiae Math.10 (2018), 56-69.10.2478/ausm-2018-0005
Search in Google Scholar Back to article
[19] H. Fukhar-ud-din, V. Berinde, A. R. Khan, Fixed point approximation of Prešić nonexpansive mappings in product of CAT (0) spaces, Carpathian J. Math.32 (2016), 315-322.10.1186/s13663-015-0483-2
Search in Google Scholar Back to article
[20] D. Göhde, Nichtexpansive und pseudokontraktive Abbildungen sternförmiger Mengen im Hilbertraum, Beiträge Anal.9 (1976), 23-25.
Search in Google Scholar Back to article
[21] G. G. Johnson, Fixed points by mean value iterations, Proc. Amer. Math. Soc.34 (1972), 193-194.10.1090/S0002-9939-1972-0291918-4
Search in Google Scholar Back to article
[22] A. Khan, A. H. Siddiqi, Nonexpansive type mappings in 2-pre-Hilbert spaces, Math. Sem. Notes Kobe Univ.11 (1983), 331-334.
Search in Google Scholar Back to article
[23] M. A. Krasnosel’skiǐ, Two remarks about the method of successive approximations (Russian), Uspehi Mat. Nauk (N.S.)10 (1955), 123-127.
Search in Google Scholar Back to article
[24] G. López, V. Martín-Márquez, H.-K. Xu, Halpern’s iteration for nonexpansive mappings, Nonlinear analysis and optimization I. Nonlinear analysis513 (2010), 211-231.10.1090/conm/513/10085
Search in Google Scholar Back to article
[25] G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl.329 (2007), 336-346.10.1016/j.jmaa.2006.06.055
Search in Google Scholar Back to article
[26] J. J. Moloney, Some fixed point theorems, Glas. Mat. Ser. III24 (44) (1989), 59-76.
Search in Google Scholar Back to article
[27] R. N. Mukherjee, T. Som, Construction of fixed points of strictly pseudocontractive mappings in a generalized Hilbert space and related applications, Indian J. Pure Appl. Math.15 (1984), 1183-1189.
Search in Google Scholar Back to article
[28] M. O. Osilike, A. Udomene, Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl.256 (2001), 431-445.10.1006/jmaa.2000.7257
Search in Google Scholar Back to article
[29] J. A. Park, Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces, J. Korean Math. Soc.31 (1994), 333-337.
Search in Google Scholar Back to article
[30] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl.14 (1966), 276-284.10.1016/0022-247X(66)90027-8
Search in Google Scholar Back to article
[31] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl.67 (1979), 274-276.10.1016/0022-247X(79)90024-6
Search in Google Scholar Back to article
[32] B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc.196 (1974), 161-176.10.1090/S0002-9947-1974-0348565-1
Search in Google Scholar Back to article
[33] J. Schu, Iterative construction of fixed points of strictly pseudocontractive mappings, Appl. Anal.40 (1991), 67-72.10.1080/00036819108839994
Search in Google Scholar Back to article
[34] J. Schu, On a theorem of C. E. Chidume concerning the iterative approximation of fixed points, Math. Nachr.153 (1991), 313-319.10.1002/mana.19911530127
Search in Google Scholar Back to article
[35] X. L. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proc. Amer. Math. Soc.113 (1991), 727-731.10.1090/S0002-9939-1991-1086345-8
Search in Google Scholar Back to article
[36] Z. Zhou, Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal.69 (2008), 456-462.10.1016/j.na.2007.05.032
Search in Google Scholar Back to article