Iterative scheme generating method for demiclosed and 2-demiclosed mappings
DOI:
https://doi.org/10.66147/lnaa.20253250Keywords:
Iterative scheme generating method, common fixed point, demiclosed mapping, 2-demiclosed mappingAbstract
This study extends iterative scheme generating methods (ISGMs) to apply to general classes of mappings. In contrast to previous studies, this study examines quasi-nonexpansive and 2-demiclosed mappings, in addition to quasi-nonexpansive and demiclosed mappings. By doing so, we can consider a class of mappings called normally 2-generalized hybrid mappings. For these extended classes of mappings, we develop the ISGMs that generate various iterative schemes to locate common fixed points.
References
[1] K. Aoyama, S. Iemoto, F. Kohsaka, and W. Takahashi, Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 11(2) (2010), 335–343.
[2] M. Asadi and E. Karapinar, Coincidence Point Theorem on Hilbert Spaces via Weak Ekeland Variational Principle and Application to Boundary Value Problem, Thai J. Math., 19(1), 1–7.
[3] M. Hojo, Attractive point and mean convergence theorems for normally generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 18(12) (2017), 2209–2120.
[4] M. Hojo, S. Takahashi and W. Takahashi, Attractive point and ergodic theorems for two nonlinear mappings in Hilbert spaces, Linear Nonlinear Anal. 3(2) (2017), 275–286.
[5] M. Hojo, W. Takahashi and I. Termwuttipong, Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces, Nonlinear Anal. 75(4) (2012), 2166–2176. DOI: https://doi.org/10.1016/j.na.2011.10.017
[6] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. DOI: https://doi.org/10.1090/S0002-9939-1974-0336469-5
[7] S. Itoh and W. Takahashi, The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. Math. 79(2) (1978), 493–508. DOI: https://doi.org/10.2140/pjm.1978.79.493
[8] P. Kocourek, W. Takahashi and J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert Spaces, Taiwanese J. Math. 14(6) (2010), 2497–2511. DOI: https://doi.org/10.11650/twjm/1500406086
[9] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. 91(2) (2008), 166–177. DOI: https://doi.org/10.1007/s00013-008-2545-8
[10] A. Kondo, Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces, Adv. Oper. Theory 7(3) Article number: 26, (2022). DOI: https://doi.org/10.1007/s43036-022-00190-5
[11] A. Kondo, Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces, Demonstr. Math. 55 (2022), 752–759. DOI: https://doi.org/10.1515/dema-2022-0167
[12] A. Kondo, Strong approximation using hybrid methods to find common fixed points of noncommutative nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal. 23(1) (2022), 33–58. DOI: https://doi.org/10.1007/s12190-021-01527-8
[13] A. Kondo, A generalization of the common fixed point theorem for normally 2-generalized hybrid mappings in Hilbert spaces, Filomat, 37(26) (2023), 9051–9062. DOI: https://doi.org/10.2298/FIL2326051K
[14] A. Kondo, Ishikawa type mean convergence theorems for finding common fixed points of nonlinear mappings in Hilbert spaces, Rend. Circ. Mat. Palermo, II. Ser, 72(2) (2023), 1417–1435. DOI: https://doi.org/10.1007/s12215-022-00742-x
[15] A. Kondo, Strong convergence theorems by Martinez-Yanes–Xu projection method for mean-demiclosed mappings in Hilbert spaces, Rendiconti di Mat. e delle Sue Appl. 44(1-2) (2023), 27–51.
[16] A. Kondo, Strong convergence to common fixed points using Ishikawa and hybrid methods for mean-demiclosed mappings in Hilbert spaces, Math. Model. Anal., 28(2) (2023), 285–307. DOI: https://doi.org/10.3846/mma.2023.15843
[17] A. Kondo, Halpern-type strong convergence theorems using a multi-step mean-valued iterative method in Hilbert spaces, J. Nonlinear Convex Anal. 25(11) (2024), 2703–2715.
[18] A. Kondo, On the iterative scheme generating methods using mean-valued sequences, Carpathian J. Math. 40(3) (2024), 819–840. DOI: https://doi.org/10.37193/CJM.2024.03.18
[19] A. Kondo, Iterative scheme generating method beyond Ishikawa iterative method, Math. Ann. 391(2) (2025), 2007–2028. DOI: https://doi.org/10.1007/s00208-024-02977-8
[20] A. Kondo and W. Takahashi, Attractive point and weak convergence theorems for normally N -generalized hybrid mappings in Hilbert spaces, Linear Nonlinear Anal. 3 (2017), 297–310.
[21] A. Kondo and W. Takahashi, Approximation of a common attractive point of noncommutative normally 2-generalized hybrid mappings in Hilbert spaces, Linear Nonlinear Anal. 5(2) (2019), 279–297.
[22] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. DOI: https://doi.org/10.1090/S0002-9939-1953-0054846-3
[23] T. Maruyama, W. Takahashi and M. Yao, Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal. 12(1) (2011), 185–197. DOI: https://doi.org/10.1186/1687-1812-2011-51
[24] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(1) (2000), 217–229. DOI: https://doi.org/10.1006/jmaa.2000.7042
[25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67(2) (1979), 274–276. DOI: https://doi.org/10.1016/0022-247X(79)90024-6
[26] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11(1) (2010), 79–88.
[27] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118(2) (2003), 417–428. DOI: https://doi.org/10.1023/A:1025407607560
[28] W. Takahashi, N.-C. Wong, and J.-C. Yao, Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal 13(4) (2012), 745–757.
[29] H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl. 62(11) (2011), 4007–4014. DOI: https://doi.org/10.1016/j.camwa.2011.09.018
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Atsumasa Kondo

This work is licensed under a Creative Commons Attribution 4.0 International License.