A faster iterative scheme for common fixed points of $ G $-nonexpansive mappings via directed graphs: application in split feasibility problems

Maryam Iqbal; Afshan Batool; Aftab Hussain; Hamed Al-Sulami; Maryam Iqbal; Afshan Batool; Aftab Hussain; Hamed Al-Sulami

doi:10.3934/math.2024583

AIMS Mathematics

2024, Volume 9, Issue 5: 11941-11957. doi: 10.3934/math.2024583

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A faster iterative scheme for common fixed points of $ G $-nonexpansive mappings via directed graphs: application in split feasibility problems

1.
Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi, Pakistan
2.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 29 January 2024 Revised: 17 March 2024 Accepted: 21 March 2024 Published: 27 March 2024
MSC : 47H09, 47H10

We have suggested a new modified iterative scheme for approximating a common fixed point of two $ G $-nonexpansive mappings. Our approach was based on an iterative scheme in the context of Banach spaces via directed graphs. First, we proved a weak convergence theorem using the Opial's property of the underlying space. A weak convergence result without the Opial's property was also given. After this, we established several strong convergence theorems using various mild conditions. We also carried out some numerical simulations to examine the main techniques. Eventually, we obtained an application of our result to solve split feasibility problems (SFP) in the context of $ G $-nonexpansive mappings.
- $ K^{\ast} $-iteration,
- common fixed points,
- $ G $-nonexpansive mappings,
- convergence,
- condition (B),
- directed graph
Citation: Maryam Iqbal, Afshan Batool, Aftab Hussain, Hamed Al-Sulami. A faster iterative scheme for common fixed points of $ G $-nonexpansive mappings via directed graphs: application in split feasibility problems[J]. AIMS Mathematics, 2024, 9(5): 11941-11957. doi: 10.3934/math.2024583

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Abstract

We have suggested a new modified iterative scheme for approximating a common fixed point of two $ G $-nonexpansive mappings. Our approach was based on an iterative scheme in the context of Banach spaces via directed graphs. First, we proved a weak convergence theorem using the Opial's property of the underlying space. A weak convergence result without the Opial's property was also given. After this, we established several strong convergence theorems using various mild conditions. We also carried out some numerical simulations to examine the main techniques. Eventually, we obtained an application of our result to solve split feasibility problems (SFP) in the context of $ G $-nonexpansive mappings.

References

[1]	F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA, 53 (1965), 1272–1276. https://doi.org/10.1073/pnas.53.6.1272 doi: 10.1073/pnas.53.6.1272
[2]	W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Mon., 72 (1965), 1004–1006. https://doi.org/10.2307/2313345 doi: 10.2307/2313345
[3]	D. Göhde, Zum prinzip der kontraktiven abbildung, Math. Nachr., 30 (1965), 251–258.
[4]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA, 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[5]	E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6 (1890), 145–210.
[6]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
[7]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5 doi: 10.1090/S0002-9939-1974-0336469-5
[8]	R. P. Agarwal, D. O'Regon, D. R. Sahu, Iterative construction of fixed points of nearly asymtotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
[9]	K. Ullah, M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces, J. Linear Topol. Algebra, 7 (2018), 87–100.
[10]	C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[11]	A. Moussaoui, N. Hussain, S. Melliani, N. Hayel, M. Imdad, Fixed point results via extended $\mathcal{FZ}$-simulation functions in fuzzy metric spaces, J. Inequal. Appl., 2022 (2022), 69. https://doi.org/10.1186/s13660-022-02806-z doi: 10.1186/s13660-022-02806-z
[12]	A. Moussaoui, S. Radenović, S. Melliani, Fixed point theorems involving $\mathcal{FZ}$-$\vartheta_{f}$-contractions in $GV$-fuzzy metrics, Filomat, 6 (2024), 1973–1985.
[13]	A. Moussaoui, S. Melliani, S. Radenović, A nonlinear fuzzy contraction principle via control functions, Filomat, 6 (2024), 1963–1972.
[14]	M. Iqbal, A. Ali, H. A. Sulami, A. Hussain, Iterative stability analysis for generalized $\alpha$-nonexpensive mappings with fixed points, Axioms, 13 (2024), 156. https://doi.org/10.3390/axioms13030156 doi: 10.3390/axioms13030156
[15]	A. Hussain, N. Hussain, D. Ali, Estimation of newly established iterative scheme for generalized nonexpansive mappings, J. Funct. Spaces, 2021 (2021), 6675979. https://doi.org/10.1155/2021/6675979 doi: 10.1155/2021/6675979
[16]	D. Ali, A. Hussain, E. Karapinar, P. Cholamjiak, Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces, Open Math., 20 (2023), 1753–1769. https://doi.org/10.1515/math-2022-0461 doi: 10.1515/math-2022-0461
[17]	A. Hussain, D. Ali, E. Karapinar, Stability data dependency and errors estimation for a general iteration method, Alex. Eng. J., 60 (2021), 703–710. https://doi.org/10.1016/j.aej.2020.10.002 doi: 10.1016/j.aej.2020.10.002
[18]	Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algor., 8 (1994), 221–239. https://doi.org/10.1007/BF02142692 doi: 10.1007/BF02142692
[19]	G. Lopez, V. Martin-Marquez, H. K. Xu, Halpern's iteration for nonexpansive mappings, Contemp. Math., 513 (2010), 211–231.
[20]	P. Debnath, Results on discontinuity at fixed point for a new class of $F$-contractive mappings, Sahand Commun. Math. Anal., 20 (2023), 21–32. https://doi.org/10.22130/scma.2023.560141.1161 doi: 10.22130/scma.2023.560141.1161
[21]	P. Debnath, New common fixed point theorems for Górnicki-type mappings and enriched contractions, São Paulo J. Math. Sci., 16 (2022), 1401–1408. https://doi.org/10.1007/s40863-022-00283-2 doi: 10.1007/s40863-022-00283-2
[22]	J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359–1373. https://doi.org/10.1090/S0002-9939-07-09110-1 doi: 10.1090/S0002-9939-07-09110-1
[23]	J. Tiammee, A. Kaewkhao, S. Suantai, On Browder's convergence theorem and Halpern iteration process for $G$-nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory Appl., 2015 (2015), 187. https://doi.org/10.1186/s13663-015-0436-9 doi: 10.1186/s13663-015-0436-9
[24]	S. M. A. Aleomraninejad, S. Rezapour, N. Shahzad, Some fixed point results on a metric space with a graph, Topol. Appl., 159 (2012), 659–663. https://doi.org/10.1016/j.topol.2011.10.013 doi: 10.1016/j.topol.2011.10.013
[25]	O. Tripak, Common fixed points of $G$-nonexpansive mappings on Banach spaces with a graph, Fixed Point Theory Appl., 2016 (2016), 83. https://doi.org/10.1186/s13663-016-0578-4 doi: 10.1186/s13663-016-0578-4
[26]	R. Suparatulatorn, W. Cholamjiak, S. Suantai, A modified $S$-iteration process for $G$-nonexpansive mappings in Banach spaces with graphs, Numer. Algor., 77 (2018), 479–490. https://doi.org/10.1007/s11075-017-0324-y doi: 10.1007/s11075-017-0324-y
[27]	T. Thianwan, D. Yambangwai, Convergence analysis for a new two-step iteration process for $G$-nonexpansive mappings with directed graphs, J. Fixed Point Theory Appl., 21 (2019), 44. https://doi.org/10.1007/s11784-019-0681-3 doi: 10.1007/s11784-019-0681-3
[28]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597. https://doi.org/10.1090/S0002-9904-1967-11761-0 doi: 10.1090/S0002-9904-1967-11761-0
[29]	S. Shahzad, R. Al-Dubiban, Approximating common fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J., 13 (2006), 529–537. https://doi.org/10.1515/GMJ.2006.529 doi: 10.1515/GMJ.2006.529
[30]	J. Schu, Weak and strong convergence to fixed points of asymtotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), 153–159. https://doi.org/10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
[31]	S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2015), 506–517. https://doi.org/10.1016/j.jmaa.2005.03.002 doi: 10.1016/j.jmaa.2005.03.002
[32]	M. G. Sangago, Convergence of iterative schemes for nonexpansive mappings, Asian-Eur. J. Math., 4 (2011) 671–682. https://doi.org/10.1142/S1793557111000551 doi: 10.1142/S1793557111000551

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