Approximation of iterative methods for altering points problem with applications
Abstract
In this paper, we consider and investigate an altering points problem involving generalized accretive mappings over closed convex subsets of a real uniformly smooth Banach space. Parallel Mann and parallel S-iterative methods are suggested to analyze the approximate solution of altering points problem. Consequently, some systems of generalized variational inclusions and generalized variational inequalities are also explored using the conceptual framework of altering points. Convergence of suggested iterative methods are verified by an illustrative numerical example.
Keyword : iterative methods, altering points problem
How to Cite
Khan, A., Akram, M., & Dilshad, M. (2023). Approximation of iterative methods for altering points problem with applications. Mathematical Modelling and Analysis, 28(1), 118–145. https://doi.org/10.3846/mma.2023.14858
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Akram, A. Khan and M. Dilshad. Convergence of some iterative algorithms for system of generalized set-valued variational inequalities. J. Function Spaces, 2021, 2021. https://doi.org/10.1155/2021/6674349
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N. Buong, N.S. Ha and N.T. Thuy. A new explicit iteration method for a class of variational inequalities. Numer. Algor., 72:467–481, 2016. https://doi.org/10.1007/s11075-015-0056-9
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N.S. Ha, N. Buong and N.T. Thuy. A new simple parallel iteration method for a class of variational inequalities. Acta Math. Vietnam., 43(2):293–255, 2017.
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W.R. Mann. Mean value methods in iteration. Proc. Amer. Math. Soc., 4:506– 510, 1953.
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Y. Qing and Songtao Lv. Strong convergence of a parallel iterative algorithm in a reflexive Banach space. Fixed Point Theory Appl., 2014(125), 2014. https://doi.org/10.1186/1687-1812-2014-125
Jr. R.E. Bruck. Nonexpansive retracts of Banach spaces. Bull. Amer. Math. Soc., 7:384–386, 1970.
D.R. Sahu. Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1):187–204, 2011.
D.R. Sahu. Altering points and applications. Nonlinear Stud., 21(2):349–365, 2014.
D.R. Sahu, S.M. Kang and A. Kumar. Convergence analysis of parallel Siteratition process for system of generalized variational inequalities. J. Function Spaces, 2017, 2017. https://doi.org/10.1155/2017/5847096
G.S. Saluja. Convergence of modified S-iteration process for two generalized asymptotically quasi-nonexpansive mappings in CAT(0) spaces. Math. Morav., 19(1):19–31, 2015. https://doi.org/10.5937/MatMor1501019S
G. Stampacchia. Formes bilineaires coercivites sur les ensembles convexes. Comptes Rendus de l’Academie des Sciences, 258:4413–4416, 1964.
G. Stampacchia. Le probleme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus. Ann. lnst. Fourier, 15(1):189–257, 1965. https://doi.org/10.5802/aif.204
Y.F. Sun, Z. Zeng and J. Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 10(2):157–164, 2020. https://doi.org/10.3934/naco.2019045
R. Suparatulatorn, W. Cholamjiak and S. Suantai. A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer. Algorithms, 77:479–490, 2018. https://doi.org/10.1007/s11075-017-0324-y
G. Tang and X. Wang. A perturbed algorithm for a system of variational inclusions involving H(·,·)-accretive operators in Banach spaces. J. Comput. Appl. Math., 272:1–7, 2014. https://doi.org/10.1016/j.cam.2014.04.023
R.U. Verma. Projection methods, algorithms, and a new system of nonlinear variational inequalities. Comput. Math. Appl., 41(7–8):1025–1031, 2001. https://doi.org/10.1016/S0898-1221(00)00336-9
X. Weng. Fixed point iteration for local strictly pseudo-contractive mappings. Proc. Amer. Math. Soc., 113:727–731, 1991. https://doi.org/10.1090/S0002-9939-1991-1086345-8
H.K. Xu. Inequalities in Banach spaces with applications. Nonlinear Anal. Theory Methods Appl., 16(12):1127–1138, 1991. https://doi.org/10.1016/0362-546X(91)90200-K
H.K. Xu and D.R. Sahu. Parallel normal S-iteration methods with applications to optimization problems. Numer. Funct. Anal. Optim., 42(16):1925–1953, 2021. https://doi.org/10.1080/01630563.2021.1950761
C. Zhao, T.Z. Huang, X.L. Zhao and L.J. Deng. Two new efficient iterative regularization methods for image restoration problems. Abst. Appl. Anal., 2013, 2013. https://doi.org/10.1155/2013/129652
X. Zhao, D.R. Sahu and C.F. Wen. Iterative methods for sytem of variational inclusions involving accretive operators and applications. Fixed Point Theory, 19(2):801–822, 2018. https://doi.org/10.24193/fpt-ro.2018.2.59
M. Akram, A. Khan and M. Dilshad. Convergence of some iterative algorithms for system of generalized set-valued variational inequalities. J. Function Spaces, 2021, 2021. https://doi.org/10.1155/2021/6674349
M. Alansari, M. Akram and M. Dilshad. Iterative algorithms for a generalized system of mixed variational-like inclusion problems and altering points problem. Stat. Optim. Inf. Comput., 8(2):549–564, 2020. https://doi.org/10.19139/soic2310-5070-884
Y. Alber and J-C. Yao. Algorithm for generalized multivalued co-variational inequalities in Banach spaces. Funct. Differ. Equ., 7:5–13, 2000.
N. Buong, N.S. Ha and N.T. Thuy. A new explicit iteration method for a class of variational inequalities. Numer. Algor., 72:467–481, 2016. https://doi.org/10.1007/s11075-015-0056-9
Q. Dong and D. Jiang. Solve the split equality problem by a projection algorithm with inertial effects. J. Nonlinear Sci. Appl., 10(3):1244–1251, 2017. https://doi.org/10.22436/jnsa.010.03.33
F. Gürsoy and V. Karakaya. A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv, p. arXiv:1403.2546, 2014.
N.S. Ha, N. Buong and N.T. Thuy. A new simple parallel iteration method for a class of variational inequalities. Acta Math. Vietnam., 43(2):293–255, 2017.
P. Hartman and G. Stampacchia. On some nonlinear elliptic differential functional equations. Acta Math., 115:271–310, 1966. https://doi.org/10.1007/BF02392210
S. Ishikawa. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44:147–150, 1974. https://doi.org/10.1090/S0002-9939-1974-0336469-5
J.U. Jeong. Convergence of parallel iterative algorithms for a system of nonlinear variational inequalities in Banach spaces. J. Appl. Math. & Informatics, 34(1 2):61–73, 2016. https://doi.org/10.14317/jami.2016.061
A. Kilicman and M. Wadai. On the solutions of three-point boundary value problems using variational-fixed point iteration method. Math. Sci., 10:33–40, 2016. https://doi.org/10.1007/s40096-016-0175-z
Q. Liu and J. Cao. A recurrent neural network based on projection operator for extended general variational inequalities. IEEE Transaction on Systems, Man and Cybernetics, Part B(Cybernetics), 40(3):928–938, 2010. https://doi.org/10.1109/TSMCB.2009.2033565
W.R. Mann. Mean value methods in iteration. Proc. Amer. Math. Soc., 4:506– 510, 1953.
W.V. Petryshyn. A characterization of strictly convexity of Banach spaces and other uses of duality mappings. J. Funct. Anal., 6(2):282–291, 1970. https://doi.org/10.1016/0022-1236(70)90061-3
J. Puangpee and S. Suantai. A new accelerated viscosity iterative method for an infinite family of nonexpansive mappings with applications to image restoration problems. Mathematics, 8(4:615), 2020. https://doi.org/10.3390/math8040615
Y. Qing and Songtao Lv. Strong convergence of a parallel iterative algorithm in a reflexive Banach space. Fixed Point Theory Appl., 2014(125), 2014. https://doi.org/10.1186/1687-1812-2014-125
Jr. R.E. Bruck. Nonexpansive retracts of Banach spaces. Bull. Amer. Math. Soc., 7:384–386, 1970.
D.R. Sahu. Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1):187–204, 2011.
D.R. Sahu. Altering points and applications. Nonlinear Stud., 21(2):349–365, 2014.
D.R. Sahu, S.M. Kang and A. Kumar. Convergence analysis of parallel Siteratition process for system of generalized variational inequalities. J. Function Spaces, 2017, 2017. https://doi.org/10.1155/2017/5847096
G.S. Saluja. Convergence of modified S-iteration process for two generalized asymptotically quasi-nonexpansive mappings in CAT(0) spaces. Math. Morav., 19(1):19–31, 2015. https://doi.org/10.5937/MatMor1501019S
G. Stampacchia. Formes bilineaires coercivites sur les ensembles convexes. Comptes Rendus de l’Academie des Sciences, 258:4413–4416, 1964.
G. Stampacchia. Le probleme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus. Ann. lnst. Fourier, 15(1):189–257, 1965. https://doi.org/10.5802/aif.204
Y.F. Sun, Z. Zeng and J. Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 10(2):157–164, 2020. https://doi.org/10.3934/naco.2019045
R. Suparatulatorn, W. Cholamjiak and S. Suantai. A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer. Algorithms, 77:479–490, 2018. https://doi.org/10.1007/s11075-017-0324-y
G. Tang and X. Wang. A perturbed algorithm for a system of variational inclusions involving H(·,·)-accretive operators in Banach spaces. J. Comput. Appl. Math., 272:1–7, 2014. https://doi.org/10.1016/j.cam.2014.04.023
R.U. Verma. Projection methods, algorithms, and a new system of nonlinear variational inequalities. Comput. Math. Appl., 41(7–8):1025–1031, 2001. https://doi.org/10.1016/S0898-1221(00)00336-9
X. Weng. Fixed point iteration for local strictly pseudo-contractive mappings. Proc. Amer. Math. Soc., 113:727–731, 1991. https://doi.org/10.1090/S0002-9939-1991-1086345-8
H.K. Xu. Inequalities in Banach spaces with applications. Nonlinear Anal. Theory Methods Appl., 16(12):1127–1138, 1991. https://doi.org/10.1016/0362-546X(91)90200-K
H.K. Xu and D.R. Sahu. Parallel normal S-iteration methods with applications to optimization problems. Numer. Funct. Anal. Optim., 42(16):1925–1953, 2021. https://doi.org/10.1080/01630563.2021.1950761
C. Zhao, T.Z. Huang, X.L. Zhao and L.J. Deng. Two new efficient iterative regularization methods for image restoration problems. Abst. Appl. Anal., 2013, 2013. https://doi.org/10.1155/2013/129652
X. Zhao, D.R. Sahu and C.F. Wen. Iterative methods for sytem of variational inclusions involving accretive operators and applications. Fixed Point Theory, 19(2):801–822, 2018. https://doi.org/10.24193/fpt-ro.2018.2.59