# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022005
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## Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces

 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  May 2021 Revised  February 2022 Early access March 2022

Fund Project: The second author is supported by International Mathematical Union (IMU) Breakout Graduate Fellowship and the third author is supported by National Research Foundation (NRF), South Africa, (Grant Number 119903)

In this paper, we present a new algorithm for solving split common fixed point problem for asymptotically demicontractive mapping in two real Hilbert spaces. Under some mild conditions, we prove that the proposed method converges strongly to a solution of the problem. We give examples to illustrate that the class of asymptotically demicontractive mappings and the class of demicontractive mappings are independent. Moreover, we give numerical experiments to show the efficiency and applicability of our method in comparison with a related method in the literature. The results obtained unify, improve and extend so many related results in the literature in this direction.

Citation: Emeka Chigaemezu Godwin, Adeolu Taiwo, Oluwatosin Temitope Mewomo. Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022005
##### References:
 [1] T. O. Alakoya, L. O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, (2021).  doi: 10.1080/02331934.2021.1895154. [2] T. O. Alakoya and O. T. Mewomo, Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems, Comput. Appl. Math., 41 (2022), Paper No. 39, 31 pp. doi: 10.1007/s40314-021-01749-3. [3] T. O. Alakoya, A. O. E. Owolabi and O. T. Mewomo, An inertial algorithm with a self-adaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions, J. Nonlinear Var. Anal., 5 (2021), 803-829.  doi: 10.1007/s40314-021-01749-3. [4] T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2. [5] T. O. Alakoya, A. Taiwo and O. T. Mewomo, On system of split generalised mixed equilibrium and fixed point problems for multivalued mappings with no prior knowledge of operator norm, Fixed Point Theory, 23 (2022), 45-74.  doi: 10.24193/fpt-ro. [6] H. Q. Ansari and A. Rehan, Split feasibility and fixed point problems, Nonlinear Analysis, Trends Math., Birkhäuser/Springer, New Delhi, Berlin, (2014) 281–322. [7] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2003), 103-120.  doi: 10.1088/0266-5611/20/1/006. [8] C. Byrne, Iterative oblique projection onto convex subsets and the split feasiblity problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310. [9] L.-C. Ceng, Q. H. Ansari and J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116-2125.  doi: 10.1016/j.na.2011.10.012. [10] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. [11] Y. Censor, Y. Elving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2353-2365.  doi: 10.1088/0266-5611/21/6/017. [12] Y. Censor and T. Elfving, A multi projection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [13] Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587600. [14] E. E. Chima and M. O. Osilike, Split common fixed point problem for class of asymptotically hemicontractive mappings, J. Nig. Math. Soc., 38 (2019), 363-389. [15] M. Eslamian and P. Eslamian, Strong convergence of a split common fixed point problem, Numer. Func. Anal. Optim., 37 (2016), 1248-1266.  doi: 10.1080/01630563.2016.1200076. [16] E. C. Godwin, C. Izuchukwu and O. T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 14 (2021), 379-401.  doi: 10.1007/s40574-020-00272-3. [17] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0. [18] C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, (2020).  doi: 10.1080/02331934.2020.1808648. [19] R. Kraikaew and S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513-524.  doi: 10.1016/j.jmaa.2014.01.068. [20] Q. H. Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26 (1996), 1835-1842.  doi: 10.1016/0362-546X(94)00351-H. [21] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z. [22] A. Moudafi, Viscocity approximation methods for fixed-point problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [23] A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087.  doi: 10.1016/j.na.2011.03.041. [24] C. Moore and B. V. C. Nnoli, Iterative sequence for asymptotically demicontractive maps in Banach spaces, J. Math. Anal. Appl., 302 (2005), 557-562.  doi: 10.1016/j.jmaa.2004.03.006. [25] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems, Optimization, (2021).  doi: 10.1080/02331934.2021.1981897. [26] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces, Demonstr. Math., (2021).  doi: 10.1515/dema-2020-0119. [27] G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 88 (2021), 1419-1456.  doi: 10.1007/s11075-021-01081-1. [28] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math., 54 (2021), 47-67.  doi: 10.1515/dema-2021-0006. [29] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings, J. Nonlinear Funct. Anal., 2021 (2021), Art. ID 10, 21 pp. doi: 10.1007/s40314-021-01749-3. [30] A. O.-E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 12 (2022), 255-278.  doi: 10.3934/naco.2021004. [31] M. O. Osilike, Iterative approximation of fixed points of asymptotically demicontractive mappings, Indian J. Pure Appl. Math., 24 (1998), 1291-1300. [32] M. O. Osilike, A. Udomene, D. I. Igbokwe and B. G. Akuchu, Demiclosedness principle and convergence theorems for $k$-strictly asymptotically pseudocontractive maps, J. Math. Anal. Appl., 326 (2007), 1334-1345.  doi: 10.1016/j.jmaa.2005.12.052. [33] M. O. Osilike and S. C. Aniagbosor, Fixed points of asymptotically demicontractive mappings in certain Banach spaces, Indian J. Pure Appl. Math., 32 (2001), 1519-1537. [34] Y. Shehu and P. Cholamjiak, Another look at the split common fixed point problem for demicontractive operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 110 (2016), 201-218.  doi: 10.1007/s13398-015-0231-9. [35] Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6. [36] A. Taiwo, T. O. Alakoya and O.T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, 86 (2021), 1359-1389.  doi: 10.1007/s11075-020-00937-2. [37] A. Taiwo, T. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with applications, Asian-Eur. J. Math., 14 (2021), Art. ID 2150137, 31 pp. doi: 10.1142/S1793557121501370. [38] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Inertial-type algorithm for solving split common fixed point problem in Banach spaces, J. Sci. Comput., (2020).  doi: 10.1007/s10915-020-01385-9. [39] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ric Mat., 69 (2020), 235-259.  doi: 10.1007/s11587-019-00460-0. [40] A. Taiwo, O. T. Mewomo and A. Gibali, Simple strong convergent method for solving split common fixed point problem, J. Nonlinear Var. Anal., 5 (2021), 777-793. [41] J. F. Tang, S. S. Chang and M. Liu, General split feasibility problems for two families of nonexpansive mappings in Hilbert spaces, Acta Math. Sci., 36B (2016), 602-613.  doi: 10.1016/S0252-9602(16)30024-8. [42] V. A. Uzor, T. O. Alakoya and O. T. Mewomo, Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems, Open Math., (2022).  doi: 10.1515/math-2022-0429. [43] T. Q. Wang and T. M. Kim, Simultaneous iterative algorithm for the split equality fixed-point problem of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 154-65.  doi: 10.22436/jnsa.010.01.15. [44] F. Wang and H. K. Xu, Weak and strong convergence of two algorithms for the split fixed point problem, Numer. Math. Theory Method Appl., 11 (2018), 770-781. [45] Y. Wang, X. Wu and C. Pan, The iterative solutions of split common fixed point problem for asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2020 (2020), Art. 18. doi: 10.1186/s13663-020-00686-w. [46] H. K. Xu, Iterarive algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 246-256.  doi: 10.1112/S0024610702003332. [47] H. K. Xu, An alternative regularization method for nonexpansive mappings with applications, Contmp. Maths., 513 (2010), 239-263.  doi: 10.1090/conm/513/10087. [48] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Stud. Comput. Math., 8 (2001), 473-504.  doi: 10.1016/S1570-579X(01)80028-8. [49] C. Yang and S. He, General alternative regularization methods for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), Art. 203, 14 pp. doi: 10.1186/1687-1812-2014-203. [50] J. Zhao, Y. Jia and H. Zhang, General alternative regularization methods for split equality common fixed-point problem, Optimization, 67 (2018), 619-635.  doi: 10.1080/02331934.2017.1412437. [51] J. Zhao and S. He, Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings, J. Appl. Math., 2012 (2012), Art. ID. 438023, 12 pp. doi: 10.1155/2012/438023.

show all references

##### References:
 [1] T. O. Alakoya, L. O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, (2021).  doi: 10.1080/02331934.2021.1895154. [2] T. O. Alakoya and O. T. Mewomo, Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems, Comput. Appl. Math., 41 (2022), Paper No. 39, 31 pp. doi: 10.1007/s40314-021-01749-3. [3] T. O. Alakoya, A. O. E. Owolabi and O. T. Mewomo, An inertial algorithm with a self-adaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions, J. Nonlinear Var. Anal., 5 (2021), 803-829.  doi: 10.1007/s40314-021-01749-3. [4] T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2. [5] T. O. Alakoya, A. Taiwo and O. T. Mewomo, On system of split generalised mixed equilibrium and fixed point problems for multivalued mappings with no prior knowledge of operator norm, Fixed Point Theory, 23 (2022), 45-74.  doi: 10.24193/fpt-ro. [6] H. Q. Ansari and A. Rehan, Split feasibility and fixed point problems, Nonlinear Analysis, Trends Math., Birkhäuser/Springer, New Delhi, Berlin, (2014) 281–322. [7] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2003), 103-120.  doi: 10.1088/0266-5611/20/1/006. [8] C. Byrne, Iterative oblique projection onto convex subsets and the split feasiblity problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310. [9] L.-C. Ceng, Q. H. Ansari and J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116-2125.  doi: 10.1016/j.na.2011.10.012. [10] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. [11] Y. Censor, Y. Elving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2353-2365.  doi: 10.1088/0266-5611/21/6/017. [12] Y. Censor and T. Elfving, A multi projection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [13] Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587600. [14] E. E. Chima and M. O. Osilike, Split common fixed point problem for class of asymptotically hemicontractive mappings, J. Nig. Math. Soc., 38 (2019), 363-389. [15] M. Eslamian and P. Eslamian, Strong convergence of a split common fixed point problem, Numer. Func. Anal. Optim., 37 (2016), 1248-1266.  doi: 10.1080/01630563.2016.1200076. [16] E. C. Godwin, C. Izuchukwu and O. T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 14 (2021), 379-401.  doi: 10.1007/s40574-020-00272-3. [17] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0. [18] C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, (2020).  doi: 10.1080/02331934.2020.1808648. [19] R. Kraikaew and S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513-524.  doi: 10.1016/j.jmaa.2014.01.068. [20] Q. H. Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26 (1996), 1835-1842.  doi: 10.1016/0362-546X(94)00351-H. [21] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z. [22] A. Moudafi, Viscocity approximation methods for fixed-point problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [23] A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087.  doi: 10.1016/j.na.2011.03.041. [24] C. Moore and B. V. C. Nnoli, Iterative sequence for asymptotically demicontractive maps in Banach spaces, J. Math. Anal. Appl., 302 (2005), 557-562.  doi: 10.1016/j.jmaa.2004.03.006. [25] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems, Optimization, (2021).  doi: 10.1080/02331934.2021.1981897. [26] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces, Demonstr. Math., (2021).  doi: 10.1515/dema-2020-0119. [27] G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 88 (2021), 1419-1456.  doi: 10.1007/s11075-021-01081-1. [28] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math., 54 (2021), 47-67.  doi: 10.1515/dema-2021-0006. [29] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings, J. Nonlinear Funct. Anal., 2021 (2021), Art. ID 10, 21 pp. doi: 10.1007/s40314-021-01749-3. [30] A. O.-E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 12 (2022), 255-278.  doi: 10.3934/naco.2021004. [31] M. O. Osilike, Iterative approximation of fixed points of asymptotically demicontractive mappings, Indian J. Pure Appl. Math., 24 (1998), 1291-1300. [32] M. O. Osilike, A. Udomene, D. I. Igbokwe and B. G. Akuchu, Demiclosedness principle and convergence theorems for $k$-strictly asymptotically pseudocontractive maps, J. Math. Anal. Appl., 326 (2007), 1334-1345.  doi: 10.1016/j.jmaa.2005.12.052. [33] M. O. Osilike and S. C. Aniagbosor, Fixed points of asymptotically demicontractive mappings in certain Banach spaces, Indian J. Pure Appl. Math., 32 (2001), 1519-1537. [34] Y. Shehu and P. Cholamjiak, Another look at the split common fixed point problem for demicontractive operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 110 (2016), 201-218.  doi: 10.1007/s13398-015-0231-9. [35] Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6. [36] A. Taiwo, T. O. Alakoya and O.T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, 86 (2021), 1359-1389.  doi: 10.1007/s11075-020-00937-2. [37] A. Taiwo, T. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with applications, Asian-Eur. J. Math., 14 (2021), Art. ID 2150137, 31 pp. doi: 10.1142/S1793557121501370. [38] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Inertial-type algorithm for solving split common fixed point problem in Banach spaces, J. Sci. Comput., (2020).  doi: 10.1007/s10915-020-01385-9. [39] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ric Mat., 69 (2020), 235-259.  doi: 10.1007/s11587-019-00460-0. [40] A. Taiwo, O. T. Mewomo and A. Gibali, Simple strong convergent method for solving split common fixed point problem, J. Nonlinear Var. Anal., 5 (2021), 777-793. [41] J. F. Tang, S. S. Chang and M. Liu, General split feasibility problems for two families of nonexpansive mappings in Hilbert spaces, Acta Math. Sci., 36B (2016), 602-613.  doi: 10.1016/S0252-9602(16)30024-8. [42] V. A. Uzor, T. O. Alakoya and O. T. Mewomo, Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems, Open Math., (2022).  doi: 10.1515/math-2022-0429. [43] T. Q. Wang and T. M. Kim, Simultaneous iterative algorithm for the split equality fixed-point problem of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 154-65.  doi: 10.22436/jnsa.010.01.15. [44] F. Wang and H. K. Xu, Weak and strong convergence of two algorithms for the split fixed point problem, Numer. Math. Theory Method Appl., 11 (2018), 770-781. [45] Y. Wang, X. Wu and C. Pan, The iterative solutions of split common fixed point problem for asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2020 (2020), Art. 18. doi: 10.1186/s13663-020-00686-w. [46] H. K. Xu, Iterarive algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 246-256.  doi: 10.1112/S0024610702003332. [47] H. K. Xu, An alternative regularization method for nonexpansive mappings with applications, Contmp. Maths., 513 (2010), 239-263.  doi: 10.1090/conm/513/10087. [48] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Stud. Comput. Math., 8 (2001), 473-504.  doi: 10.1016/S1570-579X(01)80028-8. [49] C. Yang and S. He, General alternative regularization methods for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), Art. 203, 14 pp. doi: 10.1186/1687-1812-2014-203. [50] J. Zhao, Y. Jia and H. Zhang, General alternative regularization methods for split equality common fixed-point problem, Optimization, 67 (2018), 619-635.  doi: 10.1080/02331934.2017.1412437. [51] J. Zhao and S. He, Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings, J. Appl. Math., 2012 (2012), Art. ID. 438023, 12 pp. doi: 10.1155/2012/438023.
Example 4.1: Top left: Case Ⅰ; Top right: Case Ⅱ; Bottom left: Case Ⅲ; Bottom right: Case Ⅳ
Numerical results for Example 4.1
 Algorithm 3.4 Algorithm (10) Alg. 3.2Chi Case I CPU time (sec) 0.0020 0.0024 0.0036 No. of Iter. 18 21 33 Case II CPU time (sec) 0.0033 0.0389 0.0054 No. of Iter. 19 23 35 Case III CPU time (sec) 0.0026 0.0034 0.0059 No. of Iter. 18 21 33 Case IV CPU time (sec) 0.0019 0.0195 0.0032 No. of Iter. 19 22 34
 Algorithm 3.4 Algorithm (10) Alg. 3.2Chi Case I CPU time (sec) 0.0020 0.0024 0.0036 No. of Iter. 18 21 33 Case II CPU time (sec) 0.0033 0.0389 0.0054 No. of Iter. 19 23 35 Case III CPU time (sec) 0.0026 0.0034 0.0059 No. of Iter. 18 21 33 Case IV CPU time (sec) 0.0019 0.0195 0.0032 No. of Iter. 19 22 34
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