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July  2021, 17(4): 2161-2180. doi: 10.3934/jimo.2020063

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

Kazeem Olalekan Aremu 1, , Chinedu Izuchukwu 1,2, , Grace Nnenanya Ogwo 1, and  Oluwatosin Temitope Mewomo 1,,

1. 

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
2. 

DSI-NRF Center of Excellence in, Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

* Corresponding author: O. T. Mewomo

Received  May 2019 Revised  October 2019 Published  March 2020

Fund Project: The second author is supported by the Department of Science and Innovation and National Research Foundation, Republic of South Africa, Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS), Doctoral Bursary. The third author is supported by the African Institute for Mathematical Sciences (AIMS), South Africa. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903)

In this paper, we propose and study a multi-step iterative algorithm that comprises of a finite family of asymptotically $ k_i $-strictly pseudocontractive mappings with respect to $ p, $ and a $ p $-resolvent operator associated with a proper convex and lower semicontinuous function in a $ p $-uniformly convex metric space. Also, we establish the $ \Delta $-convergence of the proposed algorithm to a common fixed point of finite family of asymptotically $ k_i $-strictly pseudocontractive mappings which is also a minimizer of a proper convex and lower semicontinuous function. Furthermore, nontrivial numerical examples of our algorithm are given to show its applicability. Our results complement a host of recent results in literature.

Keywords: $ p $-uniformly convex metric spaces, asymptotically $ k $-strictly pseudocontractive mapping, $ p $-resolvent operator, $ \Delta $-convergence.
Mathematics Subject Classification: Primary: 47H09; 47H10; 49J20; 49J40.
Citation: Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063
References:
[1]

H. A. Abass, C. Izuchukwu, F. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136.   Google Scholar

[2]

N. Akkasriworn, A. Kaewkhao, A. Keawkhao and K. Sokhuma,, Common fixed-point results in uniformly convex Banach spaces, Fixed Point Theory Appl., 2012 (2012), 171, 7 pp. doi: 10.1186/1687-1812-2012-171.  Google Scholar

[3]

M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.  doi: 10.1007/s11856-012-0091-3.  Google Scholar

[4]

M. Başarir and A. Şahin,, On the strong and $\delta$-convergence of new multi-step and s-iteration processes in a CAT(0) space, J. Inequal. Appl., 2013 (2013), 482, 13 pp. doi: 10.1186/1029-242x-2013-482.  Google Scholar

[5]

M. Başarir and A. Şahin, Two general iteration schemes for multi-valued maps in hyperbolic spaces, Commun. Korean Math. Soc., 31 (2016), 713-727.  doi: 10.4134/CKMS.c150146.  Google Scholar

[6]

K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), 463-482.  doi: 10.1007/BF01231769.  Google Scholar

[7]

R. P. Boas Jr., Some uniformly convex spaces, Bull. Amer. Math. Soc., 46 (1940), 304-311.  doi: 10.1090/S0002-9904-1940-07207-6.  Google Scholar

[8]

P. Chaipunya and P. Kumam, On the proximal point method in Hadamard spaces, Optimization, 66 (2017), 1647-1665.  doi: 10.1080/02331934.2017.1349124.  Google Scholar

[9]

B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc., 31 (2016), 845-855.  doi: 10.4134/CKMS.c150114.  Google Scholar

[10]

J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.  doi: 10.1090/S0002-9947-1936-1501880-4.  Google Scholar

[11]

S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762-772.  doi: 10.1016/j.na.2005.09.044.  Google Scholar

[12]

R. Espínola, A. Fernández-León and B. Piatek,, Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory Appl., 2010 (2010), Art. ID 169837, 16 pp. doi: 10.1155/2010/169837.  Google Scholar

[13]

C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.  Google Scholar

[14]

C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in $p$-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[15]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo,, A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems, Rendiconti del Circolo Matematico di Palermo, (2019). Google Scholar

[16]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces, Demonstr. Math., 52 (2019), 183-203.  doi: 10.1515/dema-2019-0013.  Google Scholar

[17]

F. Gürsoy, V. Karakaya and B. E. Rhoades,, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl., 2013 (2013), Art. 76, 12 pp. doi: 10.1186/1687-1812-2013-76.  Google Scholar

[18]

A. R. Khan, H. Fukhar-ud-din and M. A. A. Khan,, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl., 2012 (2012), 54, 12 pp. doi: 10.1186/1687-1812-2012-54.  Google Scholar

[19]

H. Khatibzadeh and V. Mohebbi,, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, Journal of the Australian Mathematical Society, (2019), 1–23. doi: 10.1017/S1446788719000041.  Google Scholar

[20]

H. Khatibzadeh and S. Ranjbar, A variational inequality in complete $\rm CAT(0)$ spaces, J. Fixed Point Theory Appl., 17 (2015), 557-574.  doi: 10.1007/s11784-015-0245-0.  Google Scholar

[21]

H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete Cat(0) metric spaces, J. Aust. Math. Soc., 103 (2017), 70-90.  doi: 10.1017/S1446788716000446.  Google Scholar

[22]

W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696.  doi: 10.1016/j.na.2007.04.011.  Google Scholar

[23]

P. Kumam and P. Chaipunya,, Equilibrium problems and proximal algorithms in Hadamard spaces, preprint, arXiv: 1807.10900. Google Scholar

[24]

K. Kuwae, Resolvent flows for convex functionals and $p$-harmonic maps, Anal. Geom. Metr. Spaces, 3 (2015), 46-72.   Google Scholar

[25]

E. Kreyszig,, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978.  Google Scholar

[26]

L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl., 325 (2007), 386-399.  doi: 10.1016/j.jmaa.2006.01.081.  Google Scholar

[27]

T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182.  doi: 10.1090/S0002-9939-1976-0423139-X.  Google Scholar

[28]

B. Martinet,, Régularisation d'inéquations varaiationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158.  Google Scholar

[29] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London-New York, 1970.   Google Scholar
[30]

A. Naor and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compos. Math., 147 (2011), 1546-1572.  doi: 10.1112/S0010437X11005343.  Google Scholar

[31]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete $\rm CAT(0)$ spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.  Google Scholar

[32]

N. Pakkaranang, P. Kewdee, P. Kumam and P. Borisut, The modified multi-step iteration process for pairwise generalized nonexpansive mappings in CAT(0) spaces, Studies in Computational Intelligence, 760 (2018), 381-393.  doi: 10.1007/978-3-319-73150-6_31.  Google Scholar

[33]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[34]

H. L. Royden,, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[35]

D. Ariza-Ruiz, G. López-Acedo and A. Nicolae, The asymptotic behavior of the composition of firmly nonexpansive mappings, J. Optim. Theory Appl., 167 (2015), 409-429.  doi: 10.1007/s10957-015-0710-3.  Google Scholar

[36]

A. Şahin and M. Başarir,, On the new multi-step iteration process for multi-valued mappings in a complete geodesic space, Commun. Fac. Sci. Univ. Ank. Sér A1 Math Stat., 64 (2015), 77–87.  Google Scholar

[37]

A. Şahin amd M. Başarir, Some convergence results for nearly asymptotically nonexpansive nonself mappings in CAT($\kappa$) spaces, Math Sci. (Springer), 11 (2017), 79-86.  doi: 10.1007/s40096-017-0209-1.  Google Scholar

[38]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), Art. 77, 28 pp. doi: 10.1007/s40314-019-0841-5.  Google Scholar

[39]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.  Google Scholar

[40]

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, Ricerche di Matematica, (2019). doi: 10.1007/s11587-019-00460-0.  Google Scholar

[41]

G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo,, On nonspreading-type mappings in Hadamard spaces, Bol. Soc. Paran. Mat., (2018), 23 pp. Google Scholar

[42]

G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, Proximal point algorithm involving fixed point of nonexpansive mapping in $p$-uniformly convex metric space, Adv. Pure Appl. Math., 10 (2019), 437-446.  doi: 10.1515/apam-2018-0026.  Google Scholar

[43]

G. C. Ugwunnadi, A. R. Khan and M. Abbas,, A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces, J. Fixed Point Theory Appl., 20 (2018), Art. 82, 19 pp. doi: 10.1007/s11784-018-0555-0.  Google Scholar

[44]

I. Yildirim and M. Özdemir, A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 991-999.  doi: 10.1016/j.na.2008.11.017.  Google Scholar

[45]

G. Z. Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80 (2019), 1155-1179.  doi: 10.1007/s11075-018-0521-3.  Google Scholar

show all references

References:
[1]

H. A. Abass, C. Izuchukwu, F. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136.   Google Scholar

[2]

N. Akkasriworn, A. Kaewkhao, A. Keawkhao and K. Sokhuma,, Common fixed-point results in uniformly convex Banach spaces, Fixed Point Theory Appl., 2012 (2012), 171, 7 pp. doi: 10.1186/1687-1812-2012-171.  Google Scholar

[3]

M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.  doi: 10.1007/s11856-012-0091-3.  Google Scholar

[4]

M. Başarir and A. Şahin,, On the strong and $\delta$-convergence of new multi-step and s-iteration processes in a CAT(0) space, J. Inequal. Appl., 2013 (2013), 482, 13 pp. doi: 10.1186/1029-242x-2013-482.  Google Scholar

[5]

M. Başarir and A. Şahin, Two general iteration schemes for multi-valued maps in hyperbolic spaces, Commun. Korean Math. Soc., 31 (2016), 713-727.  doi: 10.4134/CKMS.c150146.  Google Scholar

[6]

K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), 463-482.  doi: 10.1007/BF01231769.  Google Scholar

[7]

R. P. Boas Jr., Some uniformly convex spaces, Bull. Amer. Math. Soc., 46 (1940), 304-311.  doi: 10.1090/S0002-9904-1940-07207-6.  Google Scholar

[8]

P. Chaipunya and P. Kumam, On the proximal point method in Hadamard spaces, Optimization, 66 (2017), 1647-1665.  doi: 10.1080/02331934.2017.1349124.  Google Scholar

[9]

B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc., 31 (2016), 845-855.  doi: 10.4134/CKMS.c150114.  Google Scholar

[10]

J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.  doi: 10.1090/S0002-9947-1936-1501880-4.  Google Scholar

[11]

S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762-772.  doi: 10.1016/j.na.2005.09.044.  Google Scholar

[12]

R. Espínola, A. Fernández-León and B. Piatek,, Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory Appl., 2010 (2010), Art. ID 169837, 16 pp. doi: 10.1155/2010/169837.  Google Scholar

[13]

C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.  Google Scholar

[14]

C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in $p$-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[15]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo,, A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems, Rendiconti del Circolo Matematico di Palermo, (2019). Google Scholar

[16]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces, Demonstr. Math., 52 (2019), 183-203.  doi: 10.1515/dema-2019-0013.  Google Scholar

[17]

F. Gürsoy, V. Karakaya and B. E. Rhoades,, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl., 2013 (2013), Art. 76, 12 pp. doi: 10.1186/1687-1812-2013-76.  Google Scholar

[18]

A. R. Khan, H. Fukhar-ud-din and M. A. A. Khan,, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl., 2012 (2012), 54, 12 pp. doi: 10.1186/1687-1812-2012-54.  Google Scholar

[19]

H. Khatibzadeh and V. Mohebbi,, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, Journal of the Australian Mathematical Society, (2019), 1–23. doi: 10.1017/S1446788719000041.  Google Scholar

[20]

H. Khatibzadeh and S. Ranjbar, A variational inequality in complete $\rm CAT(0)$ spaces, J. Fixed Point Theory Appl., 17 (2015), 557-574.  doi: 10.1007/s11784-015-0245-0.  Google Scholar

[21]

H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete Cat(0) metric spaces, J. Aust. Math. Soc., 103 (2017), 70-90.  doi: 10.1017/S1446788716000446.  Google Scholar

[22]

W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696.  doi: 10.1016/j.na.2007.04.011.  Google Scholar

[23]

P. Kumam and P. Chaipunya,, Equilibrium problems and proximal algorithms in Hadamard spaces, preprint, arXiv: 1807.10900. Google Scholar

[24]

K. Kuwae, Resolvent flows for convex functionals and $p$-harmonic maps, Anal. Geom. Metr. Spaces, 3 (2015), 46-72.   Google Scholar

[25]

E. Kreyszig,, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978.  Google Scholar

[26]

L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl., 325 (2007), 386-399.  doi: 10.1016/j.jmaa.2006.01.081.  Google Scholar

[27]

T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182.  doi: 10.1090/S0002-9939-1976-0423139-X.  Google Scholar

[28]

B. Martinet,, Régularisation d'inéquations varaiationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158.  Google Scholar

[29] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London-New York, 1970.   Google Scholar
[30]

A. Naor and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compos. Math., 147 (2011), 1546-1572.  doi: 10.1112/S0010437X11005343.  Google Scholar

[31]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete $\rm CAT(0)$ spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.  Google Scholar

[32]

N. Pakkaranang, P. Kewdee, P. Kumam and P. Borisut, The modified multi-step iteration process for pairwise generalized nonexpansive mappings in CAT(0) spaces, Studies in Computational Intelligence, 760 (2018), 381-393.  doi: 10.1007/978-3-319-73150-6_31.  Google Scholar

[33]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[34]

H. L. Royden,, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[35]

D. Ariza-Ruiz, G. López-Acedo and A. Nicolae, The asymptotic behavior of the composition of firmly nonexpansive mappings, J. Optim. Theory Appl., 167 (2015), 409-429.  doi: 10.1007/s10957-015-0710-3.  Google Scholar

[36]

A. Şahin and M. Başarir,, On the new multi-step iteration process for multi-valued mappings in a complete geodesic space, Commun. Fac. Sci. Univ. Ank. Sér A1 Math Stat., 64 (2015), 77–87.  Google Scholar

[37]

A. Şahin amd M. Başarir, Some convergence results for nearly asymptotically nonexpansive nonself mappings in CAT($\kappa$) spaces, Math Sci. (Springer), 11 (2017), 79-86.  doi: 10.1007/s40096-017-0209-1.  Google Scholar

[38]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), Art. 77, 28 pp. doi: 10.1007/s40314-019-0841-5.  Google Scholar

[39]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.  Google Scholar

[40]

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, Ricerche di Matematica, (2019). doi: 10.1007/s11587-019-00460-0.  Google Scholar

[41]

G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo,, On nonspreading-type mappings in Hadamard spaces, Bol. Soc. Paran. Mat., (2018), 23 pp. Google Scholar

[42]

G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, Proximal point algorithm involving fixed point of nonexpansive mapping in $p$-uniformly convex metric space, Adv. Pure Appl. Math., 10 (2019), 437-446.  doi: 10.1515/apam-2018-0026.  Google Scholar

[43]

G. C. Ugwunnadi, A. R. Khan and M. Abbas,, A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces, J. Fixed Point Theory Appl., 20 (2018), Art. 82, 19 pp. doi: 10.1007/s11784-018-0555-0.  Google Scholar

[44]

I. Yildirim and M. Özdemir, A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 991-999.  doi: 10.1016/j.na.2008.11.017.  Google Scholar

[45]

G. Z. Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80 (2019), 1155-1179.  doi: 10.1007/s11075-018-0521-3.  Google Scholar

Figure 1.  Errors vs Iteration numbers(n) for Example 4.1: Case 1 (top left); Case 2 (top right); Case 3 (bottom left); Case 4 (bottom right)
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Figure 2.  Errors vs Iteration numbers(n) for Example 4.2: Case 1 (top left); Case 2 (top right); Case 3 (bottom left); Case 4 (bottom right)
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