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Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces
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 |
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.
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.
|
[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. |
[3] |
M. Bačák,
The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.
doi: 10.1007/s11856-012-0091-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. A. Clarkson,
Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.
doi: 10.1090/S0002-9947-1936-1501880-4. |
[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. |
[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. |
[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. |
[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. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Kumam and P. Chaipunya,, Equilibrium problems and proximal algorithms in Hadamard spaces, preprint, arXiv: 1807.10900. |
[24] |
K. Kuwae,
Resolvent flows for convex functionals and $p$-harmonic maps, Anal. Geom. Metr. Spaces, 3 (2015), 46-72.
|
[25] |
E. Kreyszig,, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978. |
[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. |
[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. |
[28] |
B. Martinet,, Régularisation d'inéquations varaiationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158. |
[29] |
I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London-New York, 1970.
![]() ![]() |
[30] |
A. Naor and L. Silberman,
Poincaré inequalities, embeddings, and wild groups, Compos. Math., 147 (2011), 1546-1572.
doi: 10.1112/S0010437X11005343. |
[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. |
[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. |
[33] |
R. T. Rockafellar,
Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[34] |
H. L. Royden,, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[41] |
G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo,, On nonspreading-type mappings in Hadamard spaces, Bol. Soc. Paran. Mat., (2018), 23 pp. |
[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. |
[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. |
[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. |
[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. |
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.
|
[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. |
[3] |
M. Bačák,
The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.
doi: 10.1007/s11856-012-0091-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. A. Clarkson,
Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.
doi: 10.1090/S0002-9947-1936-1501880-4. |
[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. |
[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. |
[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. |
[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. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Kumam and P. Chaipunya,, Equilibrium problems and proximal algorithms in Hadamard spaces, preprint, arXiv: 1807.10900. |
[24] |
K. Kuwae,
Resolvent flows for convex functionals and $p$-harmonic maps, Anal. Geom. Metr. Spaces, 3 (2015), 46-72.
|
[25] |
E. Kreyszig,, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978. |
[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. |
[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. |
[28] |
B. Martinet,, Régularisation d'inéquations varaiationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158. |
[29] |
I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London-New York, 1970.
![]() ![]() |
[30] |
A. Naor and L. Silberman,
Poincaré inequalities, embeddings, and wild groups, Compos. Math., 147 (2011), 1546-1572.
doi: 10.1112/S0010437X11005343. |
[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. |
[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. |
[33] |
R. T. Rockafellar,
Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[34] |
H. L. Royden,, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[41] |
G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo,, On nonspreading-type mappings in Hadamard spaces, Bol. Soc. Paran. Mat., (2018), 23 pp. |
[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. |
[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. |
[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. |
[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. |


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