Abstract
In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of
Funding statement: The second and third authors acknowledge thankfully the bursary and financial support from 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 fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF.
Acknowledgements
The authors thank the anonymous referee for valuable and useful suggestions and comments which led to the great improvement of the paper.
References
[1]
B. Ahmadi Kakavandi and M. Amini,
Duality and subdifferential for convex functions on complete
[2] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), no. 8, 2350–2360. 10.1016/j.na.2006.08.032Search in Google Scholar
[3]
K. O. Aremu, C. Izuchukwu, G. C. Ugwunnadi and O. T. Mewomo,
On the proximal point algorithm and demimetric mappings in
[4] M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math. 194 (2013), no. 2, 689–701. 10.1007/s11856-012-0091-3Search in Google Scholar
[5] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata 133 (2008), 195–218. 10.1007/s10711-008-9243-3Search in Google Scholar
[6] M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999. 10.1007/978-3-662-12494-9Search in Google Scholar
[7] K. S. Brown, Buildings, Springer, New York, 1989. 10.1007/978-1-4612-1019-1Search in Google Scholar
[8]
H. Dehghan and J. Rooin,
A characterization of metric projection in
[9] H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces, preprint (2014), https://arxiv.org/abs/1410.1137. Search in Google Scholar
[10] S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 65 (2006), no. 4, 762–772. 10.1016/j.na.2005.09.044Search in Google Scholar
[11]
S. Dhompongsa and B. Panyanak,
On Δ-convergence theorems in
[12] G. Z. Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms 80 (2019), no. 4, 1155–1179. 10.1007/s11075-018-0521-3Search in Google Scholar
[13] S. Huang and Y. Kimura, A projection method for approximating fixed points of quasinonexpansive mappings in Hadamard spaces, Fixed Point Theory Appl. 2016 (2016), Paper No. 36. 10.1186/s13663-016-0523-6Search in 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), no. 3, 909–935. 10.1007/s11075-018-0633-9Search in Google Scholar
[15] L. O. Jolaoso, K. O. Oyewole, C. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem, and for finding fixed point of nonspreading mapping in Hilbert spaces, Demonstr. Math. 51 (2018), no. 1, 211–232. 10.1515/dema-2018-0015Search in Google Scholar
[16] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995), no. 4, 659–673. 10.1007/BF02566027Search in Google Scholar
[17] W. A. Kirk, Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville 2002/2003), Colecc. Abierta 64, Universidad de Sevilla, Sevilla (2003), 195–225. Search in Google Scholar
[18] W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689–3696. 10.1016/j.na.2007.04.011Search in Google Scholar
[19] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), no. 2, 166–177. 10.1007/s00013-008-2545-8Search in Google Scholar
[20]
W. Kumam, N. Pakkaranang and P. Kumam,
Modified viscosity type iteration for total asymptotically nonexpansive mappings in
[21] Y. Kurokawa and W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010), no. 6, 1562–1568. 10.1016/j.na.2010.04.060Search in Google Scholar
[22]
W. Laowang and B. Panyanak,
Strong and Δ convergence theorems for multivalued mappings in
[23] H. Liu and Y. Li, Convergence theorems for k-strictly pseudononspreading multivalued in Hilbert spaces, Adv. Pure Math. 4 (2014), 317–323. 10.4236/apm.2014.47042Search in Google Scholar
[24] P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7–8, 899–912. 10.1007/s11228-008-0102-zSearch in Google Scholar
[25] B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle 4 (1970), no. R-3, 154–158. 10.1051/m2an/197004R301541Search in Google Scholar
[26]
C. C. Okeke and C. Izuchukwu,
A strong convergence theorem for monotone inclusion and minimization problems in complete
[27]
W. Phuengrattana,
Approximation of common fixed points of two strictly pseudononspreading multivalued mappings in
[28]
S. Ranjbar and H. Khatibzadeh,
Strong and Δ convergence to a zero monotone operator in
[29] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877–898. 10.1137/0314056Search in Google Scholar
[30]
K. Samanmit,
A convergence theorem for a finite family of multivalued k-strictly pseudononspreading mappings in
[31] 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), no. 2, Article ID 77. 10.1007/s40314-019-0841-5Search in Google Scholar
[32] 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. (2019), 10.1007/s40840-019-00781-1. 10.1007/s40840-019-00781-1Search in Google Scholar
[33] G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, On nonspreading-type mappings in Hadamard spaces, Bol. Soc. Parana. Mat. (3) (2018), 10.5269/bspm.41768. 10.5269/bspm.41768Search in Google Scholar
[34] 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), no. 4, 437–446. 10.1515/apam-2018-0026Search in Google Scholar
[35]
G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo,
Strong convergence theorem for monotone inclusion problem in
[36]
R. Wangkeeree, U. Boonkong and P. Preechasilp,
Viscosity approximation methods for asymptotically nonexpansive mapping in
[37]
R. Wangkeeree and P. Preechasilp,
Viscosity approximation methods for nonexpansive mappings in
[38] D.-J. Wen, Y.-A. Chen and Y. Tang, Strong convergence of a unified general iteration for k-strictly pseudononspreading mapping in Hilbert spaces, Abstr. Appl. Anal. 2014 (2014), Article ID 219695. 10.1155/2014/219695Search in Google Scholar
[39] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240–256. 10.1112/S0024610702003332Search in Google Scholar
[40] S. S. Zhang, Integral Equations, Chongqing Press, Chongqing, 1984. Search in Google Scholar
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