A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

C. Izuchukwu, K. O. Aremu, A. A. Mebawondu, O. T. Mewomo

Abstract

The main purpose of this paper is to introduce a viscosity-type proximal point algorithm, comprising of a nonexpansive mapping and a finite sum of resolvent operators associated with monotone bifunctions. A strong convergence of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a nonexpansive mapping is established in a Hadamard space. We further applied our results to solve some optimization problems in Hadamard spaces.


Keywords

equilibrium problems; monotone bifunctions; variational inequalities; convex feasibility problems; minimization problems; viscosity iterations; CAT(0) space

Subject classification

47H09; 47H10; 49J20; 49J40.

Full Text:

PDF

References

K. O. Aremu, C. Izuchukwu, G. C. Ugwunnadi and O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT(0) spaces, Demonstr. Math. 51 (2018), 277-294. https://doi.org/10.1515/dema-2018-0022

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

M. Bacák and S. Riech, The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces, J. Fixed Point Theory Appl. 16 (2014), 189-202. https://doi.org/10.1007/s11784-014-0202-3

I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), 195-218. https://doi.org/10.1007/s10711-008-9243-3

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim Theory Appl. 90 (1996), 31-43. https://doi.org/10.1007/BF02192244

Bridson and A. Haefliger, Metric spaces of nonpositive curvature, Springer-Verlag, Berlin, Heidelberg, New York, 1999. https://doi.org/10.1007/978-3-662-12494-9

F. Bruhat and J. Tits, Groupes réductifs sur un corp local, I. Donneés Radicielles Valuées, Institut des Hautes Études Scientifiques 41 (1972). https://doi.org/10.1007/bf02715544

P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320, no. 2 (2006), 983-987. https://doi.org/10.1016/j.jmaa.2005.08.006

V. Colao, G. López, G. Marino, V. Martín-Márquez, Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl. 388 (2012), 61-77. https://doi.org/10.1016/j.jmaa.2011.11.001

P. L. Combetes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117-136.

H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces, arXiv:1410.1137v1[math.FA]2014.

S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 64, no. 4 (2006), 762-772. https://doi.org/10.1016/j.na.2005.09.044

S. Dhompongsa and B. Panyanak, On △-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036

J. N. Ezeora and C. Izuchukwu, Iterative approximation of solution of split variational inclusion problems, Filomat, to appear.

K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Marcel Dekker, New York, (1984).

A. N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., Ser. B 116 (2009), 259-273. https://doi.org/10.1007/s10107-007-0125-5

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. Algor., to appear. https://doi.org/10.1007/s11075-018-0633-9

B. A. Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141, no. 3 (2013), 1029-1039. https://doi.org/10.1090/S0002-9939-2012-11743-5

H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math Soc. 103, no. 1 (2017), 70-90. https://doi.org/10.1017/S1446788716000446

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

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

P. Kumam and P. Chaipunya, Equilibrium problems and proximal algorithms in Hadamard spaces, arXiv: 1807.10900v1 [math.oc].

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

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

B. Martinet, Régularisation d'Inéquations Variationnelles par approximations successives, Rev.Francaise d'Inform. et de Rech. Opérationnelle 3 (1970), 154-158. https://doi.org/10.1051/m2an/197004r301541

M. A. Noor and K. I. Noor, Some algorithms for equilibrium problems on Hadamard manifolds, J. Inequal. Appl. 2012:230, 8 pp. https://doi.org/10.1186/1029-242x-2012-230

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces, Optimization Methods and Software, to appear.

https://doi.org/10.1080/10556788.2018.1472259

O. K. Oyewole, L. O Jolaoso, C. Izuchukwu and O. T. Mewomo, On approximation of common solution of finite family of mixed equilibrium problems involving μ−α relaxed monotone mapping in a Banach space, U. P. B. Sci. Bull., Series A, to appear.

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537-558. https://doi.org/10.1016/0362-546X(90)90058-O

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

S. Saejung, Halpern's iteration in CAT(0) spaces, Fixed Point Theory Appl. 2010, Art. ID 471781, 13 pp. https://doi.org/10.1155/2010/471781

Y. Song and X. Liu, Convergence comparison of several iteration algorithms for the common fixed point problems, Fixed Point Theory Appl. 2009, Art. ID 824374, 13 pp. https://doi.org/10.1155/2009/824374

R. Suparatulatorn, P. Cholamjiak and S. Suantai, On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods and Software 32 (2017), 182-192. https://doi.org/10.1080/10556788.2016.1219908

T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl. 1 (2005), 103-123. https://doi.org/10.1155/FPTA.2005.103

J. Tang, Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) Spaces, Abstr. Appl. Anal. 2014, Art. ID 389804, 9 pages. G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT(0) spaces, Afr. Mat., to appear.

G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT(0) spaces, Afr. Mat., to appear. https://doi.org/10.1007/s13370-018-0633-x

R. Wangkeeree and P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl. 2013, Art. ID 93. https://doi.org/10.1186/1029-242X-2013-93

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66, no. 1 (2002), 240-256. https://doi.org/10.1112/S0024610702003332

Abstract Views

801
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces
K. O. Aremu, L. O. Jolaoso, C. Izuchukwu, O. T. Mewomo
Ricerche di Matematica  year: 2019  
doi: 10.1007/s11587-019-00446-y



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt