A NEW ALGORITHM FOR MONOTONE INCLUSION PROBLEMS AND FIXED POINTS ON HADAMARD MANIFOLDS WITH APPLICATIONS

doi:10.1007/s10473-021-0413-9

Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1250-1262.doi: 10.1007/s10473-021-0413-9

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A NEW ALGORITHM FOR MONOTONE INCLUSION PROBLEMS AND FIXED POINTS ON HADAMARD MANIFOLDS WITH APPLICATIONS

Shih-sen CHANG¹, Jinfang TANG², Chingfeng WEN³

1. Center for General Education, China Medical University, Taichung 40402, Taiwan, China;
2. Department of Mathematics, Yibin University, Yibin 644007, China;
3. Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan, China

Received:2020-05-31 Revised:2020-09-16 Online:2021-08-25 Published:2021-09-01
Contact: Shih-sen CHANG E-mail:changss2013@163.com
Supported by:
This study was supported by the Natural Science Foundation of China Medical University, Taiwan. This work was also supported by Scientific Research Fund of SiChuan Provincial Education Department (14ZA0272).

Abstract

Abstract: In this article, we propose a new algorithm and prove that the sequence generalized by the algorithm converges strongly to a common element of the set of fixed points for a quasi-pseudo-contractive mapping and a demi-contraction mapping and the set of zeros of monotone inclusion problems on Hadamard manifolds. As applications, we use our results to study the minimization problems and equilibrium problems in Hadamard manifolds.

Key words: Monotone inclusion problem, quasi-pseudo-contractive mapping, demi-contraction mapping, maximal monotone vector field, quasi-nonexpansive mappings, Hadamard manifold

CLC Number:

49J53

Shih-sen CHANG, Jinfang TANG, Chingfeng WEN. A NEW ALGORITHM FOR MONOTONE INCLUSION PROBLEMS AND FIXED POINTS ON HADAMARD MANIFOLDS WITH APPLICATIONS[J].Acta mathematica scientia,Series B, 2021, 41(4): 1250-1262.

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References

[1] Rockafellar R T. Monotone operators and the proximal point algorithm. SIAM J Control Optim, 1976, 14(5):877-898. doi:10.1137/0314056
[2] Chang S S. Set-valued variational inclusions in banach spaces. J Math Anal Appl, 2000, 248(2):438-454. doi:10.1006/jmaa.2000.6919.
[3] Chang S S. Existence and approximation of solutions for set-valued variational inclusions in Banach spaces. Nonlinear Anal, 2001, 47(1):583-594. doi:10.1016/S0362-546X(01)00203-6
[4] Chang S S, Cho Y J, Lee B S, Jung I H. Generalized set-valued variational inclusions in Banach spaces. J Math Anal Appl, 2000, 246(2):409-422. doi:10.1006/jmaa.2000.6795
[5] Manaka H, Takahashi W. Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space. Cubo, 2011, 13(1):11-24. doi:10.4067/S0719-06462011000100002
[6] Sahu D R, Ansari Q H, Yao J C. The Prox-Tikhonov forward-backward method and applications. Taiwan J Math, 2015, 19(2):481-503. doi:10.11650/tjm.19.2015.4972
[7] Takahashi S, Takahashi W, Toyoda M. Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J Optim Theory Appl, 2010, 147(1):27-41. doi:10.1007/s10957-010-9713-2
[8] Chang S S, Lee J H W, Chan C K. Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl Math Mech Engl Ed, 2008, 29(5):571-581. doi:10.1007/s10483-008-0502
[9] Li C, Yao J C. Variational inequalities for set-valued vector fields on Riemannian manifolds:convexity of the solution set and the proximal point algorithm. SIAM J Control Optim, 2012, 50(4):2486-2514
[10] Li C, López G, Martín-Márquez V. Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J Lond Math Soc, 2009, 79(3):663-683. doi:10.1112/jlms/jdn087
[11] Li C, López G, Martín-Márquez V. Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwan J Math, 2010, 14(2):541-559
[12] Ansari Q H, Babu F, Li X. Variational inclusion problems in Hadamard manifolds. J Nonlinear Convex Anal, 2018, 19(2):219-237
[13] Suliman Al-Homidan, Qamrul Hasan Ansari, Feeroz Babu. Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds. Numerical Functional Analysis and Optimization, 2019, 40(6):621-653
[14] Qamrul Hasan Ansari, Feeroz Babu, Jen-Chih Yao. Regularization of proximal point algorithms in Hadamard manifolds. J Fixed Point Theory Appl, 2019, 21:25. https://doi.org/10.1007/s11784-019-0658-2
[15] Sakai T. Riemannian Geometry, Translations of Mathematical Monographs. Providence, RI:American Mathematical Society, 1996
[16] Ferreira O P, Oliveira P R. Proximal point algorithm on Riemannian manifolds. Optimization, 2002, 51(2):257-270. doi:10.1080/02331930290019413.
[17] Iusem A N. An iterative algorithm for the variational inequality problem. Comput Appl Math, 1994, 13:103-114
[18] Li C, López G, Martín-Márquez V, Wang J -H. Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set-Valued Anal, 2011, 19(3):361-383. doi:10.1007/s11228-010-0169-1
[19] Liu X D, Chang S S. Convergence theorems on total asymptotically demicontractive and hemicontractive mappings in CAT(0) spaces. J Ineq Appl, 2014, 436(2014)
[20] da Cruz Neto J X, Ferreira O P, Lucambio Pérez L R. Monotone point-to-set vector fields. Balkan J Geometry Appl, 2000, 5(1):69-79
[21] Udriste C. Convex Functions and Optimization Methods Manifolds. Dordrecht, The Netherlands:Kluwer Academic Publishers, 1994
[22] Colao V, López G, Marino G, Martín-Márquez V. Equilibrium problems in hadamard manifolds. J Math Anal Appl, 2012, 388(1):61-71

A NEW ALGORITHM FOR MONOTONE INCLUSION PROBLEMS AND FIXED POINTS ON HADAMARD MANIFOLDS WITH APPLICATIONS

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