Proximal Point Algorithm for Nonexpansive Mappings in Hadamard Spaces Based on SRJ Iteration Process
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2023-11-05 https://doi.org/10.14419/ijamr.v12i1.32376 -
Abstract
In this paper, We provide a new modified proximal point approach utilizing fixed point iterates of nonexpansive mappings in Hadamard space and show that the sequence created by our iterative process converges to a minimizer of a convex function and a fixed point of mappings. Finally, we present a numerical illustration for supporting our main result. Our results obtained in this paper improve, extend and unify results of Khan-Abbas [23], Cholamjiak et al. [10] and Dashputre et al. [11].
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References
- R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton’s method on Riemannian manifolds and a geometric model for human spine, IMA Journal of Numerical Analysis, 22(3)(2002), 359-390, DOI: 10.1093/imanum/22.3.359.
- L. Ambrosio, N. Gigli and G. Savare, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ´ETH Zurich, Birkh ¨ auser, Basel, (2008), DOI: 10.1007/978-3-7643-8722-8. ¨
- D. Ariza-Ruiz, L. Leu stean and G. Lopez-Acedo, Firmly nonexpansive mappings in classes of geodesic spaces, ´ Transactions of the American Mathematical Society, 366 (2014), 4299 – 4322.
- M. Bacak, Computing medians and means in Hadamard spaces, ´ SIAM Journal on Optimization, 24(3)(2014), 1542 – 1566, DOI: 10.1137/140953393.
- M. Bacak, The proximal point algorithm in metric spaces, ´ Israel Journal of Mathematics, 194(2013), 689 – 701, DOI: 10.1007/s11856-012-0091-3.
- M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer Science and Business Media, 319(2013), URL:
- https://link.springer.com/book/10.1007/978-3-662-12494-9.
- K.S. Brown and K.S Brown, Buildings, Springer, (1989).
- F. Bruhat and J. Tits, Groupes Reductifs Sur Un Corps local, ´ Publications Mathematiques de l’Institut des Hautes ´ Etudes Scientifiques, 44(1)(1972), 5–251, DOI: 10.1007/BF02715544.
- D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 33(2001), DOI: 10.1090/gsm/033.
- P. Cholamjiak, A. A. N. Abdou and Y. J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Applications, 2015(2015), 227, 1– 13 DOI: 10.1186/s13663-015-0465-4.
- S. Dashputre, R. Tiwari and J. Shrivas, A new iterative algorithm for generalized (α, β)-nonexpansive mapping in CAT (0) space, Adv. Fixed Point Theory, 13(2023), https://doi.org/10.28919/afpt/8084.
- S. Dhompongsa and B. Panyanak, On-convergence theorems in CAT(0) spaces, Computers & Mathematics with Applications, 56(10)(2008), 2572–2579, DOI: 10.1016/j.camwa.2008.05.036.
- S. Dhompongsa, W.A. Kirk, and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, Journal of Nonlinear and Convex Analysis, 8(2007), 35 – 45.
- S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear analysis: theory, methods & applications, 65(4)(2006), 762–772, DOI: 10.1016/j.na.2005.09.044.
- O. P. Ferreira and P. R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization, 51(2)(2002), 257 – 270, DOI:
- 1080/02331930290019413.
- H. Fukhar-ud-din, Strong convergence of an Ishikawa-type algorithm in CAT(0) spaces, Fixed Point Theory and Applications, 207(2013), DOI: 10.1186/1687-1812-2013-207.
- K. Goebel and R. Simeon, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Dekker, (1984).
- M. Gromov, Hyperbolic groups, In: Essays in Group Theory, S. M. Gersten (eds), Mathematical Sciences Research Institute Publications, 8)(1987), DOI: 10.1007/978-1- 4613-9586-7-3.
- O. Guler, On the convergence of the proximal point algorithm for convex minimization, ¨ SIAM Journal on Control and Optimization, 29(2)(1991), 403 – 419, DOI: 10.1137/0329022.
- B. Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society, (1967), 957–961, DOI: 10.1090/S0002-9904-1967- 11864-0.
- J. Jost, Convex functionals and generalized harmonic maps into spaces of non positive curvature, Commentarii Mathematici Helvetici, 70(1995), 659 – 673, DOI: 10.1007/BF02566027.
- S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, Computers & Mathematics with Applications, 106(2)(2000), 226 – 240, DOI: 10.1006/jath.2000.3493.
- S. H. Khan and M. Abbas, Strong and ∆-convergence of some iterative schemes in CAT(0) spaces, Computers and Mathematics with Applications, 61(1)(2011), 109 – 116, DOI: 10.1016/j.camwa.2010.10.037.
- W. A. Kirk, Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta. University Seville Secretary of Publications, Seville, Spain, 64(2003), 195 – 225.
- W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods and Applications, 68(13)(2008), 3689 – 3696, DOI: 10.1016/j.na.2007.04.011.
- T. Laokul and B. Panyanak, Approximating fixed points of nonexpansive mappings in CAT(0) spaces, International Journal of Mathematical Analysis, 3(25-28)(2009), 1305 – 1315, URL: http: //cmuir.cmu.ac.th/jspui/handle/6653943832/59721.
- W. Laowang and B. Panyanak, Strong and ∆-convergence theorems for multivalued mappings in CAT(0) spaces, Journal of Inequalities and Applications, 2009(2009), Article ID 730132, DOI: 10.1155/2009/730132.
- C. Li, G. Lopez and V. Mart ´ ´ın-Marquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, ´ Journal of the London Mathematical Society, 79(3)(2009), 663 – 683, DOI: 10.1112/jlms/jdn087.
- T. C. Lim, Remarks on some fixed point theorems, Proceedings of the American Mathematical Society, 60(1976), 179-182, DOI: 10.1090/S0002-9939- 1976-0423139-X.
- B. Martinet, Regularisation d’in ´ equations variationnelles par approximations successives, ´ Revue franc¸aise d’informatique et de recherche operationnelle ´, 4(R-3)(1970), 154 – 158,
- U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Communications in Analysis and Geometry, 6(2)(1998), 199 – 253, DOI: 10.4310/CAG.1998.v6.n2.a1.
- A. Panwar, Jyoti, P. Mor and Pinki, Proximal point algorithm based on AP iterative technique for nonexpansive mappings in CAT(0) spaces, Communications in Mathematics and Applications, 14(1)(2023), 117 – 129, DOI: 10.26713/cma.v14i1.1831
- E. A. P. Quiroz and P. R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on hadamard manifolds, Journal of Convex Analysis, 16(1)(2009), 49 – 69.
- R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14(5)(1976), 877 – 898, DOI: 10.1137/0314056.
- S. T. Smith, Optimization techniques on Riemannian manifolds, Hamiltonian and gradient flows, algorithms and control, Fields Institute Communications, 3(1994), 113 – 136.
- C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications book series, 297(1994), DOI: 10.1007/978-94-015-8390-9.
- J. H. Wang and G. Lopez, Modified proximal point algorithms on Hadamard manifolds, ´ Optimization, 60(6)(2011), 697 – 708, DOI:
- 1080/02331934.2010.505962.
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How to Cite
Dashputre, S., Tiwari, R., & Shrivas, J. (2023). Proximal Point Algorithm for Nonexpansive Mappings in Hadamard Spaces Based on SRJ Iteration Process. International Journal of Applied Mathematical Research, 12(1), 3-10. https://doi.org/10.14419/ijamr.v12i1.32376