Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

• Autorzy: Abbas Mujahid , Iqbal Hira , de la Sen Manuel

Identyfikatory

DOI

10.1515/dema-2019-0031

• Języki publikacji: EN

Abstrakty

EN

We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete R-trees without the endpoint condition.

Słowa kluczowe

EN

viscosity approximation method; variational inequality; multivalued nonexpansive mapping; CAT(0) space

PL

metoda aproksymacji; nierówność wariacyjna; odwzorowanie wielowartościowe; przestrzenie CAT(0)

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 347--360
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Abbas Mujahid

• Department of Mathematics, Government College University, Lahore 54000, Pakistan
• Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

autor

Iqbal Hira

• Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan

autor

de la Sen Manuel

• Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia) - P.O. Box 644 - Bilbao, Barrio Sarriena, 48940 - Leioa, Spain

Bibliografia

• [1] Moudafi A., Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 2008, 241, 46-55
• [2] Xu H. K., Alghamdi M. A., Shahzad N., The implicit midpoint rule for nonexpansive mappings in Banach spaces, Fixed Point Theory, 2016, 17(2), 509-517
• [3] Ke Y., Ma C., The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl.,2015, 2015:190
• [4] Abkar A., Eslamian M., Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces, Nonlinear Anal., 2012, 75(4), 1895-1903
• [5] Uddin I., Nieto J. J., Ali J., One-step iteration scheme for multivalued nonexpansive mappings in CAT (0) spaces, Mediterr. J.Math., 2016, 13(3), 1211-1225
• [6] Uddin I., Ali J., Nieto J. J., An iteration scheme for a family of multivalued mappings in CAT(0) spaces with an application toimage recovery, Rev. R. Acad. Cienc. Exactas Fés. Nat. Ser. A Math. RACSAM, 2018, 112(2), 373-384
• [7] Wu X. B., Zhao L. N., Viscosity approximation methods for multivalued nonexpansive mappings, Mediterr. J. Math., 2016,13, 2645-2657
• [8] Preechasilp P., Viscosity approximation methods for implicit midpoint rule of nonexpansive mappings in geodesic spaces, Bull. Malays. Math. Sci. Soc., 2018, 41(3), 1561-1579
• [9] Panyanak B., Suantai S., Viscosity approximation methods for multivalued nonexpansive mappings in geodesic spaces, Fixed Point Theory Appl., 2015, 2015:114
• [10] Wangkeeree R., Preechasilp P., Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl., 2013, 2013:160
• [11] Xiong T. J., Lan H. Y., Strong convergence of new two-step viscosity iterative approximation methods for set-valued nonexpansive mappings in CAT(0) spaces, J. Funct. Spaces, 2018, Article ID 1280241
• [12] Xiong T. J., Lan H. Y., New two-step viscosity approximation methods of fixed points for set-valued nonexpansive mappings associated with contraction mappings in CAT(0) spaces, J. Comput. Anal. Appl., 2019, 26(5), 899-909
• [13] Burago D. et al., A Course in Metric Geometry, vol. 33, Am. Math. Soc., Providence, 2001
• [14] Bridson M., Haefliger A., Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999
• [15] Brown K. S., Buildings, Springer, New York, 1989
• [16] Dhompongsa S., Kirk W. A., On∆-convergence theorems in CAT(0) spaces, Comput. Math Appl., 2008, 56, 2572-2579
• [17] Kirk W. A., Geodesic geometry and fixed point theory II, In: International Conference on Fixed Point Theory and Applications,Yokohama Publ., 2004, 113-142
• [18] Chaoha P., Phon-on A., A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 2006, 320(2), 983-987
• [19] Berg I.D., Nikolaev I.G., Quasi linearization and curvature of Alexandrov spaces, Geom. Dedic., 2008, 133, 195-218
• [20] Dehghan H., Rooin J., A characterization of metric projection in CAT(0) spaces, In: Proceedings of International Conferenceon Functional Equation, Geometric Functions and Applications (ICFGA), Tabriz, Iran, 2012, 41-43
• [21] Xu H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 2003, 116 , 659-678
• [22] Aksoy A. G., Khamsi M. A., A selection theorem in metric trees, Proc. Am. Math. Soc., 2006, 134, 2957-2966
• [23] Markin J., Fixed points for generalized nonexpansive mappings in R-trees, Comput. Math. Appl., 2011, 62(12), 4614-4618

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).