×

Strong and \(\Delta\)-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT\((0)\) spaces. (English) Zbl 1469.65099

Summary: Let \(K\) be a nonempty closed and convex subset of a complete CAT(0) space. Let \(T_i : K \rightarrow \text{C} \text{B} \left(K\right)\), \(i = 1,2, \dots, m\), be a family of multivalued demicontractive mappings such that \(F : = \bigcap_{i = 1}^m F(T_i) \neq \varnothing\). A Krasnoselskii-type iterative sequence is shown to \(\Delta\)-converge to a common fixed point of the family \(\left\{T_i, i = 1,2, \dots, m\right\}\). Strong convergence theorems are also proved under some additional conditions. Our theorems complement and extend several recent important results on approximation of fixed points of certain nonlinear mappings in CAT\((0)\) spaces. Furthermore, our method of the proof is of special interest.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bridson, M. R.; Haefliger, A., Metric Spaces of Non-Positive Curvature (1999), Berlin, Germany: Springer, Berlin, Germany · Zbl 0988.53001
[2] Brown, K. S., Buildings (1989), New York, NY, USA: Springer, New York, NY, USA
[3] Burago, D.; Burago, Y.; Ivanov, S., A course in metric geometry, Graduate Studies in Math., 33 (2001), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0981.51016
[4] Kirk, W. A., Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta, 64, 195-225 (2003), Sevilla, Spain: Universidad de Sevilla, Sevilla, Spain · Zbl 1058.53061
[5] Kirk, W. A., Geometry and fixed point theory II, Proceedings of the International Conference of Fixed Point Theory and Application, Yokohama Publishers · Zbl 1083.53061
[6] Chaoha, P.; Phon-on, A., A note on fixed point sets in CAT(0) spaces, Journal of Mathematical Analysis and Applications, 320, 2, 983-987 (2006) · Zbl 1101.54040
[7] Dhompongsa, S.; Kaewkhao, A.; Panyanak, B., Lim’s theorems for multivalued mappings in CAT (0) spaces, Journal of Mathematical Analysis and Applications, 312, 2, 478-487 (2005) · Zbl 1086.47019
[8] Dhompongsa, S.; Kirk, W. A.; Sims, B., Fixed points of uniformly Lipschitzian mappings, Nonlinear Analysis: Theory, Methods & Applications, 65, 4, 762-772 (2006) · Zbl 1105.47050
[9] Dhompongsa, S.; Fupinwong, W.; Kaewkhao, A., A common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 12, 4268-4273 (2009) · Zbl 1191.47077
[10] Fujiwara, K.; Nagano, K.; Shioya, T., Fixed point sets of parabolic isometries of CAT(0)-spaces, Commentarii Mathematici Helvetici, 81, 2, 305-335 (2006) · Zbl 1098.53025
[11] Leustean, L., A quadratic rate of asymptotic regularity for CAT(0)-spaces, Journal of Mathematical Analysis and Applications, 325, 1, 386-399 (2007) · Zbl 1103.03057
[12] Shahzad, N., Invariant approximations in \(C A T(0)\) spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 12, 4338-4340 (2009) · Zbl 1167.47042
[13] Shahzad, N.; Markin, J., Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces, Journal of Mathematical Analysis and Applications, 337, 2, 1457-1464 (2008) · Zbl 1137.47043
[14] Lim, T. C., Remarks on some fixed point theorems, Proceedings of the American Mathematical Society, 60, 179-182 (1976) · Zbl 0346.47046
[15] Kirk, W. A.; Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods & Applications, 68, 12, 3689-3696 (2008) · Zbl 1145.54041
[16] Dhompongsa, D.; Panyanak, B., On Δ-convergence theorems in CAT(0) spaces, Computers & Mathematics with Applications, 56, 2572-2579 (2008) · Zbl 1165.65351
[17] Chidume, C. E.; Chidume, C. O.; Djitté, N.; Minjibir, M. S., Convergence theorems for fixed points of multivalued strictly pseudocontractive mappings in Hilbert spaces, Abstract and Applied Analysis, 2013 (2013) · Zbl 1273.47109
[18] Chidume, C. E.; Ezeora, J. N., Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-conttractive mappings, Fixed Point Theory and Applications, 2014, article 111 (2014) · Zbl 1345.47035
[19] Isiogugu, F. O.; Osilike, M. O., Convergence theorems for new classes of multi-valued hemicontractive-type mapping, Fixed Point Theory and Applications, 2014, article 93 (2014) · Zbl 1332.47046
[20] Sokhuma, K., On Δ-convergence theorems for a pair of single-valued and multivalued nonexpansive mappings in CAT(0) spaces, Journal of Mathematical Analysis, 4, 2, 23-31 (2013) · Zbl 1312.47062
[21] Bruhat, F.; Tits, J., Groupes Réductifs sur un Corp Local. I. Donneés Radicielles Valuées, 41 (1972), Institut des Hautes Études Scientifiques
[22] Kopecká, E.; Reich, S., Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball, Nonlinear Analysis, 70, 9, 3187-3194 (2009) · Zbl 1166.47050
[23] Dhompongsa, S.; Kirk, W. A.; Panyanak, B., Nonexpansive set-valued mappings in metric and Banach spaces, Journal of Nonlinear and Convex Analysis, 8, 1, 35-45 (2007) · Zbl 1120.47043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.