Strong convergence theorems for fixed point of multi-valued mappings in Hadamard spaces. (English) Zbl 1514.47107
Summary: With the help of CN-inequality, we study fixed point of multi-valued mappings with closed bounded images and establish some strong convergence theorems involving a countable family of demicontractive mappings in Hadamard spaces. Furthermore, we use the established theorems to deduce some theorems involving a family of minimization problems, variational inequality problems, and monotone inclusion problems. We finally give examples to illustrate the results. The results obtained herein generalise some recent results in the literature.
MSC:
| 47J26 | Fixed-point iterations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H04 | Set-valued operators |
| 47J22 | Variational and other types of inclusions |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
Keywords:
common fixed point; demiclosedness-type property; Hadamard space; multi-valued demicontractive mappings; strong convergence; \(\Delta\)-convergenceReferences:
| [1] | Bruck, R. E., Asymptotic behavior of nonexpansive mappings, Contemp. Math., 18, 1-47 (1983) · Zbl 0528.47039 |
| [2] | Chidume, C., Some Geometric Properties of Banach Spaces (2009), Berlin: Springer, Berlin · Zbl 1167.47002 |
| [3] | Oden, J. T.; Kikuchi, N., Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Eng. Sci., 18, 10, 1173-1284 (1980) · Zbl 0444.76069 |
| [4] | Dehghan, H.; Izuchukwu, C.; Mewomo, O.; Taba, D.; Ugwunnadi, G., Iterative algorithm for a family of monotone inclusion problems in \(\operatorname{CAT}(0)\) spaces, Quaest. Math., 43, 7, 975-998 (2020) · Zbl 07311174 |
| [5] | Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3, 1, 133-181 (1922) · JFM 48.0201.01 |
| [6] | Mann, W. R., Mean value methods in iteration, Proc. Am. Math. Soc., 4, 3, 506-510 (1953) · Zbl 0050.11603 |
| [7] | Ishikawa, S., Fixed points by a new iteration method, Proc. Am. Math. Soc., 44, 1, 147-150 (1974) · Zbl 0286.47036 |
| [8] | Noor, M. A., New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 1, 217-229 (2000) · Zbl 0964.49007 |
| [9] | Xu, H. K., Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66, 1, 240-256 (2002) · Zbl 1013.47032 |
| [10] | Dhompongsa, S.; Kaewkhao, A.; Panyanak, B., On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on \(\operatorname{CAT}(0)\) spaces, Nonlinear Anal., Theory Methods Appl., 75, 2, 459-468 (2012) · Zbl 1443.47062 |
| [11] | Chaolamjiak, W.; Yambangwai, D.; Hammad, H. A., Modified hybrid projection methods with SP iterations for quasi-nonexpansive multivalued mappings in Hilbert spaces, Bull. Iran. Math. Soc., 47, 1399-1422 (2021) · Zbl 1473.54047 |
| [12] | Nadler, S. B., Some results on multi-valued contraction mappings, Set-Valued Mappings, Selections and Topological Properties of \(2^{\text{x}} , 64-69 (1970)\), Berlin: Springer, Berlin · Zbl 0211.26001 |
| [13] | Abbas, M.; Khan, S. H.; Khan, A. R.; Agarwal, R. P., Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Appl. Math. Lett., 24, 2, 97-102 (2011) · Zbl 1223.47068 |
| [14] | Chidume, C.; Chidume, C.; Djitte, N.; Minjibir, M., Convergence theorems for fixed points of multivalued strictly pseudocontractive mappings in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013) · Zbl 1273.47109 |
| [15] | Chidume, C.; Minjibir, M., Krasnoselskii algorithm for fixed points of multivalued quasi-nonexpansive mappings in certain Banach spaces, Fixed Point Theory, 17, 2, 301-311 (2016) · Zbl 1382.47015 |
| [16] | Minjibir, M. S.; Strong, S. S., Δ-Convergence theorems for a countable family of multi-valued demicontractive maps in Hadamard spaces, Nonlinear Funct. Anal. Appl., 27, 1, 45-58 (2022) · Zbl 1496.47120 |
| [17] | Tang, J.; Zhu, J., A new modified proximal point algorithm for a finite family of minimization problem and fixed point for a finite family of demicontractive mappings in Hadamard spaces, Nonlinear Funct. Anal. Appl., 25, 3, 563-577 (2020) · Zbl 1481.47096 |
| [18] | Kirk, W.; Shahzad, N., Fixed Point Theory in Distance Spaces (2014), Berlin: Springer, Berlin · Zbl 1308.58001 |
| [19] | Eskandani, G. Z.; Raeisi, M., On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80, 4, 1155-1179 (2019) · Zbl 07042044 |
| [20] | Izuchukwu, C.; Aremu, K. O.; Oyewole, O. K.; Mewomo, O. T.; Khan, S. H., On mixed equilibrium problems in Hadamard spaces, J. Math., 2019 (2019) · Zbl 07172800 |
| [21] | Bacak, M.: Old and new challenges in Hadamard spaces. arXiv preprint (2018). arXiv:1807.01355 |
| [22] | Bridson, M. R.; Haefliger, A., Metric Spaces of Non-positive Curvature (2013), Berlin: Springer, Berlin · Zbl 0988.53001 |
| [23] | Kirk, W., Fixed point theorems in \(\operatorname{CAT}(0)\) spaces and \(\mathbb{R} \)-trees, Fixed Point Theory Appl., 2004, 4, 309-316 (2004) · Zbl 1089.54020 |
| [24] | Bačák, M., The proximal point algorithm in metric spaces, Isr. J. Math., 194, 2, 689-701 (2013) · Zbl 1278.49039 |
| [25] | Zamani Eskandani, G.; Kim, J. K.; Moharami, R.; Lim, W. H., An extragradient method for equilibrium problems and fixed point problems of an asymptotically nonexpansive mapping in \(\operatorname{CAT}(0)\) spaces, J. Nonlinear Convex Anal., 23, 7, 1509-1523 (2022) · Zbl 1503.47101 |
| [26] | Sokhuma, K.; Sokhuma, K., Convergence theorems for two nonlinear mappings in \(\operatorname{CAT}(0)\) spaces, Nonlinear Funct. Anal. Appl., 27, 3, 499-512 (2022) · Zbl 07589360 |
| [27] | Kirk, W. A., Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 195-225 (2003) · Zbl 1058.53061 |
| [28] | Dhompongsa, S.; Panyanak, B., On △-convergence theorems in \(\operatorname{CAT}(0)\) spaces, Comput. Math. Appl., 56, 10, 2572-2579 (2008) · Zbl 1165.65351 |
| [29] | Cuntavepanit, A.; Phuengrattana, W., On solving the minimization problem and the fixed-point problem for a finite family of non-expansive mappings in \(\operatorname{CAT}(0)\) spaces, Optim. Methods Softw., 33, 2, 311-321 (2018) · Zbl 1489.47089 |
| [30] | Aremu, K.; Jolaoso, L.; Izuchukwu, C.; Mewomo, O., Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in \(\operatorname{CAT}(0)\) spaces, Ric. Mat., 69, 1, 13-34 (2020) · Zbl 1439.47045 |
| [31] | Chaipunya, P.; Kumam, P., On the proximal point method in Hadamard spaces, Optimization, 66, 10, 1647-1665 (2017) · Zbl 1409.90224 |
| [32] | Kumam, P., Chaipunya, P.: Equilibrium problems and proximal algorithms in Hadamard spaces. arXiv preprint (2018). arXiv:1807.10900 |
| [33] | Kumam, W.; Pakkaranang, N.; Kumam, P.; Cholamjiak, P., Convergence analysis of modified Picard-S hybrid iterative algorithms for total asymptotically nonexpansive mappings in Hadamard spaces, Int. J. Comput. Math., 97, 1-2, 175-188 (2020) · Zbl 07475967 |
| [34] | Phuengrattana, W.; Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235, 9, 3006-3014 (2011) · Zbl 1215.65095 |
| [35] | Bruhat, F.; Tits, J., Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci., 41, 5-251 (1972) · Zbl 0254.14017 |
| [36] | Goebel, K.; Simeon, R., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984), New York: Dekker, New York · Zbl 0537.46001 |
| [37] | Reich, S.; Shafrir, I., Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., Theory Methods Appl., 15, 6, 537-558 (1990) · Zbl 0728.47043 |
| [38] | Brown, K. S., Buildings, Buildings, 76-98 (1989), Berlin: Springer, Berlin · Zbl 0715.20017 |
| [39] | Dhompongsa, S.; Kirk, W. A.; Sims, B., Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., Theory Methods Appl., 65, 4, 762-772 (2006) · Zbl 1105.47050 |
| [40] | Dhompongsa, S.; Kirk, W.; Panyanak, B., Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8, 1, 35 (2007) · Zbl 1120.47043 |
| [41] | Kirk, W.; Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Anal., Theory Methods Appl., 68, 12, 3689-3696 (2008) · Zbl 1145.54041 |
| [42] | Kakavandi, B., Weak topologies in complete \(\operatorname{CAT}(0)\) metric spaces, Proc. Am. Math. Soc., 141, 3, 1029-1039 (2013) · Zbl 1272.53031 |
| [43] | Maingé, P. E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16, 7-8, 899-912 (2008) · Zbl 1156.90426 |
| [44] | Dehghan, H., Rooin, J.: Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces. arXiv preprint (2014). arXiv:1410.1137 |
| [45] | Ariza-Ruiz, D.; Leuştean, L.; López-Acedo, G., Firmly nonexpansive mappings in classes of geodesic spaces, Trans. Am. Math. Soc., 366, 8, 4299-4322 (2014) · Zbl 1524.47056 |
| [46] | Jost, J., Convex functionals and generalized harmonic maps into spaces of non positive curvature, Comment. Math. Helv., 70, 1, 659-673 (1995) · Zbl 0852.58022 |
| [47] | Khatibzadeh, H.; Ranjbar, S., A variational inequality in complete \(\operatorname{CAT}(0)\) spaces, J. Fixed Point Theory Appl., 17, 3, 557-574 (2015) · Zbl 1323.47066 |
| [48] | Dehghan, H., Rooin, J.: A characterization of metric projection in \(\operatorname{CAT}(0)\) spaces. arXiv preprint (2013). arXiv:1311.4174 |
| [49] | Aremu, K. O.; Izuchukwu, C.; Abass, H. A.; Mewomo, O. T., On a viscosity iterative method for solving variational inequality problems in Hadamard spaces, Axioms, 9, 4 (2020) · Zbl 1443.47057 |
| [50] | Khatibzadeh, H.; Ranjbar, S., Monotone operators and the proximal point algorithm in complete \(\operatorname{CAT}(0)\) metric spaces, J. Aust. Math. Soc., 103, 1, 70-90 (2017) · Zbl 1489.47093 |
| [51] | Rezaei, M.; Borjalilu, N., A dynamic risk assessment modeling based on fuzzy ANP for safety management systems, Aviation, 22, 4, 143-155 (2018) |
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