Some results on fixed points of multifunctions on abstract metric spaces. (English) Zbl 1217.54055

Summary: Recently, H. E. Kunze, D. La Torre and E. R. Vrscay [J. Math. Anal. Appl. 330, No. 1, 159-173 (2007; Zbl 1115.47043)] proved some fixed point results for multifunctions in metric spaces. Sh. Rezapour and R. H. Haghi [Numer. Funct. Anal. Optim. 30, No. 7-8, 825–832 (2009; Zbl 1171.54033)] adapted these results to the case of abstract (cone) metric spaces when the underlying cone is normal with normal constant \(M=1\). The aim of this paper is to show that these results remain valid in the case when \(M>1\). Introducing new contraction conditions, our results generalize fixed point theorems of Covitz and Nadler, Kunze et al. and Rezapour and Haghi. An example is given to distinguish our results from the known ones. In addition, the case when two mappings are considered is treated.


54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
Full Text: DOI


[1] Kantorovich, L.V., Lineare halbeordnete Räume, Mat. sb., 2, 44, 121-168, (1937) · Zbl 0016.40502
[2] Kantorovich, L.V., The majorant principle and newton’s method, Dokl. akad. nauk SSSR (NS), 76, 17-20, (1951), (in Russian)
[3] Vandergraft, J.S., Newton method for convex operators in partially ordered spaces, SIAM J. numer. anal., 4, 3, 406-432, (1967) · Zbl 0161.35302
[4] Zabreĭko, P.P., \(K\)-metric and \(K\)-normed spaces: survey, Collect. math., 48, 4-6, 825-859, (1997) · Zbl 0892.46002
[5] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag · Zbl 0559.47040
[6] Huang, L.G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. math. anal. appl., 332, 2, 1468-1476, (2007) · Zbl 1118.54022
[7] M. Asadi, H. Soleimani, S.M. Vaezpour, An order on subsets of cone metric spaces and fixed points of set-valued contractions, Fixed Point Theory Appl., vol. 2009, doi:10.1155/2009/723203. · Zbl 1187.47041
[8] Feng, Y.; Liu, S., Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. math. anal. appl., 317, 103-112, (2006) · Zbl 1094.47049
[9] Klim, D.; Wardowski, D., Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear anal., 71, 5170-5175, (2009) · Zbl 1203.54042
[10] Rezapour, Sh.; Haghi, R.H., Fixed point of multifunctions on cone metric spaces, Numer. funct. anal. optim., 30, 825-832, (2009) · Zbl 1171.54033
[11] Haghi, R.H.; Rezapour, Sh., Fixed points of multifunctions on regular cone metric spaces, Expo. math., 28, 71-77, (2010) · Zbl 1193.47058
[12] Wardowski, D., Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear anal., 71, 512-516, (2009) · Zbl 1169.54023
[13] Kunze, H.E.; La Torre, D.; Vrscay, E.R., Contraction multifunctions, fixed point inclusions and iterated multifunction system, J. math. anal. appl., 330, 159-173, (2007) · Zbl 1115.47043
[14] S. Radenović, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (2011) (in press).
[15] Du, Wei-Shih, A note on cone metric fixed point theory and its equivalence, Nonlinear anal., 72, 2259-2261, (2010) · Zbl 1205.54040
[16] Amini-Harandi, A.; Fakhar, M., Fixed point theory in cone metric spaces obtained via the scalarization method, Comput. math. appl., 59, 3529-3534, (2010) · Zbl 1197.54055
[17] Z. Kadelburg, S. Radenović, V. Rakočević, A note on equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., in press (doi:10.1016/j.aml.2010.10.030).
[18] Kreĭn, M.G., Propriétés fondamentales des ensembles conique normaux dans l’espace de Banach, C. R. acad. sci. URSS (NS), 28, 13-17, (1940) · Zbl 0024.12202
[19] Schaefer, H.H., Topological vector spaces, (1971), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0212.14001
[20] Wong, Y.C.; Ng, K.F., Ordered topological vector spaces, (1973), Clarendon Press Oxford
[21] Kreĭn, M.G.; Rutman, M.A., Linear operators leaving invariant cone in Banach spaces, Uspekhi mat. nauk (NS), 3, 1, 3-95, (1948), (in Russian); Transl. in Amer. Math. Soc. Transl. 26 (1950) · Zbl 0030.12902
[22] Wilson, W.A., On semimetric spaces, Amer. J. math., 53, 361-373, (1931) · JFM 57.0735.01
[23] Covitz, H.; Nadler, S.B., Multi-valued contraction mappings in generalized metric spaces, Israel J. math., 8, 5-11, (1970) · Zbl 0192.59802
[24] Azam, A.; Arshad, M.; Beg, I., Existence of fixed points in complete cone metric spaces, Int. J. mod. math., 5, 91-99, (2010) · Zbl 1203.54034
[25] Abbas, M.; Rhoades, B.E., Fixed and periodic point results in cone metric spaces, Appl. math. lett., 21, 511-515, (2008) · Zbl 1167.54014
[26] Ćirić, Lj.B., A generalization of banach’s contraction principle, Proc. amer. math. soc., 45, 267-273, (1974) · Zbl 0291.54056
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