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A note on the equivalence of some metric and cone metric fixed point results. (English) Zbl 1213.54067
Let $$E$$ be a Hausdorff topological vector space and $$K$$ be a proper closed convex cone of it, with nonempty interior. The following is the main result of the paper:
Theorem. Let $$(X,d)$$ be a $$K$$-metric space. Take $$e\in \text{int}(K)$$ and let $$q_e$$ be the Minkowski functional of $$[-e.e]$$. Then i) $$d_q:=q_e\circ d$$ is a standard metric on $$X$$, ii) $$d(x_1,y_1)\leq d(x_2,y_2)$$ $$\Rightarrow$$ $$d_q(x_1,y_1)\leq d_q(x_2,y_2)$$. As a consequence, most of the fixed point results for $$K$$-metric spaces are deductible from their standard versions ($$K=\mathbb R_+$$).

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces
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##### References:
 [1] Kantorovich, L.V., The majorant principle and newton’s method, Dokl. akad. nauk SSSR (NS), 76, 17-20, (1951) [2] Kantorovich, L.V., On some further applications of the Newton approximation method, Vestn. leningr. univ. ser. mat. mekh. astron., 12, 7, 68-103, (1957) · Zbl 0091.11502 [3] Vandergraft, J.S., Newton’s method for convex operators in partially ordered spaces, SIAM J. numer. anal., 4, 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] Aliprantis, C.D.; Tourky, R., () [7] 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 [8] Rezapour, Sh.; Hamlbarani, R., Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings, J. math. anal. appl., 345, 719-724, (2008) · Zbl 1145.54045 [9] P. Raja, S.M. Vaezpour, Some extensions of Banach’s contraction principle in complete metric spaces, Fixed Point Theory Appl. 2008 doi:10.1155/2008/768294. · Zbl 1148.54339 [10] Abbas, M.; Jungck, G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. math. anal. appl., 341, 416-420, (2008) · Zbl 1147.54022 [11] Vetro, P., Common fixed points in cone metric spaces, Rend. circ. mat. Palermo (2), 56, 464-468, (2007) · Zbl 1196.54086 [12] Abbas, M.; Rhoades, B.E., Fixed and periodic point results in cone metric spaces, Appl. math. lett., 21, 511-515, (2008) · Zbl 1167.54014 [13] Ilić, D.; Rakočević, V., Common fixed points for maps on cone metric space, J. math. anal. appl., 341, 876-882, (2008) · Zbl 1156.54023 [14] Ilić, D.; Rakočević, V., Quasi-contraction on a cone metric space, Appl. math. lett., 22, 728-731, (2009) · Zbl 1179.54060 [15] Kadelburg, Z.; Radenović, S.; Rakočević, V., Remarks on quasi-contraction on a cone metric space, Appl. math. lett., 22, 1674-1679, (2009) · Zbl 1180.54056 [16] Rezapour, Sh.; Haghi, R.H.; Shahzad, N., Some notes on fixed points of quasi-contraction maps, Appl. math. lett., 23, 498-502, (2010) · Zbl 1206.54061 [17] Altun, I.; Durmaz, G., Some fixed point theorems on ordered cone metric spaces, Rend. circ. mat. Palermo, 58, 319-325, (2009) · Zbl 1184.54038 [18] Altun, I.; Damnjanović, B.; Djorić, D., Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. math. lett., 23, 310-316, (2010) · Zbl 1197.54052 [19] Kadelburg, Z.; Pavlović, M.; Radenović, S., Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. math. appl., 59, 3148-3159, (2010) · Zbl 1193.54035 [20] Du, Wei-Shih, A note on cone metric fixed point theory and its equivalence, Nonlinear anal., 72, 2259-2261, (2010) · Zbl 1205.54040 [21] Chen, G.Y.; Yang, X.Q.; Yu, H., A nonlinear scalarization function and generalized quasi-vector equilibrium problem, J. global optim., 32, 451-466, (2005) · Zbl 1130.90413 [22] S. Janković, Z. Kadelburg, S. Radenović, On cone metric spaces: survey (submitted for publication). [23] Schaefer, H.H., Topological vector spaces, (1971), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0212.14001 [24] I. Beg, A. Azam, M. Arshad, Common fixed points for maps on topological vector space valued cone metric spaces, Int. J. Math. Math. Sci. (2009) doi:10.1155/2009/560264. · Zbl 1187.54032 [25] Das, K.M.; Naik, K.V., Common fixed point theorems for commuting maps on metric spaces, Proc. amer. math. soc., 77, 369-373, (1979) · Zbl 0418.54025 [26] Ćirić, Lj.B., A generalization of banach’s contraction principle, Proc. amer. math. soc., 45, 267-273, (1974) · Zbl 0291.54056 [27] Ran, A.C.M.; Reurings, M.C.B., A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. amer. math. soc., 132, 1435-1443, (2004) · Zbl 1060.47056 [28] Sönmez, A., On paracompactness in cone metric spaces, Appl. math. lett., 23, 494-497, (2010) · Zbl 1187.54022
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