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Topological vector space-valued cone metric spaces and fixed point theorems. (English) Zbl 1197.54063

Summary: We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of (normed-valued) cone metric spaces. Examples are given to distinguish our results from the known ones.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Kantorovič LV: The principle of the majorant and Newton’s method.Doklady Akademii Nauk SSSR 1951, 76: 17-20. · Zbl 0042.11901
[2] Kantorovitch LV: On some further applications of the Newton approximation method.Vestnik Leningrad University. Mathematics 1957,12(7):68-103. · Zbl 0091.11502
[3] Vandergraft JS: Newton’s method for convex operators in partially ordered spaces.SIAM Journal on Numerical Analysis 1967, 4: 406-432. 10.1137/0704037 · Zbl 0161.35302 · doi:10.1137/0704037
[4] Zabrejko PP: [InlineEquation not available: see fulltext.]-metric and [InlineEquation not available: see fulltext.]-normed linear spaces: survey Collectanea Mathematica 1997,48(4-6):825-859. · Zbl 0892.46002
[5] Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450. · Zbl 1257.47059 · doi:10.1007/978-3-662-00547-7
[6] Aliprantis CD, Tourky R: Cones and Duality, Graduate Studies in Mathematics. Volume 84. American Mathematical Society, Providence, RI, USA; 2007:xiv+279. · Zbl 1127.46002
[7] Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings.Journal of Mathematical Analysis and Applications 2007,332(2):1468-1476. 10.1016/j.jmaa.2005.03.087 · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087
[8] Rezapour S, Hamlbarani R: Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings”.Journal of Mathematical Analysis and Applications 2008,345(2):719-724. 10.1016/j.jmaa.2008.04.049 · Zbl 1145.54045 · doi:10.1016/j.jmaa.2008.04.049
[9] Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces.Journal of Mathematical Analysis and Applications 2008,341(1):416-420. 10.1016/j.jmaa.2007.09.070 · Zbl 1147.54022 · doi:10.1016/j.jmaa.2007.09.070
[10] Vetro P: Common fixed points in cone metric spaces.Rendiconti del Circolo Matematico di Palermo. Serie II 2007,56(3):464-468. 10.1007/BF03032097 · Zbl 1196.54086 · doi:10.1007/BF03032097
[11] Ilić D, Rakočević V: Common fixed points for maps on cone metric space.Journal of Mathematical Analysis and Applications 2008,341(2):876-882. 10.1016/j.jmaa.2007.10.065 · Zbl 1156.54023 · doi:10.1016/j.jmaa.2007.10.065
[12] Ilić D, Rakočević V: Quasi-contraction on a cone metric space.Applied Mathematics Letters 2009,22(5):728-731. 10.1016/j.aml.2008.08.011 · Zbl 1179.54060 · doi:10.1016/j.aml.2008.08.011
[13] Kadelburg Z, Radenović S, Rakočević V: Remarks on “Quasi-contraction on a cone metric space”.Applied Mathematics Letters 2009,22(11):1674-1679. 10.1016/j.aml.2009.06.003 · Zbl 1180.54056 · doi:10.1016/j.aml.2009.06.003
[14] Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces.Applied Mathematics Letters 2009,22(4):511-515. 10.1016/j.aml.2008.07.001 · Zbl 1167.54014 · doi:10.1016/j.aml.2008.07.001
[15] Janković, S.; Kadelburg, Z.; Radenović, S.; Rhoades, BE, Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces, 16 (2009) · Zbl 1186.54035
[16] Jungck, G.; Radenović, S.; Radojević, S.; Rakočević, V., Common fixed point theorems for weakly compatible pairs on cone metric spaces, 13 (2009) · Zbl 1190.54032
[17] Kadelburg, Z.; Radenović, S.; Rosić, B., Strict contractive conditions and common fixed point theorems in cone metric spaces, 14 (2009) · Zbl 1179.54062
[18] Włodarczyk, K.; Plebaniak, R., Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces, 32 (2010) · Zbl 1193.37101
[19] Włodarczyk, K.; Plebaniak, R., Maximality principle and general results of ekeland and caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances, 37 (2010) · Zbl 1201.54039
[20] Włodarczyk K, Plebaniak R, Doliński M: Cone uniform, cone locally convex and cone metric spaces, endpoints, set-valued dynamic systems and quasi-asymptotic contractions.Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):5022-5031. 10.1016/j.na.2009.03.076 · Zbl 1203.54051 · doi:10.1016/j.na.2009.03.076
[21] Włodarczyk K, Plebaniak R, Obczyński C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces.Nonlinear Analysis: Theory, Methods & Applications 2010,72(2):794-805. 10.1016/j.na.2009.07.024 · Zbl 1185.54020 · doi:10.1016/j.na.2009.07.024
[22] Azam A, Arshad M, Beg I: Existence of fixed points in complete cone metric spaces.International Journal of Modern Mathematics 2010,5(1):91-99. · Zbl 1203.54034
[23] Du W-S: A note on cone metric fixed point theory and its equivalence.Nonlinear Analysis: Theory, Methods & Applications 2010,72(5):2259-2261. 10.1016/j.na.2009.10.026 · Zbl 1205.54040 · doi:10.1016/j.na.2009.10.026
[24] Beg, I.; Azam, A.; Arshad, M., Common fixed points for maps on topological vector space valued cone metric spaces, 8 (2009) · Zbl 1187.54032
[25] Schaefer HH: Topological Vector Spaces. 3rd edition. Springer, New York, NY, USA; 1971:xi+294. · Zbl 0217.16002 · doi:10.1007/978-1-4684-9928-5
[26] Kadelburg Z, Pavlović M, Radenović S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces.Computers & Mathematics with Applications 2010,59(9):3148-3159. 10.1016/j.camwa.2010.02.039 · Zbl 1193.54035 · doi:10.1016/j.camwa.2010.02.039
[27] Jungck G, Rhoades BE: Fixed point theorems for occasionally weakly compatible mappings.Fixed Point Theory 2006,7(2):287-296. · Zbl 1118.47045
[28] Al-Thagafi MA, Shahzad N: Generalized -nonexpansive selfmaps and invariant approximations.Acta Mathematica Sinica 2008,24(5):867-876. 10.1007/s10114-007-5598-x · Zbl 1175.41026 · doi:10.1007/s10114-007-5598-x
[29] Rhoades BE: A comparison of various definitions of contractive mappings.Transactions of the American Mathematical Society 1977, 226: 257-290. · Zbl 0365.54023 · doi:10.1090/S0002-9947-1977-0433430-4
[30] Jungck G: Commuting mappings and fixed points.The American Mathematical Monthly 1976,83(4):261-263. 10.2307/2318216 · Zbl 0321.54025 · doi:10.2307/2318216
[31] Das KM, Naik KV: Common fixed-point theorems for commuting maps on a metric space.Proceedings of the American Mathematical Society 1979,77(3):369-373. · Zbl 0418.54025
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