Karapınar, Erdal Fixed point theorems in cone Banach spaces. (English) Zbl 1204.47066 Fixed Point Theory Appl. 2009, Article ID 609281, 9 p. (2009). Summary: A class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset \(C\) of a cone Banach space with the norm \(\| x\| _P=d(x,0)\), if there exist \(a, b, s\) and \(T:C\rightarrow C\) satisfies the conditions \(0\leq s+|a| - 2b<2(a+b)\) and \(4ad(Tx,Ty)+b(d(x,Tx)+d(y,Ty))\leq sd(x,y)\) for all \(x,y\in C\), then \(T\) has at least one fixed point. Cited in 48 Documents MSC: 47H10 Fixed-point theorems 46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.) 46B99 Normed linear spaces and Banach spaces; Banach lattices 54H25 Fixed-point and coincidence theorems (topological aspects) PDFBibTeX XMLCite \textit{E. Karapınar}, Fixed Point Theory Appl. 2009, Article ID 609281, 9 p. (2009; Zbl 1204.47066) Full Text: DOI References: [1] Rzepecki B: On fixed point theorems of Maia type.Publications de l’Institut Mathématique 1980, 28(42): 179-186. · Zbl 0482.47029 [2] Maia MG: Un’osservazione sulle contrazioni metriche.Rendiconti del Seminario Matematico della Università di Padova 1968, 40: 139-143. · Zbl 0188.45603 [3] Lin SD: A common fixed point theorem in abstract spaces.Indian Journal of Pure and Applied Mathematics 1987,18(8):685-690. · Zbl 0622.47057 [4] Khan MS, Imdad M: A common fixed point theorem for a class of mappings.Indian Journal of Pure and Applied Mathematics 1983,14(10):1220-1227. · Zbl 0533.54031 [5] 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 [6] Rezapour Sh, 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 [7] Turkoglu D, Abuloha M: Cone metric spaces and fixed point theorems in diametrically contractive mappings. to appear in Acta Mathematica Sinica · Zbl 1203.54049 [8] Turkoglu D, Abuloha M, Abdeljawad T: KKM mappings in cone metric spaces and some fixed point theorems.Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):348-353. 10.1016/j.na.2009.06.058 · Zbl 1197.54076 [9] Şahin I, Telci M: Fixed points of contractive mappings on complete cone metric spaces.Hacettepe Journal of Mathematics and Statistics 2009,38(1):59-67. · Zbl 1190.47058 [10] Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.Order 2005,22(3):223-239. 10.1007/s11083-005-9018-5 · Zbl 1095.47013 [11] Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.Acta Mathematica Sinica 2007,23(12):2205-2212. 10.1007/s10114-005-0769-0 · Zbl 1140.47045 [12] Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces.Proceedings of the American Mathematical Society 2007,135(8):2505-2517. 10.1090/S0002-9939-07-08729-1 · Zbl 1126.47045 [13] Ćirić L: Fixed Point Theory Contraction Mapping principle. Faculty of Mechanical Engineering Press, Beograd, Serbia; 2003. [14] Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings.Journal of Mathematical Analysis and Applications 2008,340(2):1088-1095. 10.1016/j.jmaa.2007.09.023 · Zbl 1140.47041 [15] Turkoglu D, Abuloha M, Abdeljawad T: Some theorems and examples of cone metric spaces. to appear in Journal of Computational Analysis and Applications · Zbl 1203.54049 [16] Abdeljawad T: Completion of cone metric spaces. to appear in Hacettepe Journal of Mathematics and Statistics · Zbl 1195.54057 [17] Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450. · Zbl 1257.47059 [18] Abdeljawad T, Karapinar E: Quasicone Metric Spaces and Generalizations of Caristi Kirk’s Theorem.Fixed Point Theory and Applications 2009, 2009: 9 page. · Zbl 1197.54051 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.