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A Kannan-like contraction in partially ordered spaces. (English) Zbl 1296.54060

Let \((X,\leq)\) be a partially ordered set; and \(d:X\times X\to [0,\infty)\) be a metric over \(X\) such that \((X,d)\) is complete. Further, let \(f:X\to X\) be a selfmap of \(X\) and set, for simplicity, \(A(x,y)=(1/2)[d(x,fx)+d(y,fy)]\), \(x,y\in X\). Finally, denote by \(S\) the class of all functions \(\beta:(0,\infty)\to [0,1)\) with [\(\beta(t_n)\to 1\) implies \(t_n\to 0\)].
The following is the main result in this paper:
{Theorem.} Suppose that \(f\) is non-decreasing and
(K1)\(d(fx,fy)\leq \beta(A(x,y))A(x,y)\) for all comparable \(x,y\in X\);
(K2)\(x_0\leq fx_0\) for at least one \(x_0\in X\).
Then the following conclusions hold:
{(I)} If, in addition, one of the extra conditions holds: {(i)} \(f\) is continuous, or {(ii)} \((x_n)\)=non-decreasing and \(x_n\to x\) implies \(x_n\leq x\) for all \(n\), then \(f\) has a fixed point in \(X\).
{(II)} If, in addition, for each \(x,y\in X\) there exists \(z\in X\), comparable with \(x\) and \(y\), such that \(z\leq fz\), then \(f\) has a unique fixed point in \(X\).
Some technical connections with other statements in the area are also discussed.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54E50 Complete metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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References:

[1] R. P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Applicable Anal. 87(1) (2008), 109-116.; · Zbl 1140.47042
[2] A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. TMA 72(5) (2010), 2238-2242.; · Zbl 1197.54054
[3] I. Altun, V. Rakočević, Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), 1145-1151; · Zbl 1201.65084
[4] B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling 54 (2011), 73-79.; · Zbl 1225.54016
[5] B. S. Choudhury, K. Das, Fixed points of generalized Kannan type mappings in generalized Menger spaces, Commun. Korean Math. Soc. 24 (2009), 529-537.; · Zbl 1231.54020
[6] J. Caballero, J. Harjani, K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed Point Theory and Appl. 2010, Article ID 916064, doi:10.1155/2010/916064.; · Zbl 1194.54057
[7] Lj. B. Ciric, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topology Appl. 156 (2009), 2838-2844.; · Zbl 1206.54039
[8] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974-979.; · Zbl 0163.17705
[9] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.; · Zbl 0245.54027
[10] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65(7) (2006), 1379-1393.; · Zbl 1106.47047
[11] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. TMA 71 (2009), 3403-3410.; · Zbl 1221.54058
[12] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal. 74 (2011), 768-774.; · Zbl 1201.54034
[13] L. Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc. 61(1) (1976), 171-175.; · Zbl 0342.54024
[14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.; · Zbl 0209.27104
[15] R. Kannan, Some results of fixed points-II, Amer. Math. Monthly 76 (1969), 405-408.; · Zbl 0179.28203
[16] M. Kikkaw, T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory and Appl. 2008 (2008), Article ID 649749.;
[17] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA 70 (2009), 4341-4349.; · Zbl 1176.54032
[18] J. J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(3) (2005), 223-239.; · Zbl 1095.47013
[19] J. J. Nieto, R. Rodriguez-Lopez, Applications of contractive-like mapping principles to fuzzy equations, Rev. Mat. Comp. 19(2) (2006), 361-383.; · Zbl 1113.26030
[20] J. J. Nieto, R. R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sin. (Engl. Ser.) 23(12) (2007), 2205-2212.; · Zbl 1140.47045
[21] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132(5) (2004), 1435-1443.; · Zbl 1060.47056
[22] D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341(2) (2008), 1241-1252.; · Zbl 1142.47033
[23] N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc. 126 (1998), 3117-3124.; · Zbl 0955.54009
[24] P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325-330.; · Zbl 0312.54048
[25] M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl. 117 (1986), 100-127.; · Zbl 0613.47037
[26] Y. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl. 341(2) (2008), 883-893.; · Zbl 1137.47044
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