Generalized contractions in metric spaces endowed with a graph
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Generalized contractions in metric spaces endowed with a graph Cristian Chifu* and Gabriela Petru¸sel *
Correspondence: [email protected] Department of Business, Babe¸s-Bolyai University Cluj-Napoca, Horea street, No.7, Cluj-Napoca, Romania
Abstract A very interesting approach in the theory of fixed points in some general structures was recently given by Jachymski (Proc. Amer. Math. Soc. 136:1359-1373, 2008) and Gwó´zd´z-Lukawska and Jachymski (J. Math. Anal. Appl. 356:453-463, 2009) by using the context of metric spaces endowed with a graph. The purpose of this article is to ´ c-Reich-Rus present some new fixed point results for graphic contractions and for Ciri´ G-contractions on complete metric spaces endowed with a graph. The particular case of almost contractions is also considered. MSC: 47H10; 54H25 Keywords: metric space; connected graph; fixed point; graphic contraction; almost contraction; generalized contraction
1 Introduction A very interesting approach in the theory of fixed points in some general structures was recently given by Jachymski [] and Gwóźdź-Lukawska and Jachymski [] by using the context of metric spaces endowed with a graph. More precisely, let (X, d) be a metric space and be the diagonal of X × X. Let G be a directed graph such that the set V (G) of its vertices coincides with X and ⊆ E(G), where E(G) is the set of edges of the graph. Assume also that G has no parallel edges, and thus, one can identify G with the pair (V (G), E(G)). By definition, an operator f : X → X is called a Banach G-contraction (see Definition . in Jachymski []) if and only if: (a) for each x, y ∈ X with (x, y) ∈ E(G), we have (f (x), f (y)) ∈ E(G); (b) there exists α ∈ ], [ such that for each x, y ∈ X, the following implication holds: ((x, y) ∈ E(G) implies d(f (x), f (y)) ≤ αd(x, y)). If x and y are vertices of G, then a path in G from x to y of length k ∈ N is a finite sequence (xn )n∈{,,,...,k} of vertices such that x = x, xk = y and (xi– , xi ) ∈ E(G) for i ∈ {, , . . . , k}. Notice that a graph G is connected if there is a path between any two vertices, and it is ˜ is connected, where G ˜ denotes the undirected graph obtained from weakly connected if G G by ignoring the direction of edges. Denote by G– the graph obtained from G by reversing the direction of edges. Thus, E G– = (x, y) ∈ X × X : (y, x) ∈ E(G) .
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© 2012 Chifu and Petru¸sel; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Chifu and Petru¸sel Fixed Point Theory and Applications 2012, 2012:161 http://www.fixedpointtheoryandapplications.com/content/2012/1/161
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˜ as a directed graph for which the set of its edges is Since it is more convenient to treat G symmetric, under this convention, we have that ˜ = E(G) ∪ E G– . E(G)
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