Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph
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RESEARCH
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Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph Cristian Chifu1 , Gabriela Petru¸sel1* and Monica-Felicia Bota2 *
Correspondence: [email protected] 1 Department of Business, Babe¸s-Bolyai University Cluj-Napoca, Horea street, No. 7, Cluj-Napoca, Romania Full list of author information is available at the end of the article
Abstract The purpose of this paper is to present some strict fixed point theorems for multivalued operators satisfying a Reich-type condition on a metric space endowed with a graph. The well-posedness of the fixed point problem is also studied. MSC: 47H10; 54H25 Keywords: fixed point; strict fixed point; metric space endowed with a graph; well-posed problem
1 Preliminaries A new approach in the theory of fixed points was recently given by Jachymski [] and Gwóźdź-Lukawska and Jachymski [] by using the context of metric spaces endowed with a graph. Other recent results for single-valued and multivalued operators in such metric spaces are given by Nicolae, O’Regan and Petruşel in [] and by Beg, Butt and Radojevic in []. Let (X, d) be a metric space and let 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 the 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)). 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) .
(∗)
˜ 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)
(∗∗)
© 2013 Chifu et al.; 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 et al. Fixed Point Theory and Applications 2013, 2013:203 http://www.fixedpointtheoryandapplications.com/content/2013/1/203
If G is such that E(G) is symmetric, then for x ∈ V (G), the symbol [x]G denotes the equivalence class of the relation defined on V (G) by the rule: yz
if there is a path in G from y to z.
Let us consider the following families of subsets of a metric space (X, d): Pb (X) := Y ∈ P(X) | Y is bounded ; P(X) := Y ∈ P (X) | Y = ∅ ; Pcp (X) := Y ∈ P(X) | Y is compact . Pcl (X) := Y ∈ P(X) | Y is closed ;
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