Fixed point of set-valued graph contractions in metric spaces

Document Type : Research Paper


Department of Mathematics, Faculty of Basic Science,Iran University of Science and Technology, Narmak, Tehran,Iran


In this paper, we introduce the (G-$\psi$) contraction in a metric space by using a graph. Let $T$ be a multivalued mappings on $X.$ Among other things, we obtain a fixed point of the mapping $T$ in the metric space $X$ endowed with a graph $G$ such that the set of vertices of $G,$ $V(G)=X$ and the set of edges of $G,$ $E(G)\subseteq X\times X.$


[1] N.A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math. 43 (1972) 533-562.
[2] T. G. Bahaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393.
[3] I. Beg and A. Rashid Butt, Fixed point of set-valued graph contractive mappings, J. Inequal. Appl. 2013 (2013) 252.
[4] I. Beg, A. Rashid Butt and S. Radojevic, The contraction principle for set-valued mappings on a metric space with a graph, Comput. Math. Appl. 60 (2010) 1214-1219.
[5] J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, 1998.
[6] J. Jamchymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008) 1359-1373.
[7] S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475-487.
[8] M. Ozturk and E. Girgin, On some fixed-point theorems for ψ-contraction on metric space involving a graph, J. Inequal. Appl. 2014 (2014) 39.
[9] A. Petrusel and I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418.
Volume 12, Issue 1
May 2021
Pages 741-747
  • Receive Date: 17 October 2015
  • Revise Date: 03 September 2016
  • Accept Date: 18 December 2016