Document Type: Research Paper

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran.

2 Primary School ''Kneginja Milica", Beograd, Serbia.

Abstract

In this paper, we study the existence and uniqueness of fixed points for mappings with respect to a $wt$-distance in $b$-metric spaces endowed with a graph. Our results are significant, since we replace the condition of continuity of mapping with the condition of orbitally $G$-continuity of mapping and we consider $b$-metric spaces with graph instead of $b$-metric spaces, under which can be generalized, improved, enriched and unified a number of recently announced results in the existing literature. Additionally, we elicit all of our main results by a non-trivial example and pose an interesting two open problems for the enthusiastic readers.

Keywords

Main Subjects

###### ##### References

[1] R.P. Agarwal, E. Karapinar, D. O'Regan, and A.F. Roldan-Lopez-de-Hierro, Fixed Point Theory in Metric Type Spaces, Springer-International Publishing, Switzerland, 2015.

[2] A. Aghanians and K. Nourouzi, Fixed points for Kannan type contractions in uniform spaces endowed with a graph, Nonlinear Anal. Model. Control., 21 (2016), pp. 103-113.

[3] I.A. Bakhtin, The contraction mapping principle in almost metric space, Functional Analysis., 30 (1989), pp. 26-37.

[4] B. Bao, S. Xu, L. Shi, and V. Cojbasic Rajic, Fixed point theorems on generalized $c$-distance in ordered cone $b$-metric spaces, Int. J. Nonlinear Anal. Appl., 6 (2015), pp 9-22.

[5] F. Bojor, Fixed points of Kannan mappings in metric spaces endowed with a graph, An. St. Ovidius. Constanta., 20 (2012), pp. 31-40.

[6] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008.

[7] M. Bota, A. Molnar, and C. Varga, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory., 12 (2011), pp. 21-28.

[8] Y.J. Cho, R. Saadati, and S.H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl., 61 (2011), pp. 1254-1260.

[9] Lj.B. Ciric, On contraction type mappings, Math. Balkanica., 1 (1971), pp. 52-57.

[10] A.S. Cvetkovic, M.P. Stanic, S. Dimitrijevic, and S. Simic, Common fixed point theorems for four mappings on cone metric type space, Fixed Point Theory Appl., (2011), 2011:589725.

[11] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), pp. 5-11.

[12] K. Fallahi, $G$-asymptotic contractions in metric spaces with a graph and fixed point results, Sahand Commun. Math. Anal., 7 (2017), pp. 75-83.

[13] K. Fallahi, A. Petrusel, and G. Soleimani Rad, Fixed point results for pointwise Chatterjea type mappings with respect to a $c$-distance in cone metric spaces endowed with a graph, U.P.B. Sci. Bull. (Series A)., 80 (2018), pp. 47-54.

[14] K. Fallahi and G. Soleimani Rad, Fixed point results in cone metric spaces endowed with a graph, Sahand Commun. Math. Anal., 6 (2017), pp. 39-47.

[15] L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), pp. 1467-1475.

[16] N. Hussain, R. Saadati, and R.P. Agrawal, On the topology and $wt$-distance on metric type spaces, Fixed Point Theory Appl., (2014), 2014:88.

[17] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), pp. 1359-1373.

[18] O. Kada, T. Suzuki, and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), pp. 381-391.

[19] Z. Kadelburg and S. Radenovic, Coupled fixed point results under tvs-cone metric spaces and $w$-cone-distance, Advances Fixed Point Theory and Appl., 2 (2012), pp. 29-46.

[20] M.A. Khamsi and N. Hussain, $KKM$ mappings in metric type spaces, Nonlinear Anal., 73 (2010), pp. 3123-3129.

[21] A. Petrusel and I.A. Rus, Fixed point theorems in ordered $L$-spaces, Proc. Amer. Math. Soc., 134 (2006), pp. 411-418.

[22] G. Soleimani Rad, H. Rahimi, and C. Vetro, Fixed point results under generalized $c$-distance with application to nonlinear fourth-order differential equation, Fixed Point Theory., in press.

[23] W.A. Wilson, On semi-metric spaces, Amer. Jour. Math., 53 (1931), pp. 361-373.