Multivalued FG-Contraction Mappings on Directed Graphs

Download PDF


DOI: 10.46793/KgJMat2206.943N


In this paper, we study generalized ????????-contraction conditions for a pair of mappings defined on a family of subsets of a metric space endowed with a directed graph, and discuss coincidence and common fixed point results relaxing the continuity of mappings. The given notions and results are exemplified by suitable models. We apply our results to the problem of existence of solutions of a Fredholm integral inclusion.


Multivalued contraction, metric space with directed graph, Fredholm integral inclusion.


[1]   M. Abbas, M. R. Alfuraidan, A. R. Khan and T. Nazir, Fixed point results for set-contractions on metric spaces with a directed graph, Fixed Point Theory Appl. 2015(14) (2015), 1–9.

[2]   M. Abbas, T. Nazir, T. A. Lampert and S. Radenović, Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph, Comp. Appl. Math. 36 (2017), 1607–1622.

[3]   M. Abbas, T. Nazir, B. Popović and S. Radenović, On weakly commuting set-valued mappings on a domain of sets endowed with directed graph, Results Math. 71 (2017), 1277–1295.

[4]   S. M. A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Some fixed point results on a metric space with a graph, Topology Appl. 159 (2012), 659–663.

[5]   I. Beg and A. Butt, Fixed point of set-valued graph contractive mappings, J. Inequal. Appl. 2013(52) (2013), 1–7.

[6]   F. Bojor, Fixed point of φ-contraction in metric spaces endowed with a graph, An. Univ. Craiova Ser. Mat. Inform. Ser. 37 (2010), 85–92.

[7]   F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal. 75 (2012), 3895–3901.

[8]   Lj. Ćirić, Some Recent Results in Metrical Fixed Point Theory, University of Belgrade, Beograd, 2003.

[9]   G. Gwozdz-Lukawska and J. Jachymski, IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl. 356 (2009), 453–463.

[10]   J. Jachymski and I. Jozwik, Nonlinear contractive conditions: a comparison and related problems, Banach Center Publ. 77 (2007), 123–146.

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

[12]   G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4 (1996), 199–215.

[13]   E. Malkowsky and V. Rakočević, Advanced Functional Analysis, CRS Press, Taylor and Francis Group, Boca Raton, FL, 2019.

[14]   S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.

[15]   H. K. Nashine, R. P. Agarwal and Z. Kadelburg, Solution to Fredholm integral inclusions via (F,δb)-contractions, Open Math. 14 (2016), 1053–1064.

[16]   V. Parvaneh, N. Hussain and Z. Kadelburg, Generalized Wardowski type fixed point theorems via α-admissible FG-contractions in b-metric spaces, Acta Math. Sci. 36B (2016), 1445–1456.

[17]   A. Petrusel, Integral inclusions. Fixed point approaches, Annales Societatis Mathematicae Polonae. Seria I. Commentationes Mathematicae 40 (2000), 147–158.

[18]   A. Petrusel, G. Petrusel and J-C. Yao, A study of a system of operator inclusions via a fixed point approach and applications to functional-differential inclusions, Carpathian J. Math. 32 (2016), 349–361.

[19]   J. Vujaković, S. Mitrović, S. Radenović and M. Pavlović, On recent results concerning F-contraction in generalized metric spaces, Mathematics 8 (2020), DOI 10.3390/math8050767

[20]   D. Wardowski, Fixed points of new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012(94) (2012), 1–6.