# Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph

- 331 Downloads
- 3 Citations

## Abstract

The aim of this paper is to present common fixed point results of set-valued graphic *F*-contraction mappings on a family of sets endowed with a graph. Some examples are presented to support the results proved herein. Our results unify, generalize and extend various results in the existing literature.

## Keywords

Set-valued mapping Domain of sets Common fixed point Graph*F*-contraction pair Directed graph

## Mathematics Subject Classification

47H10 54E50 54H25## Notes

### Acknowledgments

The authors are grateful to the referees for useful suggestion and remarks which helped to improve the presentation of the paper. The third author is thankful for the support to the Serbian Ministry of Education, Science and Technological Development (Grant No. 174033).

### Conflict of interest

The author declare that they have not competing interests.

### Author’s contributions

TN, MA, TAL and SR have worked together on each section of the paper such as the literature review, results and examples. All authors read approved the final manuscript.

## References

- Abbas M, Alfuraidan MR, Khan AR, Nazir T (2015) Fixed point results for set-contractions on metric spaces with a directed graph. Fixed Point Theory Appl 14:1–9zbMATHMathSciNetGoogle Scholar
- Abbas M, Alfuraidan MR, Nazir T (2016) Common fixed points of multivalued \(F\)-contractions on metric spaces with a directed graph. Carpath. J. Math. 32(1):1–12zbMATHMathSciNetGoogle Scholar
- Abbas M, Ali B, Petrusel G (2014) Fixed points of set-valued contractions in partial metric spaces endowed with a graph. Carpath. J. Math. 30(2):129–137zbMATHMathSciNetGoogle Scholar
- Abbas M, Nazir T (2013) Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl 20:1–8zbMATHMathSciNetGoogle Scholar
- Aleomraninejad SMA, Rezapoura Sh, Shahzad N (2012) Some fixed point results on a metric space with a graph. Topol Appl 159:659–663CrossRefzbMATHMathSciNetGoogle Scholar
- Alfuraidan MR, Khamsi MA (2014) Caristi fixed point theorem in metric spaces with a graph. Abstr Appl Anal 1–5 (Article ID 303484)Google Scholar
- Amini-Harandi A, Emami H (2010) A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal 72:2238–2242CrossRefzbMATHMathSciNetGoogle Scholar
- Assad NA, Kirk WA (1972) Fixed point theorems for setvalued mappings of contractive type. Pacific J Math 43:533–562CrossRefGoogle Scholar
- Beg I, Butt AR (2013) Fixed point theorems for set valued mappings in partially ordered metric spaces. Int J Math Sci 7(2):66–68Google Scholar
- Beg I, Butt AR (2013) Fixed point of set-valued graph contractive mappings, J Inequ Appl 52:7Google Scholar
- Berinde M, Berinde V (2007) On a general class of multivalued weakly Picard mappings. J Math Anal Appl 326:772–782CrossRefzbMATHMathSciNetGoogle Scholar
- Berinde V (2007) Iterative approximation of fixed points. Springer-Verlag, Berlin-HeidelbergzbMATHGoogle Scholar
- Bojor F (2010) Fixed point of \(\varphi \)-contraction in metric spaces endowed with a graph. Ann Univ Craiova Math Comp Sci Series 37(4):85–92Google Scholar
- Bojor F (2012) Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal 75:3895–3901CrossRefzbMATHMathSciNetGoogle Scholar
- Bojor F (2012) On Jachymski’s theorem. Ann Univ Craiova Math Comp Sci Series 40(1):23–28Google Scholar
- Chifu CI, Petrusel GR (2012) Generalized contractions in metric spaces endowed with a graph. Fixed Point Theory Appl 161:1–9zbMATHMathSciNetGoogle Scholar
- Edelstein M (1961) An extension of Banach’s contraction principle. Proc Am Math Soc 12:7–10zbMATHMathSciNetGoogle Scholar
- Gwozdz-Lukawska G, Jachymski J (2009) IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem. J Math Anal Appl 356:453–463CrossRefzbMATHMathSciNetGoogle Scholar
- Harjani J, Sadarangani K (2009) Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal 71:3403–3410CrossRefzbMATHMathSciNetGoogle Scholar
- Jachymski J, Jozwik I (2007) Nonlinear contractive conditions: a comparison and related problems. Banach Center Publ 77:123–146CrossRefzbMATHMathSciNetGoogle Scholar
- Jachymski J (2008) The contraction principle for mappings on a metric space with a graph. Proc Am Math Soc 136:1359–1373CrossRefzbMATHMathSciNetGoogle Scholar
- Nadler SB (1969) Multivalued contraction mappings. Pacific J Math 30:475–488CrossRefzbMATHMathSciNetGoogle Scholar
- Nieto JJ, López RR (2005) Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22:223–239CrossRefzbMATHMathSciNetGoogle Scholar
- Nicolae A, O’Regan D, Petrusel A (2011) Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph. J Georgian Math Soc 18:307–327zbMATHMathSciNetGoogle Scholar
- Ran ACM, Reurings MCB (2004) A fixed point theorem in partially ordered sets and some application to matrix equations. Proc Am Math Soc 132:1435–1443CrossRefzbMATHMathSciNetGoogle Scholar
- Sgroi M, Vetro C (2013) Multi-valued \(F\)-contractions and the solution of certain functional and integral equations. Filomat 27(7):1259–1268CrossRefzbMATHMathSciNetGoogle Scholar
- Shukla S, Radenović S (2013) Some common fixed point theorems for \(F\)-complete partial metric spaces. J Math 2013(Article ID 878730):7Google Scholar
- Shukla S, Radenović S, Kadelburg Z (2014) Some fixed point theorems for ordered \(F\)-generalized contractions in 0-f-orbitally complete partial metric spaces. Theor Appl Math Comput Sci 4(1):87–98zbMATHGoogle Scholar
- Wardowski D (2012) Fixed points of new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl 94:1–6zbMATHMathSciNetGoogle Scholar