Multivalued relation-theoretic graph contraction principle with applications

Document Type : Research Paper


1 Department of Mathematics, Graphic Era Hill University, Uttarakhand, India

2 Government Degree college, Purwala Dogi, Uttarakhand, India

3 Department of Mathematics, HNB Garhwal University, Srinagar(Garhwal), Uttarakhand, India

4 Department of Mathematics, Graphic Era (Deemed to be) University, Dehradun, Uttarakhand, India

5 Department of Mathematics, HNB Garhwal University, Campus Pauri, Uttarakhand, India


In this paper, we present a new generalization of Nadler's fixed point theorem for multivalued relation-theoretic graph contractions on relational metric spaces. Our results extend and generalize the result of Shukla and Rodriguez-Lopez (Questiones Mathematiae, (2019) 1-16), Nadler (Pacific J. Math. 30 (1969), 475-488), Alam and Imdad (J. Fixed Point Theory and Appl., 17(4) (2015), 693-702) and many others in the existing literature of fixed point theory. Some illustrative examples are also provided to illustrate the usefulness of our main results. Moreover, we have an application to generalized coupled fixed point problems.


[1] A. Alam and M. Imdad , Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17 (2015), 693–702.
[2] A. Alam and M. Imdad, Nonlinear contractions in metric spaces under locally T-transitive binary relations, Fixed
Point Theory 19 (2018), 13–23.
[3] H. Baghani and M. Ramezani, A fixed point theorem for a new class of set-valued mappings in R-complete (not
necessarily complete) metric spaces, Filomat 31 (2017), 3875–3884.
[4] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales, Fund.
Math. 3 (1922), 133–181.
[5] H. Ben-El-Mechaiekh , The Ran-Reurings fixed point theorem without partial order: a simple proof, J. Fixed Point
Theory Appl. 16 (2014), 373–383.
[6] D. Khantwal, S. Aneja, G. Prasad and U.C. Gairola , A generalization of relation-theoretic contraction principle,
TWMS J. App. Eng. Math. (Accepted).
[7] S.B. Nadler Jr, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.
[8] J.J. Nieto and R. Rodr´─▒guez-L´opez, Contractive mapping theorems in partially ordered sets and applications to
ordinary differential equations, Order 22 (2005), 223–239.
[9] A. Petru¸sel, G. Petru¸sel and J. Yao, Multi-valued graph contraction principle with applications, Optim. 69 (2020),
[10] G. Prasad and R. C. Dimri, Fixed point theorems for weakly contractive mappings in relational metric spaces with
an application, J. Anal. 26 (2018), 151–162.
[11] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix
equations, Proc. Amer. Math. Soc. 132 (2004), 1435–1443.
[12] B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and
applications, Commun. Math. Anal. 13 (2012), 82–97.
[13] S. Shukla and R. Rodr´─▒guez-L´opez, Fixed points of multi-valued relation-theoretic contractions in metric spaces
and application, Q. Math. 43 (2020), 409–424.
[14] A. Tomar, M. Joshi, S.K. Padaliya, B. Joshi and A. Diwedi, Fixed point under set-valued relation-theoretic
nonlinear contractions and application, Filomat 33 (2019), 4655–4664.
[15] M. Turinici, Nieto-Lopez theorems in ordered metric spaces, Math. Stud. 81 (2012), 219–229.
Volume 13, Issue 2
July 2022
Pages 2961-2971
  • Receive Date: 02 February 2021
  • Revise Date: 11 June 2022
  • Accept Date: 24 July 2022
  • First Publish Date: 24 July 2022