Document Type : Research Paper


Department of Mathematics, Lovely Professional University, Phagwara-144411, Punjab, India.


This paper explores certain fixed point results for multivalued mapping in a metric space endowed with an arbitrary binary relation $\mathrm{R}$, briefly written as $\mathrm{R}$-metric space. The fixed point results proved  are subjected to contraction conditions corresponding to the multivalued counterpart of $F$-contraction and $F$-weak contraction in $\mathrm{R}$-metric space. The main results unify, extend and generalize the results on multivalued and single-valued mapping in the literature. To support the conclusion, several examples have been provided.


[1] A. Alam and M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), pp. 693-702.
[2] A. Alam, R. George, M. Imdad and M. Hasanuzzaman, Fixed Point Theorems for Nonexpansive Mappings under Binary Relations, Mathematics, 9 (2021), Article No. 2059.
[3] A. Alam, M. Imdad, M. Asim and S. Sessa, A Relation-Theoretic Formulation of Browder–Göhde Fixed Point Theorem, Axioms, 10 (2021), Article No. 285.
[4] B.S. Choudhury, N. Metiya and C. Bandyopadhyay, Fixed points of multivalued $\alpha $-admissible mappings and stability of fixed point sets in metric spaces, Rend. Circ. Mat. Palermo, 64 (2015), pp. 43-55.
[5] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl., 334 (2007), pp. 132-139.
[6] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (1) (2012), Artice No. 94.
[7] D. Wardowski and N. Van Dung, Fixed points of F-weak contractions on complete metric spaces, Demonstr. Math., 47 (2014), pp. 146-155.
[8] G. Prasad, R.C. Dimri and A. Bartwal, Fixed points of Suzuki contractive mappings in relational metric spaces, Rendiconti del Circolo Mat. di Palermo, 2 (2019), pp. 1347–1358.
[9] I. Altun, G. Durmaz, G. Mınak and S. Romaguera, Multivalued Almost $F$-contractions on Complete Metric Spaces, Filomat, 30 (2016), pp. 441-448.
[10] I. Altun, G. Minak and H. Dag, Multivalued $F$-contractions on complete metric spaces, J. Nonlinear Convex Anal., 28 (2015), pp. 659-666.
[11] L. Ćirić, Fixed point theorems for multi-valued contractions in complete metric spaces, J. Math. Anal. Appl., 348 (2008), pp. 499-507.
[12] M. Imdad, Q.H. Khan, W.M. Alfaqih and R. Gubran, A relation theoretic $(F, R)$-contraction principle with applications to matrix equation, Bull. Math. Anal. Appl., 10 (2018), pp. 1-12.
[13] M. Imdad and W.M. Alfaqih, A relation-theoretic expansion principle, Acta Univ. Apulensis., 54 (2018), pp. 55-69.
[14] O. Acar, G. Durmaz and G. Minak, Generalized multivalued F-contractions on complete metric spaces, Bull. Iranian Math. Soc., 40 (2014), pp. 1469-1478.
[15] R. K. Sharma and S. Chandok, Multivalued problems, orthogonal mappings, and fractional integro-differential equation, J. Math., (2020), Article ID 6615478.
[16] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922), pp. 133-181.
[17] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena., 46 (1998), pp. 263-276.
[18] S. Khalehoghli, H. Rahimi and M.E. Gordji, Fixed point theorems in $\mathrm{R}$-metric spaces with applications, AIMS Math., 5 (2020), pp. 3125-3137.
[19] S. Lipschutz, Schaum's outline of theory and problems of set theory and related topics, McGraw-Hill, New York, 1964.
[20] S.B. Nadler Jr, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), pp. 475-488.