A relation-theoretic set-valued version of Prešić-Ćirić theorem and applications

In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above.