We establish fixed point theorems for nonlinear contractions on a metric space (not essentially complete) endowed with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those contained in Samet and Turinici [Commun. Math. Anal. 13, 82-97 (2012)] and Alam and Imdad [J. Fixed Point Theory Appl. 17(4), 693-702 (2015)]. Interestingly a corollary to one of our main results proved under symmetric closure of any binary relation remains a sharpened version of a theorem due to Samet and Turinici. Finally, we use examples to highlight the realized improvements in the results proved in this paper.

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We observe that all the results involving a-type F-contractions are not correct in their present forms. In this article, we prove some fixed point results for extended Fweak contraction mappings in metric and ordered-metric spaces. Our observations and the usability of our results are substantiated by using suitable examples. As an application, we prove an existence and uniqueness result for the solution of a first-order ordinary differential equation satisfying periodic boundary conditions in the presence of either its lower or upper solution.

In this paper, we begin with some observations on F-contractions. Thereafter, we introduce the notion of (F, R) g-contractions and utilize the same to prove some coincidence and common fixed point results in the setting of metric spaces endowed with binary relations. An example is also given to exhibit the utility of our results. We also deduce some consequences in the setting of ordered metric spaces. As an application, we investigate the existence and uniqueness of a solution of integral equation of Volterra type.

In this paper, we introduce a new class of contractions which remains a mixed type of weak and F-contractions but not any of them.