In this paper, we give some remarks on the recent papers on generalized F-contractions. Our results unify and generalize the previous results in the existing literature. Moreover, we give an example to support our results. As an application, we give the existence and uniqueness of a solution to a class of differential equations.

In this study, we initiate the concept of fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction and establish some fixed point results involving a $\mathcal{G}$-transitive binary relation and fuzzy $\mathcal{L}$-simulation functions, by employing suitable hypotheses on a fuzzy metric space endowed with a binary relation. The presented results unify, generalize, and improve various previous findings in the literature.

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli–Arzelà theorem and Mönch’s fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results.

In this article, we give sharp two-sided bounds for the generalized Jensen functional $J_n(f,g,h;$p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean $A_n(h;$p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.