In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature.

This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point results in complete metric spaces, both in a continuous and a non-continuous sense. Additionally, we provide an explicit example to validate our findings. Furthermore, we discuss a general model for fractional differential equations using the Caputo derivative. Finally, we outline the benefits of our study and suggest potential areas for future research.

In this work, we deduce a new criterion that guarantees the oscillation of solutions to linear Sturm–Liouville delay noncanonical dynamic equations; these results emulate the criteria of the Hille and Ohriska types for canonical dynamic equations, and these results also solve an open problem in many works in the literature. Several examples are offered, demonstrating that the findings achieved are precise, practical, and adaptable.

This article focuses on the problem of integral boundary value for Riemann-Liouville derivatives equipped with $ \left(p, q\right) $-difference calculus in Banach space. To provide further clarification, our focus lies in establishing the existence of a solution to our problem using the measure of noncompactness (m.n.) and the Mönch's fixed point theorem. Our investigation in the Banach space encompasses two nonlinear terms with two distinct orders of derivatives. Our paper concludes with an illustrative example and conclusion.

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, $\varsigma$ and $\tau$, such that $\varsigma \in \left(1,2\right]$ and $\tau \in \left(0,\varsigma \right]$. Our paper ends with a conclusion.

Using the monophonic paths in the theory of directed graphs, this paper constructs a new topology called the out mondirected topology, which characterizes the graphs that induce indiscrete or discrete topology. We give and study some of the relations and properties, such as the relationship between the isomorphic relation, in directed graphs and the homeomorphic property in out mondirected topological spaces, compactness, ${\mathbb{D}}_{\pm }$-connectedness, connectedness and ${\mathbb{D}}_{\pm }$-discrete properties. Finally, we apply our results of out mondirected topological spaces in the nervous system of the human body, such as in the messenger signal network, in diagrams of sensory neuron cells and in models of two distinct nicotinic receptor types based on the second messenger signal.