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Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p = 2 to results in the L p space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. [ABSTRACT FROM AUTHOR]

Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method. [ABSTRACT FROM AUTHOR]

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in L p space by using Jensen's, Grönwall–Bellman's, Hölder's, and Burkholder–Davis–Gundy's inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ -Caputo derivative. We generalize the two situations of p = 2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. [ABSTRACT FROM AUTHOR]

The main goals of this paper are to provide an introduction to the idea of interval-valued n -polynomial s -type convex functions and to investigate the algebraic properties of this type of function. This new generalization aims to show the existence of new Hermite–Hadamard inequalities for the recently presented class of interval-valued n -polynomials of s -type convex describing the φ -fractional integral operator. In the classical sense, some special cases are figured out, and the two examples are also given. There are some recently discovered inequalities for interval-valued functions that are regulated by fractional calculus applicable to interval-valued n -polynomial s -type convexity. The results obtained show that future research will be simple to implement, highly efficient, feasible, and extremely precise in its investigation. It could also help solve modeling problems, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables. [ABSTRACT FROM AUTHOR]

Discrete fractional calculus (DFC) is suggested to interpret neural schemes with memory impacts. This study seeks to formulate some discrete fractional nonlinear inequalities with ξ ̂ fractional sum operators that are used with some conventional and forthright inequalities. Taking into consideration, we recreate the explicit bounds of Gronwall-type inequalities by observing the principle of DFC for unknown functions here. Such inequalities are of new version relative to the current literature findings so far and can be used as a helpful method to evaluate the numerical solutions of discrete fractional differential equations. We show a few uses of the rewarded inequalities to mirror the advantages of our work. Regarding applications, we can apply the acquainted results to discuss boundedness, uniqueness, and continuous dependency on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations. The leading consequences may be proven by the usage of the analysis process and the methodology of the mean value theorem. These variations can be used as an advantageous device in the subjective examination solutions of discrete fractional difference equations. [ABSTRACT FROM AUTHOR]

This content replicates some discrete nonlinear fractional inequalities by virtue of the fractional sum operator Ψ ¯ on time scales. Through the recognition of the principle of discrete fractional calculus, we are able to recover the precise estimates for unknown functions of inequalities of the Gronwall type. The resultant inequalities are of unique structure comparative with the latest reviewing disclosures and can be described as a complementary tool for numerically testing the solutions of discrete partial differential equations. The foremost consequences are probably confirmed via handling of assessment procedure and technique of mean value speculation. We display few examples of the proposed inequalities to represent the incentives of our effort. [ABSTRACT FROM AUTHOR]

By means of ς fractional sum operator, certain discrete fractional nonlinear inequalities are replicated in this text. Considering the methodology of discrete fractional calculus, we establish estimations of Gronwall type inequalities for unknown functions. These inequalities are of a new form comparative with the current writing discoveries up until this point and can be viewed as a supportive strategy to assess the solutions of discrete partial differential equations numerically. We show a couple of employments of the compensated inequalities to reflect the benefits of our work. The main outcomes might be demonstrated by the use of the examination procedure and the approach of the mean value hypothesis. [ABSTRACT FROM AUTHOR]

In this article we have considered a fractional order impulsive integro-differential equation (IDE) in Caputo's sense for the unique solution and data dependence results. We take help of the Banach fixed point theory and basic literature of fractional calculus. The results are examined with the help of an expressive numerical example for an application of the results. [ABSTRACT FROM AUTHOR]

The celebrated Montgomery identity has been studied extensively since it was established. We found a novel version of the Montgomery identity when we were working inside the framework of p- and q-calculus. We acquire a Montgomery identity through a definite (p , q) -integral from these results. Consequently, we establish specific Ostrowski-type (p , q) -integral inequalities by using Montgomery identity. In addition to the well-known repercussions, this novel study provides an opportunity to set up new boundaries in the field of comparative literature. The research that is being proposed on the (p , q) -integral includes some fascinating results that demonstrate the superiority and applicability of the findings that have been achieved. This highly successful and valuable strategy is anticipated to create a new venue in the contemporary realm of special relativity and quantum theory. These mathematical inequalities and the approaches that are related to them have applications in the areas that deal with symmetry. Additionally, an application to special means is provided in the conclusion. [ABSTRACT FROM AUTHOR]