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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]

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian's and He's polynomials, thereby offering a more efficient way of solving nonlinear problems. [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 the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively. [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]

An interval-valued q-rung orthopair fuzzy set ( I V q r O F S ) is a robust and esteemed model in managing imprecise information, utilizing a controllable parameter q ≥ 1 . Expanding its applicability, we propose a refinement to it, termed the interval-valued p,q-rung orthopair fuzzy set, denoted I V p , q r O F S . This advancement enables the handling of scenarios where varied assessment levels for membership and non-membership grades are necessary, a challenge unaddressed by existing extensions of interval-valued intuitionistic fuzzy sets. We establish the fundamental mathematical operations for the I V p , q r O F S and explore their key properties. To compare interval-valued p,q-rung orthopair fuzzy ( I V p , q r O F ) numbers, a novel score function is put forward and compared to the known I V q r O F S -based score functions to determine its viability. Moreover, we present I V p , q r O F aggregation operators (AOs), accompanied by rigorous proof of their essential characteristics, including boundedness, monotonicity, and symmetry. This study introduces the interaction of I V p , q r O F criteria through the inter-criteria correlation (CRITIC) approach. This method determines criteria weights based on the proposed distance measurements and the score function. The CRITIC method is integrated with the exponential TODIM approach (exp-TODIM), resulting in an innovative decision-making framework that is less susceptible to parameter fluctuations and showcases a high stability. An illustrative example of selecting a suitable supplier is provided to demonstrate the practical application of the developed exp-TODIM approach. Finally, a sensitivity analysis is conducted to exhibit the method's stability, and a comparative analysis is performed to elucidate its strengths and advantages. [ABSTRACT FROM AUTHOR]

The concept of linear Diophantine fuzzy set (LDFS) theory with its control parameters is a strong model for machine learning and data-driven multi-criteria decision making (MCDM). The sine-trigonometric function (STF) has two significant features, periodicity and symmetry about the origin that are very useful tools for information analysis. Keeping in view the characteristics of both STF and LDFS theory, this article introduces the sine-trigonometric operations for linear Diophantine fuzzy numbers (LDFNs). These operational laws lay a foundation for developing new linear Diophantine fuzzy sine-trigonometric aggregation operators (LDFSTAOs). The integration of Industry 4.0 technology into healthcare has the potential to revolutionize patient care. One of the most challenging tasks is the selection of efficient suppliers for the healthcare supply chain (HSC). The traditional suppliers are not efficient in accordance with Industry 4.0, with particular uncertainties. A new MCDM framework is presented based on LDFSTAOs to examine the HSC performance in industry 4.0. A credibility test, sensitivity analysis and comparative analysis are performed to express the novelty, reliability, and efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided. [ABSTRACT FROM AUTHOR]

This research work is devoted to undertaking a dynamical system representing SARS-CoV-19 disease under the concept of piecewise fractional-order derivative using the Caputo concept since long-memory and short-memory terms are not well explained by ordinary fractional differential equations. It has been found that for such disruption, piecewise operators of fractional derivatives have been found useful in many cases. Therefore, we study a compartmental model of susceptible and infected individuals under the concept of piecewise derivative. We establish the existence theory of the considered model by using some Banach and Schauder fixed-point theorems. Keeping the importance of stability, a pertinent result related to the said area is also developed. The said concept of stability is based on the concept given by Ulam and Hyers. Further, to derive the numerical results, we use the Euler method to develop a numerical scheme for the considered model. Using real available data, we have presented various graphical presentations of two compartments against different fractional orders and various values of isolation parameters. The crossover behaviors in the dynamics can be clearly observed, which is explained by the piecewise operators, not the usual fractional-order derivative. [ABSTRACT FROM AUTHOR]

The main objective of this study is the use of Taylor’s series method for approximate solution of HIV-1 infection model. This method explores to solve a system of ODE expressed as an infinite series. These series components are easily determined. The presented method’s effectiveness and reliability are shown using a numerical example, and the consequences are evaluated to those acquired from different techniques in the research using tables and graphs. The proposed method has no assumptions about small or large parameters, and the technique’s accuracy increases when the order of approximation is increased. The results reveal that the approximate solution obtained through the use of Taylor's series method is more reliable and accurate. [ABSTRACT FROM AUTHOR]

Four-dimensional continuous chaotic models with Caputo fractional derivative are presented. Fixed point theory is used to investigate the existence and uniqueness of complex systems. The dynamical properties are studied, including the Lyapunov exponent, phase portrait, and time series analysis. The hyperchaotic nature of each system is revealed by the positive exponents. The numerical method is introduced to describe the influence of the order of the Caputo fractional derivative. The phase portraits are presented to investigate the behavior and effect of some key parameters and fractional orders on model dynamics. The systems approach fixed point attractors for fractional-order and increase the visibility of the attractor by decreasing fractional order. This means that a change in fractional order has a significant impact on the dynamics of the models. When the order of the derivative is equal to one, both systems oscillate frequently. However, as the fractional order is reduced, the system oscillations decrease as compared to the integer-order, and the system moves towards its fixed point, reveals the hidden attractors inherent in the system, and enabling it to develop to a stable state more efficiently. [ABSTRACT FROM AUTHOR]

We investigate a general sequential hybrid class of fractional differential equations in the Caputo and Atangana–Baleanu fractional senses of derivatives. We consider the existence and uniqueness of solutions and the Hyers–Ulam (H-U) stability for a general class. We use the Banach and Leray–Schauder alternative theorems for the existence criteria. With the help of nonnegative Green's functions, the fractional-order class is turned into m-equivalent integral forms. As an application of our problem, a fractional-order smoking model in terms of the Atangana–Baleanu derivative is presented as a particular case. [ABSTRACT FROM AUTHOR]

This manuscript investigates fractal-fractional order smoking models with relapse and harmonic mean type incidence rate under the Caputo derivative. We derive the existence and unique results about the solution for the considered model via fixed point theory. For the stability of the considered system, Ulam-Hyers (UH) approach is used. We compute the numerical solution by using fractional Adams-Bashforth method. For the simulation of the model, we consider different values of fractional order δ and fractal dimension θ by using some real values of the parameters. The proposed scheme is used to simulate the available data for some smoking community including potential, light, and quit smokers. Various graphical presentations are given to understand the dynamics of the model at various fractional orders. [ABSTRACT FROM AUTHOR]

We investigate a nonlinear system of pantograph-type fractional differential equations (FDEs) via Caputo-Hadamard derivative (CHD). We establish the conditions for existence theory and Ulam-Hyers-type stability for the underlying boundary value system (BVS) of FDE. We use Krasnoselskii's and Banach's fixed point theorems to obtain the desired results for the existence of solution. Stability is an important aspect from a numerical point of view we investigate here. To justify the main work, relevant examples are provided. [ABSTRACT FROM AUTHOR]

Our manuscript is devoted to investigating a class of impulsive boundary value problems under the concept of the Riemann-Liouville fractional order derivative. The subject problem is of implicit type. We develop some adequate conditions for the existence and puniness of a solution to the proposed problem. For our required results, we utilize the classical fixed point theorems from Banach and Scheafer. It is to be noted that the impulsive boundary value problem under the fractional order derivative of the Riemann-Liouville type has been very rarely considered in literature. Finally, to demonstrate the obtained results, we provide some pertinent examples. [ABSTRACT FROM AUTHOR]

We construct a subclass of Copson's integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations r > 1 as well as r < 1. We demonstrate the inequalities with the guidance of basic comparison, Holder's inequality, and the integration by parts approach. Moreover, some new variations of Hardy's integral inequality are also presented with the utilization of Steklov operator. We also formulate many remarks and two examples to show the novelty and authenticity of our results. [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]

The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses. [ABSTRACT FROM AUTHOR]

The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to set up some new estimates of a specific class of dynamic integral inequalities (DII) which are partially linear on a time scale T with two independent variables. These, from the one hand, sum up and, on the other hand, offer a helpful method for both the qualitative and quantitative study of dynamic equations on time scales. Some applications are taken into consideration to show the validity of the fundamental results. [ABSTRACT FROM AUTHOR]

In this paper, we build up some generalizations of nonlinear integral inequalities and recreate the results of some Pachpatte's inequalities on time scales. We not just settle new estimated bounds of a particular class of nonlinear retarded dynamic inequalities, but additionally determine and unify continuous analogs alongside a subjective time scale T . We demonstrate applications of the treated inequalities to reflect the benefits of our work. The key effects will be proven by using the analysis procedure and the standard time-scale comparison theorem technique. [ABSTRACT FROM AUTHOR]

By the utilization of diamond-α derivatives, certain new generalizations of Ou-Iang type of dynamic integral inequalities of one independent variable on time scales are examined. The resulting inequalities are significant in the study of various fields of dynamic equations. A few mathematical applications are also presented. [ABSTRACT FROM AUTHOR]

Several inequalities were proved in 2018 by Boudeliou, in 2015 by Abdeldain and El-Deeb and in 1998 by Pachpatte. It is our aim in this paper to generalize these inequalities to time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study the boundedness of some problems by applying our results. [ABSTRACT FROM AUTHOR]

We investigate a certain class of nonlinear Gronwall–Bellman–Pachpatte type of integral inequalities involving retarded term and nonlinear integrals. Our results derive some new inequalities as well as give an upper bound estimation of the unknown function. The significance of these inequalities originate from the truth that it is far applicable in specific situations in which different available inequalities do not apply legitimately. Our consequences additionally can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is also illustrated to show the validity of our established theorems. [ABSTRACT FROM AUTHOR]

The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations. [ABSTRACT FROM AUTHOR]

In this work, we introduce new definitions of left and right-sides generalized conformable K -fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for ϕ -preinvex functions. Moreover, we use these new identities to prove some bounds for the Hermite-Hadamard-Fejér type inequality for generalized conformable K -fractional integrals regarding ϕ -preinvex functions. Finally, we also present some applications of the generalized definitions for higher moments of continuous random variables, special means, and solutions of the homogeneous linear Cauchy-Euler and homogeneous linear K -fractional differential equations to show our new approach. [ABSTRACT FROM AUTHOR]

Endophytic associations with plants have a beneficial effect for many different plant species with some of them being host specific. Several endophytes isolated from poplar and willow were screened for their effects on commercially important crops including corn, tomato, pepper, squash, sunflower, and grasses. Most of these endophytes produce growth hormones such as indoleacetic acid (IAA) and have the nitrogenase gene required for nitrogen fixation. The effects of these isolates on plant growth and yield were evaluated under greenhouse conditions. We found that inoculated plants not only had better viability and earlier flowering and fruiting, they also had increased plant growth and fruit yields when grown in nitrogen-limited soil. In a particular variety of perennial rye grass, the endophytes increased the total nitrogen content of the plants, indicative of nitrogen fixation, in addition to promoting plant growth. The use of specific endophytes may be preferable to the use of chemical fertilizers because of the monetary and environmental costs, contributing to more sustainable agricultural systems. [ABSTRACT FROM AUTHOR]

Endophytic bacteria associated with sweet potato plants ( Ipomoea batatas (L.) Lam.) were isolated, identified and tested for their ability to fix nitrogen, produce indole acetic acid (IAA), and exhibit stress tolerance. Eleven different strains belonging to the genera, Enterobacter, Rahnella, Rhodanobacter, Pseudomonas, Stenotrophomonas, Xanthomonas and Phyllobacterium, were identified. Four strains were shown to produce IAA (a plant growth hormone) and one strain showed the ability to grow in nitrogen free medium and had the nitrogenase subunit gene, nifH. To determine if IAA production by the endophytes had any role in protecting the cells against adverse conditions, different stress tests were conducted. The IAA producer grew well in the presence of some antibiotics, UV and cold treatments but the response to pH, osmotic shock, thermal and oxidative treatments was the same for both the IAA producer and the no IAA producer. To determine if IAA produced by the strains was biologically relevant to plants, cuttings of poplar were inoculated with the highest IAA producing strain. The inoculated cuttings produced roots sooner and grew more rapidly than uninoculated cuttings. These studies indicate that endophytes of sweet potato plants are beneficial to plant growth. [ABSTRACT FROM AUTHOR]

The main purpose of this research is to concentrate on the development of new definitions for the weighted geometric fractional integrals of the left-hand side and right-hand side of the function ℵ with regard to an increasing function used as an integral kernel. Moreover, the newly developed class of left-hand side and right-hand side weighted geometric fractional integrals of a function ℵ, by applying an additional increasing function, identifies a variety of novel classes as special cases. This is a development of the previously established fractional integrals by making use of the class of geometrically convex functions. Geometrically convex functions in weighted fractional integrals of a function ℵ in the form of another rising function yield the Hermite–Hadamard inequality type. We also establish a novel midpoint identity and the associated inequalities for a class of weighted fractional integral functions known as geometrically convex with respect to an increasing function and symmetric with respect to the geometric mean of the endpoints of the interval. In order to demonstrate the validity of our research, we present examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains. [ABSTRACT FROM AUTHOR]

In this paper, we present some new refinement mappings associated with the Hermite–Hadamard type inequalities that are constructed for GA-convex mappings. Our investigation of the mappings leads to the discovery of several interesting features as well as the development of some inequalities for the Hermite–Hadamard type inequalities, which have already been established for GA-convex functions, as well as refining the relationship between the middle, rightmost, and leftmost elements of the function. Some applications to special means of positive real numbers are also given. [ABSTRACT FROM AUTHOR]

In the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities. As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena. [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]

Tyrosinase is an important component of the enzyme polyphenol oxidase, which upon contact with the phenolic substrates forms the pigment melanin and induces undesirable food browning. The phenolic and triterpenoid compounds that naturally occur in plants are well known as tyrosinase inhibitors. Combretum micranthum (CM) leaves, Euphorbia hirta (EH) plant, and Anacardium occidentale (AO) fruits are traditionally known to have potential anti-tyrosinase activities. The aim of this study was to optimize the ultrasound-assisted extraction of secondary metabolites from these matrices, and to evaluate in tubo the antityrosinase activity of these extracts. Efforts were also taken to profile the secondary metabolites, mainly the phenolic and triterpenoid compounds, in order to understand their probable association with tyrosinase inhibition. The optimal ultrasound-assisted extraction conditions for simultaneous extraction of phenolic, and triterpenoid compounds were determined. The aqueous fraction of these extracts showed significant antityrosinase activity, with the CM leaves exhibiting the strongest inhibitory effect (IC50 of 0.58 g·L−1). The predominant metabolic compounds from these natural extracts were putatively identified by using a high-resolution quadrupole-time of flight (QToF) LC-MS instrument. The high-resolution accurate mass-based screening resulted in identification of 88 predominant metabolites, which included dihydrodaidzein-7-O-glucuronide, micromeric acid, syringic acid, morin, quercetin-3-O-(6″-malonyl-glucoside), 4-hydroxycoumarin, dihydrocaffeic acid-3-O-glucuronide, to name some, with less than 5 ppm of mass error. [ABSTRACT FROM AUTHOR]

This paper is committed to introducing some Bihari type inequalities for scalar functions of one independent variable under an initial condition associated with an arbitrary time scale T . The integrals involve the maximum of an unknown function over a past time interval. We not only solve some new estimated bounds of a specific class of retarded and nonlinear dynamic inequalities but also derive and unify continuous inequalities along with the corresponding discrete analogs of some known results with 'maxima' on time scales. We illustrate some applications of the considered inequalities to represent the advantages of our work. The main results will be proved by utilizing some examination procedures and the basic technique of Keller's chain rule on time scales. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. [ABSTRACT FROM AUTHOR]

In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results. [ABSTRACT FROM AUTHOR]

In the present paper, we study a mathematical model of an imperfect testing infectious disease model in the sense of the Mittage-Leffler kernel. The Banach contraction principle has been used for the existence and uniqueness of solutions of the suggested model. Furthermore, a numerical method equipped with Lagrangian polynomial interpolation has been utilized for the numerical outcomes. Diagramming and discussion are used to clarify the effects of related parameters in the fractional-order imperfect testing infectious disease model. [ABSTRACT FROM AUTHOR]

In this paper, we establish some dynamic Hilbert-type inequalities in two independent variables on time scales by using the Fenchel–Legendre transform. We also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases. Our results give more general forms of several previously established inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method to unify and extend some nonlinear dynamic double integral inequalities of two independent variables involving pairs of time scales via nabla derivative. The acquired results give explicit bounds which can be utilized to consider the subjective and quantitative properties of specific classes of dynamic conditions on time scales. An application to prove the validity of our established results is also given. [ABSTRACT FROM AUTHOR]

In this article, we obtain some new dynamic inequalities of Hardy type on time scales. The main results are derived using Fubini's theorem and the chain rule on time scales. We apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases. [ABSTRACT FROM AUTHOR]

In this article, we establish several integral majorization type and generalized Favard's inequalities for the class of strongly convex functions. Our results generalize and improve the previous known results. [ABSTRACT FROM AUTHOR]

In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of cr -convex interval-valued functions and deal with differintegrals of the p + s 2 type. We believe this will be an important contribution to spurring additional research. [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]

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function Υ ˘ using an another function ϑ (ζ ˙) . As an additional observation, it is noted that the new class of weighted interval-valued fractional integrals of a function Υ ˘ by employing an additional function ϑ (ζ ˙) characterizes a variety of new classes as special cases, which is a generalization of the previous class. Secondly, we prove a new version of the Hermite-Hadamard-Fejér type inequality for h-convex interval-valued functions using weighted interval-valued fractional integrals of a function Υ ˘ according to another function ϑ (ζ ˙). Finally, by using weighted interval-valued fractional integrals of a function Υ ˘ according to another function ϑ (ζ ˙) , we are establishing a new Hermite-Hadamard-Fejér type inequality for harmonically h-convex interval-valued functions that is not previously known. Moreover, some examples are provided to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

This paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

We consider some new Steffensen-type dynamic inequalities on an arbitrary time scale by utilizing the diamond-α dynamic integrals, which are characterized as a combination of the delta and nabla integrals. These inequalities expand some known dynamic inequalities on time scales, bind together and broaden some integral inequalities and their discrete analogs. [ABSTRACT FROM AUTHOR]

In the article, we establish an inequality for Csiszár divergence associated with s-convex functions, present several inequalities for Kullback–Leibler, Renyi, Hellinger, Chi-square, Jeffery's, and variational distance divergences by using particular s-convex functions in the Csiszár divergence. We also provide new bounds for Bhattacharyya divergence. [ABSTRACT FROM AUTHOR]

In this paper, we present a Hermite-Hadamard-Fejér inequality for conformable fractional integrals by using symmetric preinvex functions. We also establish an identity associated with the right hand side of Hermite-Hadamard inequality for preinvex functions; then by using this identity and preinvexity of functions and some well-known inequalities, we find several new Hermite-Hadamard type inequalities for conformal fractional integrals. [ABSTRACT FROM AUTHOR]