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This article is concerned with the stability issue of PI-based load frequency control (LFC) systems with time-varying delays and load interference. Firstly, a novel augmented Lyapunov–Krasovskii functional (LKF) is developed, in which the single integral term includes the augmented s-dependent integral term of delay-partition. For the purpose of coordinating with the constructed LKF effectively, a generalised free-matrix-based integral inequality and quadratic inequality are utilised to estimate the functional derivatives accurately, so that the stability criterion of less conservative is obtained. Finally, the relationship between the PI gains and the delay margins is presented, and the effects of time delays and PI gains on the system dynamic performance are discussed simultaneously by simulation. The simulation results verify the validity of the proposed method. [ABSTRACT FROM AUTHOR]

The problem of robust tracking and model following is reconsidered for a class of uncertain nonlinear systems with some uncertainties and dead-zone input constraints. In this paper, the system uncertainties are regarded as some nonlinear functions which are not required to must satisfy the matching conditions. That is, being different from traditional robust control theory, the matched structures of uncertainties are sufficient, but not necessary, for being able always to design some types of robust tracking control schemes. In addition, the external disturbances of dynamical systems are assumed to be any bounded functions which is not also required to satisfy the matching condition. By making use of the integral inequality, instead of Lyapunov function, a class of robust tracking control schemes is constructed to guarantee the uniform exponential boundedness of the model tracking errors. Moreover, the upper bounds of nonlinear uncertainties are not required to be known, and it is also unnecessary to know or estimate the characteristic parameters of dead-zone functions. Therefore, the resulting robust tracking control schemes have a simple structure. Actually, the resulting robust tracking control scheme is a linear feedback control one with a self-tuning control gain. Finally, an illustrative numerical example is provided to demonstrate the validity of the presented theoretical results. [ABSTRACT FROM AUTHOR]

This article examines the problem of estimating the states of Markovian jumping competitive neural networks, where the estimation is done using stochastic sampled-data control with time-varying delay. Instead of continuously measuring the states, the network relies on sampled measurements, and a sampled-data estimator is proposed. The estimator uses probabilistic sampling during two sampling periods, following a Bernoulli distribution. The article also takes into account the possibility of actuator failure in real systems. To ensure the exponentially mean-square stability of the delayed neural networks, the article constructs a Lyapunov-Krasovskii functional (LKF) that includes information about the bounds of the delay. The sufficient conditions for stability are derived in the form of linear matrix inequalities (LMIs) by employing modified free matrix-based integral inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with new generalizations and refinements of the diamond-α integral Cauchy–Schwartz inequality on time scales. The new generalization and refinement of the Cauchy–Schwartz inequality related to diamond integral on time scales are also considered. These obtained inequalities unify continuous inequalities and their corresponding discrete forms associated with Cauchy–Schwartz inequality. [ABSTRACT FROM AUTHOR]

Motivated by certain results known from the literature, in this paper we give several new dynamic inequalities for B-monotone functions with respect to time scales nabla calculus. If the time scale represents the set of real numbers, our results reduce to integral inequalities known from the literature. On the other hand, in the setting of positive integers, we obtain new discrete inequalities for B-monotone sequences. [ABSTRACT FROM AUTHOR]

Let A be a positive bounded operator on a Hilbert space H and let ‖ T ‖ A , w A (T) , and m A (T) denote the A -operator seminorm, the A -numerical radius, and the A -minimum modulus of an operator T in the semi-Hilbertian space (H , ‖ ⋅ ‖ A) , respectively. In this paper, we present new improvements of certain A -Cauchy–Schwarz type inequalities and as applications of our results, we provide refinements of some A -numerical radius inequalities for semi-Hilbertian space operators. It is shown, among other inequalities, that w A (T) ≤ (1 − 1 2 inf λ ∈ C ⁡ m A 2 (I − λ T)) ‖ T ‖ A , where I is the identity operator on H. A refinement of the triangle inequality for semi-Hilbertian space operators is also given. [ABSTRACT FROM AUTHOR]

This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen's inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen's inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen's inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds. [ABSTRACT FROM AUTHOR]

This work investigates the less conservative stability conditions for linear systems with a time-varying delay. At first, augmented Lyapunov–Krasovskii functionals(LKFs) are constructed with state vectors that have not been utilized in the existing works, and an augmented zero equality that can be derived according to the augmented vector is proposed. By utilizing them, a stability condition is proposed in the form of a linear matrix inequality. And, by using novel delay-dependent LKFs and the introduced ones, improved results are obtained than the previous result. The addition of the delay-dependent LKFs increases the number of decision variables in the results. Therefore, any vectors of integral inequalities utilized in the proposed criterion are appropriately adjusted to reduce computational complexity. To check the excellence and validity of the proposed results, several numerical examples are applied. • The improved stability conditions for the linear system with a time-varying delay are proposed. • The proposed Lyapunov-Krasovskii functionals(LKFs) have more extended augmented vectors to obtain less conservative results. And LKFs based on an integral inequality (Lee et al., 2023) is utilized. • The novel LKFs which are the delay-dependent forms are proposed. The superiority of the results is checked according to the presence or absence of the forms. • The computational complexity is reduced by adjusting any vector of the integral inequality utilized in the proposed criteria. [ABSTRACT FROM AUTHOR]

We primarily examined the effect of leakage delays on finite-time stability problems for genetic regulatory networks with interval time-varying delays. Since leakage delays can occur within the negative feedback components of networks and significantly impact their dynamics, they may potentially cause instability or suboptimal performance. The derived criteria encompass both leakage delays and discrete interval time-varying delays through the construction of a LyapunovKrasovskii function. We employed the estimation of various integral inequalities and a reciprocally convex technique. Additionally, these models consider lower bounds on delays, which may be either positive or zero, and allow for the derivatives of delays to be either positive or negative. Consequently, new criteria for genetic regulatory networks with interval time-varying delays under the effect of leakage delays are expressed in the form of linear matrix inequalities. Ultimately, a numerical example is presented to show the effect of leakage delays and to emphasize the significance of our theoretical findings. [ABSTRACT FROM AUTHOR]

Integral inequalities in mathematical interpretations are a substantial and ongoing body of research. Because fractional calculus techniques are widely used in science, a lot of research has recently been done on them. A key concept in fractional calculus is the Caputo-Fabrizio fractional integral. In this work, we focus on using the Caputo-Fabrizio fractional integral operator to build a multi-parameter fractional integral identity. Using the obtained integral identity, certain generalized estimates of Bullen-type fractional inequalities have been generated. By establishing certain inequalities, this study advances the fields of fractional calculus and convex function research. Both graphical and numerical statistics are provided to show the correctness of our results. We finally provide applications to modified Bessel functions, h-divergence measures, and probability density functions. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized n-fractional polynomial s-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowskitype inequalities are presented for generalized n-fractional polynomial s-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized n-fractional polynomial s-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the RiemannLiouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

The relationship is studied here between the 3D incompressible Brinkman–Forchheimer problem with delay and its generalized steady state. First, with some restrictive condition on the delay term, the global well-posedness of 3D Brinkman–Forchheimer problem and its steady state problem are obtained by compactness method and Brouwer fixed point method respectively. Then the global L p (2 ≤ p < ∞) decay estimates are established for weak solution of non-autonomous Brinkman–Forchheimer equations with delay by using a retarded integral inequality. The global decay estimates can be proved for strong solution as well. Finally, the exponential stability property is investigated for weak solution of the 3D non-autonomous Brinkman–Forchheimer problem by a direct approach and also for the autonomous system by using a retarded integral inequality. Furthermore, the Razumikhin approach is utilized to achieve the asymptotic stability for strong solution of autonomous system under a relaxed restriction. [ABSTRACT FROM AUTHOR]

In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities. [ABSTRACT FROM AUTHOR]

Let p (z) be a polynomial of degree n having no zero in | z | < k , k ≤ 1 , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ (z) | ≤ n 1 + k n max | z | = 1 | p (z) | , provided | p ′ (z) | and | q ′ (z) | attain their maxima at the same point on the circle | z | = 1 , where q (z) = z n p (1 z ‾) ‾. In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results. [ABSTRACT FROM AUTHOR]

Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parameter p. Finally, we present a Milne-type inequality in the fractional context. [ABSTRACT FROM AUTHOR]

The authors of the paper present a method to examine some Newton‐type inequalities for various function classes using Riemann‐Liouville fractional integrals. Namely, some fractional Newton‐type inequalities are established by using convex functions. In addition, several fractional Newton‐type inequalities are proved by using bounded functions by fractional integrals. Moreover, we construct some fractional Newton‐type inequalities for Lipschitzian functions. Furthermore, several Newton‐type inequalities are acquired by fractional integrals of bounded variation. Finally, we provide our results by using special cases of obtained theorems and examples. [ABSTRACT FROM AUTHOR]

In this paper, fractional order integral operators are applied to the study of inequalities, which is also a research hotspot of fractional order integral operators in recent years. After introducing some basic concepts, we establish some new Volterra-Fredholm-type fractional integral inequalities. They can be used as an effective tool to study the properties of solutions of fractional differential equations and fractional integral equations. In addition, several examples are given to illustrate the effectiveness of the research results. [ABSTRACT FROM AUTHOR]

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p -Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski's inequality, trapezoidal inequality, midpoint inequality, Simpson's inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ. [ABSTRACT FROM AUTHOR]

Establishing a new integral inequality for the Riemann–Liouville fractional integral operator is the main objective of this paper. For twice differentiable s- (κ , H) -convex functions, we present a number of new inequalities that are connected to the Hermite–Hadamard integral inequality. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of (η 1 , η 2) -convex stochastic processes on coordinates and establish Hermite-Hadamard-type inequality for these stochastic processes. Moreover, we prove new integral inequality of Hermite-Hadamard-Fejér type for newly defined coordinated η -convex stochastic processes on a rectangle. The results presented in this article would provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

This document is a summary of a special issue of the journal "Fractal & Fractional" titled "Mathematical Inequalities in Fractional Calculus and Applications." The special issue contains fifteen articles that explore mathematical inequalities involving fractional derivatives and fractional integral operators and their applications in various mathematical and related fields. The articles cover a range of topics, including convexity, quantum inequalities, fuzzy-interval-valued functions, integral operators, and symmetric derivatives. The authors of the summary express their gratitude to the contributing authors and reviewers for their valuable contributions to the special issue. [Extracted from the article]

This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j ∈ N) positively continuous and decaying functions in the finite interval a ≤ t ≤ x , the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. [ABSTRACT FROM AUTHOR]

This article deals with the stability and H∞$$ {H}_{\infty } $$ control synthesis for a class of linear sampled‐data systems. First, a novel integral inequality with the cubic term of integral interval length instead of reciprocal one of integral interval length is constructed by non‐orthogonal polynomials. This can make full use of slack matrix variables and additional information about sawtooth structural sampling pattern. Then, on the basis of the constructed integral inequality, a cubic‐term‐dependent discontinuous exponent Lyapunov–Krasovskii Functional (LKF) is developed containing additional information about system states to design an H∞$$ {H}_{\infty } $$ sampled‐data state feedback controller. To better estimate the upper bound of the derivative of LKF, a kind of multiple convex function approximation approach is developed, which can be used to deal with the two‐variable polynomial negative definite issue by dividing the variable interval into multiple subintervals. By the constructed integral inequality and multiple convex function approximation approach, sufficient conditions with less conservatism are deduced for the feasibility of H∞$$ {H}_{\infty } $$ sampled‐data state feedback controller. Moreover, by the inner convex approximation solution technique and proposed iterative algorithm, the bilinear matrix inequalities can be transformed into linear matrix ones, which can be easily tackled, to obtain the desired sampled‐data controller gain. Finally, by comparing with existing results, the advantages and superiority of our proposed approaches can be confirmed via two numerical examples. [ABSTRACT FROM AUTHOR]

• Memory sampled-data control is applied for the uncertain rigid spacecraft model. • Randomly missing measurement in the system output obeys the Bernoulli distribution. • It tolerates uncertainty and disturbances using the H ∞ technique. • A novel looped and delay-dependent LKF is constructed for the proposed systems. • It gives the minimum disturbance attenuation level compared with existing works. This paper focuses on stabilizing the attitude of the rigid spacecraft under the sampled-data control design technique with constant communication delay. The system dynamics with model uncertainty (perturbation) and missing measurements are considered. The aim is to design a desirable attitude control that ensures the stabilization of the rigid spacecraft with an optimal H ∞ performance level. A novel Lyapunov–Krasovskii functional is developed, incorporating looped characteristics, for the proposed study on the rigid spacecraft model by developing relevant new terms. Making use of the Lyapunov function as well as free-matrix-based integral inequality, sufficient stability criteria are established to assure the stability of the rigid spacecraft model. Furthermore, the desired sampled-data control gain matrices are acquired through the solution of linear matrix inequalities, which guarantees the asymptotic stability of the rigid spacecraft model. Finally, the numerical simulation demonstrates the efficacy of the theoretical study proposed for the rigid spacecraft model. [ABSTRACT FROM AUTHOR]

This paper centers on stochastic Hopfield neural networks with variable coefficients and infinite delay. First, we propose an integral inequality that improves and extends some existing works. Second, by employing some inequalities and stochastic analysis techniques, some suffcient conditions for ensuring pth moment generalized exponential stability are established. Our results do not necessitate the construction of a complex Lyapunov function or rely on the assumption of bounded variable coeffscients, and our results expand some existing works. At last, to illustrate the efficacy of our result, we present several simulation examples. [ABSTRACT FROM AUTHOR]

Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized $ n $-fractional polynomial $ s $-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowski-type inequalities are presented for generalized $ n $-fractional polynomial $ s $-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized $ n $-fractional polynomial $ s $-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the Riemann-Liouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

Abstract Let p ( z ) $p(z)$ be a polynomial of degree n having no zero in | z | < k $|z|< k$ , k ≤ 1 $k\leq 1$ , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ ( z ) | ≤ n 1 + k n max | z | = 1 | p ( z ) | , $$ \max _{|z|=1}|p'(z)|\leq \frac{n}{1+k^{n}}\max _{|z|=1}|p(z)|, $$ provided | p ′ ( z ) | $|p'(z)|$ and | q ′ ( z ) | $|q'(z)|$ attain their maxima at the same point on the circle | z | = 1 $|z|=1$ , where q ( z ) = z n p ( 1 z ‾ ) ‾ . $$ q(z)=z^{n}\overline{p\bigg(\frac{1}{\overline{z}}\bigg)}. $$ In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.

This paper is dedicated to the numerical solution of a mathematical model that describes frictional quasistatic contact between an elastic body and a moving foundation, with the wear effect on the contact interface of the moving foundation due to friction. The mathematical problem is a system consisting of a time-dependent quasi-variational inequality and an integral equation. The numerical method is based on the use of the virtual element method (VEM) for the spatial discretization of the variational inequality and a variable step-size left rectangle integration formula for the integral equation. The existence and uniqueness of a numerical solution are shown, and optimal order error estimates are derived for both the displacement and the wear function for the lowest order VEM. Numerical results are presented to demonstrate the efficiency of the method and to illustrate the numerical convergence orders. [ABSTRACT FROM AUTHOR]

This article presents new classes of ζ -uniformly q-starlike and qconvex analytic functions of order υ using the principle of q-calculus. The renowned classes of starlike and convex functions are utilized to establish these classes. The investigation of these uniformly classes leads to the study of some geometric notions, including coefficient estimates, sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, some subordination properties involving integral means inequalities and subordinate sequences are examined. [ABSTRACT FROM AUTHOR]

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F , f) , on ℝ n in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley g -functions, Lusin area functions, Littlewood–Paley g λ ∗ -functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley g -functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ℝ n in the aforementioned function spaces. [ABSTRACT FROM AUTHOR]

This article explores integral inequalities within the framework of local fractional calculus, focusing on the class of generalized α$$ \alpha $$‐convex functions. It introduces a novel extension of the Hermite‐Hadamard inequality and derives numerous fractal inequalities through a novel multiparameterized identity. The primary aim is to generalize existing inequalities, highlighting that previously established results can be obtained by setting specific parameters within the main inequalities. The validity of the derived results is demonstrated through an illustrative example, accompanied by 2D and 3D graphical representations. Lastly, the paper discusses potential practical applications of these findings. [ABSTRACT FROM AUTHOR]

This paper employs two scalar parameters to extend auxiliary function-based integral inequality into a new formulations, say a parametrised auxiliary function-based integral inequality. Using these formulations, the constituent signals are utilised more efficiently by exploiting the interaction between them to reduce conservatism. Numerical example for stability analysis of linear systems with time-varying delay shows the improved performance of the proposed new formulations in terms of maximum delay bounds and decision variables. [ABSTRACT FROM AUTHOR]

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler's lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems. [ABSTRACT FROM AUTHOR]

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying some new reverse Hölder-type inequalities associated with the dynamic integral called diamond on time scale, which is expressed as an ’approximate’ symmetric integral on time scales. Some related inequalities are also presented. The obtained inequalities extend known results related with discrete and continuous forms. [ABSTRACT FROM AUTHOR]

This article delves into the intricate matter of achieving robust consensus tracking of multi‐agent systems with structured uncertainty, one‐sided Lipschitz nonlinearity and time‐varying delay. It is important to note that the time delay is both time‐varying and bounded. To solve the consensus tracking problem of multi‐agent systems, we transform it into the stability problem of an error system. Then, a Lyapunov–Krasovskii functional, incorporating the free‐weighting matrix technique and integral inequality approach, is designed to investigate the stability of the obtained error system. The Leibnitz–Newton formula and its associated relationships among terms are also taken into account. Then, two distributed control strategyies that respectively utilizes relative state and output information of neighboring agents are designed. This approach enables us to find a feasible solution to the consensus tracking problem. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature. [ABSTRACT FROM AUTHOR]

In this article, we begin by deriving a new identity with the help of twice‐differentiable convex functions for the proportional Caputo‐hybrid operator. Then, using this newly uncovered identity, we obtain various integral inequalities associated with the Simpson's integral inequality for proportional Caputo‐hybrid operator. Moreover, we indicate that the acquired results improve and refine certain existing discoveries in the realm of integral inequalities. Finally, for a better understanding of the newly obtained inequalities, we establish illustrative examples and visualize them through their corresponding graphs. [ABSTRACT FROM AUTHOR]

This paper introduces and establishes new Hermite‐Hadamard type inequalities specifically tailored for m$$ m $$‐preinvex functions within the (p,q)$$ \left(p,q\right) $$‐calculus framework. These newly developed inequalities come with accompanying left‐right estimates, which enhance their practical utility. The primary objective of this research is to investigate the properties of (p,q)$$ \left(p,q\right) $$‐differentiable m$$ m $$‐preinvex functions and derive inequalities that extend and generalize existing results in the domain of integral inequalities. The techniques employed in this study hold broader implications, finding relevance in various fields where symmetry is paramount. The findings presented in this paper make a significant contribution to the field of analytic inequalities, offering valuable insights into the behavior and characteristics of m$$ m $$‐preinvex functions. Moreover, the established results demonstrate the wider applicability and generalization of analogous findings from prior literature. The techniques and inequalities introduced herein pave the way for further exploration and research in the realm of integral inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we present Katugampola fractional parameterized inequalities for s-convex function in the fourth sense. In the first place, with the help of integration by parts method, we provide a very important identity that will be of great significance for the next research. In addition, by comprehensively making use of some important concepts such as s-convex function in the fourth sense, Hölder's integral inequality, and the well-known power-mean inequality, we establish three new parameterized inequalities. Finally, we provide some corollaries to help readers gain a deeper understanding of the theorems and the relevant theories. [ABSTRACT FROM AUTHOR]

In this manuscript, we obtain refinements of the Hermite-Hadamard type inequalites for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is trigonometrically ρ-convex. We also show that our results coincide with previous results. [ABSTRACT FROM AUTHOR]

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right ℏ -pre-invex interval-valued mappings ( C · L · R - ℏ -pre-invex Ι · V - M ), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte's type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes. [ABSTRACT FROM AUTHOR]

In this study, we introduce some new mappings in connection with Hermite-Hadamard and Fej'er type integral inequalities which have been proved using the GA-convex functions. As a consequence, we obtain certain new inequalities of the Fej'er type that provide refinements of the Hermite-Hadamard and Fej'er type integral inequalities that have already been obtained. [ABSTRACT FROM AUTHOR]

The paper concentrates on the issue of distributed disturbance observer-based nonfragile bipartite consensus within nonlinear delayed multiagent systems, encompassing both leaderless and leader-following structures. The delays under consideration are nonuniform, manifesting in the state, the nonlinearity, and the communication processes. To suppress the external disturbances and the observer gain perturbations, distributed nonfragile disturbance observers pertaining to relative output and communication delays are developed to estimate the external disturbances for each agent. Employing the developed disturbance observer, distributed control protocols for nonfragile bipartite consensus are constructed incorporating states, estimated disturbances, and communication delays. These protocols can ensure bipartite consensus, compensate for external disturbances, and tolerate uncertainties in control gain. New augmented Lyapunov-Krasovskii functions are formulated by introducing the triple integral term and the augmented vector. The new bipartite consensus criteria for the studied multiagent systems are established with less conservatism by employing the techniques on second-order Bessel-Legendre integral inequality, reciprocally convex combination, and free weight matrix. Finally, numerical simulations and comparisons are performed for both leaderless and leader-following scenarios, thereby validating and enhancing the theoretical outcomes. • Multiagent systems with nonlinearity, disturbance and multiple delay, is considered. • Both cooperative and antagonistic relationships are all considered in consensus. • The perturbations in both observer and bipartite consensus protocol. • New Lyapunov-Krasovskii functions, and Bessel-Legendre inequality are employed. [ABSTRACT FROM AUTHOR]

This paper revisits the issue of stability analysis of neural networks subjected to time-varying delays. A novel approach, termed a compound-matrix-based integral inequality (CPBII), which accounts for delay derivatives using two adjustable parameters, is introduced. By appropriately adjusting these parameters, the CPBII efficiently incorporates coupling information along with delay derivatives within integral inequalities. By using CPBII, a novel stability criterion is established for neural networks with time-varying delays. The effectiveness of this approach is demonstrated through a numerical illustration. [ABSTRACT FROM AUTHOR]

This paper explored the topic of extended dissipativity analysis for Markovian jump neural networks (MJNNs) that were influenced by time-varying delays. A distinctive Lyapunov functional, distinguished by a non-zero delay-product types, was presented. This was achieved by combining a Wirtinger-based double integral inequality with a flexible matrix set. This novel methodology addressed the limitations of the slack matrices found in earlier research. As a result, a fresh condition for extended dissipativity in MJNNs was formulated, utilizing an exponential type reciprocally convex inequality in conjunction with the newly introduced nonzero delay-product types. A numerical example was included to demonstrate the effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

The focal point of this investigation is the exploration of solutions for Caputo-Hadamard fractional differential equations with boundary conditions, and it follows the initial formulation of a model that is intended to address practical problems. The research emphasizes resolving the challenges associated with determining precise solutions across diverse scenarios. The application of the Burton-Kirk fixed-point theorem and the Kolmogorov compactness criterion in Lp-spaces ensures the existence of the solution to our problem. Banach’s theory is crucial for the establishment of solution uniqueness, and it is complemented by utilizing the Hölder inequality in integral analysis. Stability ¨ analyses from the Ulam-Hyers perspective provide key insights into the system’s reliability. We have included practical examples, tables, and figures, thereby furnishing a comprehensive and multifaceted examination of the outcomes. [ABSTRACT FROM AUTHOR]