Integral Inequalities Research Articles

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

A New Differentiable Sphere Theorem and Its Applications

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

Some new integral inequalities for F-convex functions via ABK-fractional operator

In this paper, we present Hermite-Hadamard type and midpoint-type integral inequalities for F-convex function via ABK-fractional integral operators. In the context of differentiable mapping Ψ, we provide multiple new identities and prove new results for F-convex functions using ABK-fractional integral operators whose derivatives in the absolute values are F-convex. We establish correlations between our results and a number of significant results in the literature. The results presented in this paper might encourage more study in this field.

Structural and Parametric Synthesis of a Hybrid Repetitive Control System for Uncertain Multi-Mode Plant

Article deals with the problem of constructing a hybrid adaptive-robust repetitive control system for one class of a MIMO plants which operating with structural and parametric uncertainty and constant action of an external disturbances. A procedure for structural and parametric synthesis of a control system is proposed. At the structural synthesis stage with the help of hyperstability criterion the authors are developing a decentralized combined controller. After that using the method of continuous models the authors are built a continuous-discrete control system. The main feature of a control loop structure synthesis stage is to ensure the validity of V. M. Popov’s integral inequality. For this purpose, the authors determining the special estimates that guarantee positivity for nonlinear non-stationary part of the system under consideration. At the system’s parametric synthesis stage the authors are use the environment of engineering and technical calculations "Matlab — Simulink" to perform optimization modeling of the continuous and hybrid repetitive control systems. This action is carried out using one of the functional optimization methods — a genetic algorithm. Authors use the criterion of generalized operation for automatic control systems to form the functionality for assessing the quality of the proposed system. At the simulation stage initially for continuous repetitive system it was searched its controller parameters which ensure the minimum value of the specified functional. Then, with a given discretization step of the control loop elements, the hybrid periodic system was optimized in order to improve the quality of its operation.

A family of quadrature formulas with their error bounds for convex functions and applications

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.

Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices

This note aims to reduce the conservatism of stability criteria for neural networks with time-varying delay. To this goal, on the one hand, we construct an augmented Lyapunov–Krasovskii functional (LKF), incorporating some delay-product terms that capture more information about neural states. On the other hand, when dealing with the derivative of the LKF, we introduce several parameter-dependent slack matrices into an affine integral inequality, zero equations, and the S-procedure. As a result, more relaxed stability criteria are obtained by employing the so-called Lyapunov–Krasovskii Theorem. Two numerical examples show that the proposed stability criteria are of less conservatism compared with some existing methods.

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

A matrix-separation-based integral inequality for aperiodic sampled-data synchronization of delayed neural networks considering communication delay

This paper achieves the synchronization of delayed neural networks (DNNs) considering aperiodic sampled-data control and communication delay. First of all, based on the master-slave DNNs with aperiodic sampling synchronization controller, a synchronization error system is constructed. Then, an augmented functional containing both the error state and its derivative is constructed. Compared with the existing researches, the augmented functional introduces more cross information of error states to the criterion. Next, an integral inequality based on the separation of internal integral variable and matrix is developed. Compared to the inequalities that treat the internal variable and the matrix as unified ones, the developed inequality provides a tighter estimate of the derivative of the augmented functional. On this basis, a criterion with less conservative is developed for the aperiodic sampled-data synchronization of DNNs considering communication delay. Finally, to indicate the superiority of the developed method on improving the acceptable sampling upper bound of synchronization, three numerical examples are provided.

Tensorial and Hadamard Product Integral Inequalities for Convex Functions of Continuous Fields of Operators in Hilbert Spaces

Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫_{Ω}1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (A_{τ})_{τ∈Ω} is a continuous field of positive operators in B(H) such that Sp(A_{τ}) ⊂I for each τ∈Ω and B and operator such that Sp(B)⊂I, then we have ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)⊗1-∫_{Ω}f′(A_{τ})dμ(τ)⊗B ≥∫_{Ω}f(A_{τ})dμ(τ)⊗1-1⊗f(B) ≥(∫_{Ω}A_{τ}dμ(τ)⊗1-(1⊗B))(1⊗f′(B)) and the Hadamard product inequality ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)∘1-∫_{Ω}f′(A_{τ})dμ(τ)∘B ≥∫_{Ω}f(A_{τ})dμ(τ)∘1-1∘f(B) ≥∫_{Ω}A_{τ}dμ(τ)∘f′(B)-1∘(f′(B)B).

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial

Let p(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(z)$\\end{document} be a polynomial of degree n having no zero in |z|<k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|< k$\\end{document}, k≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\leq 1$\\end{document}, then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max|z|=1|p′(z)|≤n1+knmax|z|=1|p(z)|,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\max _{|z|=1}|p'(z)|\\leq \\frac{n}{1+k^{n}}\\max _{|z|=1}|p(z)|, $$\\end{document} provided |p′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|p'(z)|$\\end{document} and |q′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|q'(z)|$\\end{document} attain their maxima at the same point on the circle |z|=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|=1$\\end{document}, where q(z)=znp(1z‾)‾.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(z)=z^{n}\\overline{p\\bigg(\\frac{1}{\\overline{z}}\\bigg)}. $$\\end{document} 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.

Optimal power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

In this paper, we prove that the inequalities H p ( K ( r ) , E ( r ) ) < π 2 , L q ( K ( r ) , E ( r ) ) < π 2 hold for all r ∈ ( 0 , 1 ) if and only if p ≤ − log ⁡ 3 / log ⁡ ( π / 2 ) and q<−1, where H p ( a , b ) and L q ( a , b ) are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b, and K ( r ) and E ( r ) are respectively the complete elliptic integrals of the first and second kinds.

Mode-dependent observer-based secured hybrid-driven reliable control for semi-Markov jump cyber–physical systems with DoS attacks

This study focuses on the issue of secured observer based hybrid-triggered quantized reliable control design for semi-Markov jump cyber–physical systems (S-MJCPSs) with general uncertain transition rates, quantization, actuator faults, denial-of-service (DoS) attacks and disturbances. In particular, a mode-dependent observer is developed to estimate the unmeasurable states of the considered system. Further, to minimize unnecessary data transmissions within the network channel, we have implemented a hybrid-triggered mechanism in which, the swap process between the time-triggered and the event-triggered mechanisms is governed by a Bernoulli distribution. Moreover, an observer-based hybrid-triggered quantized reliable controller is devised by utilizing the state estimation information obtained from the observer, which assists the S-MJCPSs to achieve primary intent of this study. Subsequently, the control input signals are quantized before being transmitted into the actuator due to the limited capacity of the data transmission medium. Concurrently, periodic DoS jamming attacks are taken into consideration, which have a significant impact on control performances. A set of adequate criteria is established in the context of linear matrix inequalities by employing the mode-dependent time-varying piecewise Lyapunov-Krasovskii functional and the integral inequalities, guaranteeing that the undertaken system accomplishes the global exponential stability and meet a specific H∞ performance level. Following that, by means of the deduced adequate criteria, the values of the controller and observer gain are computed. At last, the efficacy and applicability of the derived theoretical findings are demonstrated through two practical examples.

Some New Estimates for Integral Inequalities and Their Applications

Some New Estimates for Integral Inequalities and Their Applications

Internal/Boundary Control-Based Fixed-Time Synchronization for Spatiotemporal Networks.

This article is concerned about fixed-time (FT) synchronization of spatiotemporal networks (STNs) with the Robin boundary condition. Above all, a switching-type FT stability theorem and an integral inequality are established, which provide a novel theoretical tool for the rigorous analysis of FT control in STNs. Subsequently, three kinds of nontrivial power-law controllers are developed which are separately acted on the interior, the boundary, and the whole of the spatial domain. Based on these control schemes and Lyapunov-like method, several flexible criteria are obtained to achieve FT synchronization of STNs, and the upper bound of the synchronization time is explicitly estimated. Note that, the derived results here are also perfectly applicable to STNs with Neumann or Dirichlet boundary condition. Several illustrate examples are presented at final to confirm the developed controllers and criteria.

KATUGAMPOLA FRACTIONAL PARAMETERIZED INEQUALITIES FOR S-CONVEX FUNCTION IN THE FOURTH SENSE

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.

Some Classical Inequalities Associated with Generic Identity and Applications

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.

Resilient sampled-data control for fractional-order PMVG-based WTS with actuator saturation and probabilistic faults using fuzzy Lyapunov function method

The goal of this study is to present a new fuzzy Lyapunov function-based resilient sampled-data control method for a fractional-order permanent magnet vernier generator (PMVG)-based wind turbine system (WTS) that addresses actuator saturation and probabilistic faults. To achieve this, we first model the dynamical fractional model of the PMVG-based WTS as a T-S fuzzy model. Next, we introduce a fuzzy Lyapunov function within a unified framework that incorporates actuator saturation, probabilistic faults, and uncertainty information from control gain matrices. We then present an actuator fault model that represents actuator faults as stochastic variables with a specific probability distribution. Subsequently, we formulate a robust asymptotic stability criterion for closed-loop systems using linear matrix inequality (LMI), which integrates the fuzzy Lyapunov function and membership function information with fractional integral inequality techniques based on sampling time. We also propose an LMI approach to identify the domain of attraction that meets the necessary actuator saturation criteria. Finally, we apply the proposed method to a PMVG-based WTS, demonstrating its superiority and practical applicability through a comparison with existing methods using a numerical example of a nonlinear truck trailer system.

Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals and (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} derivatives, respectively. Then we investigate some implicit integral inequalities for (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}-convex functions defined on the non-negative part of the real line.

Fractional Newton‐type integral inequalities by means of various function classes

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.