Dynamic Inequalities Research Articles

On dynamic inequalities in two independent variables on time scales and their applications for boundary value problems

Our work is based on the multiple inequalities illustrated by Boudeliou and Khalaf in 2015. With the help of the Leibniz integral rule on time scales, we generalize a number of those inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete, quantum, and continuous calculus. These inequalities may be of use in the analysis of some kinds of partial dynamic equations on time scales and their applications in environmental phenomena, physical and engineering sciences described by partial differential equations.

A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht’s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (when T ≪ ℕ ) are essentially new.

On some dynamic inequalities of Ostrowski, trapezoid, and Grüss type on time scales

In the present manuscript, we prove some new extensions of the dynamic Ostrowski inequality and its companion inequalities on an arbitrary time scale by using two parameters for functions whose second delta derivatives are bounded. In addition to these generalizations, some integral and discrete inequalities are obtained as special cases of main results.

On Some Important Dynamic Inequalities of Hardy–Hilbert-Type on Timescales

In this article, by using some algebraic inequalities, nabla Hölder inequalities, and nabla Jensen’s inequalities on timescales, we proved some new nabla Hilbert-type dynamic inequalities on timescales. These inequalities extend some known dynamic inequalities on timescales and unify some continuous inequalities and their corresponding discrete analogues. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

On Some Generalizations of Reverse Dynamic Hardy Type Inequalities on Time Scales

In the present paper, we prove some new reverse type dynamic inequalities on T. Our main inequalities are proved by using the chain rule and Fubini’s theorem on time scales T. Our results extend some existing results in the literature. As special cases, we obtain some new discrete inequalities, quantum inequalities and integral inequalities.

On Some Important Class of Dynamic Hilbert’s-Type Inequalities on Time Scales

In this important work, we discuss some novel Hilbert-type dynamic inequalities on time scales. The inequalities investigated here generalize several known dynamic inequalities on time scales and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using some algebraic inequalities, Hölder inequality, and Jensen’s inequality on time scales.

Diamond Alpha Hilbert-Type Inequalities on Time Scales

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales.

Development of an autonomous fog computing platform using control-theoretic approach for robot-vision applications

This paper presents the dynamic modelling and linear matrix inequality (LMI) based controller design of a distributed fog computing framework for real-time robot vision applications. A mobile robot vision system acquires the images from an application environment such as a warehouse, where articles are stacked in numerous racks. We characterise the mobile robot vision data (MRVD) using frames per second (FPS) and the image resolution. From the MRVD, object detection is performed by an open-source deep learning (DL) platform for detecting and localising various objects. However, with higher FPS together with high-resolution images, the processing time by the DL algorithm increases significantly. This necessitates the deployment of a distributed computing platform with several computing nodes. In this work, we deploy a distributed fog computing environment (DFCE) for the real-time object detection in an application environment.The processing time required to handle the MRVD is called the service time. However, for efficient auto-scaling performance, the mathematical model of the DFCE, taking into consideration the characteristics of the MRVD is necessary. In this context, we envisage the application of control theory to build the dynamic model of the DFCE. A Linear Parameter Varying (LPV) model is proposed for the DFCE with the service time as the output, the number of fog nodes as the input, and the characteristics of MRVD as the time-varying parameters. At first, an LPV model for the DFCE is derived using system identification, and the model is validated using the real-time test data. The LPV model is converted to a Polytopic LPV (PLPV) model for LMI based controller design. Finally, we develop and validate a Linear Matrix Inequality (LMI) based LPV controller to meet the service time constraints for a given application environment.For localisation and trajectory tracking with obstacle avoidance in the application environment, the mobile robot implements an Extended Kalman Filter (EKF) based simultaneous localisation and mapping (SLAM) and a bug-based path planning algorithm respectively. Finally, we present, detailed controller validation results illustrating the mobile robot navigation together with the auto-scaling control for the fog computing platform to modulate the service time.

Hardy-Leindler-Type Inequalities via Conformable Delta Fractional Calculus

In this article, some fractional Hardy-Leindler-type inequalities will be illustrated by utilizing the chain law, Hölder’s inequality, and integration by parts on fractional time scales. As a result of this, some classical integral inequalities will be obtained. Also, we would have a variety of well-known dynamic inequalities as special cases from our outcomes when α = 1 .

Dissipative Dynamical Systems With Set-Valued Feedback Loops: Well-Posed Set-Valued Lur’e Dynamical Systems

Passivity and dissipativity, in Jan Willems’s sense, are known to be powerful tools for stability analysis and feedback control design. For instance, passive systems in negative feedback interconnection with slope-restricted, static, single-valued, smooth nonlinearities, known as Lur’e systems, have been thoroughly studied in automatic control, yielding the so-called absolute stability problem (with the famous Popov and circle criteria). On the other hand, large classes of nonsmooth systems (complementarity dynamical systems, relay systems, projected dynamical systems, evolution and differential variational inequalities, Moreau’s sweeping processes, and maximal monotone differential inclusions), with applications in circuits, mechanics, and economics, are interpreted as set-valued Lur’e systems, in which the feedback nonlinearity is a multivalued mapping. Therefore, the closed-loop system is a differential inclusion of a certain type, the well posedness of which must be analyzed as a prerequisite for stability and control. This introductory article focuses on the well posedness of such set-valued feedback systems. It is shown how the existence and uniqueness of solutions to these specific differential inclusions benefit a lot from the passivity of the system and the maximal monotonicity (which is a form of incremental passivity) of the feedback set-valued mapping. Available results are reviewed, many illustrative examples are given, and some open issues are highlighted.

Self-triggered MPC with performance guarantee for tracking piecewise constant reference signals

This paper considers a self-triggered MPC controller design strategy for tracking piecewise constant reference signals. The proposed triggering scheme is based on the relaxed dynamic programming inequality and the idea of reference governor; such a scheme computes both the updated control action and the next triggering time. The resulting self-triggered tracking MPC control law preserves stability and constraint satisfaction and also satisfies certain a priori chosen performance requirements without the need to impose stabilizing terminal conditions. An illustrative example shows the effectiveness of this self-triggered tracking MPC implementation.

Two Weighted Higher-Order Dynamic Inequalities of Opial Type with Two Functions

Two Weighted Higher-Order Dynamic Inequalities of Opial Type with Two Functions

Some new dynamic Gronwall\u2013Bellman\u2013Pachpatte type inequalities with delay on time scales and certain applications

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results.

Applications of the novel diamond alpha Hardy–Copson type dynamic inequalities to half linear difference equations

This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are complements of the classical ones obtained for and their applications to difference equations. We obtain two kinds of diamond alpha Hardy–Copson type inequalities for , one of which is mixed type and established by the convex linear combinations of the related delta and nabla inequalities while the other one is new and is obtained by using time scale calculus rather than algebra. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy–Copson type inequalities obtained for into one diamond alpha Hardy–Copson type inequalities and offer new types of diamond alpha Hardy–Copson type inequalities which have the same directions as the classical ones and can be considered as complementary inequalities. Moreover the application of these inequalities in the oscillation theory of half linear difference equations provides several nonoscillation criteria for such equations.

A Variety of Nabla Hardy’s Type Inequality on Time Scales

The primary goal of this research is to prove some new Hardy-type ∇-conformable dynamic inequalities by employing product rule, integration by parts, chain rule and (γ,a)-nabla Hölder inequality on time scales. The inequalities proved here extend and generalize existing results in the literature. Further, in the case when γ=1, we obtain some well-known time scale inequalities due to Hardy inequalities. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities and new classical conformable fractional integral inequalities.

Important Study on the ∇ Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel–Legendre transform, submultiplicative functions, and Hölder’s and Jensen’s inequality on time scales. We obtain some well-known time scale inequalities due to Hardy–Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic inequalities

Diamond alpha Hardy-Copson type dynamic inequalities

In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.

Novel Diamond Alpha Bennett–Leindler Type Dynamic Inequalities and Their Applications

Novel Diamond Alpha Bennett–Leindler Type Dynamic Inequalities and Their Applications

Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right χ-pre-invex interval-valued functions (LR-χ-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings.

On Refinements of Multidimensional Inequalities of Hardy‐Type via Superquadratic and Subquadratic Functions

By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy‐type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.