Differential Inequalities Research Articles

Agonistic freedoms: enabling disaffected working-class students to turn-back-around to mainstream school

ABSTRACT The sifting and sorting mechanisms of Australia’s education system continue to work to the detriment of groups put at a disadvantage. For mainstream schools serving working-class communities in particular, the rejection of the offers and advantages of schooling continue to result in differential class consequences and inequalities for working-class young people. In response, this paper pursues a more hopeful project of schooling by considering how schools can work to circumvent the alienation of working-class students from the mainstream secondary school system. Foucault’s conceptualisations of ‘power relations’, ‘agonistic freedom’ and ‘technologies of the self’ are used alongside Kamler and Comber’s notion of ‘turn-around pedagogies’ to draw attention to the complex relationships between students’ subjectivities, their shifting dispositions towards school and eventual enticement to turn-back-around to mainstream school.

Optimal control of differential quasivariational-hemivariational inequalities with applications

Optimal control of differential quasivariational-hemivariational inequalities with applications

Finite-time stability of fractional-order quaternion-valued memristive neural networks with time delay

This study addresses the finite-time stability (FTS) problem of fractional-order quaternion-valued memristive neural networks (FQMNNs) with time delay. Firstly, by leveraging set-valued mapping, differential inclusion, and contracting mapping principle, the sufficient condition is provided to assure the existence and uniqueness of the equilibrium point. Secondly, given the fractional order differential inequality and Gronwall inequality, a finite-time stability (FTS) criterion is obtained for the delayed FQMNNs with the fractional order between 0 and 1. Additionally, a sufficient condition is proposed to guarantee the FTS of FQMNNs with the fractional order between 1 and 2. Compared with existing results, this study not only extends the fractional order between 0 and 2, but also incorporates quaternions, which significantly enhances the FTS theory. Finally, two numerical examples demonstrate the practicality of the obtained results.

Aperiodically Intermittent Event-Triggered Optimal Average Consensus for Nonlinear Multi-Agent Systems.

This article is concerned with average consensus of multi-agent systems via intermittent event-triggered strategy. First, a novel intermittent event-triggered condition is designed and the corresponding piecewise differential inequality for the condition is established. Using the established inequality, several criteria on average consensus are obtained. Second, the optimality has been investigated based on average consensus. The optimal intermittent event-triggered strategy in the sense of Nash equilibrium and corresponding local Hamilton-Jacobi-Bellman equation are derived. Third, the adaptive dynamic programming algorithm for the optimal strategy and its neural network implementation with actor-critic architecture are also given. Finally, two numerical examples are presented to show the feasibility and effectiveness of our strategies.

On the positive invertibility of a differential operator on a graph

In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.

Some Properties and Graphical Applications of a New Subclass of Harmonic Functions Defined by a Differential Inequality

Some Properties and Graphical Applications of a New Subclass of Harmonic Functions Defined by a Differential Inequality

Exponential synchronization of stochastic coupled neural networks with stochastic delayed impulsive effect

This paper studies the synchronization of stochastic coupled neural networks (SCNNs) with stochastic delayed impulses, which are more realistic and complex than the deterministic ones. Unlike previous works, we incorporate two distinct types of stochastic factors into the model: random noise affecting the systems continuous dynamics, and stochastic impulsive intensity applied at discrete impulsive instances. We extend the differential inequality with stochastic delayed impulses to stochastic systems without requiring that impulsive delay to be smaller than impulsive interval. We also model the impulsive intensity as a discrete-time Markov chain. By using the generalized inequality, graph theory, and stochastic analysis techniques, we derive a synchronization criterion for SCNNs with stochastic delayed impulses. We present numerical examples to demonstrate the effectiveness of our results.

Extinction and non-extinction of solutions for nonlocal fractional [formula omitted]-Kirchhoff problem with logarithmic nonlinearity

In this paper, we discuss the extinction and non-extinction properties of solutions for a class of Kirchhoff type parabolic problems involving fractional p-Laplacian and logarithmic nonlinearity. Based on energy estimates, embedding theorems, and certain ordinary differential inequalities, the global existence of solutions, and the extinction and non-extinction properties of global weak solutions are obtained. The results of this paper extend and complement previous research findings on this type of equation.

Synchronization of Intermittently Coupled Neural Networks With Coupling Delay.

In recent years, the synchronization of coupled neural networks (CNNs) has been extensively studied. However, existing results heavily rely on assuming continuous couplings, overlooking the prevalence of intermittent couplings in reality. In this article, we address for the first time the synchronization challenge posed by intermittently CNNs (ICNNs) with coupling delay. To overcome the difficulties arising from intermittent couplings, we put forward a general piecewise delay differential inequality to characterize the dynamics during both coupled intervals and decoupled intervals. Based on the proposed inequality, we establish delay-independent synchronization criteria (DISCs) for ICNNs, enabling them to tackle general coupling delay. Notably, unlike previous studies, the achievement of synchronization in our approach does not rely on external control. Furthermore, for ICNNs that synchronize only under small delays, we formulate non-linear matrix inequality (LMI)-based delay-dependent synchronization criteria (DDSCs) that are computationally efficient and do not require delay differentiability. Finally, we provide illustrative examples to demonstrate our theoretical results.

Bipartite quasi synchronization of stochastic delayed cooperation–competition networks with gain-waving intermittent control

Given the cooperation and competition bi-directional mechanisms between subsystems, this paper aims at addressing bipartite quasi synchronization (BQS) of stochastic delayed cooperation–competition networks (SDCNs). We propose a gain-waving intermittent control policy, that includes more of the existing works, to assist the BQS, in an effort to cope with external interferences and the possibility of actuator damage caused by rapid switching. Intermittent gain is to be quantized legitimately via the postulate of waving average gain. Within this context, novel Lyapunov inequalities concerning bounded waving gain and unbounded one are advanced to ensure the BQS of SDCNs in conjunction with graph theory, since the present differential inequalities are incapable of being inoperative. Then we develop two kinds of BQS criteria and the bounds for the BQS error are estimated where the scenarios of bounded waving gain and unbounded one are involved separately. Besides, corresponding criteria about complete bipartite synchronization are presented and several design methods of gain-waving intermittent control are introduced by specific examples. Finally, the validity of the theoretical results is examined by an application on Chua’s circuit with numerical simulations being shown.

Existence and Stability of a Stationary Solution in a Two-Dimensional Reaction-Diffusion System with Slow and Fast Components

In the paper, the existence of a stable stationary solution in a reaction-diffusion system with slow and fast components in a two-dimensional spatial variable case is investigated. The theorem of the existence of a stationary solution with boundary layers in the case of Dirichlet boundary conditions is proven, its asymptotic approximation is constructed, and conditions for Lyapunov asymptotic stability of this solution are obtained. The research is based on the asymptotic method of differential inequalities, applied to a new class of problems. This result is practically important both for various applications described by similar systems and for the application of numerical stationing methods when solving elliptical boundary value problems.

Asymptotic spatial behaviour in linearised thermoelasticity for non-compact regions

A quasi-static approximation is studied of linearised nonhomogeneous anisotropic compressible thermoelasticity on a non-compact region. Differential inequalities are constructed which under appropriate conditions lead to algebraic spatial growth and decay estimates for various cross-sectional and volume measures of the temperature, displacement and their spatial gradients.

Minimum Principles for Sturm–Liouville Inequalities and Applications

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

Stable Recovery of Coefficients in an Inverse Fault Friction Problem

We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.

Applications of Differential Inequalities Employing A New ConvolutedOperator Constructed by The Supertrigonometric Function

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea ofsubordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem.

Stability and optimal control for differential constrained variational-hemivariational inequalities

Stability and optimal control for differential constrained variational-hemivariational inequalities

On a subclass of close-to-convex functions

In this paper, we define a new subclass of close-to-convex functions with respect to certain univalent functions studied by [Z. Peng and G. Zhong, Some properties for certain classes of univalent functions defined by differential inequalities, Acta Math. Sci. Ser. B (Engl. Ed.) 37(1) (2017) 69–78]. We find certain examples of functions belong to the class. Further, we give coefficient estimates, distortion theorem and some more results for this new subclass. Finally, we obtain radius of convexity for functions belong to the class.

Exponential stability of periodic solutions for a hematopoiesis model with two time delays

In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

Optimal control, well-posedness and sensitivity analysis for a class of generalized evolutionary systems

In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective of this paper is four fold. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set to the FDVHVI. The second aim is to study the optimal control problems described by the FDVHVI. The third goal is to establish the well-posedness results of the FDVHVI, including the existence, uniqueness, and stability. Furthermore, the sensitivity analysis of a perturbed problem associated to the FDVHVI with respect to the initial state and the two parameters is also obtained. Finally, a comprehensive fractional model is given to illustrate the validity of our main results.