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In this study, a class of neutral-type Rayleigh equations incorporating $$D$$ D operators and $$T$$ T -periodic time-varying coefficients are introduced. By utilising differential inequality techniques, some new sufficient criteria are derived to guarantee the existence and global exponential stability of $$T$$ T -periodic solutions for the addressed equations. Through numerical experiments, the validity of the main findings is verified. [ABSTRACT FROM AUTHOR]

In this work, a class of neutral-type Rayleigh equations involving $$D$$ D operators are proposed, where the damping item is a continuous $$T$$ T -anti-periodic function. Several new stability theorems are obtained to assure the $$T$$ T -anti-periodic exponential stability of the addressed equations by applying differential inequality analyses. Moreover, an illustrative example incorporating its graphical simulations is furnished to illustrate the validity of the derived findings. [ABSTRACT FROM AUTHOR]

In this paper, the projective synchronization problem of nonautonomous neural networks with time delay and Caputo derivative is researched. First, by introducing time delay and variable coefficient into the known neural network model, the neural network that can more accurately describe the interaction between neurons is given. Second, based on the improved neural network model, two global synchronization schemes are achieved, respectively. Finally, by constructing two novel Lyapunov functions and utilizing the properties of delay fractional-order differential inequalities, the asymptotic stability of the zero equilibrium point of the error system obtained from the master–slave systems is proved by some new developing analysis methods, respectively, and some criteria for global projective synchronization of delayed nonautonomous neural networks with Caputo derivatives are obtained, respectively, under two new synchronous controllers. In addition, the correctness of the theoretical results obtained in this paper is verified by some numerical simulation. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for a class of n -dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem. [ABSTRACT FROM AUTHOR]

In fault detection, feedback control, and other fields, real-time differential estimation of a given signal in a complex noise environment is an important but challenging task. In this paper, a discrete-time fast nonlinear tracking differentiator (FNTD) based on hyperbolic tangent functions was proposed. To start, the differential signal acquisition problem was equated to the time-optimal control (TOC) law for constructing a double-integral system using a state feedback approach. Next, the FNTD algorithm based on the hyperbolic tangent function was presented by utilizing the isochronic region (IR) method in the discrete time domain. Then, the frequency-domain characteristics of the FNTD were analyzed and the rule for tuning the parameters was provided by the frequency scan test method. Finally, the simulation results demonstrated that the proposed FNTD had fast and accurate tracking performance, as well as excellent filtering and differential extraction capability compared with other differentiators. [ABSTRACT FROM AUTHOR]

In this paper, the exponential synchronization problem of a class of neural networks with mixed delays under impulsive control is studied. Combining the impulsive comparison principle and the concept of an average impulsive interval, two impulsive differential inequalities with mixed delays are discussed, and the sufficient conditions for the existence of exponential decay are obtained. Based on two different impulsive control strategies, and then by means of the Lyapunov function, the inequality technique, and these two new inequalities, a set of sufficient conditions are derived to ensure the synchronization of the drive and response systems. In order to prove the effectiveness of the proposed control scheme, two numerical examples are given to prove its practicability and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for some existence results for nonlinear impulsive differential equations involving causal operators. Our method is based on the monotone iterative technique, a new differential inequality, and the Schauder fixed point theorem. Moreover, we consider three impulsive differential equations as applications to verify our theoretical results. [ABSTRACT FROM AUTHOR]

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complexexponential forms, and also in the light of certain special information in [1], [20], [17] and [38], an extensive result together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out. [ABSTRACT FROM AUTHOR]

This paper deals with the following attraction-repulsion chemotaxis system$ \begin{align} \begin{cases} u_t = \Delta u - \chi\nabla \cdot (u\nabla v)+\xi \nabla\cdot(u\nabla w), &x\in \Omega, t>0, \\ \tau_1v_t = \Delta v -\beta v+\alpha u, &x\in \Omega, t>0, \\ \tau_2 w_t = \Delta w-\delta w +\gamma u, &x\in \Omega, t>0, \\ u(x, t = 0) = u_0 (x) , \tau_1 v(x, t = 0) = \tau_1v_0 (x) , & x\in \Omega, \\ \tau_2 w(x, t = 0) = \tau_2 w_0 (x), & x\in\Omega, \end{cases} \end{align} $where the parameters $ \chi $, $ \xi $, $ \alpha $, $ \beta $, $ \gamma $ and $ \delta $ are positive, $ \tau_1, \tau_2 = 0, 1 $, and $ \Omega = B_1(0)\subset\mathbb{R}^2 $ is a unit ball supplemented with homogenous Neumann boundary conditions. By developing a differential inequality for the energy functional, we derive a precise criterion for finite-time blow-up of solution to the system with $ \tau_1 = 1, \tau_2 = 0 $ if attraction dominates (i.e. $ \theta = \chi\alpha-\xi\gamma>0 $). Moreover, finite-time blow-up solutions also are constructed for the system with $ \tau_1 = \tau_2 = 1 $ although the energy functional can not be expected to exist. [ABSTRACT FROM AUTHOR]

This paper examines the notion of normalcy within the discourse of the European Union (EU), with a focus on its response to transformative dynamics and challenges, especially post-2008. I analyse how the EU, facing diminishing ideological supremacy, has reinforced its institutional power through the framework of normalcy. By emphasizing constitutionalisation and integration through legal means, bodies such as the European Court of Justice (ECJ) have expanded their roles, promoting "integration through law" to counter alternative political projects invariably labelled as populist, thereby giving rise to anti-populism as a discursive tool. Drawing on crisis discourse and diagnostic practice, this paper explores how normalcy legitimises and perpetuates existing power structures within EU governance, subtly coercing member states and citizens into accepting its norms and values, thus shaping perceptions of normalcy and also inevitability. At the core of this framework lies the rule of law (RoL), which establishes legal boundaries in the first place, but in the same move shapes political contexts. My paper argues that the exclusive focus on legalistic interpretations obscures the underlying structural factors perpetuating power dynamics and economic disparities among member states, constraining adaptive responses. I examine the narrative of "growing up to democracy" and its impact on the European project discourse, particularly in Central and Eastern European (CEE) countries. I then scrutinise the roles of experts and the "juristocracy" in reinforcing this narrative while simultaneously masking underlying inequalities and power differentials. Furthermore, the paper explores the strategic deployment of language and discourse by EU institutions during crises, highlighting their implications for public understanding, political action, and outcomes. Finally, it investigates the EU's strategic use of crisis discourse and diagnostic practices, focusing particularly on the ascendancy of the judiciary, but also highlighting how this trajectory may also have negative influence of legislative and executive bodies as their role diminishes in the process. [ABSTRACT FROM AUTHOR]

For μ∈C1(I)\mu \in {C}^{1}\left(I), μ>0\mu \gt 0, and λ∈C(I)\lambda \in C\left(I), where II is an open interval of R{\mathbb{R}}, we consider the set of functions f∈C2(I)f\in {C}^{2}\left(I) satisfying the second-order differential inequality ddtμdfdt+λf≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in II. The considered set includes several classes of generalized convex functions from the literature. In particular, if μ≡1\mu \equiv 1 and λ=k2\lambda ={k}^{2}, k>0k\gt 0, we obtain the class of trigonometrically kk-convex functions, while if μ≡1\mu \equiv 1 and λ=−k2\lambda =-{k}^{2}, k>0k\gt 0, we obtain the class of hyperbolic kk-convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic kk-convex functions.

In the framework of the first-order differential structure introduced by Gigli, we obtain a Gauss–Green formula on regular bounded open sets in doubling metric measure spaces supporting a weak Poincaré inequality, valid for BV functions and vector fields with integrable divergence. Then, we study least gradient functions in metric measure spaces and provide an Euler–Lagrange-type formulation of the least gradient problem, using this formula as the main tool. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a fractional differential equation   0 C D α u (t) + q (t) u (t) = 0 with boundary value conditions u (0) = δ u (1) , u ′ (0) = γ u ′ (1) . The paper begins by pointing out the inadequacies of the study conducted by Ma and Yangin establishing Lyapunov-type inequalities. It then discusses the properties of its Green's function and investigates extremum problems related to several linear functions. Finally, thorough classification and analysis of various cases for parameters δ and γ are conducted. As a result, a comprehensive solution corresponding to the Lyapunov-type inequality is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present a method to achieve exponential stability in a class of impulsive delayed neural networks containing parameter uncertainties, time-varying delays, and impulsive effect and reaction–diffusion terms. By using an integro-differential inequality with impulsive initial conditions and employing the M-matrix theory and the nonlinear measure approach, some new sufficient conditions ensuring the global exponential stability and global robust exponential stability of the considered system are derived. In particular, the results obtained are presented by simple algebraic inequalities, which are certainly more concise than the previous methods. By comparisons and examples, it is shown that the results obtained are effective and useful. [ABSTRACT FROM AUTHOR]

There are numerous articles dealing with first and second order differential inequalities and differential subordinations and only three articles which are related to third order differential inequalities and subordinations. In this paper we generalise these inequalities for N-th order differential inequalities for functions belonging to class of analytic functions f such that f(0) = 0. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study a class of damped p$$ p $$‐ biharmonic$$ biharmonic $$ type wave equation with homogeneous Dirichlet boundary condition. First of all, we prove the local existence of weak solutions by Galerkin method. Besides, when the energy level is low E(0)

This paper investigates new sufficient conditions for the stability, asymptotic stability, and global asymptotic stability of nonlinear autonomous systems, specifically in cases where the first derivative of the Lyapunov function candidate may have both positive and negative values on its domain. The main contribution of this approach is the introduction of a new auxiliary function that relaxes the stability conditions, allowing the first derivative of the Lyapunov function candidate to be less than or equal to a nonnegative function. The suggested auxiliary function should be integrable within our first theorem. Meanwhile, our first corollary presents a technique that simplifies the task by establishing specific conditions related to differential inequalities. This weaker condition in the proposed results enables the establishment of stability properties in cases where the Lyapunov function candidate is not well chosen or finding a Lyapunov function is not straightforward. Additionally, it is proven that the original Lyapunov method for autonomous systems is a special case of our first theorem. Furthermore, it is demonstrated that assumptions in previous studies, such as Matrosov's theorem or results on higher-order derivatives of the Lyapunov function, guarantee the existence of our auxiliary function. Finally, lemmas are provided to construct these auxiliary functions, and examples are presented to demonstrate the effectiveness of this approach. This work will contribute to the development of stability analysis techniques for nonlinear autonomous systems. [ABSTRACT FROM AUTHOR]

Though extensively studied, the longstanding pinning synchronization problem of dynamical networks with coupling delay has not been well solved until now. In this paper, we further investigate this problem. By proposing a system of functional differential inequalities, we derive synchronization criteria for dynamical networks with coupling delay under linear pinning control, where the threshold of the admissible delay and the control gain threshold are estimated. Since the estimated control gain threshold could be very large when the delay draws close to the delay threshold, we also use the adaptive pinning control scheme to avoid control gain estimation. Pinning synchronization criteria of networks under adaptive control are derived and the delay threshold is given. This is the first time that general coupling delay has been addressed in the field of pinning control. Finally, two numerical examples are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the bidispersive flow with nonlinear boundary conditions in a bounded region. By using the differential inequality technique, we get the bound for the L 4 -norm of the salinity which plays an important role. The continuous dependence and the convergence results on the Soret coefficient are established. [ABSTRACT FROM AUTHOR]

Consider k (≥ 2) number of independent populations, following two parameter exponential distributions, sharing a common location parameter and unequal scale parameters. The location and scale parameters are assumed to be unknown. We focus into the study of estimation of the hazard rate of a selected population with respect to entropy loss function. Define W i = Z i − Y , where Z i denotes the sample mean of the i-th sample and Y represents the minimum observation of all the samples, for i = 1 , ... , k. We select the population with the largest W i . In order to obtain improved estimator, Brewster-Zidek technique is implemented. Further, dominating estimators upon the improved ones are obtained using differential inequality approach. A numerical study of the risk improvements for the proposed estimators has been carried out. [ABSTRACT FROM AUTHOR]

We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case. [ABSTRACT FROM AUTHOR]

The existence of stationary solutions of singularly perturbed systems of reaction–diffusion–advection equations is studied in the case of fast and slow reaction–diffusion–advection equations with nonlinearities containing the gradient of the squared sought function (KPZ nonlinearities). The asymptotic method of differential inequalities is used to prove the existence theorems. The boundary layer asymptotics of solutions are constructed in the case of Neumann and Dirichlet boundary conditions. The case of quasimonotone sources and systems without the quasimonotonicity requirement is also considered. [ABSTRACT FROM AUTHOR]

We consider a periodic problem for a singularly perturbed parabolic reaction–diffusion–advection equation of the Burgers type with the modulus advection; it has a solution in the form of a moving front. We formulate conditions for the existence of such a solution and construct its asymptotic approximation. We pose a control problem where the required front propagation law is implemented by a specially chosen boundary condition. We construct an asymptotic solution of the boundary control problem. Using the asymptotic method of differential inequalities, we estimate the accuracy of the solution of the control problem. We propose an original numerical algorithm for solving singularly perturbed problems involving the modulus advection. [ABSTRACT FROM AUTHOR]

This article addresses the robust fixed‐time synchronization of fuzzy shunting‐inhibitory cellular neural networks (FSICNNs) with delays by utilizing two different types of control strategies. First, a feedback controller is proposed to achieve fixed‐time synchronization of FSICNNs. Secondly, a novel adaptive controller is designed to guarantee fixed‐time synchronization of FSICNNs, automatically adjusting all control gains without the need for advanced settings. The use of differential inequality techniques and the Lyapunov function method yields several sufficient conditions to ensure fixed‐time synchronization for the considered FSICNNs. Finally, an example, along with its numerical simulation, is presented to demonstrate the validity of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

We consider the class of functions u ∈ C²((0, ∞)) satisfying second-order differential inequalities in the form u'(x) + k/x²u(x) ≥ 0 for all x > 0. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases (k = 1/4 and 0 < k < 1/4 ). We next extend our obtained results to the two-dimensional case. In the limit case k → 0+ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed. [ABSTRACT FROM AUTHOR]

This paper deals with the infinite blow‐up phenomena for a class of damped plate equations with logarithmic nonlinearity under the Navier boundary condition. Combining potential well method and modified differential inequality technique, we establish the infinite blow‐up result of solutions with arbitrary initial energy. In particular, it is not necessary to suppose that the initial velocity and the initial displacement should have the same sign in the sense of the L2${L^2}$ inner product, that is, the solution may blow up at infinity even ∫Ωu0u1dx<0$\int _\Omega {{u_0}{u_1}dx} < 0$, more precisely, ∫Ωu0u1dx>−12u0∗2$\int _\Omega {{u_0}{u_1}dx} > - \frac{1}{2}\left\Vert {{u_0}} \right\Vert _*^2$. [ABSTRACT FROM AUTHOR]

In this technical note, the problem of the extended dissipative control for interval type-2 fuzzy model with nonlinear network systems in the event-triggered framework is investigated. A novel dynamic event-triggered strategy is developed to ascertain the instants of activation. It has the advantage of reducing data transmission and optimising the utilisation of communication resources. On the basis of the proposed mechanism, the stochastic Lyapunov function including internal dynamic variables is constructed. Based on this function, coupled with the inequality scaling technique and the application of Dynkin's formula, we derive a sufficient condition for the system's stochastic asymptotic stability and extended dissipativity. The design of the dynamic event-triggered extended dissipative controller becomes challenging because the new sufficient condition is nonlinear for some matrix variables. However, utilising the Projection Lemma and the co-design approach, such a controller can be developed based on the obtained analysis result. The methodology is demonstrated through a rigorous numerical simulation case study. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided. [ABSTRACT FROM AUTHOR]

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p = 2 to results in the L p space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic stability of the trajectories governed by some delay differential inequalities on time scales. Based on time scale theory and the fixed-point theorem, some sufficient conditions are obtained for guaranteeing asymptotic stability. It is interesting that the inequalities studied in this paper include the generalized Halanay inequalities. Due to the fact that dynamic systems on a time scale unify discrete and continuous systems, the results of this paper have wider application value. Furthermore, some numerical examples verify the main results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterratype integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order d, 0 < d < 1. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a Liouville theorem of F-harmonic maps between complete Riemannian manifolds with moderate divergent F-energy. We assume that F is a concave function and satisfies a differential inequality. We employ Ara's F-stress-energy tensor and the Hessian comparison theorem to prove the main result. [ABSTRACT FROM AUTHOR]

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A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the subclass of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well known results are illustrated. [ABSTRACT FROM AUTHOR]

We consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flowmeasure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates formean-square cross-sectional measures of the time-weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time-weighted temperature. In particular, when the base heat flow measure is positive, the timeweighted temperature becomes pointwise unbounded at large spatial distance. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of a class of evolutionary variational–hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modeling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb's friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient μ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable α and the slip rate | u · τ |. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a dynamic adhesive contact model where the body is composed of a viscoelastic material with long memory. The adhesion process is modeled by a bonding field on part of the contact surface and on which the contact and the friction are modeled by nonmonotone Clarke subdifferential conditions with adhesion. Meanwhile, on another part of the contact surface, the contact is comprised of a normal compliance contact condition while the memory effects of the obstacle are considered, the friction is modeled by a version of Coulomb's law of dry friction and the friction bound is dependent on the normal stress. The variational form of the model generates a system of a doubly-history dependent variational–hemivariational inequality as well as a differential equation. Then, on the basis of the proof of abstract variational–hemivariational inequality, the existence and uniqueness results of the contact problem are established. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of octonion‐valued stochastic shunting inhibitory cellular neural networks with delays. First, we give an estimate of the distance between two different moments of finite‐dimensional distributions of a stochastic process. Then, based on this and by using fixed point theorems and inequality techniques, we establish the existence and global exponential stability of Besicovitch almost periodic (B$$ \mathcal{B} $$‐almost periodic for short) solutions in finite‐dimensional distributions for this kind of networks. Our results are new even if the networks we consider in the paper are real‐valued ones. At the same time, the method proposed in this paper can be applied to study the existence of Besicovitch almost periodic solutions in finite‐dimensional distributions for other types of stochastic neural networks. Finally, an example is given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this article, we study a class of differential quasi-variational–hemivariational inequalities involving time delays. We establish new systems and prove the solvability and the existence of decay solutions. Moreover, we are concerned with long-time behavior of solutions by showing the existence of a compact global attractor to m-semiflow associated with delay differential quasi-variational–hemivariational inequalities. An application to the contact problems driven by dynamic systems is discussed to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality (DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space. First, by applying surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient, we show the nonempty of the solution set for the parabolic hemivariational inequality. Then, some topological properties of the solution set are established such as boundedness, closedness and convexity. Furthermore, we explore the upper semicontinuity of the solution mapping. Finally, we prove the solution set of the system (DEHVI) is nonempty and the set of all trajectories of (DEHVI) is weakly compact in C(I, X). [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality. [ABSTRACT FROM AUTHOR]

This paper deals with a 4th-order parabolic PDE involving $ p $-Laplacian in a bounded domain, subject to Neumann boundary conditions. Firstly, we show the proof of the local existence and uniqueness of weak solutions and the energy identity. The blow-up time estimates are discussed for the corresponding blow-up solutions. Secondly, we define different Nehari manifolds and the energy functional and then determine the existence of blow-up and global solutions in the subcritical, critical, super critical energy cases. Thirdly, we combine some ordinary differential inequalities and different auxiliary functions to obtain the conditions on the existence of extinction and non-extinction solutions with extinction rate and time. [ABSTRACT FROM AUTHOR]

We consider pseudo-parabolic equations with r(x)-Kirchhoff coefficient and logarithmic nonlinear term subject to Dirichlet boundary condions...Using a method based on differential inequalities, we prove that the solutions become unbounded at a finite time T, and, we ascertain an upper limit for this time in the case of negative initial energy. Additionally, we determine a lower limit for the time at which blow-up occurs. [ABSTRACT FROM AUTHOR]

Let (M n , F , m) be a compact Finsler manifold. This paper will obtain a global estimate and a differential Harnack inequality for positive solutions to the Newell–Whitehead–Segal equation ∂ t u (x , t) = Δ m u (x , t) + a u − b u 3 ,   (x , t) ∈ M × [ 0 , T ] , on compact Finsler manifolds, where a , b > 0 are real constants. [ABSTRACT FROM AUTHOR]