Theory Of Differential Inequalities Research Articles

A compliant visco-plastic particle contact model based on differential variational inequalities

This work describes an approach to simulate contacts between three-dimensional shapes with compliance and damping using the framework of the differential variational inequality theory. Within the context of non-smooth dynamics, we introduce an extension to the classical set-valued model for frictional contacts between rigid bodies, allowing contacts to experience local compliance, viscosity, and plasticization. Different types of yield surfaces can be defined for various types of contact, a versatile approach that contains the classic dry Coulomb friction as a special case. The resulting problem is a differential variational inequality that can be solved, at each integration time step, as a variational inequality over a convex set.

Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

Dynamical Behaviors of the Stochastic Hopfield Neural Networks with Mixed Time Delays

This paper investigates dynamical behaviors of the stochastic Hopfield neural networks with mixed time delays. The mixed time delays under consideration comprise both the discrete time-varying delays and the distributed time-delays. By employing the theory of stochastic functional differential equations and linear matrix inequality (LMI) approach, some novel criteria on asymptotic stability, ultimate boundedness, and weak attractor are derived. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

Asymptotic solution of nonlinear nonlocal singularly perturbed reaction diffusion problems with two parameters

A class of differential-difference reaction diffusion equations initial boundary problem with a small time delay is considered. Under suitable conditions and by using method of the stretched variable, the formal asymptotic solution is constructed. And then, by using the theory of differential inequalities the uniformly validity of solution is proved.

Singularly perturbed boundary value problems for a class of second order turning point on infinite interval

This article solved the asymptotic solution of a singularly perturbed boundary value problem with second order turning point, encountered in the dissipative equilibrium vector field of the coupled convection disturbance kinetic equations under the constrained filed and the gravity. Using the matching of asymptotic expansions, the formal asymptotic solution is constructed. By using the theory of differential inequality the uniform validity of the asymptotic expansion for the solution is proved.

Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach

Comparison principles for general impulsive stochastic functional differential systems are established. Employing the comparison principles and the theory of differential inequalities, stability and instability, involving two measures, of impulsive stochastic functional differential systems are investigated. Several stability and instability criteria are obtained, and two examples are also given to illustrate our results.

Some generalizations of comparison results for fractional differential equations

In this paper, by using the technique of upper and lower solutions together with the theory of strict and nonstrict fractional differential inequalities involving Riemann–Liouville differential operator of order q , 0 < q < 1 , some necessary comparison results for further generalizations of several dynamical concepts are obtained. Furthermore, these results are extended to the finite systems of fractional differential equations.

Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach

Lyapunov’s second method is widely recognized as a fundamental tool not only in the theory of stability but also in studying other qualitative properties of solutions of differential equations. The main characteristic of this method is the utilization of a function, namely the Lyapunov function, together with the theory of differential and integral inequalities to develop a very general comparison principle under much less restrictive conditions to investigate qualitative and quantitative properties of nonlinear differential equations. The most important advantage of this technique is that it does not require the explicit knowledge of solutions. This paper is devoted to extend this technique to investigate the practical stability of the solution process of stochastic hybrid parabolic partial differential equations of Itô type under Markovian structural perturbations. The jump nonlinear system is a hybrid system with state vector that has two components x ( t ) and η ( t ) . Here x ( t ) is referred to as the state and η ( t ) is referred as the mode. During the operation, the system can jump from one mode to another in a random way, which makes this class of systems a stochastic one. The switching between the modes is governed by a Markov process with discrete and finite state space. When the system mode is fixed it evolves like a deterministic nonlinear system. This kind of system can be used to describe abrupt phenomena, such as component and interconnection failures.

Singularly perturbed reaction diffusion equations with time delay

A class of initial boundary value problems of differential-difference equations for reaction diffusion with a small time delay is considered. Under suitable conditions and by using the stretched variable method, a formal asymptotic solution is constructed. Then, by use of the theory of differential inequalities, the uniform validity of the solution is proved.

Asymptopic solution for a class of semilinear singularly perturbed fractional differential equation

This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation. Under the suitable conditions, first, the outer solution of the original problem is obtained; secondly, using the stretched variable and the composing expansion method the boundary layer is constructed; finally, using the theory of differential inequalities the asymptotic behaviour of solution for the problem is studied and the uniformly valid asymptotic estimation is discussed.

Asymptotic solution of quasi-linear singularly perturbed boundary value problems with singular equation

A class of quasi-linear singularly perturbed boundary value problems with singular equation is considered. Under suitable conditions, the existence and asymptotic behavior of a solution for the boundary value problem are studied by using the theory of differential inequalities, and the uniformly valid asymptotic expansion of solution with boundary layer is obtained.

Impulsive effect on global exponential stability of BAM fuzzy cellular neural networks with time-varying delays

In this article, a class of impulsive bidirectional associative memory (BAM) fuzzy cellular neural networks (FCNNs) with time-varying delays is formulated and investigated. By employing delay differential inequality and M-matrix theory, some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM FCNNs with time-varying delays are obtained. In particular, a precise estimate of the exponential convergence rate is also provided, which depends on system parameters and impulsive perturbation intention. It is believed that these results are significant and useful for the design and applications of BAM FCNNs. An example is given to show the effectiveness of the results obtained here.

A singularly perturbed reaction diffusion problem for the nonlinear boundary condition with two parameters

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.

Singularly perturbed differential-difference reaction diffusion equations with time delay

A class of differential-difference reaction diffusion equations with a small time delay is considered. Under suitable conditions and by using the method of the stretched variable, the formal asymptotic solution is constructed. And then, by using the theory of differential inequalities the uniformly validity of solution is proved.

Asymptotic solution for a perturbed mechanism of western boundary undercurrents in the Pacific

This paper consider a class of perturbed mechanism for the western boundary undercurrents in the Pacific. The model of generalized governing equations is studied. Using the perturbation method, it constructs the asymptotic solution of the model. And the accuracy of asymptotic solution is proved by the theory of differential inequalities. Thus the uniformly valid asymptotic expansions of the solution are obtained.

Dynamics of a three-species ratio-dependent diffusive model

Abstract In this paper, we study a nonautonomous ratio-dependent diffusive model of three species. By using the fixed point theorem of Brouwer and the theory of differential inequality and constructing a suitable Lyapunov function, sufficient conditions are obtained which guarantee the existence, uniqueness and stability of positive periodic solution. At last, some results are proved with the technology of numerical simulation.

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

A class of singularly perturbed initial boundary value problems for semilinear reaction diffusion equations with two parameters is considered. Under suitable conditions and using the theory of differential inequalities, the existence and the asymptotic behavior of the solution to the initial boundary value problem are studied.

Hyperbolic problems in three independent variables: Mathematical analysis and constructive methods

We consider initial-boundary value problems associated with nonlinear hyperbolic partial differential equations in three independent variables in a unified setting, wherein the forcing function is a sum of two monotonic functions. After deriving the required mathematical theory of hyperbolic partial differential inequalities, we employ natural lower–upper solutions and two types of coupled lower–upper solutions, to derive linear iterative schemes which converge uniformly and monotonically to the minimal–maximal solutions or the coupled minimal–maximal solutions of the nonlinear problem. Examples are given to illustrate the results.

Asymptotic solution for a class of weakly nonlinear singularly perturbed reaction diffusion problem

Under appropriate conditions, with the perturbation method and the theory of differential inequalities, a class of weakly nonlinear singularly perturbed reaction diffusion problem is considered. The existence of solution of the original problem is proved by constructing the auxiliary functions. The uniformly valid asymptotic expansions of the solution for arbitrary mth order approximation are obtained through constructing the formal solutions of the original problem, expanding the nonlinear terms to the power in small parameter c and comparing the coefficient for the same powers of c. Finally, an example is provided, resulting in t he error of O(e^2).

Existence and Location Results for Fully Nonlinear Boundary Value Problem of nth-Order Nonlinear System

By appropriate bounding function pair and modified functions, using the theory of differential inequalities, this paper presents the existence and location criteria of solutions for the system of general n th-order differential equations with nonlinear boundary conditions. We give an example showing that the results are sharp. Our results extend many existing results.