Integro-differential Inequalities Research Articles

Comparison principles for a class of general integro-differential inequalities with applications

Comparison principles for fractional differential equations have been investigated in many papers using different types of fractional integral and derivative operators. We here prove the strongest such results so far, for a very broad class of operators that is even more general than those with Sonine kernels. Starting from inequalities valid at global extrema, we obtain comparison principles for these general operators, which are applied to prove bounds on solutions to related integro-differential equations. Many results in the literature will be considered as particular cases of the current study.

Quasi-invariant and attracting sets of competitive neural networks with time-varying and infinite distributed delays

This paper focuses on the quasi-invariant set (QIS), global attracting set (GAS) and global exponential attracting set (GEAS) of competitive neural networks (CNNs) with time-varying and infinite distributed delays. For these purposes, based on the characteristics of nonnegative matrix and M-matrix, a new bidirectional delay integral inequality and a novel integro-differential inequality are first established. From the founded integral inequality, the existence conditions of the QIS and the GAS of the discussed system are obtained. Besides, the existence conditions of the GEAS are also given by the proposed integro-differential inequality, which gets rid of the construction of complex Lyapunov functions and functionals. The frameworks of the QIS, GAS and GEAS are also given. A numerical example is analyzed to confirm the validity of the obtained results in the end.

Stability for some types of multi control integro differential inequalities

This paper is concerned with the stability of some classes of integro-differential inequalities with multi-control inputs. Namely, some estimators’ formulations for the stability of solutions for some multi-control problems are established. The illustrative problems show that the stability technique, considered in the present paper, is efficient and robust. Moreover, it can be used with other classes of integro-differential inequalities.

Periodicity of Nonautonomous Fuzzy Neural Networks with Reaction-Diffusion terms and Distributed Time Delays

In this paper, the periodicity of a class of nonautonomous fuzzy neural networks with impulses, reaction-diffusion terms, and distributed time delays are investigated. By establishing an integro-differential inequality with impulsive initial conditions and time-varying coefficients, employing the M -matrix theory, Poincar mappings, and fixed point theory, several new sufficient conditions to ensure the periodicity and global exponential stability of the formulated system are obtained. It is worthwhile to mention that our technical methods are practical, in the sense that all new stability conditions are stated in simple algebraic forms, and an optimization method is provided to estimate the exponential convergence rate, so their verification and applications are straightforward and convenient. The validity and generality of our methods are illustrated by two numerical examples.

Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms

Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with \(t\). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time \(T^\ast\). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span \(T^\ast\) is derived by the means of integro-differential inequality techniques.

Uniform stability of integro-differential inequalities with nonlinear control inputs and delay

In this paper, the uniform stability for the solution of integro-differential inequalities, with nonlinear control inputs and delay functions, is investigated by using some inequality estimator conditions. Moreover, we apply the obtained results on the solutions of some proposed classes of integro-differential inequalities with nonlinear control input functions as problem formulations examples. The results show that the stability technique used in this work is efficient and robust and it can be applied to a general class and various types of integro-differential inequalities.

Periodicity of impulsive Cohen–Grossberg-type fuzzy neural networks with hybrid delays

In this paper, the periodicity of a class of impulsive Cohen–Grossberg-type fuzzy neural networks with hybrid delays is investigated. By applying integro-differential inequality with delays, the Lyapunov function, the Poincare map and the fixed point theory, we obtained some new sufficient conditions which guarantees the existence of periodic solutions and global exponential stability of the formulated system. Three numerical examples are given to show the effectiveness and generality of our method.

Impulsive delayed integro-differential inequality and its application on IMNNs with discrete and distributed delays

Impulsive differential inequality is crucial for nonlinear impulsive differential system. In this paper, an impulsive delayed integro-differential inequality is firstly presented and studied. New impulsive inequality is given by M-matrix, proof by contradiction and recursive inequality. Then, the discrete-distributed delayed memristive neural network model is constructed and its exponential stability criteria are obtained by the novel impulsive delayed inequality. Finally, two numerical examples are given to illustrate the effectiveness of exponential stability criteria.

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACTThe main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.

Invariant and Attracting Sets of Complex-Valued Neural Networks with Both Time-Varying and Infinite Distributed Delays

In this paper, we investigate the asymptotic property of non-autonomous complex-valued neural networks with time-varying delays and infinite distributed delays. By using the property of M-matrix, an integro-differential inequality is established. Based on the inequality and some sufficient conditions, we obtain the attracting and invariant sets of non-autonomous complex-valued neural networks with time-varying delays and infinite distributed delays. An example with numerical simulation is given out to illustrate the effectiveness of our results.

Robust exponential stability of switched interval interconnected systems with unbounded delay

In this paper, a class of switched interval interconnected systems with unbounded delay were investigated. On the assumption that the interconnected functions of the systems satisfied the global Lipschitz condition, by using vector Lyapunov methods and M-matrix theory, the integrodifferential inequalities with unbounded delay were constructed. By the stability analysis of the integrodifferential inequalities, the sufficient conditions to ensure the robust exponential stability of the interval interconnected systems were obtained. By using average dwell time approach, conditions for guaranteeing the robust exponential stability of the switched delay interval interconnected systems were derived. Finally, two numerical examples were given to illustrate the correction and effectiveness of the proposed theory.

Analytic Solutions For Integro-Differential Inequalities Using Modified Adomian Decomposition Method

This paper applies the Modified Adomian Decomposition Method (MADM) for solving Integro-Differential Inequality, this method is one of effective to construct analytic approximate solutions for linear and nonlinear integro-differential inequalities without solving many integrals and transformed or discretization. Several examples are presented, the analytic results show that this method is a promising and powerful for solving these problems.

An Error Analysis of a Finite Element Method with IMEX-Time Semidiscretizations for Some Partial Integro-differential Inequalities Arising in the Pricing of American Options

The aim of this article is to develop and analyze a finite element method, combined with implicit-explicit (IMEX) time semidiscretizations, for pricing American options under Merton's and Kou's jum...

An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

AbstractWe consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

Numerical simulation of elastohydrodynamic lubrication for an elliptical contact

An elastohydrodynamic problem on elliptical contact is formulated and numerically solved. The mathematical model is described by a system of integro-differential equations and inequalities with boundary conditions. The direction of the rolling velocity vector with respect to the axes of the ellipse has a marked influence on the distribution of pressure and the lubricating film thickness in the contact area. The elastohydrodynamic contact parameters are determined depending on the compression force applied thereto. These results are useful for analyzing processes occurring in lubricated concentrated contacts.

Integrodifferential Inequalities Arising in the Theory of Differential Equations

The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations.

A CLASS OF QUASISTATIC CONTACT PROBLEMS FOR VISCOELASTIC MATERIALS WITH NONLOCAL COULOMB FRICTION AND TIME-DELAY

In this paper, a mathematical model which describes the explicit time dependent quasistatic frictional contact problems is introduced and studied. The material behavior is described with a nonlinear viscoelastic constitutive law with time-delay and the frictional contact is modeled with nonlocal Coulomb boundary conditions. A variational formulation of the mathematical model is given, which is called a quasistatic integro-differential variational inequality. Using the Banach's fixed point theorem, an existence and uniqueness theorem of the solution for the quasistatic integro-differential variational inequality is proved under some suitable assumptions. As an application, an existence and uniqueness theorem of the solution for the dual variational formulation is also given.

Employee Stock Option Valuation under Levy Models

Employee stock options (ESOs) are American-style call options that can be terminated early due to employment shock. This paper studies an ESO valuation framework that accounts for job termination risk and jumps in the company stock price. Under general Levy stock price dynamics, we show that a higher job termination risk induces the ESO holder to voluntarily accelerate exercise, which in turn reduces the cost to the company. The holder's optimal exercise boundary and ESO cost are determined by solving an inhomogeneous partial integro-differential variational inequality (PIDVI). We apply Fourier transform to simplify the variational inequality and develop accurate numerical methods. Furthermore, when the stock price follows a geometric Brownian motion, we provide closed-form formulas for both the vested and unvested perpetual ESOs. Our model is also applied to evaluate the probabilities of ESO cost exceedance and contract termination.

On Some Fundamental Integrodifferential Inequalities

The aim of this present paper is to establish some new integrodifferential inequalities of Gronwall type involving functions of one independent variable which provide explicit bounds on unknown functions. The inequalities given here can be used in the analysis of a class of differential equations as handy tools.

Inequalities for Integro-differential equations involving derivatives of order between zero and two

We obtain bounds for integro-differential inequalities involving derivatives of orders between 0 and 2. We give applications and examples demonstrating the use of these bounds in analyzing the existence and asymptotic behavior of solutions for a class of singular fractional differential equations.