Second-order elliptic integro-differential equations: viscosity solutions' theory revisited

The aim of this work is to revisit viscosity solutions' theory for second-order elliptic integro-differential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new Jensen–Ishii's lemma for integro-differential equations, which is stated for solutions with no restriction on their growth at infinity. The proof of this result, which is of course a key ingredient to prove comparison principles, relies on a new definition of viscosity solution for integro-differential equation (equivalent to the two classical ones) which combines the approach with test-functions and sub-superjets.     

Sobolev-Volterra投影与积分微分方程有限元数值分析

本文提出一类称之为Sobolev-Volterra投影的有限元投影,研究了有关性质并将之应用于伪抛物型积分微分方程有限元方法、伪双曲型积分微分方程有限元方法以及三维伪双曲型积分微分方程交替方向有限元方法的数值分析.     

A note on the fractional Cauchy problems with nonlocal initial conditions

Of concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered.     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

Comparison results for initial value problems of second-order impulsive integro-differential inclusions

In this paper, we present new maximum and minimum principles for the initial value problem for a class of second-order nonlinear impulsive integro-differential inclusions with discontinuous multivalued functions on the right hand side in a real Banach space. This allows us to lead on to the development of the comparison principle and monotone iterative technique for a nonlinear second-order impulsive integro-differential inclusion.     

脉冲积分-微分系统零解稳定性的新的Razumikhin型定理

为研究积分-微分系统的稳定性,运用Lyapunov函数直接方法并借助Razumikhin技巧的思想,通过减弱Lyapunov函数沿系统解的导数须常负或定负的限制条件,给出了判断脉冲积分-微分系统零解稳定性的新的直接判定定理.     

Global exponential stability of impulsive integro-differential equation

In this paper, an impulsive integro-differential equation is considered. By establishing an integro-differential inequality with impulsive initial conditions and using the properties of M-cone and eigenspace of the spectral radius of nonnegative matrices, some new sufficient conditions for global exponential stability of impulsive integro-differential equation are obtained. The results extend and improve the earlier publications. An example is given to demonstrate the effectiveness of the theory.     

Stability of a system of Volterra integro-differential equations

Using new and known forms of Lyapunov functionals, this paper proposes new stability criteria for a system of Volterra integro-differential equations.     

序Banach空间不连续脉冲积分-微分方程初值问题的解

讨论了序Banach空间不连续脉冲积分-微分方程初值问题,通过建立一个新的比较定理,在比较弱的条件下推广了相关文献的主要结果.并在比较广泛的上控制条件而且只有一个上解或下解的假设下,获得了唯一解的存在性定理,而且给出了迭代序列的误差估计.从而推广并改进了最近某些文献中的相应结果.     

A Reproducing Kernel Hilbert Space Method for Solving Integro-Differential Equations of Fractional Order

In this article, we implement a relatively new analytical technique, the reproducing kernel Hilbert space method (RKHSM), for solving integro-differential equations of fractional order. The solution obtained by using the method takes the form of a convergent series with easily computable components. Two numerical examples are studied to demonstrate the accuracy of the present method. The present work shows the validity and great potential of the reproducing kernel Hilbert space method for solving linear and nonlinear integro-differential equations of fractional order.     

An example of application of the Nielsen theory to integro-differential equations

A new nontrivial example of an application of the Nielsen fixed-point theory is presented, this time, to integro-differential equations. The emphasis is on the parameter space so that no subdomain becomes invariant under the related solution (Hammerstein) operator. Thus, at least three (harmonic) periodic solutions are established to a planar integro-differential system.

     

Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory.     

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

This paper is mainly concerned with a new class of fractional impulsive partial stochastic integro-differential equations with state-dependent delay and optimal controls in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and uniqueness of mild solutions are proved by means of stochastic analysis theory, fractional calculus and the fixed point technique combined with solution operator. The existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. Finally, an example is given for demonstration.     

Volterra积分微分方程周期正解的一个新的存在性理论

该文通过使用锥不动点定理,研究了一类非自治Volterra积分微分方程周期正解的一个新的存在性理论,把一般结果应用于几类具时滞的生物数学模型时,改进了一些已知结果,并得到了一些新的结果.     

Banach空间二阶非线性脉冲微分-积分方程的极值解

通过建立一个新的比较定理研究了B anach空间二阶非线性混合型脉冲微分-积分方程初值问题的极值解,所得结果推广和改进了现有文献的主要结果.     

Local and global existence of solutions for a class of integro-differential equations with delay

In this article we consider a class of first order nonlinear integro-differential equations with delay. We propose new approach for investigating local and global existence of the solutions of its Cauchy problem. This approach gives new results.     

The singularly perturbed problem of vector integro-differential equations

The singularly perturbed boundary value problem of scalar integro-differential equations has been studied extensively by the differential inequality method . However, it does not seem possible to carry this method over to a corresponding nonlinear vector integro-differential equation. Therefore , for n-dimensional vector integro-differential equations the problem has not been solved fully. Here, we study this nonlinear vector problem and obtain some results. The approach in this paper is to transform the appropriate integro-differential equations into a canonical or diagonalized system of two first-order equations.     

Maximum principles for the periodic boundary value problem for impulsive integro-differential equations

This paper studies a first-order periodic boundary value problem for impulsive integro-differential equations and establishes some new maximum principles. Some points of a recent paper are also corrected and extended.