The monotone iterative technique for impulsive hybrid set valued integro-differential equations

We develop the monotone method for impulsive hybrid set integro-differential equations in all its generality. Some interesting observations are presented.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Periodic solutions of integro-differential equations in vector-valued function spaces

Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces Lp, the Besov spaces (and related spaces such as the Hölder-Zygmund spaces Cs) and the Triebel-Lizorkin spaces . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations.     

The improvement of Razumikhin type theorems for impulsive functional differential equations

In this paper, by using Lyapunov functions and Razumikhin techniques, the stability of impulsive functional differential equations is investigated. The obtained results avoid the difficulty of constructing a P function in Razumikhin’s condition and they generalize and improve the existing theorems. An example is given to illustrate the importance of the obtained results.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.     

Extermal solutions of terminal value problems for nonlinear impulsive integro-differential equations in Banach spaces

Abstract. The comparison principle is first established ,and then the lower and upper solutionmethod and the monotone iterative technique are employed to the study of terminal value prob-lems for the first order nonlinear impulsive integro-differential equations in Banach spaces. Fi-nally ,the existence theorem on the maximal and minimal solutions is obtained.     

Global exponential stability in DCNNs with distributed delays and unbounded activations

We study delayed cellular neural networks (DCNNs) whose state variables are governed by nonlinear integrodifferential differential equations with delays distributed continuously over unbounded intervals. The networks are designed in such a way that the connection weight matrices are not necessarily symmetric, and the activation functions are globally Lipschitzian and they are not necessarily bounded, differentiable and monotonically increasing. By applying the inequality pa^p-1b?(p-1)a^p+b^p${\mathit{pa}}^{p-1}b?(p-1)a^p+b^p$, where p   denotes a positive integer and a,b$a,b$ denote nonnegative real numbers, and constructing an appropriate form of Lyapunov functionals we obtain a set of delay independent and easily verifiable sufficient conditions under which the network has a unique equilibrium which is globally exponentially stable. A few examples added with computer simulations are given to support our results.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Qualitative analysis and applications of a kind of state-dependent impulsive differential equations

This paper analyzes a certain type of impulsive differential equations (IDEs). Several useful theorems for its periodic solutions and their stabilities are given. The key idea is that a periodically time-dependent IDE can be transformed into the state-dependent IDE. As applications of our theory, the optimization problems in population dynamics are studied. That is, the maximum sustainable yields of single population models with periodically impulsive constant harvesting are discussed. Furthermore, we apply these results to the studies of the order-1 periodic solutions and their stability of a single population model with stage structure in which the mature is impulsively proportionally harvested while the immature is impulsively added with the constant.     

Impulsive stability for systems of second order retarded differential equations

This paper is concerned with systems of impulsive second order delay differential equations. We prove that unstable systems can be stabilized by imposition of impulsive controls. The main tools used are Lyapunov functionals, stability theory and control by impulses.     

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

Almost periodic solutions for impulsive neural networks with delay

In the present paper the problems of existence and uniqueness of almost periodic solutions for impulsive cellular neural networks with delay are considered.     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^– with 0<<1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of . This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

A priori bounds for degenerate and singular evolutionary partial integro-differential equations

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional p-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi’s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.