Asymptotically exponential solutions in nonlinear integral and differential equations

In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age-dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results.     

Mean-square exponential dichotomy of numerical solutions to stochastic differential equations

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes     

Bounded solutions for fuzzy differential and integral equations

We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.     

The exponential characteristic of first-order linear differential equations and Volterra integral equations of the second kind

The form of the exponential characteristic of a first-order linear differential equation in a Banach space with a bounded, generally nonperiodic, operator coefficient is determined. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

An exponential approximation for solutions of generalized pantograph-delay differential equations

In this paper, a new matrix method based on exponential polynomials and collocation points is proposed for solutions of pantograph equations with linear functional arguments under the mixed conditions. Also, an error analysis technique based on residual function is developed for the suggested method. Some examples are given to demonstrate the validity and applicability of the method and the comparisons are made with existing results.     

Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks

The author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. Previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations. He applies fixed point theorems to prove the existence of the traveling waves. Then, he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves.     

Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.     

Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in banach spaces

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.     

Boundedness of solutions for impulsive differential equations with integral jump conditions

The boundedness of solutions for certain nonlinear impulsive differential equations are obtained, the jumping conditions at discontinuous points are related to the integral of the past states, rather than a left hand limit at the discontinuous points. These results are obtained by new built impulsive integral inequalities with integral jumping conditions using the method of successive iteration.     

Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.     

Measures of noncompactness in the study of solutions of nonlinear differential and integral equations

The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.     

Decay integral solutions for neutral fractional differential equations with infinite delays

Our aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Complexity of differential and integral equations

We are interested in the intrinsic difficulty (or complexity) of computing an approximate solution of the linear operator equation Lu = f. Practical examples of such problems include the cases where L is a known partial differential or integral operator. Problems of the form Lu = f are typically solved under the constraint that only partial information about f is available, such as the values of a finite number of inner products, or the values of f at a finite number of points. It is of interest to determine when algorithms which are in wide use are optimal algorithms, i.e., algorithms which produce an approximation with minimal cost. We are especially interested in determining conditions which are necessary and sufficient for the finite element method (FEM) to be optimal. For the cases of elliptic partial differential equations and of Fredholm integral equations of the second kind, we describe such a condition, in the form of an inequality involving the order of the problem and the degree of the finite element subspace. Suppose this inequality is violated; is the nonoptimality of the FEM inherent in the information used by the FEM, or is it because the FEM uses this information in a nonoptimal manner? The latter is the case; there always exists an algorithm using this information which is optimal. We also discuss the situation in which the information used by the finite element method (which consists of inner products) is not available. Suppose that the only admissible information about f consists of evaluations of f. In the case of the Fredholm problem of the second kind, this information is optimal; moreover, a finite element method in which the inner products are approximated by quadrature rules is an optimal algorithm. However there exist elliptic problems of positive order for which this new information is nonoptimal.