Solution estimates for abstract nonlinear time-variant differential-delay equations

Semilinear time-variant functional-differential equations in a Banach space are considered. A solution estimate is derived. By that estimate, explicit conditions for boundedness of solutions and stability are established.     

The freezing method for abstract nonlinear difference equations

The freezing method for ordinary differential systems is extended to a class of semilinear difference equations in a Hilbert space, whose linear parts have slowly varying coefficients and nonlinearities satisfying local Lipschitz conditions. The main methodology is based on a combined use of recent norm estimates for operator-valued functions with the freezing method as well as the multiplicative representation of solutions. Thus, explicit stability and boundedness conditions are derived. Applications to infinite dimensional delay difference systems are discussed.     

Some properties of a class of abstract stationary equations

We study a class of abstract nonlinear equations in a separable Hilbert space for which we prove some generic properties of the set of solutions. The results apply, in particular, in several models of hydrodynamics, such as magneto-micropolar equations, micropolar fluid equations, Boussinesq and Navier–Stokes equations.     

Exponential stability and inequalities of solutions of abstract functional differential equations

With the Lyapunov second method, we study the abstract functional differential equation, . We obtain inequalities of solutions and exponential stability with conditions like:

(i)

,

(ii)

.

Applications in ordinary and partial functional differential equations are given.     

Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family on a Banach space X we associate an abstract differential operator G through the integral equation on a Banach space of X-valued functions on . We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Contin. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 3–8, January, 1998.     

Stability criteria for a nonlinear nonautonomous system with delays

This paper further develops a method, originally introduced by Mori et al., for proving local stability of steady states in linear systems of delay differential equations. A nonlinear nonautonomous system of delay differential equations with several delays is considered. Explicit delay-independent sufficient conditions for global attractivity of the solutions with an extremely simple form are provided. The above-mentioned conditions make the stability test quite practical. We illustrate application of this test to the Hopfield neural network models. The results obtained were also applied to a new marine protected areas model with delay that describes the ecological linkage between the reserve and fishing ground.     

Asymptotically almost periodic solutions of nonlinear Volterra difference equations with unbounded delay

The existence of almost periodic solutions of nonlinear Volterra difference equations with unbounded delay is obtained by using uniform stability properties of a bounded solution. An example is also given to illustrate obtained results.     

Iterative solution of linear equations with unbounded operators

A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space, including equations with unbounded, closed, densely defined linear operators. The method is proved to be stable towards small perturbation of the data. Some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators. The dynamical systems method (DSM) is justified for unbounded, closed, densely defined linear operators. The stopping time is chosen by a discrepancy principle. Equations with selfadjoint operators are considered separately. Numerical examples, illustrating the efficiency of the proposed method, are given.     

On some qualitative behaviors of certain differential equations of fourth order with multiple retardations

In this paper, we give sufficient conditions to guarantee the asymptotic stability and boundedness of solutions to a kind of fourth-order functional differential equations with multiple retardations. By using the Lyapunov-Krasovskii functional approach, we establish two new results on the stability and boundedness of solutions, which include and improve some related results in the literature     

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

In this paper, we study the existence of -periodic solutions for the problem

where is a -periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

     

Near operators theory and fully nonlinear elliptic equations

We give a short survey of the Campanato near operators theory and of its applications to fully nonlinear elliptic equations.     

带有热源项的非线性扩散方程的精确解

讨论了带有热源项的非线性扩散方程.通过一种直接简洁的方法得到了几种精确解.该方法可用于更高阶演化方程的求解问题.