Absolute stability of the solutions of linear differential equations with time lags in Banach space

The connection between the exponential stability of the solutions of linear differential equations in space of multidimensional bounded vector sequences and the absolute asymptotic stability of the solutions of differential equations with several time lags is investigated.     

Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type (1) $$\frac{{dx}}{{dt}} = A_0 x(t) + A_1 x(t - \Delta _1 ) + ... + A_n x(t - \Delta _n )$$ we establish the followingTHEOREM. If $$\overline {\left| {z_1 } \right| = ...\underline{\underline \cup } \left| z \right|_n = 1\sigma \left( {A_0 + \sum\nolimits_{k = 1}^n {z_k A_k } } \right)} \subset \left\{ {\lambda :\operatorname{Re} \lambda< 0} \right\}$$ then system (1) is absolutely asymptotically stable.     

On continuous dependence of solutions of linear differential equations on a parameter in a Banach space

We prove the reduction theorem for linear differential equations in a Banach space in the case where the convergence is strong. This result is used to obtain the necessary and sufficient conditions for continuous dependence of solutions on a parameter. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 275–280, February, 1999.     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with an unbounded operator in the linear part

For linear and weakly linear differential equations in a Banach space, we obtain necessary and sufficient conditions for the existence of bounded solutions on the entire real line under the assumption that the homogeneous equation dx/dt = A(t)x(t) admits an exponential dichotomy on the half-lines.     

Properties of theL p-solutions of linear impulsive equations in a Banach space

Summary In the paper conditions for the existence ofL _p-conditions (1 p ) of linear impulsive equations in a Banach space are found.     

Reducibility of linear differential equations in a Banach space with quasiperiodic coefficients

Ukrainian Mathematical Journal -     

On solutions of elliptic-type differential equations in a Banach space on a semiaxis

We describe all classical solutions of an abstract m-harmonic equation on (0,∞) and study their properties on this interval and in the neighborhood of the singular point 0.     

Differential equations in a Banach space

A systematic survey of the theory of linear evolution equations in Banach spaces, reviewed in the period 1968–1982 in Ref. Zh. Matematika, is presented.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 21, pp. 130–264, 1983.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

On stability and robust stability of positive linear Volterra equations in Banach lattices

We study positive linear Volterra integro-differential equations in Banach lattices. A characterization of positive equations is given. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, we deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given.     

Extremal solutions and comparison principle for nonlinear integro-differential equations in a Banach space

In this paper, we consider an initial value problem for nonlinear integro-differential equations in a Banach space. First, we give a comparison result between the under and over functions and some comparison principles. Then, using these results and the Kuratowski measure of noncompactness, we establish the existence theorem of extremal solutions between the under and over functions, and prove that there exists a unique solution between the lower and upper solutions under an additional Lipschitz's condition. Project supported by the Natural Science Foundation of Shandong Province.