On a linear nth-order differential equation with special right-hand side in a Banach space

We show that the classical algorithm for finding a particular solution of a scalar linear differential equation can be applied to linear differential equations with bounded operator coefficients in a Banach space.     

On a nonlinear Volterra integral equation in a Banach space

The equation u(t) + ∝₀^tk(t ? s)g(s) ds?f(t), t ? 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g(t) and the function f. The case when k is realvalued, g(t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy.     

A multiparametric family of solutions to a singular Volterra integral equation in a Banach space

We construct a multiparametric set of solutions to a singular Volterra integral equation of the first kind with a sufficiently smooth kernel in the space of integrable functions whose values belong to a Banach space.     

An abstract Volterra integral equation in a reflexive Banach space

This paper discusses the existence, uniqueness, and asymptotic behavior of solutions to the equation u(t) + ∝₀^ta(t ? s) Au(s) ds = f(t), where A is a maximal monotone operator mapping the reflexive Banach space V into its dual V′.     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

On the approximation of a bounded solution of a linear differential equation in a Banach space

We investigate the problem of approximation of a bounded solution of a linear differential equation by solutions of the corresponding difference equations in a Banach space. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1268–1271, September, 1998.     

Some remarks on a linear stochastic differential equation

In this note we study the properties of the solution of a two-parameter linear stochastic differential equation with constant coefficients. We prove, in particular, that this solution may be negative with positive probability.     

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.     

Multi-term fractional differential equation in reflexive Banach space

In this paper, we investigate the solvability of a multi-term fractional differential equation in a reflexive Banach space relative to weak topology. A particular case of our main result is treated.     

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

Ukrainian Mathematical Journal -     

On a family of solutions of a linear second-order differential equation with constant unbounded operator coefficients in a Banach space

In a Banach space E, we study the equation 1 $$ u''(t) + Bu'(t) + Cu(t) = f(t), 0 \leqslant t < \infty $$ , where f(t) ∈ C([0,∞);E), B,C ∈ N(E), and N(E) is the set of closed unbounded linear operators from E to E with dense domain in E. We find a two-parameter family of solutions of Eq. (1) in two cases: (a) the operator discriminant D = B ² ? 4C of Eq. 1 is zero; (b) D = F ², where F is some operator in N(E). We suggest a method for increasing the smoothness of such solutions by imposing more restrictive conditions on the input data W = (B,C,f(t)) and the parameters x ₁, x ₂ ∈ E.     

Stability of stochastic differential equation with linear fractal noise

We study a class of stochastic differential equation with linear fractal noise. By an auxiliary stochastic differential equation, we prove the existence and uniqueness of the solution under some mild assumptions. We also give some estimates of moments of the solution. The exponential stability of the solution is discussed.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

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.     

Existence and regularity results for an integral equation with infinite delay in a Banach space

Some integrodifferential equations with infinite delay in Banach spaces are studied.