Convergence estimates of a projection-difference method for an operator-differential equation

This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given.     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

Galerkin method for a nonstationary equation with a monotone operator

In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic equation of higher order.     

Projection Method for Cauchy Problem for an Operator-Differential Equation

In the current paper, we study a projection method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in a Hilbert space. The projection subspaces are linear spans of eigenvectors of an operator similar to A(t). It is assumed that the operators A(t) and K(t) are sufficiently smooth. Error estimates for the approximate solutions and their derivatives are obtained. The application of the developed method for solving the initial boundary value problems is given.     

Error estimates for projection-difference methods for differential equations with differentiable operators

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to A(0).     

Galerkin method for a third-order differential-operator equation

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.     

Periodic solutions for doubly nonlinear evolution equations

We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u^′(t))+∂?(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂? in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided.     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

An abstract doubly nonlinear equation with a measure as initial value

The solvability of the abstract implicit nonlinear nonautonomous differential equation (A(t)u^′(t))+B(t)u(t)+C(t)u(t)∋f(t) will be investigated in the case of a measure as an initial value. It will be shown that this problem has a solution if the inner product of A(t)x and B(t)x+C(t)x is bounded below.     

An abstract Szeg? theorem

Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

Periodic solutions of semilinear equations of evolution of compact type

The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.     

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′.     

Stabilization of unbounded bilinear control systems in Hilbert space

We consider bilinear control systems of the form y^′(t)=Ay(t)+u(t)By(t) where A generates a strongly continuous semigroup of contraction (etA)_t?0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. We suppose that this system is unbounded in the sense that the linear operator B is unbounded from the state Y into itself. Tacking into account eventual control saturation, we study the problem of stabilization by (possibly nonquadratic) feedback of the form u(t)=−f(〈By(t),y(t)〉). Applications to the heat equation is considered.     

A two-parameter family of pension contribution functions and stochastic optimization

In a previous article the author has suggested a linear function of A(t) (present value of future benefits) and F(t) (fund) as pension contribution function in place of the form given in Trowbridge (1963) which is a one-parameter family of funding methods. Here we provide some theoretical justification for such a method by showing that, in the simplified model of this paper, the optimal solution of a stochastic control problem yields, as contribution function, an affine function of A(t) and F(t).     

An abstract semilinear hyperbolic Volterra integrodifferential equation

We consider semilinear integrodifferential equations of the form u′(t) + A(t) u(t) = ∝₀^tg(t, s, u(s)) ds + f(t), u(0) = u₀. For each t ? 0, the operator A(t) is assumed to be the negative generator of a strongly continuous semigroup in a Banach space X. The domain D(A(t)) of A(t) is allowed to vary with t. Thus our models are Volterra integrodifferential equations of “hyperbolic type.” These types of equations arise naturally in the study of viscoelasticity. Our main results are the proofs of existence, uniqueness, continuation and continuous dependence of the solutions.     

A Massera type theorem for almost automorphic solutions of differential equations

In this note, we present a Massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x^′(t)=A(t)x(t)+f(t), where A(t) is unbounded linear operator depending on t periodically and generates a τ-periodic evolutionary process, f is almost automorphic. The main results are stated in terms of the almost automorphy of solutions and their Carleman spectra.     

Generalized solutions of differential-operator equations with singular white noise

In various distribution spaces, we study the Cauchy problem for the equation u′(t) = Au(t)+B $\mathbb{W}$ (t), t ≥ 0, with a singular white noise $\mathbb{W}$ and an operator A generating various regularized semigroups in a Hilbert space. Depending on the properties of the operator A, we construct solutions generalized separately and jointly with respect to the time, random, and “space” variables.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set Λ of parameters t = (t ₁, ..., t _n ) for which the equation A(t)u = 0 has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set Λ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set E(k) in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.