Uniformly monotonic explicit approximate methods for first-order differential operator equations in Hilbert space

The properties of contractivity, monotonicity, and sign constancy of approximate methods of solution of the initial problem are studied for linear and nonlinear differential operator equations in a complex Hilbert space. Explicit methods of first and second order of accuracy are presented that are monotonc and sign constant in the corresponding classes of problems for any value of the mesh size.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1593–1598, December, 1990.     

Continuous regularization of linear operator equations in a Hilbert space

For the linear operator equation Au=f we offer a continuous regularization based on the stabilization of solutions of differential equations in a Hilbert space H. We assume that A is a positive operator and that the equation Au=f has a solution in H.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 503–510, November, 1968.     

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

Optimization of direct methods for the approximate solution of weakly singular integral equations

For a class of weakly singular integral equations with power and logarithmic singularities, we establish the optimal order of the error of direct methods and indicate the procedure which realizes this order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1246–1254, September, 1994.     

Origins of oscillation criteria of operator differential equations in Hilbert space

It is shown that, for a nonempty ergodic family {T_t: t?J} of continuous linear operators on a topological vector space, the solution by iteration of the simultaneous linear functional equations x ? T_t(x) = h, t?J, is a special case of mean ergodic theory for a semigroup of linear operators.     

多元Fredholm积分方程自适应直接方法的优化

考虑了以各向异性光滑函数为核及自由项的多元Fredholm积分方程类的自适应直接方法的优化,得到误差阶的精确估计及相应的最优算法。     

On direct methods for solution of regularized equations

We prove that the application of so-called adaptive direct methods to approximation of Fredholm equations of the first kind leads to a more economical way of finite-dimensional approximation as compared with traditional approaches.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1231–1242, September, 1995.     

On the solvability in Hilbert space of certain nonlinear operator equations depending on parameters

The equation (*) Au − λTu + μCu = f is studied in a real separable Hilbert space H. Here, λ, μ > 0 are fixed constants and f ε H is fixed. The operators A: D H → H, C: D H → H are monotone and compact, respectively, where D denotes a closed ball in H. The operator T: H → H is linear, compact, self-adjoint and positive-definite. Degree-theoretic arguments are used for the existence of solutions of (*) and extensions of recent results of Kesavan are established. For example, it is shown, under additional assumptions, that there exists a constant μ₀ > 0 such that (*) is solvable for all μ μ₀ and all λ ε R.     

Iterative methods for solving nonlinear irregular operator equations in banach space

We present an approach to constructing stable methods for solving nonlinear operator equations in Banach space without anyassumptions on the regularity of the operator. The approach is based on the linearization of the equation and the use of aregularization algorithm to find an approximate solution of the linearized equation at each iteration. The local convergence of proposed methods is proved and the estimations of the rate of convergence are established, provided that solution satisfies a sourcewise representation condition. The case of noisy data is also analysed.     

Optimal control by direct inversion of a positive-definite operator in a Hilbert space

A well-known perturbation theorem for linear operators is used to provide a series expansion for the inverse of a positive-definite operator in a Hilbert space. With this expansion, feasible computational procedures are obtained for solving some optimal control problems with quadratic cost functionals. The methods presented here are compared to alternate methods available in the literature.     

Direct methods for the approximate solution of operator equations with nonzero kernel

We justify direct methods for the approximate solution of linear operator equations with nonzero kernels and apply these methods to the justification of projective methods for the approximate solution of standard singular integral equations with Cauchy kernels and positive index on the unit disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1202–1213, September, 1998.     

Generalized Hilbert operator on the Dirichlet-type space

The boundedness of the generalized Hilbert operator on the Dirichlet-type space S_p when 0<p<1 is investigated.