Аналоги монотонного методу Ньютона

In this article there are investigated close to the method of Newton algorithms for equations with monotone operators.Дослiдженi близькi до методу Ньютона алгоритми для рiвнянь з монотонними операторами

The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established

The nonlocal problem for the 2n2n differential equations with unbounded operator coefficients and the involution

We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained

Unconventional analogs of single-parametric method of iterational aggregation

When we solve practical problems that arise, for example, in mathematical economics, in the theory of Markov processes, it is often necessary to use the decomposition of operator equations using methods of iterative aggregation. In the studies of these methods for the linear equation $x=Ax+b$ the most frequent are the conditions of positiveness of the operator $A$, constant $b$ and the aggregation functions, and also the implementation of the inequality $\rho(A) <1$ for the spectral radius $\rho(A)$ of the operator $A$. In this article for an approximate solution of a system composed of the equation $x = Ax + b$ represented in the form $x = A_1x + A_2x + b,$ where $b \in E,$ $E$ is a Banach space, $A_1, A_2$ are linear continuous operators that act from $E$ to $E$ and the auxiliary equation $y = \lambda y - (\varphi, A_2x) - (\varphi, b)$ with a real variable $y$, where $(\varphi, x)$ is the value of the linear functional $\varphi \in E ^ *$ on the elements $x \in E$, $E^*$ is conjugation with space $E$, an iterative process is constructed and investigated \begin{equation*} \begin{split} x^{(n+1)}&=Ax^{(n)}+b+\frac{\sum\limits_{i=1}^{m}A^i_1x^{(n)}}{(\varphi, x^{(n)})\sum\limits_{i=0}^{m}\lambda^i}(y^{(n)}-y^{(n+1)}) \quad (m<\infty),\\ y^{(n+1)}&=\lambda y^{(n+1)}-(\varphi,A_2x^{(n)})-(\varphi,b). \end{split} \end{equation*} The conditions are established under which the sequences ${x ^ {(n)}}, {y ^ {(n)}}$, constructed with the help of these formulas, converge to $x ^ *, y ^ *$ as a component of solving the system constructed from equations $x = A_1 x + A_2 x + b$ and the equation $y = \lambda y - (\varphi, A_2 x) - (\varphi, b)$ not slower than the rate of convergence of the geometric progression with the denominator less than $1$. In this case, it is required that the operator $A$ be a compressive and constant by sign, and that the space $E$ is semi-ordered. The application of the proposed algorithm to systems of linear algebraic equations is also shown

Interpolational (L,M)(L,M)-rational integral fraction on a continual set of nodes

In the paper, an integral rational interpolant on a continual set of nodes, which is the ratio of a functional polynomial of degree $L$ to a functional polynomial of degree $M$, is constructed and investigated. The resulting interpolant is one that preserves any rational functional of the resulting form

On applications of iteration algorithms and Skorobagatko's branching fractions to approximation of roots of polynomials in Banach algebras

Iteration algorithms for approximate factorization of some classes of polynomials with coefficients from a Banach algebra are investigated. These algorithms may be considered as methods of construction of analogues of V.Ya. Skorobagatko's branching fractions in Banach algebras

Aggregation-iterative analogues and generalizations of projection-iterative methods

Aggregation-iterative algorithms for linear operator equations are constructed and investigated. These algorithms cover methods of iterative aggregation and projection-iterative methods. In convergence conditions there is neither requirement for the corresponding operator of fixed sign no restriction to the spectral radius to be less than one

Aggregation-iterative analogues and generalizations of projection-iterative methods

Aggregation-iterative algorithms for linear operator equations are constructed and investigated. These algorithms cover methods of iterative aggregation and projection-iterative methods. In convergence conditions there is neither requirement for the corresponding operator of fixed sign no restriction to the spectral radius to be less than one

Диференцiальнi нерiвностi з односторонньою лiпшицiєвicтю

New results on differential inequalities under assumptions, which are weaker than the Lipshitz conditions, are obtained.Отримано нові результати про диференціальні нерівності за припущень, які є слабшими за умову Ліпшиця

CONVERGENCE INVESTIGATION OF ITERATIVE AGGREGATION METHODS FOR LINEAR EQUATIONS IN A BANACH SPACE

The sufficient conditions of convergence for a class of multi-parameter iterativeaggregation methods are established. These conditions do not contain the requirements ofpositivity for the operators and aggregating functionals. Moreover, it is not necessary thatthe corresponding linear continuous operators are compressing.</span