Аналоги двостороннiх методiв Курпеля для диференцiальних рiвнянь з пiслядiєю

It is studied analogues of bilateral Kurpel's methods for differential equations with aftereffect in which the right parts tend heteroton. Estimates of convergence of the algorithms are established.Дослiдженi аналоги двостороннiх методiв Курпеля для диференцiальних рiвнянь з пiслядiєю, в яких правi частини мають властивiсть гетеротонностi. Встановленi оцiнки збiжностi алгоритмiв

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

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

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

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

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

Analogues of bilateral Kurpel's methods for differential equations with aftereffect

It is studied analogues of bilateral Kurpel's methods for differentialequations with aftereffect in which the right parts tendheteroton. Estimates of convergence of the algorithms established

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

An application of analogues of two-sided Kurpel's methods to ordinary differential equation

An analogues of two-sided Kurpel's methods of approximate solution of ordinary differentia lequation that give possibility to get above-linear convergence in the case of nondifferential rightpart are constructed and investigated

Two-sided inequalities with nonmonotone sublinear operators

The theorems on solutions and their two-sided estimates for one class of nonlinear operator equations $x=Fx$ with nonmonotone operators