arXiv subjects
Publications and source records attributed to G. A. Grigorian.
Extencion of Krein's special method for solving of integral equation to that method for solving of systems of integral equations is established. Generalizations of formulae for solution of integral equations are obtained. The result obtained is demonstrated by examples.
We use a new approach with a matrix transformation to obtain a new global solvability criterion for matrix Riccati equations. The proven theorem completes an well known result in directions of extension of classes of coefficient of equations and of extension of initial values,under which the solutions of equations are continuable to $\infty$.
The Riccati equation method is used to establish new oscillation criteria for extended linear matrix Hamiltonian systems. This method allows to obtain results in in a new direction, which is to break the positive definiteness condition, imposed on one of coefficients of the system. Some examples are provided for comparing the obtained results with each other and with the result of K. I. Al - Dosary, H. Kh. Abdullah and D. Husein.
A new approach with the Riccati equation method is used to obtain a non oscillation criterion for extended quasi linear Hamiltonian systems.
In this paper we use the comparison method for investigation of first order polynomial differential equations. We prove two comparison criteria for these equations. The proved criteria we use to obtain some global solvability criteria for first order polynomial differential equations. On the basis of these criteria we prove some criteria for existence of a closed solution (of closed solutions) of for first order polynomial differential equations. The results obtained we compare with some known results.
The Wiener-Hopf integral equations of 1-st kind relates to the class of Wiener-Hopf equations of non normal type, to which the classical Wiener-Hopf method is not applicable, but is completely applicable the special factorization method. In this paper we use the special factorization method to obtain solvability conditions for Wiener-Hopf equations of 1-st kind.
Three comparison criteria are obtained for second order Riccati equations. On the basis of these criteria some global existence theorems are proved mentioned equations. The results obtained are used to derive a non oscillation criterion for three dimensional linear systems of ordinary differential equations.
The Riccati equation method is used to obtain a generalization of the Gronvall-Bellman lemma the obtained result is used to generalize a result of Lyapunov.
In this article we use a method of finding the index of a complex-valued function by determined number of arithmetic operations to describe an algorithm of localization of roots of square-free polynomials. We give an estimation of the number of arithmetic operations for the described algorithm.
Three comparison criteria for the Abel equation of 1es kind are proved. The results obtained are used to obtain global solvability criteria and some criteria of existence of closed solutions for the mentioned equation. The results obtained are demonstrated by examples.
The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's oscillation criterion.
A new approach is used to obtain a global solvability criterion for matrix Riccati equations. It is shown that the obtained result is an extension of a result derived from a comparison theorem for matrix Riccati equations. Two corollaries were drawn from the obtained result as well.
The Riccati equation method and an approach of the use of unknown factors is used to establish oscillation, suboscillation and nonoscillation criteria for linear systems of ordinary differential equations. A necessary condition for Lyapunov (asymptotic) stability for these systems is obtained.
In this paper convolution type integral equations in the conservative case are studied. The conservative case of convolution type of equations relates to the case of non normal type of equations and is that of the corresponding symbols degenerate at some points of the real line, and the classical Furier transformation method meets difficulties with its application to the studying equations. To the study in the conservative case of convolution type equations in this paper we use the special factorization method.
The Riccati equation method is used to establish new oscillation criteria for linear matrix Hamiltonian systems. New approaches allow to extend and completed a result, obtained by S. Kumary and S. Umamaheswaram. The oscillation problem for linear matrix Hamiltonian systems in a new direction, which is to break the positive definiteness condition, imposed on one of the coeffcients of the system, is investigated. Some examples are provided for comparing the obtained results with each other and with the result of S. Kumary and S. Umamaheswaram, as well as to illustrate the applicability of these results.
In this paper we study the conditions, under which the quaternionic Riccati equations have periodic solutions. The obtained result we compare with one recently obtained important one.
The Riccati equation method is used to establish oscillation and non-oscillation criteria for second order linear nonhomogeneous functional-differential equations.We show that the obtained oscillation criterion is a generalization of J. S. W. Wong's oscillation criterion for second order linear nonhomogeneous ordinary differential equations. Two examples, demonstrating the aptitude of the obtained criteria, are presented.
In this paper we study properties of regular solutions of matrix Riccati equations. The obtained results are used to study the asymptotic behavior of solutions of linear systems of ordinary differential equations.