Operator equation and related questions of stability of differential equations

2020 ◽  
Author(s):  
Lyubov' Orlik ◽  
Galina Zhukova
Keyword(s):  
Functional Analysis ◽  
Differential Equations ◽  
Operator Equation ◽  
Electromagnetic Waves ◽  
Special Kind ◽  
Stability Criteria ◽  
Boundedness Of Solutions ◽  
Operator Equations ◽  
Electrical Systems ◽  
Volterra Operators

The monograph is devoted to the application of methods of functional analysis to the problems of qualitative theory of differential equations. Describes an algorithm to bring the differential boundary value problem to an operator equation. The research of solutions to operator equations of special kind in the spaces polutoratonny with a cone, where the limitations of the elements of these spaces is understood as the comparability them with a fixed scale element of exponential type. Found representations of the solutions of operator equations in the form of contour integrals, theorems of existence and uniqueness of such solutions. The spectral criteria for boundedness of solutions of operator equations and, as a consequence, sufficient spectral features boundedness of solutions of differential and differential-difference equations in Banach space. The results obtained for operator equations with operators and work of Volterra operators, allowed to extend to some systems of partial differential equations known spectral stability criteria for solutions of A. M. Lyapunov and also to generalize theorems on the exponential characteristic. The results of the monograph may be useful in the study of linear mechanical and electrical systems, in problems of diffraction of electromagnetic waves, theory of automatic control, etc. It is intended for researchers, graduate students functional analysis and its applications to operator and differential equations.

scholarly journals On Stability of Linear Barbashin Type Integrodifferential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Homogeneous Equation ◽  
Type Equation ◽  
Integrodifferential Equations ◽  
Stability Test ◽  
Boundedness Of Solutions ◽  
Operator Equations ◽  
Freezing Method ◽  
Solution Estimates

We consider the Barbashin type equation∂u(t,x)/∂t=c(t,x)u(t,x)+∫01k(t,x,s)u(t,s)ds+f(t,x)   (t>0; 0≤x≤1),wherec(·, ·),k(·, ·, ·), andf(·, ·)are given real functions andu(·, ·)is unknown. Conditions for the boundedness of solutions of this equation are suggested. In addition, a new stability test is established for the corresponding homogeneous equation. These results improve the well-known ones in the case when the coefficients are differentiable in time. Our approach is based on solution estimates for operator equations. It can be considered as the extension of the freezing method for ordinary differential equations.

scholarly journals The unique solution of some operator equations via fractional differential equations

2022 ◽  
Vol 40 ◽  
pp. 1-9
Author(s):  
Hojat Afshari ◽  
L. Khoshvaghti
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Unique Solution ◽  
Positive Solutions ◽  
Operator Equation ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Ordered Banach Space ◽  
Operator Equations ◽  
Fractional Differential

In this paper we consider the existence and uniqueness of positive solutions to the following operator equation in an ordered Banach space $E$$$A(x,x)+B(x,x)=x,~x\in P,$$where $P$ is a cone in $E$. We study an application for fractional differential equations.

scholarly journals Stability Criteria of Solutions for Stochastic Set Differential Equations

2012 ◽  
Vol 03 (04) ◽  
pp. 354-359 ◽  
Author(s):  
Ho Vu ◽  
Nguyen Ngoc Phung ◽  
Ngo Van Hoa ◽  
Nguyen Dinh Phu
Keyword(s):  
Differential Equations ◽  
Stability Criteria

scholarly journals On the solvability of nonlinear noncompact operator equations

1987 ◽  
Vol 43 (1) ◽  
pp. 103-126 ◽  
Author(s):  
E. U. Tarafdar ◽  
H. B. Thompson ◽  
R. O. Vyborny
Keyword(s):  
Differential Equations ◽  
Operator Equations

AbstractThe notion of (p, k)-epi mappings is introduced. The properties of such mappings are studied and the results obtained are applied to some differential equations.

scholarly journals Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Cemil Tunç ◽  
Muzaffer Ateş
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Cauchy Formula ◽  
Boundedness Of Solutions ◽  
Constant Coefficients ◽  
Particular Solution

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

scholarly journals Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium

1936 ◽  
Vol 155 (885) ◽  
pp. 235-257 ◽  
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Electron Density ◽  
Isotropic Medium ◽  
Electromagnetic Waves ◽  
Mathematical Theory ◽  
Stratified Medium ◽  
Oblique Propagation ◽  
Isotropic Media ◽  
Mathematical Justification

The influence of the earth’s magnetic field on the propagation of wireless waves in the ionosphere has stimulated interest in the problem of the propagation of electromagnetic waves through a non-isotropic medium which is stratified in planes. Although the differential equations of such a medium have been elegantly deduced by Hartree,f it appears that no solution of them has yet been published for a medium which is both non-isotropic and non-homogeneous. Thus the work of Gans and Hartree dealt only with a stratified isotropic medium, while in the mathematical theory of crystal-optics the non-isotropic medium is always assumed to be homogeneous. In the same way Appleton’s magneto-ionic theory of propagation in an ionized medium under the influence of a magnetic field is confined to consideration of the “ characteristic ”waves which can be propagated through a homogeneous medium without change of form. In applying to stratified non-isotropic media these investigations concerning homogeneous non-isotropic media difficulty arises from the fact that the polarizations of the characteristic waves in general vary with the constitution of the medium, and it is not at all obvious that there exist waves which are propagated independently through the stratified medium and which are approximately characteristic at each stratum. The existence of such waves has usually been taken for granted, although for the ionosphere doubt has been cast upon this assumption by Appleton and Naismith, who suggest that we might “ expect the components ( i. e ., characteristic waves) to be continually splitting and resplitting”, even if the increase of electron density “ takes place slowly with increase of height”. It is clear that, until the existence of independently propagated approximately characteristic waves has been established, at any rate for a slowly-varying non-isotropic medium, no mathematical justification exists for applying Appleton's magnetoionic theory to the ionosphere. It is with the provision of this justification that we are primarily concerned in the present paper. This problem has been previously considered by Försterling and Lassen,f but we feel that their work does not carry conviction because they did not base their calculations on the differential equations for a non-homo-geneous medium, and were apparently unable to deal with the general case in which the characteristic polarizations vary with the constitution of the medium.

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

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