scholarly journals Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Alexander Boichuk ◽  
Martina Langerová ◽  
Jaroslava Škoríková
Keyword(s):  
Small Parameter ◽  
Real Axis ◽  
Laurent Series ◽  
Existence Of Solutions ◽  
Bounded Solutions ◽  
Impulsive Systems ◽  
Entire Real Axis ◽  
Existence Conditions ◽  
Generating System

The weakly perturbed linear nonhomogeneous impulsive systems in the formẋ=A(t)x  +  εA1(t)x  +  f(t),  t∈R,  t∉T:={τi}Z,Δx|t=τi=γi+εA1ix(τi-),  τi∈T⊂R,  γi∈Rn, andi∈Zare considered. Under the assumption that the generating system (forε=0) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameterεwith finitely many terms with negative powers ofε, and we suggest an algorithm of construction of these solutions.

scholarly journals Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

2015 ◽  
Vol 63 (1) ◽  
pp. 73-87
Author(s):  
Ivanna Bondar
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Boundary Value Problems ◽  
Laurent Series ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Impulsive Action ◽  
Differential Systems

Abstract The weakly perturbed BVP's for impulsive integro-differential systems are considered. Under the assumption that the generating problem (for ε = 0) does not have solutions on the space W12[a,b] for some inhomogeneity and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space D2([a,b]{τi}I) in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.

On Bounded Solutions of Systems of Linear Functional Differential Equations

1999 ◽  
Vol 6 (5) ◽  
pp. 429-440
Author(s):  
R. Hakl
Keyword(s):  
Differential Equations ◽  
Real Axis ◽  
Existence And Uniqueness ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Functional Differential

Abstract Sufficient conditions of the existence and uniqueness of bounded on real axis solutions of systems of linear functional differential equations are established.

Internal layers in high-dimensional domains

1998 ◽  
Vol 128 (2) ◽  
pp. 359-401 ◽  
Author(s):  
Kunimochi Sakamoto
Keyword(s):  
Free Boundary ◽  
Small Parameter ◽  
Existence Of Solutions ◽  
Elliptic Partial Differential Equations ◽  
High Dimensionality ◽  
High Dimensional ◽  
Internal Layers ◽  
One Dimensional ◽  
Semilinear Elliptic ◽  
Bifurcation Phenomena

For a system of semilinear elliptic partial differential equations with a small parameter, denned on a bounded multi-dimensional smooth domain, we show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.

scholarly journals Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real Axis

2010 ◽  
Vol 2010 (1) ◽  
pp. 494379 ◽  
Author(s):  
Alexandr Boichuk ◽  
Martina Langerová ◽  
Jaroslava Škoríková
Keyword(s):  
Real Axis ◽  
Differential Systems ◽  
Entire Real Axis ◽  
Impulsive Differential Systems

25.—Spectrum of a Third-order Differential Operator with Large Coefficients

1975 ◽  
Vol 72 (4) ◽  
pp. 299-305
Author(s):  
K. Unsworth
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Order Differential Operator ◽  
Third Order ◽  
Minimal Operator ◽  
The Third ◽  
Entire Real Axis

SynopsisThis paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

scholarly journals Existence of Solutions for a Wave Equation with Non-monotone Nonlinearity and a Small Parameter

2011 ◽  
Vol 79 (1) ◽  
pp. 207-220 ◽  
Author(s):  
José F. Caicedo ◽  
Alfonso Castro ◽  
Rodrigo Duque
Keyword(s):  
Wave Equation ◽  
Small Parameter ◽  
Existence Of Solutions

scholarly journals Existence of Solutions of a Riccati Differential System from a General Cumulant Control Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Stanley R. Liberty ◽  
Libin Mou
Keyword(s):  
Control Problem ◽  
Differential System ◽  
Existence Of Solutions ◽  
Riccati Equations ◽  
Minimal Cost ◽  
Cost Variance ◽  
Risk Sensitive ◽  
Existence Conditions

We study a system of infinitely many Riccati equations that arise from a cumulant control problem, which is a generalization of regulator problems, risk-sensitive controls, minimal cost variance controls, and k-cumulant controls. We obtain estimates for the existence intervals of solutions of the system. In particular, new existence conditions are derived for solutions on the horizon of the cumulant control problem.

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