Asymptotic stability and asymptotic solutions of second-order differential equations

Hovhannisyan, Gro

doi:10.1016/j.jmaa.2006.03.076

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…Note that the error functions ( ) in representation (11) may be estimated via the characteristic function (see, e.g., Theorem 2.2 in [6]). …”

Section: Theorem 1 Assume There Exist Complex-valued Functions ( ) ∈mentioning

confidence: 99%

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Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Hovhannisyan

2013

ISRN Mathematical Analysis

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We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

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“…Note that the error functions ( ) in representation (11) may be estimated via the characteristic function (see, e.g., Theorem 2.2 in [6]). …”

Section: Theorem 1 Assume There Exist Complex-valued Functions ( ) ∈mentioning

confidence: 99%

“…Theorem 1 may be used also for the applications mentioned in [5], the stability theory, and the Dirac equation with complex coefficients (see, e.g., [6][7][8]). …”

Section: Theorem 1 Assume There Exist Complex-valued Functions ( ) ∈mentioning

confidence: 99%

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Hovhannisyan

2013

ISRN Mathematical Analysis

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On nonautonomous Dirac equation

Hovhannisyan

2009

Journal of Mathematical Physics

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We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman–Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz’s pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.

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Linear perturbations of a nonoscillatory second-order dynamic equation

Böhner

Stević

2009

Journal of Difference Equations and Applications

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Asymptotic stability and asymptotic solutions of second-order differential equations

Cited by 4 publications

References 12 publications

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

On nonautonomous Dirac equation

Linear perturbations of a nonoscillatory second-order dynamic equation

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