7,804 research outputs found
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients
For a mixed (advanced--delay) differential equation with variable delays and
coefficients
where explicit
nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with
Application
Variational electrodynamics of Atoms
We generalize Wheeler-Feynman electrodynamics by the minimization of a finite
action functional defined for variational trajectories that are required to
merge continuously into given past and future boundary segments. We prove that
the boundary-value problem is well-posed for two classes of boundary data and
show that the well-posed solution in general has velocity discontinuities,
henceforth broken extrema. Along regular segments, broken extrema satisfy the
Euler-Lagrange neutral differential delay equations with state-dependent
deviating arguments. At points where velocities are discontinuous, broken
extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta
are continuous. The electromagnetic fields of the variational trajectories are
derived quantities that can be extended only to a bounded region B of
space-time. For extrema with a finite number of velocity discontinuities,
extended fields are defined for all point in B with the exception of sets of
zero measure. The extended fields satisfy the integral laws of classical
electrodynamics for most surfaces and curves inside B. As an application, we
study globally bounded trajectories with vanishing far-fields for the
hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses
solutions of the neutral differential delay equations along regular segments
and a variational approximation for the collisional segments. Each hydrogenoid
model predicts a discrete set of finitely measured neighbourhoods of orbits
with vanishing far-fields at the correct atomic magnitude and in quantitative
and qualitative agreement with experiment and quantum mechanics, i.e., the
spacings between consecutive discrete angular momenta agree with Planck's
constant within thirty-percent, while orbital frequencies agree with a
corresponding spectroscopic line within a few percent.Comment: Full re-write using same equations and back to original title
(version 18 compiled with the wrong figure 5). A few commas introduced and
all paragraphs broken into smaller ones whenever possibl
Existence of minimal and maximal solutions to first--order differential equations with state--dependent deviated arguments
We prove some new results on existence of solutions to first--order ordinary
differential equations with deviating arguments. Delay differential equations
are included in our general framework, which even allows deviations to depend
on the unknown solutions. Our existence results lean on new definitions of
lower and upper solutions introduced in this paper, and we show with an example
that similar results with the classical definitions are false. We also
introduce an example showing that the problems considered need not have the
least (or the greatest) solution between given lower and upper solutions, but
we can prove that they do have minimal and maximal solutions in the usual
set--theoretic sense. Sufficient conditions for the existence of lower and
upper solutions, with some examples of application, are provided too
On structure of solutions of 1-dimensional 2-body problem in Wheeler-Feynman electrodynamics
The problem of 1-dimensional ultra-relativistic scattering of 2 identical
charged particles in classical electrodynamics with retarded and advanced
interactions is investigated.Comment: 16 pages, 14 figure
Numerical Methods for the 3-dimensional 2-body Problem in the Action-at-a-Distance Electrodynamics
We develop two numerical methods to solve the differential equations with
deviating arguments for the motion of two charges in the action-at-a-distance
electrodynamics. Our first method uses St\"urmer's extrapolation formula and
assumes that a step of integration can be taken as a step of light ladder,
which limits its use to shallow energies. The second method is an improvement
of pre-existing iterative schemes, designed for stronger convergence and can be
used at high-energies.Comment: 17 pages, 11 figure
Oscillatory theorems of a class of even-order neutral equations
AbstractA class of even-order nonlinear neutral differential equations with distributed deviating arguments is studied, and oscillatory criteria for solutions of such equations are established
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