Geodesic
Oscillation for equations with positive and negative coefficients and distributed delay. II: Applications
Electronic journal of differential equations, Tome 2003 (2003)
We study a scalar delay differential equation with a bounded distributed delay,

$ \dot{x}(t)+ \int_{h(t)}^t x(s)\,d_s R(t,s) - \int_{g(t)}^t x(s)\,d_s T(t,s)=0, $

where

$ R(t,h(t))=T(t,g(t))=0, \quad R(t,s) \geq T(t,s). $

We establish a connection between non-oscillation of this differential equation and the corresponding differential inequalities, and between positiveness of the fundamental function and the existence of a nonnegative solution for a nonlinear integral inequality that constructed explicitly. We also present comparison theorems, and explicit non-oscillation and oscillation results. In a separate publication (part II), we will consider applications of this theory to differential equations with several concentrated delays, integrodifferential, and mixed equations.
Classification : 34K11, 34K15
Keywords: oscillation, non-oscillation, distributed delay, comparison theorems 
@article{EJDE_2003__2003__a168,
     author = {Berezansky,  Leonid and Braverman,  Elena},
     title = {Oscillation for equations with positive and negative coefficients and distributed delay. {II:} {Applications}},
     journal = {Electronic journal of differential equations},
     year = {2003},
     volume = {2003},
     zbl = {1045.34040},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a168/}
}
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AU  - Braverman,  Elena
TI  - Oscillation for equations with positive and negative coefficients and distributed delay. II: Applications
JO  - Electronic journal of differential equations
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VL  - 2003
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%A Braverman,  Elena
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%J Electronic journal of differential equations
%D 2003
%V 2003
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%G en
%F EJDE_2003__2003__a168
Berezansky,  Leonid; Braverman,  Elena. Oscillation for equations with positive and negative coefficients and distributed delay. II: Applications. Electronic journal of differential equations, Tome 2003 (2003). http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a168/