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Abstract:
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The subject of this dissertation is the investigation of asymptotic properties of
solutions for di erential equations of Emden-Fowler type and their generalizations.
The eld to which this dissertation belongs is a Qualitative theory of ordinary di e-
rential equations.
Emden-Fowler di erential equation has the form
(t u0(t))0 t u (t) = 0 ;
where ; ; 2 R. With some changes of the variables, this di erential equation
can be reduced to the equations y00 xay = 0 and y00 + xay = 0. Firstly, in this
dissertation, the di erential equation
y00 = xay ; where a; 2 R
was observed. The conditions, which provide that this equation has in nitely many
solutions de ned in some neighborhood of zero, were described here, both with the
conditions, which guarantee the existence of in nitely many solutions with certain
asymptotic behavior. Also, a complete picture of asymptotic behavior of solutions
of equation along the positive parts of both axes is given. The conditions, which
assure existence and unique solvability of solution of the Cauchy problem for this
equation, were shown in the cases when the familiar theory can't be applied. In
some cases, asymptotic formulas for solutions were obtained.
The di erential equation
y00 = xay ; where a 2 R i < 0 ;
has also been taken into consideration. The conditions, which assure the existence
of in nitely many solutions of observed equation tending to zero as x ! 0+, were
obtained.
The conditions, which assure the unique solvability of the Cauchy problem for
generalized Emden-Fowler equation
y00 = q(x)f(y(x)); lim
x!0+
y(x) = 0; lim
x!0+
y0(x) = ;
were described, for any > 0 and functions f and q which satisfy certain conditions.
The given results generalize the results both for sublinear Emden-Fowler di erential
equation (i.e. case when 0 < < 1) and the case when < 0.
In literature, it is very rare to nd the conditions for di erent values of the para-
metar which appears in the equations of Emden-Fowler type. In this dissertation,
the results for sublinear and superlinear di erential equation Emden-Fowler, as well
as the case when < 0, are presented. Therefore, the story of the asymptotic be-
havior of solutions of the observed equation is "almost complited". |