# Oscillating differential equation

An ordinary differential equation which has at least one oscillating solution. There are different concepts of the oscillation of a solution. The most widespread are oscillation at a point (usually taken to be $+ \infty$) and oscillation on an interval. A non-zero solution of the equation

$$\tag{1 } u ^ {(} n) = f( t, u , u ^ \prime \dots u ^ {(} n- 1) ),\ n \geq 2 ,$$

where $f( t, 0 \dots 0) = 0$, is called oscillating at the point $+ \infty$( or on an interval $I$) if it has a sequence of zeros which converges to $+ \infty$( respectively, there are at least $n$ zeros in $I$ counted according to their multiplicity). Equation (1) is oscillating at $+ \infty$ or on an interval $I$ if its solutions are oscillating (at $+ \infty$, respectively, on $I$).

Among equations which are oscillatory at $+ \infty$ the equations which possess the properties $A$ or $B$, i.e. which are compatible in a specific sense with one of the equations

$$u ^ {(} n) = - u \ \textrm{ or } \ u ^ {(} n) = u ,$$

are distinguished. Equation (1) is said to possess property $A$ if all its solutions defined in a neighbourhood of $+ \infty$ are oscillating when $n$ is even; when $n$ is odd, they should either be oscillating or satisfy the condition

$$\tag{2 } \lim\limits _ {t \rightarrow + \infty } u ^ {(} i- 1) ( t) = 0,\ i = 1 \dots n.$$

If every solution of equation (1) defined in a neighbourhood of $+ \infty$, when $n$ is even, is either oscillating, or satisfies condition (2) or

$$\tag{3 } \lim\limits _ {t \rightarrow + \infty } | u ^ {(} i- 1) ( t) | = + \infty ,\ \ i = 1 \dots n,$$

while when $n$ is odd, it is either oscillating or satisfies condition (3), then the equation possesses property $B$.

The linear equation

$$\tag{4 } u ^ {(} n) = a( t) u$$

with a locally summable coefficient $a: [ t _ {0} , + \infty ) \rightarrow \mathbf R$ possesses property $A$( property $B$) if

$$a( t) \leq 0 \ ( a( t) \geq 0) \ \textrm{ when } t \geq t _ {0}$$

and either

$$\int\limits _ {t _ {0} } ^ { {+ } \infty } t ^ {n- 1- \epsilon } | a( t) | dt = + \infty$$

or

$$a( t) \leq \frac{\mu _ {n} - \epsilon }{t ^ {n} } \ \left ( a( t) \geq \frac{\nu _ {n} + \epsilon }{t ^ {n} } \right )$$

when $t \geq t _ {0}$, where $\epsilon > 0$ and $\mu _ {n}$ is the smallest ( $\nu _ {n}$ is the largest) of the local minima (maxima) of the polynomial $x( x- 1) \dots ( x- n+ 1)$( see [1][5]).

An equation of Emden–Fowler type

$$\tag{5 } u ^ {(} n) = a( t) | u | ^ \lambda \mathop{\rm sign} u ,\ \ \lambda > 0,\ \lambda \neq 1,$$

with a locally summable non-positive (non-negative) coefficient $a: [ t _ {0} , + \infty ) \rightarrow \mathbf R$ possesses property $A$( property $B$) if and only if

$$\int\limits _ {t _ {0} } ^ { {+ } \infty } t ^ \mu | a( t) | dt = + \infty ,$$

where $\mu = \min \{ n- 1, ( n- 1) \lambda \}$( see [4], [6], [7]).

In a number of cases the question of the oscillation of equation (1) can be reduced to the same question for the standard equations of the form (4) and (5) using a comparison theorem (see [11]).

In studying the oscillatory properties of equations with deviating argument, certain specific features arise. For example, if $n$ is odd, $\Delta > 0$, and if for large $t$ the inequality

$$a( t) \leq a _ {0} < - n! \Delta ^ {-} n$$

is fulfilled, then all non-zero solutions of the equation

$$u ^ {(} n) ( t) = a( t) u( t - \Delta )$$

are oscillatory at $+ \infty$( see [10], [11]). At the same time, if $a$ is non-positive and $n$ is odd, the non-retarded equation (4) always has a non-oscillating solution.

The concepts of oscillation and non-oscillation on an interval are generally studied for linear homogeneous equations. They are of fundamental value in the theory of boundary value problems (see [12]).

#### References

 [1] A. Kneser, "Untersuchungen über die reellen Nullstellen der Integrale linearer Integralgleichungen" Math. Ann. , 42 (1893) pp. 409–435 [2] J.G. Mikusinksi, "On Fite's oscillation theorems" Colloq. Math. , 2 (1951) pp. 34–39 [3] V.A. Kondrat'ev, "The oscillatory character of solutions of the equation " Trudy Moskov. Mat. Obshch. , 10 (1961) pp. 419–436 (In Russian) [4] I.T. Kiguradze, "On the oscillatory character of solutions of the equation " Mat. Sb. , 65 : 2 (1964) pp. 172–187 (In Russian) [5] T.A. Chanturiya, "On a comparison theorem for linear differential equations" Math. USSR Izv. , 10 : 5 (1976) pp. 1075–1088 Izv. Akad. Nauk. SSSR Ser. Mat. , 40 : 5 (1976) pp. 1128–1142 [6] I. Ličko, M. Švec, "La charactère oscillatoire des solutions de l'équation , " Chekhosl. Mat. Zh. , 13 (1963) pp. 481–491 [7] I.T. Kiguradze, "On the oscillatory and monotone solutions of ordinary differential equations" Arch. Math. , 14 : 1 (1978) pp. 21–44 [8] C.A. Swanson, "Comparison and oscillation theory of linear differential equations" , Acad. Press (1968) [9] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [10] A.D. Myshkis, "Linear differential equations with retarded argument" , Moscow (1972) (In Russian) [11] R.G. Koplatadze, T.A. Chanturiya, "On the oscillatory properties of differential equations with deviating argument" , Tbilisi (1977) (In Russian) [12] A.Yu. Levin, "Non-oscillation of the solutions of the equation " Russian Math. Surveys , 24 : 2 (1969) pp. 43–99 Uspekhi Mat. Nauk , 24 : 2 (1969) pp. 43–96