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 ) and oscillation on an interval. A non-zero solution of the equation

(1)

where , is called oscillating at the point (or on an interval ) if it has a sequence of zeros which converges to (respectively, there are at least zeros in counted according to their multiplicity). Equation (1) is oscillating at or on an interval if its solutions are oscillating (at , respectively, on ).

Among equations which are oscillatory at the equations which possess the properties or , i.e. which are compatible in a specific sense with one of the equations

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

(2)

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

(3)

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

The linear equation

(4)

with a locally summable coefficient possesses property (property ) if

and either

or

when , where and is the smallest ( is the largest) of the local minima (maxima) of the polynomial (see [1]–[5]).

An equation of Emden–Fowler type

(5)

with a locally summable non-positive (non-negative) coefficient possesses property (property ) if and only if

where (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 is odd, , and if for large the inequality

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

are oscillatory at (see [10], [11]). At the same time, if is non-positive and 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

References

[a1]	J.K. Hale, "Ordinary differential equations" , Wiley (1969)
[a2]	W.T. Reid, "Sturmian theory for ordinary differential equations" , Springer (1980)