Oscillating solution

A solution $x ( t)$ of a differential equation

$$\tag{* } x ^ {(n)} = \ f ( t , x , x ^ \prime \dots x ^ {(n- 1)} ),\ \ t \in [ t _ {0} , \infty ) ,$$

with the property: There exists for any $t _ {1} \geq t _ {0}$ a point $t _ {2} > t _ {1}$ such that $x ( t)$ changes sign on passing through it. In many applied problems there arises the question of the existence of an oscillating solution or the question whether all the solutions of equation (*) oscillate. Many sufficient conditions are known under which equation (*) has an oscillating solution (see [1], [2], [3]). For example, any non-trivial solution of the equation $x ^ {\prime\prime} + 2 \delta x ^ \prime + \omega ^ {2} x = 0$ with constant coefficients is oscillating if $\delta ^ {2} < \omega ^ {2}$; every non-trivial solution of the equation

$$x ^ {\prime\prime} + p ( t) x ^ \prime + q ( t) x = 0$$

with $\omega$- periodic coefficients is oscillating if

$$\int\limits _ { 0 } ^ \infty dt \int\limits _ { t } ^ { {t } + \omega } q ( s) \mathop{\rm exp} \left ( - \int\limits _ { s } ^ { t } p ( r) dr \right ) ds \geq$$

$$\geq \ - \frac{1}{2} \left ( 1 - \mathop{\rm exp} \int\limits _ { 0 } ^ \omega p ( t) dt \right ) \int\limits _ { 0 } ^ \omega p ( t) dt$$

and $q ( t) \not\equiv 0$ on $[ 0 , \omega ]$.

In a number of applications there arises the question of the existence of oscillating solutions (in the above sense) of a system of ordinary differential equations. For example, in control theory one studies the oscillation relative to a given hyperplane $\sum _ {i=1} ^ {n} c _ {i} x _ {i} = 0$ of the solutions $x ( t) = ( x _ {1} ( t) \dots x _ {n} ( t) )$ of the system of equations $x ^ \prime = f ( t , x )$, that is, the question whether the function $\sigma ( t) = \sum _ {i=1} ^ {n} c _ {i} x _ {i} ( t)$ oscillates. $[ \alpha , \beta ]$- oscillating solutions are also studied; a bounded solution $x ( t)$ of the system $x ^ \prime = f ( t , x )$ is called $[ \alpha , \beta ]$- oscillating if $\sigma ( t)$ is oscillating and for any $t _ {1} \geq t _ {0}$ there are points $t _ {2}$ and $t _ {3}$ such that $t _ {1} < t _ {2} < t _ {3}$, $\sigma ( t _ {2} ) < \alpha$, $\sigma ( t _ {3} ) > \beta$, where $\alpha < 0 < \beta$. For the system $x ^ \prime = f ( x , t )$ there also exist other definitions of an oscillating solution.

References

[1]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[2]	C.A. Swanson, "Comparison and oscillation theory of linear differential equations" , Acad. Press (1968)
[3]	I.T. Kiguradze, "Some singular boundary value problems for ordinary differential equations" , Tbilisi (1975) (In Russian)

References

[a1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17