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