# Boundary value problem, ordinary differential equations

The problem of finding a solution to an equation

$$\tag{1 } \frac{dx }{dt } = \ f (t, x),\ \ t \in J,\ \ x \in \mathbf R ^ {n} ,$$

lying in a given subset $D$ of the space $D (J, \mathbf R ^ {n} )$ of functions depending on $t$ that are absolutely continuous on $J$ and that assume values in $\mathbf R ^ {n}$:

$$\tag{2 } x ( \cdot ) \in D.$$

It is assumed that $f (t, x)$ is a function defined on $J \times \mathbf R ^ {n}$ with values in $\mathbf R ^ {n}$ and satisfying the Carathéodory conditions; $J$ is an interval on the real line $\mathbf R$.

1) The boundary value problem (1), (2) is said to be linear if

$$f (t, x) \equiv \ A (t) x + b (t),$$

where the functions $A (t)$ and $b (t)$ are summable on every compact interval in $J$ and the set $D$ is a linear manifold in $D (J, \mathbf R ^ {n} )$. In particular, one might have

$$J = \ [t _ {0} , t _ {1} ],$$

$$D = \left \{ x ( \cdot ) \in D (J, \mathbf R ^ {n} ): \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0 \right \} ,$$

where $\Phi (t)$ is a function of bounded variation. A linear boundary value problem gives rise to a linear operator

$$Lx (t) \equiv \ x ^ \prime - A (t) x,\ \ x ( \cdot ) \in D,$$

the eigen values of which are precisely those values of the parameter $\lambda$ for which the homogeneous boundary value problem

$$x ^ \prime - A (t) x = \lambda x,\ \ x ( \cdot ) \in D,$$

has non-trivial solutions. These non-trivial solutions are the eigen functions of the operator $L$. If the inverse operator $L ^ {-1}$ exists and has an integral representation

$$x (t) = \ L ^ {-1} b (t) \equiv \ \int\limits _ { J } G (t, s) b (s) ds,\ \ t \in J,$$

then $G (t, s)$ is called a Green function.

2) Let $J = (- \infty , \infty )$, let $f (t, x)$ be almost-periodic in $t$ uniformly in $x$ on every compact subset of $\mathbf R ^ {n}$ and let $D$ be the set of almost-periodic functions in $t$ that are absolutely continuous on $J$. Then problem (1), (2) is known as the problem of almost-periodic solutions.

3) In control theory one considers boundary value problems with a functional parameter: a control. For example, consider the equation

$$\tag{3 } \frac{dx }{dt } = \ f (t, x, u),\ \ t \in J = [t _ {0} , t _ {1} ],\ \ x \in \mathbf R ^ {n} ,$$

with set of admissible controls $U$ and two sets $M _ {0} , M _ {1} \subset \mathbf R ^ {n}$. Let $D$ be the set of absolutely continuous functions in $t$ such that $x (t _ {0} ) \in M _ {0}$, $x (t _ {1} ) \in M _ {1}$. The boundary value problem is to find a pair $(x _ {0} ( \cdot ), u _ {0} ( \cdot ))$ such that $u _ {0} ( \cdot ) \in U$ and the solution $x _ {0} (t)$ of equation (3) at $u = u _ {0} (t)$ satisfies the condition $x _ {0} ( \cdot ) \in D$.

4) There is a wide range of diverse necessary and sufficient conditions for the existence and uniqueness of solutions to various boundary value problems, and of methods for constructing an approximate solution (see [4][7]). For example, consider the problem

$$\tag{4 } \left . \begin{array}{c} x ^ \prime = A (t) x + f (t, x), \\ \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0, \end{array} \ \right \}$$

in which

$$\| f (t, x) \| \leq a + b \ \| x \| ^ \alpha$$

for certain constants $a > 0, b > 0, \alpha \geq 0$. Suppose that the homogeneous problem

$$\tag{5 } x ^ \prime = A (t) x,\ \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0$$

is regular, i.e. its only solution is the trivial one. Then problem (4) has at least one solution, provided either $\alpha < 1$, or $\alpha \geq 1$ and $b$ is sufficiently small. It is fairly complicated to determine whether problem (5) is regular. However, the linear (scalar) boundary value problem

$$x ^ {\prime\prime} + q (t) x ^ \prime + p (t) x = 0,\ \ x (t _ {0} ) = 0,\ \ x (t _ {1} ) = 0,$$

for example, is regular if whenever $| q (t) | \leq 2m$ there exists a $k \in \mathbf R$ such that

$$\int\limits _ { t _ {0} } ^ { {t _ 1 } } [p (t) - k] _ {+} dt < \ 2 [F (k, m) - m],$$

where

$$F (k, m) = \ \left \{ \begin{array}{l} \sqrt {k - m ^ {2} } \mathop{\rm cotg} \ { \frac{(t _ {1} - t _ {0} ) \sqrt {k - m ^ {2} } }{2} } , \\ \ \ m ^ {2} < k \leq m ^ {2} + \frac{\pi ^ {2} }{(t _ {1} - t _ {0} ) ^ {2} } , \\ \frac{2}{t _ {1} - t _ {0} } ,\ \ k = m ^ {2} , \\ \sqrt {m ^ {2} - k } \ \mathop{\rm cotg} \ { \frac{(t _ {1} - t _ {0} ) \sqrt {m ^ {2} - k } }{2} } ,\ \ k < m ^ {2} . \\ \end{array} \right .$$

#### References

 [1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [2] M.A. Krasnosel'skii, V.Sh. Burd, Yu.S. Kolesov, "Nonlinear almost-periodic oscillations" , Wiley (1973) (Translated from Russian) [3] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Interscience (1962) (Translated from Russian) [4] N.N. Krasovskii, "Theory of control of motion. Linear systems" , Moscow (1968) (In Russian) [5] V.I. Zubov, "Lectures in control theory" , Moscow (1975) (In Russian) [6] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) [7] I.T. Kiguradze, "Some singular boundary value problems for ordinary differential equations" , Tbilisi (1975) (In Russian)