# 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 ). 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

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

How to Cite This Entry:
Boundary value problem, ordinary differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_value_problem,_ordinary_differential_equations&oldid=46134
This article was adapted from an original article by Yu.V. KomlenkoE.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article