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

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Braun,   "Differential equations and their applications" , Springer  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)  pp. §§3.6, 3.51, 4.7, A.5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.K. Jackson,   "Boundary value problems for ordinary differential equations"  J.K. Hale (ed.) , ''Studies in ordinary differential equations'' , Math. Assoc. Amer.  (1977)  pp. 93–127</TD></TR></table>