Boundary value problem, ordinary differential equations
The problem of finding a solution to an equation
![]() | (1) |
lying in a given subset of the space
of functions depending on
that are absolutely continuous on
and that assume values in
:
![]() | (2) |
It is assumed that is a function defined on
with values in
and satisfying the Carathéodory conditions;
is an interval on the real line
.
1) The boundary value problem (1), (2) is said to be linear if
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where the functions and
are summable on every compact interval in
and the set
is a linear manifold in
. In particular, one might have
![]() |
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where is a function of bounded variation. A linear boundary value problem gives rise to a linear operator
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the eigen values of which are precisely those values of the parameter for which the homogeneous boundary value problem
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has non-trivial solutions. These non-trivial solutions are the eigen functions of the operator . If the inverse operator
exists and has an integral representation
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then is called a Green function.
2) Let , let
be almost-periodic in
uniformly in
on every compact subset of
and let
be the set of almost-periodic functions in
that are absolutely continuous on
. 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
![]() | (3) |
with set of admissible controls and two sets
. Let
be the set of absolutely continuous functions in
such that
,
. The boundary value problem is to find a pair
such that
and the solution
of equation (3) at
satisfies the condition
.
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
![]() | (4) |
in which
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for certain constants . Suppose that the homogeneous problem
![]() | (5) |
is regular, i.e. its only solution is the trivial one. Then problem (4) has at least one solution, provided either , or
and
is sufficiently small. It is fairly complicated to determine whether problem (5) is regular. However, the linear (scalar) boundary value problem
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for example, is regular if whenever there exists a
such that
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where
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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) |
Comments
References
[a1] | M. Braun, "Differential equations and their applications" , Springer (1975) |
[a2] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5 |
[a3] | 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 |
Boundary value problem, ordinary differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_value_problem,_ordinary_differential_equations&oldid=17771