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A linear ordinary differential equation $l ( y) = 0$ that coincides with the adjoint differential equation $l ^ {*} ( y) = 0$. Here

$$l ( y) \equiv a _ {0} ( t) y ^ {(} n) + \dots + a _ {n} ( t) y,$$

$$l ^ {*} ( y) \equiv (- 1) ^ {n} ( \overline{a}\; _ {0} ( t) y) ^ {(} n) + \dots + (- 1) ^ {0} \overline{a}\; _ {n} ( t) y,$$

where

$$y ^ {( \nu ) } = \ \frac{d ^ \nu y }{dt ^ \nu } ,\ \ y ( \cdot ) \in C ^ {n} ( I),\ \ a _ {k} ( \cdot ) \in C ^ {n - k } ( I),$$

$$a _ {0} ( t) \neq 0,\ t \in I,$$

$C ^ {m} ( I)$ is the space of $m$- times continuously-differentiable complex-valued functions on $I = ( \alpha , \beta )$, and the bar denotes complex conjugation.

The left-hand side of every self-adjoint differential equation $l ( y) = 0$ is a sum of expressions of the form

$$l _ {2m} ( y) = ( p _ {m} y ^ {(} m) ) ^ {(} m) ,$$

$$l _ {2m - 1 } ( y) = { \frac{1}{2} } [( iq _ {m} y ^ {( m - 1) } ) ^ {(} m) + ( iq _ {m} y ^ {(} m) ) ^ {( m - 1) } ],$$

where $p _ {m} ( t)$ and $q _ {m} ( t)$ are sufficiently-smooth real-valued functions and $i ^ {2} = - 1$. A self-adjoint differential equation with real coefficients is necessarily of even order, and has the form

$$( p _ {0} y ^ {(} m) ) ^ {(} m) + ( p _ {1} y ^ {( m - 1) } ) ^ {( m - 1) } + \dots + p _ {m} y = 0$$

(see [1][3]).

A linear system of differential equations

$$L ( x) = 0,\ \ L ( x) \equiv \dot{x} + A ( t) x,\ \ t \in I,$$

with a continuous complex-valued $( n \times n)$- matrix $A ( t)$, is called self-adjoint if $A ( t) = - A ^ {*} ( t)$, where $A ^ {*} ( t)$ is the Hermitian conjugate of $A ( t)$( see [1], [4], and Hermitian operator). This definition is not consistent with the definition of a self-adjoint differential equation. For example, the system

$$\dot{x} _ {1} - x _ {2} = 0,\ \ \dot{x} _ {2} + p ( t) x _ {1} = 0,$$

which is equivalent to the self-adjoint differential equation

$$\dot{y} dot + p ( t) y = 0,$$

is self-adjoint as a linear system if and only if $p ( t) \equiv 1$.

The boundary value problem

$$\tag{1 } l ( y) = 0,\ \ t \in \Delta = \ [ t _ {0} , t _ {1} ],$$

$$\tag{2 } U _ {k} ( y) = 0 ,\ k = 1 \dots n,$$

where the $U _ {k} : C ^ {(} n) ( \Delta ) \rightarrow \mathbf R ^ {1}$ are linear and linearly independent functionals describing the boundary conditions, is called self-adjoint if it coincides with the adjoint boundary value problem, that is, (1) is a self-adjoint differential equation and $U _ {k} ( y) = U _ {k} ^ {*} ( y)$ for all $y ( \cdot ) \in C ^ {n} ( \Delta )$ and for all $k = 1 \dots n$( see [1][3], [5]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see Green formulas)

$$\int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \xi \; l ( y) dt = \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline{l}\; ( \xi ) y dt$$

holds for any pair $y ( \cdot ), \xi ( \cdot ) \in C ^ {(} n) ( \Delta )$ that satisfy the boundary conditions (2).

All the eigenvalues of the self-adjoint problem

$$l ( y) = \lambda y,\ \ U _ {k} ( y) = 0,\ \ k = 1 \dots n,$$

are real, and the eigenfunctions $\phi _ {1} , \phi _ {2}$ corresponding to distinct eigenvalues $\lambda _ {1} , \lambda _ {2}$ are orthogonal:

$$\int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \phi \; _ {1} \phi _ {2} ( t) dt = 0.$$

The linear boundary value problem

$$\tag{3 } L ( x) \equiv \dot{x} + A ( t) x = 0,\ \ U ( x) = 0,\ \ t \in \Delta ,$$

where $A ( t)$ is a continuous complex-valued $( n \times n)$- matrix and $U$ is an $n$- vector functional on the space $C _ {n} ^ {1} ( \Delta )$ of continuous complex-valued functions $x: \Delta \rightarrow \mathbf R ^ {n}$, is called self-adjoint if it coincides with its adjoint boundary value problem

$$L ^ {*} ( x) = 0,\ \ U ^ {*} ( x) = 0,\ \ t \in \Delta ,$$

that is,

$$L ( x) = - L ^ {*} ( x),\ \ U ( x) = U ^ {*} ( x)$$

for all $x ( \cdot ) \in C _ {n} ^ {1} ( \Delta )$. A self-adjoint boundary value problem has properties analogous to those of the problem (1), (2) (see [4]).

The concepts of a self-adjoint differential equation and of a self-adjoint boundary value problem are closely connected with that of a self-adjoint operator [6] (cf. also Spectral theory of differential operators). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [5], [7]).

References

 [1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) [2] M.A. Naimark, "Linear differential operators" , 1–2 , Harrap (1968) (Translated from Russian) [3] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 [4] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) [5] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [6] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) [7] V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)