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to an ordinary linear differential equation $l (y) = 0$

The ordinary linear differential equation $l ^ {*} ( \xi ) = 0$, where

$$\tag{1 } l (y) \equiv a _ {0} (t) y ^ {(n)} + \dots + a _ {n} (t) y ,$$

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

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

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

$$\tag{2 } l ^ {*} ( \xi ) \equiv \ ( - 1 ) ^ {n} ( \overline{a} _ {0} \xi ) ^ {(n)} + ( - 1 ) ^ {n-1} ( \overline{a} _ {1} \xi ) ^ {(n-1)} + \dots + \overline{a} _ {n} \xi ,$$

$$\xi \in C ^ {n} (I)$$

(the bar denotes complex conjugation). It follows at once that

$$( l _ {1} + l _ {2} ) ^ {*} = \ l _ {1} ^ {*} + l _ {2} ^ {*} ,\ \ ( \lambda l ) ^ {*} = \ \overline \lambda l ^ {*} ,$$

for any scalar $\lambda$. The adjoint of the equation $l ^ {*} ( \xi ) = 0$ is $l (y) = 0$. For all $n$-times continuously-differentiable functions $y (t)$ and $\xi (t)$, Lagrange's identity holds:

$$\overline{ {\xi l }} (y) - \overline{ {l ^ {*} ( \xi ) }} y = \frac{d}{dt} \left \{ \sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi ) ^ {(j)} y ^ {( k - j - 1 ) } \right \} .$$

It implies Green's formula

$$\int\limits _ { s } ^ \tau [ \overline \xi l (y) - \overline{ {l ^ {*} ( \xi ) }} y ] d t =$$

$$= \ \left . \sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi ) ^ {(j)} y ^ {( k - j - 1 ) } \right | _ {t=s} ^ {t = \tau } .$$

If $y (t)$ and $\xi (t)$ are arbitrary solutions of $l (y) = 0$ and $l ^ {*} ( \xi ) = 0$, respectively, then

$$\sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi ) ^ {(j)} y ^ {(k-j-1)} \equiv \textrm{ const } ,\ \ t \in I .$$

A knowledge of $m ( \leq n )$ linearly independent solutions of the equation $l ^ {*} ( \xi ) = 0$ enables one to reduce the order of the equation $l (y) = 0$ by $m$ (see ).

For a system of differential equations

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

where $A (t)$ is a continuous complex-valued $( n \times n )$-matrix, the adjoint system is given by

$$L ^ {*} ( \psi ) \equiv - \dot \psi + A ^ {*} (t) \psi = \ 0 ,\ t \in I$$

(see , ), where $A ^ {*} (t)$ is the Hermitian adjoint of $A (t)$. The Lagrange identity and the Green formula take the form

$$( \overline \psi , L (x) ) - ( \overline{ {L ^ {*} ( \psi ) }} , x ) = \frac{d}{dt} ( \overline \psi , x ) ,\ \$$

$$\left . \int\limits _ { s } ^ \tau [ ( \overline \psi , L (x) ) - ( \overline{ {L ^ {*} ( \psi ) }} , x ) ] d t = \ ( \overline \psi , x ) \right | _ {t=s} ^ {t = \tau } ;$$

where $( \cdot , \cdot )$ is the standard scalar product (the sum of the products of coordinates with equal indices). If $x (t)$ and $\psi (t)$ are arbitrary solutions of the equations $L (x) = 0$ and $L ^ {*} ( \psi ) = 0$, then

$$( \overline \psi (t) , x (t) ) \equiv \textrm{ const } ,\ t \in I .$$

The concept of an adjoint differential equation is closely connected with the general concept of an adjoint operator. Thus, if $l$ is a linear differential operator acting on $C ^ {n} (I)$ into $C (I)$ in accordance with (1), then its adjoint differential operator $l ^ {*}$ maps the space $C ^ {*} (I)$ adjoint to $C (I)$ into the space $C ^ {n*} (I)$ adjoint to $C ^ {n} (I)$. The restriction of $l ^ {*}$ to $C ^ {n} (I)$ is given by formula (2) (see ).

Adjoints are also defined for linear partial differential equations (see , ).

Let $\Delta = [ t _ {0} , t _ {1} ] \subset I$, and let $U _ {k}$ be linearly independent linear functionals on $C ^ {n} ( \Delta )$. Then the boundary value problem adjoint to the linear boundary value problem

$$\tag{3 } l (y) = 0 ,\ \ t \in \Delta ,\ \ U _ {k} (y) = 0 ,\ \ k = 1, \dots, m ,\ \ m < 2 n ,$$

is defined by the equations

$$\tag{4 } l ^ {*} ( \xi ) = 0 ,\ \ U _ {j} ^ {*} ( \xi ) = 0 ,\ \ j = 1, \dots, 2 n - m .$$

Here the $U _ {j} ^ {*}$ are linear functionals on $C ^ {n} ( \Delta )$ describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see Green formulas)

$$\int\limits _ {t _ {0} } ^ { {t } _ {1} } [ \overline \xi l (y) - \overline{ {l ^ {*} ( \xi ) }} y ] d t = 0$$

holds for any pair of functions $y , \xi \in C ^ {n} ( \Delta )$ that satisfy the conditions $U _ {k} (y) = 0$, $k = 1, \dots, m$; $U _ {j} ^ {*} ( \xi ) = 0$, $j = 1, \dots, 2 n - m$.

If

$$U _ {k} (y) \equiv \ \sum _ { p=1 } ^ { n } [ \alpha _ {kp} y ^ {(p-1)} ( t _ {0} ) + \beta _ {kp} y ^ {(p-1)} ( t _ {1} ) ]$$

are linear forms in the variables

$$y ^ {(p-1)} ( t _ {0} ) ,\ \ y ^ {(p-1)} ( t _ {1} ) ,\ \ p = 1, \dots, n ,$$

then $U _ {j} ^ {*} ( \xi )$ are linear forms in the variables

$$\xi ^ {(p-1)} ( t _ {0} ) ,\ \ \xi ^ {(p-1)} ( t _ {1} ) ,\ \ p = 1, \dots, n .$$

Examples. For the problem

$$\ddot{y} + a (t) y = 0 ,\ \ 0 \leq t \leq 1 ,$$

$$y (0) + \alpha y (1) + \beta \dot{y} (1) = 0 ,$$

$$\dot{y} (0) + \gamma y (1) + \delta \dot{y} (1) = 0 ,$$

with real $a (t) , \alpha , \beta , \gamma , \delta$, the adjoint boundary value problem has the form

$$\ddot \xi + a (t) \xi = 0 ,\ \ 0 \leq t \leq 1 ,$$

$$\alpha \xi (0) + \gamma \dot \xi (0) + \xi (1) = 0 ,$$

$$\beta \xi (0) + \delta \dot \xi (0) + \dot \xi (1) = 0 .$$

If problem (3) has $k$ linearly independent solutions (in this case the rank $r$ of the boundary value problem is equal to $n-k$), then problem (4) has $m - n + k$ linearly independent solutions (its rank is $r ^ \prime = 2n - m - k$). When $m = n$, problems (3) and (4) have an equal number of linearly independent solutions. Therefore, when $m = n$, problem (3) has only a trivial solution if and only if the adjoint boundary value problem (4) has the same property. The Fredholm alternative holds: The semi-homogeneous boundary value problem

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

has a solution if $f ( t )$ is orthogonal to all non-trivial solutions $\xi ( t )$ of the adjoint boundary value problem (4), i.e. if

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

(see , ).

For the eigenvalue problem

$$\tag{5 } l ( y ) = \lambda y ,\ \ U _ {k} (y) = 0 ,\ \ k = 1, \dots, n ,$$

the adjoint eigenvalue problem is defined as

$$\tag{6 } l ^ {*} ( \xi ) = \mu \xi ,\ \ U _ {j} ^ {*} ( \xi ) = 0 ,\ \ j = 1, \dots, n .$$

If $\lambda$ is an eigenvalue of (5), then $\mu = \overline \lambda$ is an eigenvalue of (6). The eigenfunctions $y (t) , \xi ( t )$ corresponding to eigenvalues $\lambda , \mu$ of (5), (6), respectively, are orthogonal if $\lambda \neq \mu$ (see ):

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

For the linear boundary value problem

$$\tag{7 } L (x) \equiv \dot{x} + A (t) x = 0 ,\ \ U (x) = 0 ,\ t \in \Delta ,$$

where $U$ is an $m$- dimensional vector functional on the space $C _ {n} ( \Delta )$ of continuously-differentiable complex-valued $n$-dimensional vector functions with $m < 2n$, the adjoint boundary value problem is defined by

$$\tag{8 } L ^ {*} ( \psi ) = 0 ,\ \ U ^ {*} ( \psi ) = 0 ,\ \ t \in \Delta$$

(see ). Here $U ^ {*}$ is a $( 2n - m )$-dimensional vector functional defined such that the equation

$$\left . ( \psi ( t ) , x ( t ) ) \right | _ {t = t _ {0} } ^ {t = t _ {1} } = 0$$

holds for any pair of functions $x , \psi \in C _ {n} ^ {1} ( \Delta )$ satisfying the conditions

$$U (x) = 0 ,\ U ^ {*} ( \psi ) = 0.$$

The problems (7), (8) possess properties analogous to those listed above (see ).

The concept of an adjoint boundary value problem is closely connected with that of an adjoint operator . Adjoint boundary value problems are also defined for linear boundary value problems for partial differential equations (see , ).

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