Difference between revisions of "Disconjugacy"
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{{MSC|34L}} | {{MSC|34L}} | ||
− | An | + | An $ n $ |
+ | th order homogeneous [[Linear differential operator|linear differential operator]] (equation) | ||
− | + | $$ \tag{a1 } | |
+ | Ly \equiv y ^ {( n ) } + p _ {1} ( x ) y ^ {( n - 1 ) } + \dots + p _ {n} ( x ) y = 0 | ||
+ | $$ | ||
− | is called disconjugate on an interval | + | is called disconjugate on an interval $ I $ |
+ | if no non-trivial solution has $ n $ | ||
+ | zeros on $ I $, | ||
+ | multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on $ I $; | ||
+ | cf. also [[Oscillating solution|Oscillating solution]]; [[Oscillating differential equation|Oscillating differential equation]].) If (a1) has a solution with $ n $ | ||
+ | zeros on an interval, then the infimum of all values $ c $, | ||
+ | $ c > a $, | ||
+ | such that some solution has $ n $ | ||
+ | zeros on $ [ a,c ] $ | ||
+ | is called the conjugate point of $ a $ | ||
+ | and is denoted by $ \eta ( a ) $. | ||
+ | This infimum is achieved by a solution which has a total of at least $ n $ | ||
+ | zeros at $ a $ | ||
+ | and $ \eta ( a ) $ | ||
+ | and is positive on $ ( a, \eta ( a ) ) $. | ||
+ | If the equation has continuous coefficients, the conjugate point $ \eta ( a ) $ | ||
+ | is a strictly increasing, continuous function of $ a $. | ||
+ | The adjoint equation has the same conjugate point as (a1). For general properties, see [[#References|[a1]]], [[#References|[a7]]]. | ||
There are numerous explicit sufficient criteria for the equation (a1) to be disconjugate. Many of them are of the form | There are numerous explicit sufficient criteria for the equation (a1) to be disconjugate. Many of them are of the form | ||
− | + | $$ | |
+ | \sum ^ {n _ {k} = 1 } c _ {k} ( b - a ) ^ {k} \left \| {p _ {k} } \right \| < 1, | ||
+ | $$ | ||
− | where | + | where $ \| {p _ {k} } \| $ |
+ | is some [[Norm|norm]] of $ p _ {k} $, | ||
+ | $ I = [ a,b ] $ | ||
+ | and $ c _ {k} $ | ||
+ | are suitable constants. These are "smallness conditions" which express the proximity of (a1) to the disconjugate equation $ y ^ {( n ) } = 0 $. | ||
+ | See [[#References|[a12]]]. | ||
− | + | $ L $ | |
+ | is disconjugate on $ [ a,b ] $ | ||
+ | if and only if it has there the Pólya factorization | ||
− | + | $$ | |
+ | Ly \equiv \rho _ {n} { | ||
+ | \frac{d}{dx } | ||
+ | } \left ( \rho _ {n - 1 } \dots { | ||
+ | \frac{d}{dx } | ||
+ | } \left ( \rho _ {1} { | ||
+ | \frac{d}{dx } | ||
+ | } ( \rho _ {0} y ) \right ) \dots \right ) , \rho _ {i} > 0, | ||
+ | $$ | ||
or the equivalent Mammana factorization | or the equivalent Mammana factorization | ||
− | + | $$ | |
+ | Ly = \left ( { | ||
+ | \frac{d}{dx } | ||
+ | } + r _ {n} \right ) \dots \left ( { | ||
+ | \frac{d}{dx } | ||
+ | } + r _ {1} \right ) y. | ||
+ | $$ | ||
− | Among the various Pólya factorizations, the most important is the Trench canonical form [[#References|[a11]]]: If | + | Among the various Pólya factorizations, the most important is the Trench canonical form [[#References|[a11]]]: If $ L $ |
+ | is disconjugate on $ ( a,b ) $, | ||
+ | $ b \leq \infty $, | ||
+ | then there is essentially one factorization such that $ \int ^ {b} {\rho _ {i} ^ {- 1 } } = \infty $, | ||
+ | $ i = 1 \dots n - 1 $. | ||
− | Disconjugacy is closely related to solvability of the [[De la Vallée-Poussin multiple-point problem|de la Vallée-Poussin multiple-point problem]] | + | Disconjugacy is closely related to solvability of the [[De la Vallée-Poussin multiple-point problem|de la Vallée-Poussin multiple-point problem]] $ Ly = g $, |
+ | $ y ^ {( i ) } ( x _ {j} ) = a _ {ij } $, | ||
+ | $ i = 0 \dots r _ {j} - 1 $, | ||
+ | $ \sum _ {1} ^ {m} r _ {j} = n $. | ||
+ | The Green's function of a disconjugate operator $ L $ | ||
+ | and the related homogeneous boundary value problem satisfies | ||
− | + | $$ | |
+ | { | ||
+ | \frac{G ( x,t ) }{( x - x _ {1} ) ^ {r _ {1} } \dots ( x - x _ {m} ) ^ {r _ {m} } } | ||
+ | } > 0 | ||
+ | $$ | ||
− | for < | + | for $ x _ {1} \leq x \leq x _ {m} $, |
+ | $ x _ {1} < t < x _ {m} $[[#References|[a7]]]. Another interesting boundary value problem is the focal boundary value problem $ y ^ {( i ) } ( x _ {j} ) = 0 $, | ||
+ | $ i = r _ {j - 1 } \dots r _ {j} - 1 $, | ||
+ | $ j = 1 \dots m $, | ||
+ | $ 0 = r _ {0} < r _ {1} < \dots < r _ {m} = n - 1 $. | ||
− | For a second-order equation, the Sturm separation theorem (cf. [[Sturm theorem|Sturm theorem]]) yields that non-oscillation (i.e., no solution has a sequence of zeros converging to | + | For a second-order equation, the Sturm separation theorem (cf. [[Sturm theorem|Sturm theorem]]) yields that non-oscillation (i.e., no solution has a sequence of zeros converging to $ + \infty $) |
+ | implies that there exists a point $ a $ | ||
+ | such that (a1) is disconjugate on $ [ a, \infty ) $. | ||
+ | For equations of order $ n > 2 $ | ||
+ | this conclusion holds for a class of equations [[#References|[a2]]] but not for all equations [[#References|[a4]]]. | ||
Particular results about disconjugacy exist for various special types of differential equations. | Particular results about disconjugacy exist for various special types of differential equations. | ||
Line 35: | Line 111: | ||
1) The Sturm–Liouville operator (cf. [[Sturm–Liouville equation|Sturm–Liouville equation]]) | 1) The Sturm–Liouville operator (cf. [[Sturm–Liouville equation|Sturm–Liouville equation]]) | ||
− | + | $$ \tag{a2 } | |
+ | ( p y ^ \prime ) ^ \prime + q y = 0, p > 0, | ||
+ | $$ | ||
− | has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the [[Riccati equation|Riccati equation]] | + | has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the [[Riccati equation|Riccati equation]] $ z ^ \prime + q + { {z ^ {2} } / p } = 0 $. |
+ | It is also closely related to the positive definiteness of the quadratic functional $ \int _ {a} ^ {b} {( p y ^ {\prime 2 } - q y ^ {2} ) } $. | ||
+ | See [[#References|[a10]]], [[#References|[a1]]], [[#References|[a5]]]. For example, (a2) is disconjugate on $ [ a,b ] $ | ||
+ | if $ \int _ {a} ^ {b} {p ^ {- 1 } } \times \int _ {a} ^ {b} {| q | } < 4 $. | ||
2) Third-order equations are studied in [[#References|[a3]]]. | 2) Third-order equations are studied in [[#References|[a3]]]. | ||
− | 3) For a [[Self-adjoint differential equation|self-adjoint differential equation]] | + | 3) For a [[Self-adjoint differential equation|self-adjoint differential equation]] $ \sum _ {i = 0 } ^ {m} ( p _ {m - i } y ^ {( i ) } ) ^ {( i ) } = 0 $, |
+ | the existence of a solution with two zeros of multiplicity $ m $ | ||
+ | has been studied. Their absence is called $ ( m,m ) $- | ||
+ | disconjugacy. | ||
− | 4) Disconjugacy of the analytic equation | + | 4) Disconjugacy of the analytic equation $ w ^ \prime + p ( z ) w = 0 $ |
+ | in a complex domain is connected to the theory of univalent functions. See [[#References|[a8]]], [[#References|[a5]]] and [[Univalent function|Univalent function]]. | ||
− | 5) Many particularly elegant result are available for two-term equations | + | 5) Many particularly elegant result are available for two-term equations $ y ^ {( n ) } + p ( x ) y = 0 $ |
+ | and their generalizations $ Ly + p ( x ) y = 0 $[[#References|[a6]]], [[#References|[a2]]]. | ||
Disconjugacy has also been studied for certain second-order linear differential systems of higher dimension [[#References|[a1]]], [[#References|[a9]]]. In the historical prologue of [[#References|[a9]]], the connection to the calculus of variations (cf. also [[Variational calculus|Variational calculus]]) is explained. The concepts of disconjugacy and oscillation have also been generalized to non-linear differential equations and functional-differential equations. | Disconjugacy has also been studied for certain second-order linear differential systems of higher dimension [[#References|[a1]]], [[#References|[a9]]]. In the historical prologue of [[#References|[a9]]], the connection to the calculus of variations (cf. also [[Variational calculus|Variational calculus]]) is explained. The concepts of disconjugacy and oscillation have also been generalized to non-linear differential equations and functional-differential equations. |
Revision as of 19:35, 5 June 2020
2020 Mathematics Subject Classification: Primary: 34L [MSN][ZBL]
An $ n $ th order homogeneous linear differential operator (equation)
$$ \tag{a1 } Ly \equiv y ^ {( n ) } + p _ {1} ( x ) y ^ {( n - 1 ) } + \dots + p _ {n} ( x ) y = 0 $$
is called disconjugate on an interval $ I $ if no non-trivial solution has $ n $ zeros on $ I $, multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on $ I $; cf. also Oscillating solution; Oscillating differential equation.) If (a1) has a solution with $ n $ zeros on an interval, then the infimum of all values $ c $, $ c > a $, such that some solution has $ n $ zeros on $ [ a,c ] $ is called the conjugate point of $ a $ and is denoted by $ \eta ( a ) $. This infimum is achieved by a solution which has a total of at least $ n $ zeros at $ a $ and $ \eta ( a ) $ and is positive on $ ( a, \eta ( a ) ) $. If the equation has continuous coefficients, the conjugate point $ \eta ( a ) $ is a strictly increasing, continuous function of $ a $. The adjoint equation has the same conjugate point as (a1). For general properties, see [a1], [a7].
There are numerous explicit sufficient criteria for the equation (a1) to be disconjugate. Many of them are of the form
$$ \sum ^ {n _ {k} = 1 } c _ {k} ( b - a ) ^ {k} \left \| {p _ {k} } \right \| < 1, $$
where $ \| {p _ {k} } \| $ is some norm of $ p _ {k} $, $ I = [ a,b ] $ and $ c _ {k} $ are suitable constants. These are "smallness conditions" which express the proximity of (a1) to the disconjugate equation $ y ^ {( n ) } = 0 $. See [a12].
$ L $ is disconjugate on $ [ a,b ] $ if and only if it has there the Pólya factorization
$$ Ly \equiv \rho _ {n} { \frac{d}{dx } } \left ( \rho _ {n - 1 } \dots { \frac{d}{dx } } \left ( \rho _ {1} { \frac{d}{dx } } ( \rho _ {0} y ) \right ) \dots \right ) , \rho _ {i} > 0, $$
or the equivalent Mammana factorization
$$ Ly = \left ( { \frac{d}{dx } } + r _ {n} \right ) \dots \left ( { \frac{d}{dx } } + r _ {1} \right ) y. $$
Among the various Pólya factorizations, the most important is the Trench canonical form [a11]: If $ L $ is disconjugate on $ ( a,b ) $, $ b \leq \infty $, then there is essentially one factorization such that $ \int ^ {b} {\rho _ {i} ^ {- 1 } } = \infty $, $ i = 1 \dots n - 1 $.
Disconjugacy is closely related to solvability of the de la Vallée-Poussin multiple-point problem $ Ly = g $, $ y ^ {( i ) } ( x _ {j} ) = a _ {ij } $, $ i = 0 \dots r _ {j} - 1 $, $ \sum _ {1} ^ {m} r _ {j} = n $. The Green's function of a disconjugate operator $ L $ and the related homogeneous boundary value problem satisfies
$$ { \frac{G ( x,t ) }{( x - x _ {1} ) ^ {r _ {1} } \dots ( x - x _ {m} ) ^ {r _ {m} } } } > 0 $$
for $ x _ {1} \leq x \leq x _ {m} $, $ x _ {1} < t < x _ {m} $[a7]. Another interesting boundary value problem is the focal boundary value problem $ y ^ {( i ) } ( x _ {j} ) = 0 $, $ i = r _ {j - 1 } \dots r _ {j} - 1 $, $ j = 1 \dots m $, $ 0 = r _ {0} < r _ {1} < \dots < r _ {m} = n - 1 $.
For a second-order equation, the Sturm separation theorem (cf. Sturm theorem) yields that non-oscillation (i.e., no solution has a sequence of zeros converging to $ + \infty $) implies that there exists a point $ a $ such that (a1) is disconjugate on $ [ a, \infty ) $. For equations of order $ n > 2 $ this conclusion holds for a class of equations [a2] but not for all equations [a4].
Particular results about disconjugacy exist for various special types of differential equations.
1) The Sturm–Liouville operator (cf. Sturm–Liouville equation)
$$ \tag{a2 } ( p y ^ \prime ) ^ \prime + q y = 0, p > 0, $$
has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the Riccati equation $ z ^ \prime + q + { {z ^ {2} } / p } = 0 $. It is also closely related to the positive definiteness of the quadratic functional $ \int _ {a} ^ {b} {( p y ^ {\prime 2 } - q y ^ {2} ) } $. See [a10], [a1], [a5]. For example, (a2) is disconjugate on $ [ a,b ] $ if $ \int _ {a} ^ {b} {p ^ {- 1 } } \times \int _ {a} ^ {b} {| q | } < 4 $.
2) Third-order equations are studied in [a3].
3) For a self-adjoint differential equation $ \sum _ {i = 0 } ^ {m} ( p _ {m - i } y ^ {( i ) } ) ^ {( i ) } = 0 $, the existence of a solution with two zeros of multiplicity $ m $ has been studied. Their absence is called $ ( m,m ) $- disconjugacy.
4) Disconjugacy of the analytic equation $ w ^ \prime + p ( z ) w = 0 $ in a complex domain is connected to the theory of univalent functions. See [a8], [a5] and Univalent function.
5) Many particularly elegant result are available for two-term equations $ y ^ {( n ) } + p ( x ) y = 0 $ and their generalizations $ Ly + p ( x ) y = 0 $[a6], [a2].
Disconjugacy has also been studied for certain second-order linear differential systems of higher dimension [a1], [a9]. In the historical prologue of [a9], the connection to the calculus of variations (cf. also Variational calculus) is explained. The concepts of disconjugacy and oscillation have also been generalized to non-linear differential equations and functional-differential equations.
References
[a1] | W.A. Coppel, "Disconjugacy" , Lecture Notes in Mathematics , 220 , Springer (1971) |
[a2] | U. Elias, "Oscillation theory of two-term differential equations" , Kluwer Acad. Publ. (1997) |
[a3] | M. Gregus, "Third order linear differential equations" , Reidel (1987) |
[a4] | Gustafson, G. B., "The nonequivalence of oscillation and nondisconjugacy" Proc. Amer. Math. Soc. , 25 (1970) pp. 254–260 |
[a5] | E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1968) |
[a6] | I.T. Kiguradze, T.A. Chanturia, "Asymptotic properties of solutions of nonautonomous ordinary differential equations" , Kluwer Acad. Publ. (1993) (In Russian) |
[a7] | A.Yu. Levin, "Non-oscillation of solutions of the equation " Russian Math. Surveys , 24 (1969) pp. 43–99 (In Russian) |
[a8] | Z. Nehari, "The Schwarzian derivative and schlicht functions" Bull. Amer. Math. Soc. , 55 (1949) pp. 545–551 |
[a9] | W.T. Reid, "Sturmian theory for ordinary differential equations" , Springer (1980) |
[a10] | C.A. Swanson, "Comparison and oscillatory theory of linear differential equations" , Acad. Press (1968) |
[a11] | W.F. Trench, "Canonical forms and principal systems for general disconjugate equation" Trans. Amer. Math. Soc. , 189 (1974) pp. 319–327 |
[a12] | D. Willet, "Generalized de la Vallée Poussin disconjugacy tests for linear differential equations" Canadian Math. Bull. , 14 (1971) pp. 419–428 |
Disconjugacy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disconjugacy&oldid=34876