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Mingarelli identity

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In the field of ordinary differential equations, the Mingarelli identity is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order.

Contents

The identity

Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [a, b]:

( p i ( t ) x i ′ ) ′ + q i ( t ) x i = 0 , x i ( a ) = 1 , x i ′ ( a ) = R i where i = 1 , 2 , … , n .

Let Δ denote the forward difference operator, i.e.

Δ x i = x i + 1 − x i .

The second order difference operator is found by iterating the first order operator as in

Δ 2 ( x i ) = Δ ( Δ x i ) = x i + 2 − 2 x i + 1 + x i , ,

with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the xi(t) ≠ 0 on (a, b], there holds the identity,

x n − 1 2 Δ n − 1 ( p 1 r 1 ) ] a b = ∫ a b ( x n − 1 ′ ) 2 Δ n − 1 ( p 1 ) − ∫ a b x n − 1 2 Δ n − 1 ( q 1 ) − ∑ k = 0 n − 1 C ( n − 1 , k ) ( − 1 ) n − k − 1 ∫ a b p k + 1 W 2 ( x k + 1 , x n − 1 ) / x k + 1 2 ,

where

  • r i = x i ′ / x i is the logarithmic derivative,
  • W ( x i , x j ) = x i ′ x j − x i x j ′ , is the Wronskian determinant,
  • C ( n − 1 , k ) are binomial coefficients.

    • When n = 2 this equality reduces to the Picone identity.

    An application

    The above identity leads quickly to the following comparison theorem for three linear differential equations, which extends the classical Sturm–Picone comparison theorem.

    Let pi, qi i = 1, 2, 3, be real-valued continuous functions on the interval [a, b] and let

    1. ( p 1 ( t ) x 1 ′ ) ′ + q 1 ( t ) x 1 = 0 , x 1 ( a ) = 1 , x 1 ′ ( a ) = R 1
    2. ( p 2 ( t ) x 2 ′ ) ′ + q 2 ( t ) x 2 = 0 , x 2 ( a ) = 1 , x 2 ′ ( a ) = R 2
    3. ( p 3 ( t ) x 3 ′ ) ′ + q 3 ( t ) x 3 = 0 , x 3 ( a ) = 1 , x 3 ′ ( a ) = R 3

    be three homogeneous linear second order differential equations in self-adjoint form, where

  • pi(t) > 0 for each i and for all t in [a, b] , and
  • the Ri are arbitrary real numbers.

    • Assume that for all t in [a, b] we have,

    Δ 2 ( q 1 ) ≥ 0 , Δ 2 ( p 1 ) ≤ 0 , Δ 2 ( p 1 ( a ) R 1 ) ≤ 0 .

    Then, if x1(t) > 0 on [a, b] and x2(b) = 0, then any solution x3(t) has at least one zero in [a, b].

    References

    Mingarelli identity Wikipedia
    (Text) CC BY-SA