Mingarelli identity

Template:Short description 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.

The identity

Consider the Template:Mvar solutions of the following (uncoupled) system of second order linear differential equations over the Template:Mvar–interval Template:Math:

$(p_i(t)x_i^{\prime }{)}^{\prime }+q_i(t)x_i=0,x_i(a)=1,x_i^{\prime }(a)=R_i$ where $i=1,2,\dots ,n$.

Let $\mathrm{\Delta }$ denote the forward difference operator, i.e.

$\mathrm{\Delta }{x}_i=x_{i+1}-x_i.$

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

${\mathrm{\Delta }}^2(x_i)=\mathrm{\Delta }(\mathrm{\Delta }{x}_i)=x_{i+2}-2x_{i+1}+x_i,$,

with a similar definition for the higher iterates. Leaving out the independent variable Template:Mvar for convenience, and assuming the Template:Math on Template:Math, there holds the identity,

$\begin{array}{cc}{x}_{n-1}^2{\mathrm{\Delta }}^{n-1}(p_1{r}_1){]}_a^b=& \underset{a}{\overset{b}{\int }}(x_{n-1}^{\prime }{)}^2{\mathrm{\Delta }}^{n-1}(p_1)-\underset{a}{\overset{b}{\int }}{x}_{n-1}^2{\mathrm{\Delta }}^{n-1}(q_1)\\ & -\sum _{k=0}^{n-1}C(n-1,k)(-1{)}^{n-k-1}\underset{a}{\overset{b}{\int }}{p}_{k+1}{W}^2(x_{k+1},x_{n-1})/x_{k+1}^2,\end{array}$

where

• $r_i=x_i^{\prime }/x_i$ is the logarithmic derivative,
• $W(x_i,x_j)=x_i^{\prime }{x}_j-x_i{x}_j^{\prime }$, is the Wronskian determinant,
• $C(n-1,k)$ are binomial coefficients.

When Template:Math 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 Template:Mvar, Template:Mvar Template:Math, be real-valued continuous functions on the interval Template:Math and let

1. $(p_1(t)x_1^{\prime }{)}^{\prime }+q_1(t)x_1=0,x_1(a)=1,x_1^{\prime }(a)=R_1$
2. $(p_2(t)x_2^{\prime }{)}^{\prime }+q_2(t)x_2=0,x_2(a)=1,x_2^{\prime }(a)=R_2$
3. $(p_3(t)x_3^{\prime }{)}^{\prime }+q_3(t)x_3=0,x_3(a)=1,x_3^{\prime }(a)=R_3$

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

• Template:Math for each Template:Mvar and for all Template:Mvar in Template:Math , and
• the Template:Mvar are arbitrary real numbers.

Assume that for all Template:Math in Template:Math we have,

${\mathrm{\Delta }}^2(q_1)\ge 0$,

${\mathrm{\Delta }}^2(p_1)\le 0$,

${\mathrm{\Delta }}^2(p_1(a)R_1)\le 0$.

Then, if Template:Math on Template:Math and Template:Math, then any solution Template:Math has at least one zero in Template:Math.

References

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