** МАТЕМАТИКА. МЕТОДИКА ПРЕПОДАВАНИЯ МАТЕМАТИКИ**

**B. K. Kaldybekova ***

**SOME ANALOG OF THE STURM SEPARATION THEOREM FOR **
**EQUATION ON GRAPH**

*( Almaty, Kazakh-British Technical University, PhD student) *

**Abstract. ** In this paper we prove a universal version of Sturm’s separation theorem for ordinary *differential equations of second order on geometric graph. Universal means valid for all natural *
*transmission conditions. Our result generalizes some previous results. The admissibility property of the *
*transmission functionals will be used to prove of the Sturm’s separation theorem. As an application of *
*Sturm’s theorem we give a geometric approach to estimation of multiplicities of eigenvalues in the Sturm *
*Liouville problem on graph. *

**Keywords**: universal, admissible, S-zone.

1. Preliminaries

A geometrical graph *G* is defined to be a connected subset of *R*^{n}, consisting of a finite
number of points *v*_{1},*v*_{2},*v*_{3},*v*_{4},...,*v*_{m}., called vertices of *G*, and of some intervals *e*_{k} (called
edges of *G*):

, ( , ): ,

###

0, (1)

*t* *l*

*v*
*v*

*v*
*t* *v*
*v*
*x*
*v*
*v*
*R*
*x*
*e*

*i*
*j*

*i*
*j*
*i*
*j*

*i*
*n*
*k*

where *l* stands for Euclidean distance between *v*_{i} and *v*_{j}.

The graph *G* is assumed to be divided into two subsets: *G*_{0} and *G*_{0}, where *G*_{0} is some
connected subset of *G*, consisting of some vertices of *G* and all its edges (see [1]). Assuming
the set of all vertices is denoted by *V*, let us denote by *V*_{0} the subset of *V*, consisting of those
vertices, which lie in *G*_{0}. These vertices are referred to as internal vertices of *G*_{0}. The
remaining part of *V*, i.e. *V* \*V*_{0}is denoted by *G*_{0}, its elements are referred to as boundary
vertices.

We need some sets of functions on *G*. The following gives a full list of these sets.

1. *C*_{e}(*G*) consists of all functions *u*:*G**R*^{n} which are uniformly continuous on each
edge. In the sequel *u*_{i}(*v*)(*v**V*) stands for the limits

)
(
lim*u*_{i} *x*

*e*
*x*

*v*
*x*

*i*

where *u*_{i} denotes the restriction of *u* on the edge *e*_{i}. This limits exist due to uniform continuity
of *u**C*_{e}(*G*) on each edge.

2. *C*(*G*) is the set of all continuous functions on *G*. Obviously *C*(*G*)*C*_{e}(*G*). The
reverse is true if *u*_{i}(*v*) *u*(*v*) for all *i**I*(*v*), where *I*(*v*)

###

*i*:

*e*

_{i}

*v*

###

. The notation*e*

_{i}

*v*means that the edge

*e*

_{i}adjoins the vertex

*v*.

3. *C*_{e}^{1}(*G*) is a set of all functions *u*:*G**R*, such that their restrictions have uniformly
continuous derivatives *u*_{k} on *e*_{k}. Here *u*_{k}(*x*) is defined to be a derivative

47

*i*
*j*

*i*
*j*
*i*
*x*
*t*

*t* *v* *v*

*v*
*t* *v*
*v*
*d* *u*

*d*

) (

where *t* is a natural parameter, introduced by (1). We assume that the parameterization (1) is
fixed for all *e*_{k}. In what follows *u*_{k}(*v*) (*v**V* ) stands for the limit

###

1 lim*u*

_{i}(

*x*),

*e*
*x*

*v*
*x*

*i*

where 0, if *e*_{k} parameterized in the direction from *v* to the interior of *e*_{k}. Otherwise, 1
. It means that *u*_{k}(*v*) is a directional derivative of *u*_{k} with respect to internal direction (from *v*
to the interior of *e*_{k}).

4. *C*_{e}^{2}(*G*) is a set of all functions which have twice differentiable restrictions *u*_{k} .
For each internal vertex *v**V*_{0} we fix a collection of linear functionals *l*_{1},*l*_{2}, ...,*l*_{n},.*l*_{}_{}_{1}
( is a multiplicity of *v*, i.e. the number of edges, attached to *v*):

###

( ) ( )###

, )( ) (

) (

###

*v**j*

*I*
*i*

*j*
*i*
*ij*
*j*
*i*
*ij*
*j*

*j*

*j* *u* *u* *v* *u* *v* *u* *v*

*l*

called transmission conditions. The set of all transmission functionals, associated with the
vertex *v* is denoted by _{v}. A collection of all _{v} will be denoted by .

Now we are ready to define an ordinary differential equation on *G*_{0}. Namely, we shall
consider a special class of linear differential equation, which may be presented as a couple of
relations:

*f*
*qu*
*u*

*p* )
(

(2) .

, 0 )

(*u* *l*

*l* (3)
Our assumptions about the coefficients and the right-hand side *f* are quite standard:

) ( ,

),

1(

*G*
*C*
*f*
*q*
*G*
*C*

*p* _{e} _{e} and *p* is strictly positive. Strict positivity of the function *f* on *G* means
here that *f*(*x*)0 on *G*, *f*(*x*)0 on *E*(*G*) (the union of all edges in *G*) and *u*_{i}(*v*)0 for all

*I**v*

*i* and for all vertices.

A function *u**C*_{e}^{2}(*G*) is said to satisfy (2) if each restriction *u*_{k} satisfies

*k*
*k*
*k*
*k*

*k**u* *q* *u* *f*

*p* )

( .

**2. ** **An analog of the Sturm separation theorem **

To formulate our main result we need some additional notations and agreements. We say

*v* is admissible set of transmission functionals at the vertex *v* if the following property is
fulfilled: if *u* satisfies (2), *l*(*u*)0 for each *l*_{v}, *u*0 in some neighborhood of *v* and at
least one of the numbers *u*_{i}(*v*),(*i**I*_{v}) vanishes, then *u* vanishes on the "star", consisting of *v*
and all edges satisfying *e*_{i} *v*.

To illustrate this notion let us define the set _{v} as follows:

1) ( ) ( ) ( )

) (

1

0 *u* *k* *u* *v* *u* *v*

*l*

*v*

*j*
*j*

*j*

###

•,

2) *l*_{i}(*u*)*u*_{i}(*v*)*u*(*v*), *i* 1,2,...(*v*)
with positive *k*_{j} and non-negative .

**МАТЕМАТИКА. МЕТОДИКА ПРЕПОДАВАНИЯ МАТЕМАТИКИ **

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It is easy to see that the corresponding _{v} is admissible.

Another example gives a following collection _{v}:
1) *l*_{0}(*u*)*u*(*v*)

2) ( ) ( ) ^{ }

###

( ) ( )###

, 1,2,... ( )1

*v*
*i*

*v*
*u*
*v*
*u*
*k*
*v*

*u*
*p*
*u*
*l*

*v*

*m*

*m*
*i*

*im*
*j*

*i*
*i*

*i* ^{}

###

with positive *k*_{kj}.

Checking admissibility here is a little bit more complicated problem, than in the previous example.

A set of all transmission conditions is said to be admissible if _{v} is admissible for all
internal vertices.

Next definition plays a central role in this paper.

**Definition 1.** *A connected open subset S**G*_{0}* is said to be an S-zone of the function *
*R*

*G*_{0} * if *

*a) u is positive on S**G*_{0}*and has positive limit values u*_{i}(*v*)* for all the vertices lying in *
*S, and for all i**I*_{v}*. *

*b) u*(*x*)0* on **S**S* \*S. *

Here the words "connected" and "open" are interpreted in terms of topology, induced on
*G*0 by the standard topology of *R*^{n}.

Now we are ready to announce our main result - an analog of the Sturm separation theorem (see [2], [3]).

**Theorem 1.** * Let u and w be the solutions to (2),(3) with admissible set ** of transmission *
*functionals. Then w changes the sign in each S-zone S of the function u provided u and w are *
*linearly independent on S. *

The change of sign of the function *u* at the point *X* is interpreted by usual way: if *X* is a
point of some edge, while in the case *X * is an internal vertex the change of sign means

0 ) ( )

(*X* *u* *X*

*u*_{i} _{j} for all least two indices *i* and *j*.

To prove this theorem we need the following auxiliary assertion:

**Lemma 1.** *Let u be a solution to (2),(3). If ** is admissible set of transmission functionals *
*then for each open subset * *S**G the following statement is true: if u is non-trivial and *
*nonnegative on S then u is strictly positive on S. *

That is an easy consequence of admissibility of . It should be noted that the strict
positivity of *u* on *S* does not imply its strict positivity on the closure *S*.

Now we are ready to prove our analog of the Sturm separation theorem.

Arguing by contradiction let us assume that *w* keeps its sign. To fix ideas let us assume
*w* is nonnegative in *S*. We can conclude, using the previous lemma, that *w* is strictly positive on
*S*. If there exists a boundary point of the set *S* where ( )lim ( )0

*i*

*i* *x* *w*

*u* (here *x* approaches

along some edge *e*_{i} lying constantly in *S*), then ( )lim ( )0

*x* *i*

*i* *w*

*u* . Otherwise *w*_{i} 0

according to uniqueness theorem, which contradicts a strict positivity of *w* on *S*. This property
of *w* guarantees that there exists some satisfying *u*(*x*)*w*(*x*) on *E*(*G*)*S*. Let _{0} be the
greatest lower bound of such . Then *z*(*x*)_{0}*w*(*x*)*u*(*x*)0on *S*. If *z*0 on *S* then *u* and
*w* are linear dependent, which contradicts our assumptions about *u* and *w*. Otherwise, *z* is strictly
positive according to above lemma. Since *z* has the properties similar to those of *v*, we can find

49

satisfying *z*(*x*)*u*(*x*) on *S*. The last inequality is equivalent to (_{0}*w*(*x*)*u*(*x*))*u*(*x*),
which may be transformed to

)
(
)
1 ^{0}*w*(*x* *u* *x*

But the last inequality contradicts to minimality of _{0} (_{0} is the greatest lower bound of
the set of all satisfying *w*(*x*)*u*(*x*)). This completes the proof.

**3. ** **Application of the Sturm - Liouville problem on graph **

An analog of the Sturm-Liouville theorem on graph is the following boundary value problem:

*u*
*qu*
*u*

*p* )

( , (4)

*l*
*u*
*l*( ) 0,

(5) , 0

0 )

(*v* *v*

*u* (6)
If these relations admit a nontrivial *u* satisfying them, then the corresponding is called
an eigenvalue of (4)-(5).

A geometric multiplicity of the eigenvalue is defined to be a dimension of the linear space of all eigenfunctions (with respect to usual operations of summation and multiplication on scalars, see [4], [5]).

The main assertion of this section is the following one:

**Theorem 2.** * Let the problem (4)-(5) admit a solution, which has no zeroes in the internal *
*vertices and in all circles of G, (except boundary vertices, containing in the circles (6)). Then *
*the geometric multiplicity *()* of corresponding eigenvalues ** does not exceed N*1*, where *
*N is the number of circles in G, provided ** is admissible set of transmission conditions. *

The result of this kind may be found in [4], [5]. It has been proved there for so-called
non-oscillating operators. To explain this property let us represent (4)-(6) in the form *Lu**u*
. *L* is said to be non-oscillating if each solution to the equation does not admit any *S-*zone in *G*.

To this moment the relationships between non-oscillation property and admissibility of the set

are not stated. Anyway, our proof is clearer with the geometric point of view. Besides, checking admissibility of is much easier problem, then checking non-oscillation.

Remind that the number of circles *N* in the graph *G* is a minimal number of cuts of their
edges, which is necessary to transform it into some tree (a graph without circles). Another way
to define this notion is to use famous Euler’s formula for connected graphs: *V**E**N*2,
where *V* and *E* are the number of vertices and the number of edges in the graph respectively.

To prove theorem 2 we need two auxiliary assertions:

**Lemma 2.** * Let the solutions u and v of the problem (4)-(6) have the same S-zones on G, *