• Ешқандай Нәтиже Табылған Жоқ


B. K. Kaldybekova *


( 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 Rn, consisting of a finite number of points v1,v2,v3,v4,...,vm., called vertices of G, and of some intervals ek (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 vi and vj.

The graph G is assumed to be divided into two subsets: G0 and G0, where G0 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 V0 the subset of V, consisting of those vertices, which lie in G0. These vertices are referred to as internal vertices of G0. The remaining part of V, i.e. V \V0is denoted by G0, 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. Ce(G) consists of all functions u:GRn which are uniformly continuous on each edge. In the sequel ui(v)(vV) stands for the limits

) ( limui x

e x

v x


where ui denotes the restriction of u on the edge ei. This limits exist due to uniform continuity of uCe(G) on each edge.

2. C(G) is the set of all continuous functions on G. Obviously C(G)Ce(G). The reverse is true if ui(v) u(v) for all iI(v), where I(v)


. The notation eiv means that the edge ei adjoins the vertex v.

3. Ce1(G) is a set of all functions u:GR, such that their restrictions have uniformly continuous derivatives uk on ek. Here uk(x) is defined to be a derivative




 

i j

i j i x t

t v v

v t v v d u


) (

where t is a natural parameter, introduced by (1). We assume that the parameterization (1) is fixed for all ek. In what follows uk(v) (vV ) stands for the limit

 

1 limui(x),

e x

v x


 

where 0, if ek parameterized in the direction from v to the interior of ek. Otherwise,  1 . It means that uk(v) is a directional derivative of uk with respect to internal direction (from v to the interior of ek).

4. Ce2(G) is a set of all functions which have twice differentiable restrictions uk . For each internal vertex vV0 we fix a collection of linear functionals l1,l2, ...,ln,.l1 ( is a multiplicity of v, i.e. the number of edges, attached to v):

( ) ( )

, )

( ) (

) (


I i

j i ij j i ij 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 G0. 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 )


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

) ( ,



G C f q G C

pee 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 ui(v)0 for all


i and for all vertices.

A function uCe2(G) is said to satisfy (2) if each restriction uk satisfies

k k k k

ku 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 ui(v),(iIv) vanishes, then u vanishes on the "star", consisting of v and all edges satisfying eiv.

To illustrate this notion let us define the set v as follows:

1) ( ) ( ) ( )

) (


0 u k u v u v



j j


 


2) li(u)ui(v)u(v), i 1,2,...(v) with positive kj and non-negative  .




It is easy to see that the corresponding v is admissible.

Another example gives a following collection v: 1) l0(u)u(v)

2) ( ) ( )  

( ) ( )

, 1,2,... ( )


v i

v u v u k v

u p u l



m i

im j

i i

i   

  

with positive kkj.

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 SG0 is said to be an S-zone of the function R

G0 if

a) u is positive on SG0and has positive limit values ui(v) for all the vertices lying in S, and for all iIv.

b) u(x)0 on SS \S.

Here the words "connected" and "open" are interpreted in terms of topology, induced on G0 by the standard topology of Rn.

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

ui 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 SG 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 x w

u (here x approaches

 along some edge ei lying constantly in S), then ( )lim ( )0

x i

i w

u . Otherwise wi 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)0w(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


 satisfying z(x)u(x) on S. The last inequality is equivalent to (0w(x)u(x))u(x), which may be transformed to

) ( ) 1 0w(xu 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 )


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: VEN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,