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Sturm-Liouville eigenvalue problems on time scales
AP~ED ~ATHEMATr._~
AND
C~TATUON
ELSEVIER
Applied Mathematics and Computation 99 (1999) 153-166
Sturm-Liouville eigenvalue problems on time scales Ravi P. Agarwal a,., Martin Bohner Patricia J.Y. Wong b,2
a,1,
a Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Division of Mathematics, Nanyang Technological University, 469 Bukit Timah Road, Singapore 259756, Singapore
Abstract For Sturm-Liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish Rayleigh's principle. Our results not only unifiy the corresponding theories for differential and difference equations, but are also new in the discrete case. © 1999 Published by Elsevier Science Inc. All rights reserved. AMS classification: 39A10; 39A12; 34B24; 34L15; 93C70 Keywords." Sturm-Liouville eigenvalueproblem; Time scale; Rayleigh's principle; Generalized zero; Oscillation theorem; Comparison theorem
1. Introduction A time scale T is a closed subset o f ~, a n d for a f u n c t i o n f : T ~ • it is possible to define a derivative fA in such a m a n n e r that fA(t)=f'(t)
for all t c ~
if T =
*Corresponding author. E-mail: [email protected]. i E-mail: [email protected]. 2 E-mail: [email protected]. 0096-3003199/$ - see front matter © 1999 Published by Elsevier Science Inc. All rights reserved. PII: S0096-3003(98)00004-6
ILP. Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
154 and
f~(t)=Af(t)=f(t+l)-f(t)
forall tET/ ifT=Z.
Moreover, j u m p operators a, p : T ---, T can be introduced so that
a(t)=p(t)=t
for all t E ~
if T =
tr(t)=t+l,
p(t)=t-1
thrall tE7/
and if T = 7 / .
F u r t h e r details o f time scales will be provided in Section 2; but for reading this introduction it is sufficient to k n o w the particular cases R and 7 / o f T. In this paper we shall consider a Sturm-Liouville eigenvalue problem with separated b o u n d a r y conditions on an arbitrary time scale T o f the f o r m (E)
L(y) + 2y ° = O, ~a(Y) := :~b(Y) = O,
where (we put y~ = y o ~r and yaA = oA)A)
L(y) = yaA + qy, with q : T ~ R continuous, a, b E T with a < b, and
:~,(v) = ~ty(p(a)) + flya(p(a)), ~b(Y) = 7y(b) + 6ya(b) with ( ~ 2 + fl2)(72 + 62) ~ 0. Eigenvalues and eigenfunctions o f (E) are defined in the usual manner. O u r main result is an oscillation theorem for the eigenvalue problem (E), and to state this theorem, we need the concept o f generalized zeros o f solutions o f L ( y ) + ~.y~= 0. F o r this, a solution y o f L(y) + 2y ~ = 0 is said to have a zero at t E T if y(t) = 0, and it has a node at (t + a ( t ) ) / 2 if y(t)y(~r(t)) < 0 (observe that there are no nodes in the case T = ~). A generalized zero o f y is then defined as its zero or its node. O u r main result reads as follows:
Theorem 1 (Oscillation Theorem). The eigenvalues of (E) may be arranged as - c ~ < 21 < 22 < 23 < .... and an eigenfunction corresponding to 2k+1 has exactly k generalized zeros in the open interval (a, b). By introducing a scalar p r o d u c t and the corresponding n o r m
(x,y) = ~(b)x(t)y(t) At
and
Ilxll--
o n the set o f so-called rd-continuous functions (the integral will be defined in
the next section; however, for the m o m e n t we note that it is the " u s u a l " integral f r o m a to b if T = ~ and the sum f r o m a - 1 to b - 1 if T = 7/), we shall also consider the Rayleigh quotient (L(x),x g(x) -
Ilx ll =
,
x
O.
tLP. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
155
Our main result concerning this Rayleigh quotient then reads as the following theorem. Theorem 2 (Rayleigh's Principle). We have 2k+l =
min R(x), ~o(x)=~ (x)=0
where yv are the normalized eigenfunctions corresponding to 2~, 1 <~v <~k. The plan of this paper is as follows: Section 2 gives an introduction of the concept of time scales and provides some preliminaries of the eigenvalue problem (E) such as a characterization of eigenvalues and Lagrange's identity. In Section 3 we state and prove the extended Pieone's identity on time scales. This identity is the essential tool for proving most of the results in this paper. The proofs of Theorems 1 and 2 are contained in the final Section 4. Here, we shall also provide estimates for the Rayleigh quotient in terms of eigenvalues and eigenfunctions of (E*)
L(y) +
= 0,
o(y) = y ( b ) = O.
A comparison result between eigenvalues of (E) and (E*) is also given. Finally, we review the literature on this subject. The concept of time scales has been introduced by Hilger [1] in order to unify continuous and discrete calculus (see also [2]). An up-to-date information of the subject is available in the recently published monograph [3]. Section 4.7 deals with Sturm-Liouville equations on time scales (see also [4]), and we will use some of their results. Further results that we need are contained in [5,6]. The methods we apply are rather similar to those in the recent monograph "Quadratic Functionals in Variational Analysis and Control Theory" [7] ch. 0. However, a great deal of additional effort has to be made which is due to the existence of nodes in the case of general time scales (ofcourse there are no nodes in the case of T = ~). We also wish to add that many of our results are new even for the case T = Y. However, for the discrete case, Theorem 1 can be found (with fl -- 6 = 0) in [8], Theorem 7.6 while [8] Theorem 7.7 gives a result related to Theorem 2 (with fl = 6 = k = 0). Further references for the case T -- 72 are contained in [9-12]. 2. Preliminaries on time scales and eigenvalue problems We let T c ~ be closed so that the jump operators
a(t)=inf{sET:
s>t}
and
p(t)= sup{sCT: s
(supplemented by inf~ := sup T and sup 0 := inf T) are well-defined and map T into T. We call a point t E T left-dense, left-scattered, right-dense,
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156
right-scattered if p(t) = t, p(t) < t, a(t) = t, ~r(t) > t, respectively. The graininess/t : T ~ ~ - is defined by #(t) = ~r(t) - t. W e put T ~ = T if T is u n b o u n d e d a b o v e and T ~ = T \ ( p ( m a x T ) , m a x T ] ) otherwise. A function f : T ~ R is called differentiable at t c T ~ if f a ( t ) := l i m f ( a ( t ) ) - f ( s ) -
s
where s - * t,
s E T\{tT(t)}
exists, and it is called differentiable on T provided it is differentiable at each t C T ~. A function F : T ~ ~ with FA(t) = f ( t ) for all t E T ~ is called an antiderivative o f f on T, and in this case we define
ftf(r)Ar=F(t)-F(s)
for all s , t E T.
Moreover, f is called rd-continuous on T if it is continuous at all right-dense points and has left-sided limits at all left-dense points t c T. T h e following results f r o m [3], ch. 1 (see also [5], L e m m a 1) will be needed. L e m m a 1. Let f , g : T ~ R and t E T ~. Then the following holds:
(i) l f fA(t) exists, then f is continuous at t; (ii) if t is right-scattered and f is continuous at 0c(a(t)) - f(t))/Ft(t); (iii) if f 6 ( t ) exists, then f ( a ( t ) ) = f ( t ) + p(t)fA(t); (iv) if f6(t),gZX(t) exist and (fg)(t) is defined, f(~r(t))gA(t) + fA(t)g(t); (V) if f A exists on T ~ and f is invertible on T, then ( f - l ) a on T~; (vi) if f is rd-continuous on T, then it has an antiderivative
t,
then f A ( t ) =
then
(fg)A(t)=
= _(F)-lf6f-i
on T.
N o w we proceed to give some preliminaries on Sturm-Liouville eigenvalue problems. F o r x , y ~ Crd(T), the set o f twice differentiable functions with rdcontinuous second derivative, we define the Wronskian
w(x, y) = xy A - yx a. L e m m a 2 (Lagrange's Identity, G r e e n ' s F o r m u l a ) . For x , y E C2d(T) we have (i) L(y)x ~ - L ( x ) y ~ = wa(x,y) on [p(a),p(b)] N T ;
(ii) (L(y),x') - (L(x),y ~) = w(x,y)(b) - w(x,y)(p(a)). Proof. W e refer to [4], L e m m a 3.9.
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R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
157
It is easy to see that w ( x , y ) ( p ( a ) ) = O if ~ a ( X ) = ~ a ( y ) = 0 and w(x,y)(b) = 0 if ~b(X) = ~b(Y) = 0. Hence the problem (E) is self-adjoint and all eigenvalues are real. In what follows we shall always assume without loss of generality that
fl > ~l~(p(a)) if fl ~ ~It(p(a)), 6>0if6¢0,
~ = - 1 if fl = ~kt(p(a)),
7= 1 if6=0.
(1)
We will also assume throughout that if a is right-scattered,
then it is also left-scattered, and (2)
if b is left-scattered,
then it is also right-scattered.
Now we shall provide a characterization for the eigenvalues of (E). For this, we denote the unique solutions (see [4]) of the initial value problems
y(p(a)) = fl,
L(y) + 2y~ = 0,
yA(p(a)) = - ~
by y(., ).), where 2 E ~, and we put A(2) = .~b(Y(', )0). With this notation, we have the following lemmas. Lemma 3. 2 is an eigenvalue o f (E) if and o n l y / f A(2) = 0. Proof. If ~b(Y(', 2)) = 0, then y = y(., 2) satisfies L(y) + & o = 0,
&(y) =
= 0,
i.e., 2 is an eigenvalue of (E). Conversely, let 2 E ~ be an eigenvalue of (E) with corresponding eigenfunction y. Then because of the unique solvability of initial value problems (observe So (y) --- 0),
y=cy(.,2)
with
Hence ~b(Y(', 2)) = 0.
c = fly(p(a) ) - ~ya(p(a) ) []
The proof of Lemma 3 also shows that all eigenvalues of (E) are simple. Furthermore, in view of Lemma 2, eigenfunctions x,y corresponding to different eigenvalues are always orthogonal in the sense
x~ L y ~,
i.e.,
(x°,y~)=0.
Lemma 4. We have for a / / t E [p(a), b] fq T and 2 E R
Y(t'2) O y a ( t ' 2 ) - Y a ( t ' 2 )
Oy(t'2) = -
a)
158
ILP. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Proof. Let v, 2 E ~ with v # 4. Then
[y(t, v){yA(t, 4) - ya(t, v)} - ya(t, v){y(t, 4) - y ( t , v)}] a = [y(t, v)yA(t, 4) - yA(t, v)y(t, 4)] a = y~(t, v)yaa(t, 4) --yAa(t, v)y~(t, 2) = (V -- 2)y~(t, V)y(t, 4).
We divide both sides by 2 - v, let v ~ 2, and integrate from p(a) to t (observe that ( O / 0 2 ) y ( p ( a ) , 2 ) = (O/02)ya(p(a),2) ----0) to obtain our desired equation. [] F r o m Lemma 4 it follows that (yA(t, 2))/(y(t, 2)) is for each t c (a, b] N T strictly decreasing in 2 whenever y(t, 4) # 0. Note that L e m m a 4 implies (O/02)y(b, 2) # 0 if y(b, 2) = 0, and it also implies (0/02){7 + 6(yA(b, 4))/ (y(b, 2 ) ) } # 0 . This t o g e t h e r with L e m m a 3 (observe A ( 2 ) = y ( b , 2) × {7 + 6(Ya( b, 2))/(y(b, 2))}) shows that all eigenvalues of (E) are isolated. Our main concern in this paper is to give the so-called oscillation number for any eigenfunction of (E). We define n(2) to be the number of generalized zeros of y(-, 4) in the open interval (a, b). The following result is needed later, and since its p r o o f is analogous to the p r o o f for the continuous case (in view of Lemma 1 (i), y(t, 4) is continuous in t, whereas its continuity with respect to 2 follows from [3], Section 2.6), it is omitted here. Lemma 5. lim s u p ~
n(v) <<.n(2) + 1 .for all 2 E ~; and if equality holds, then
y(b, 4) = 0.
Lemma 6. Put 4" = - { 1 + maxt~[p(a),b]nTq(t)}. Let 2 ~< 4" and y be a nontrivial solution o f L(y) + 2y ~ = O. Then we have the following: (i) I f y has for some tl E [p(a),b] N T a generalized zero in J = {tl } U(tl, a(tt)), then ya has no generalized zero in J ; (ii) if y has a generalized zero in J , then yyA has a generalized zero in J also; (iii) yya strictly increases on [p(a), b] fq T; ( i v ) / f (yya)(t2) t> Ofor some t2 c [p(a), b] fq T, then y has no generalized zeros in (t2, b]; (v) [p(a), b] contains at most one generalized zero of y; (vi) if n(2) = O, then lim . . . . . n(v) = O. Proof. Assertion (i) follows from (use L e m m a 1 (iii))
ya (tl)yA (a(tl)) = ya (tl) {y~ (tl) + #(tl)yaA (tl) } = (yA(h))Z -- (;o + q(t,))It(tl)y(a(t|))yA(h) = (ya(fi))2 _ (). + q(h))y(tr(tl))(y(¢r(tt)) - y(t,)} = (yA(h))2 -- (). + q(h))(y(~r(h))) 2 + (4 + q(h))y(h)y(tr(tl)) >~ (ya(h))2 + (y(tr(fi))) 2 - y ( t l ) y ( a ( t l ) ) > O.
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
159
For (ii), 0y')(tl) ~< 0 implies (Y~y~°)(tt) > 0 by (i), and these two inequalities of course force (yya(yyA)~)(h) ~<0. Next, 0ya)a(t) = (yA(t))z + )/xA(t)y~(t) = (yA(t))2 -- (2 + q(t))(f(t)) 2 1> (y6(t))2 + (ya(/))2 > 0 proves (iii). The identity (use L e m m a 1 (iii))
y(t)y~(t) = y(t){y(t) + p(t)ya(t)} = y2(t) + p(t)y(t)yA(t) takes care of (iv), while (v) follows from (ii), (iii) and (iv). Finally assume that n(2) = 0 and put ~c = mint~[p(~),~(b)]nTy2(t, 2). Then 1< > 0 and
y(t, v)yA(t, V) = --aft + >
a)
(y6(z, V)) 2 -- (V + q(z))(y'(z, 1:))2 AT
-- a/~ + (~* - v ) ~ ( t - - p ( a ) ) > 0
for all sufficiently small v so that (vi) follows from (iv).
[]
Theorem 3. We have lim~-~_oo n(2) = 0 and lim~_oo A(2) = oo. Proof. Let 2* be as in L e m m a 6 and 2 ~<2*. Then n(2*) ~ {0, 1} according to L e m m a 6 (v). If n(2*) = 0, then lim~,~_~ n(v) = 0 because of L e m m a 6 (vi). Hence we assume n ( 2 * ) = 1. Thus aft > 0 (observe y(p(a), 2*)y~(p(a),2 *) = - a f t and use L e m m a 6 (iv)), and [p(a), a] does not contain a generalized zero of y(., 2*) (observe L e m m a 6 (v)). Therefore there exists some t E (p(a), b] n T such that [p(a), ~ contains no generalized zero ofT(., 2*). Put 5- = (p(a), ~ N T. Hence 5- ¢ 0. Therefore yA(t, 2"______<~) 0 y(t, 2")
for all 6 Y
because of L e m m a 6 (iv). We now assume that
y(t, a---Tc \ y(t, ~ ) ' o
for all t E 3- and all 2 < 2*.
This implies for all t E 5" and all 2 < 2* (use L e m m a 1 (iii)) 1 + p ( t ) ya(t'2) 1> 1 +It(t) yA(t'2*) _y~(t,)~*) I > 0
y(t, 2)
y(t, 2*)
y(t, 2*)
160
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
and hence for all 2 < 2"
y(t, 2)
. p+. ~
/> --fi +
fp(La{ (4" )
. )
. (2+q(t))
\
y(t, 4)
t
-~- txl't)
(YA(t'£)~el2
4) -- (4* + q(t)) - k y(t, 2*)
1} y(t,2) , }
,t.~ y%,;.)
1
At
+ , ;i, y%,,~U At ~,~ ) ~
- y(t, x*) + which tends to infinity as 2 ~ - 2 , a contradiction. Hence and because (ya/y)(t, .) is strictly decreasing according to Lemma 4, there must exist an s ¢ Y- and 2 < 2" such that yA(s, 2) = 0. Then y(s, 2) # 0 and y(., 3) has no generalized zero in (s, b) according to Lemma 6 (iv). If y(., 2-) had a generalized zero in [p(a),s), then (yyA)(6,).)~>0 for some tl E [p(a), s) fq T because of Lemma 6 (ii), and this contradicts Lemma 6 (iii). Hence n(2-) = 0 and lim . . . . . n(v) = 0 due to Lemma 6 (vi). Finally, using what we have shown so far (note also - y A ( b , 2 ) = ~+ f~b(a)(2+q(t))ya(t)At ) and from the proof of Lemma 6 (vi) it follows that ~limy(b, 2) = ~lira .
.
.
.
ya(b, 2) =
lim YA(b' )~) -- oc.
....
y ( b , 4)
Hence lim ~_+_ a(2)-----
.fim_~{y(b,2)(~+6Ya(b'2))}=~
according to our assumption (1).
y(b,2)
[]
F r o m the results obtained so far, we know that the eigenvalues of (E) may be arranged as - 2 < ),1 < 22 < -.. We shall write 2p = cx~ for allp > k if only k eigenvalues exist.
3. The extended Picone's identity
Besides elementary results in Section 2, the p r o o f of our main results requires appropriate applications of the extended Picone's identity on time scales, which we shall derive in this section. Let us introduce some notation which is used below. By Ker M we denote the kernel of the indicated matrix M, and we put d e f M = d i m Ker M. Moreover, we let c t = l/c i f c # 0 and 01 = 0. Finally, for a set S we abbreviate the number of elements of S by iS I.
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161
Theorem 4 (Picone's Identity). L e t t E [p(a), p(b)] fq T. S u p p o s e Kery(t) C Kerx(t)
and
Kery*(t) C Kerx*(t),
w h e r e x E C~d(T ) a n d y solves L ( y ) + 2y ~ = O. T h e n at t,
-- {L(x) + 2xa}x ~ = [ x y t w ( x , y ) ] a + ( y y a ) t w Z ( x , y ) .
ProoL First s u p p o s e t h a t (yy~)(t) ¢ 0. T h e n L e m m a s 1 (iv), (v) a n d 2 p r o v i d e at t, Xa
o = y~XV{LCv~x o . - L(x)y ~} = -{L(x)
Next,
suppose
+ ;d}x
w2(x,y) yy~
~°)tw2(x,y).
~ -
t h a t t is right-scattered
with
(yJ)(t)=
0. T h e n
we h a v e
(xx")(t) = 0 a n d hence at t, w(x,y) = xy A - yx ~ = x
y° - y
- y
#
x~ - x #
xy a - yx ~ - - - - 0 . #
I f t is left-scattered, then x y t w ( x , y ) is o b v i o u s l y c o n t i n u o u s at t, a n d if t is leftdense, then it is also c o n t i n u o u s at t, since a c c o r d i n g to l ' H o p i t a l rule (see [5], T h e o r e m 3), lira [xytw(x,y)] (s) = ~ w ( x , y ) ( t )
= o = [xytw(x,y)] (,).
s~t-
H e n c e we m a y a p p l y L e m m a s 1 (ii) a n d 2 (i) to o b t a i n at t, t
[xy w(x,y)]
A
-
[xy t w ( x , y ) ]
a
--xytw(x,y)
[xy t w ( x , y ) ]
/t
= x°(y~)tw~(x,y)
/z =
x°(y~)*{t(y)x o - L(x)y ~}
= xaCv~)ty~{ -- 2x ° -- L(x)} = - x ~ { L ( x ) + 2x ~} = -xa{L(x)
+ 2x a} - Cvy~)twZ(x,y).
Finally, if t is right-dense with y ( t ) = y ( a ( t ) ) = O , x(t) = x ( a ( t ) ) = 0, a n d l ' H o p i t a l rule yields [xy*w(x, y)] a = lim
[xytw(x, Y)I (o-(t)) -
then
we
have
[xytw(x, y)] (t)
a(t) - s
= lim [ x j w ( x , y)] (s) _ lim [xy*] (s)lism w ( x , y ) ( s ) - w ( x , y ) ( t ) s~t s - t ,~t ~ s - t XA(t) -- y a ( t ) W a ( X , y ) ( t ) ----0 = - { L ( x ) + 2x°}(t) - O y " ) t w 2 ( x , y ) ( t ) .
This p r o v e s o u r result.
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162
For a special case of Theorem 4 we refer to [6], Theorem 3. With the aid of Theorem 2 it is now possible to prove the following key result: Theorem 5 (Extended Picone's Identity). Let y be a solution o f L(y) + 2y ~ = 0 and k E ~. Let x~ be solutions o f L(x~) + p~x~ = 0 with p~ E ff~, 1 <~v <~k. Moreover, /et x E Cr2d(T) satisfy x ° I x~, 1 <~v <~k, and suppose x~ ± x ~ k IIx ll=t, Put ~ j = ~ v = l l ~ x v with fl+,ER, l <~v <~k, and Yc = x + Yq. Suppose Kery(t) c K e r x ( t ) holds for all t E [p(a), b] fq T. Then
- { L ( x ) + 2x~,x~) =
[+[
]
(yy~)+w2(~:,y)(t)At
o)
k
+ ~,B~(2 - p,) + T(b) - T(p(a)), ','=1
where r =
+
Proof. We apply Theorem 2 and Lemma 2 (ii) to obtain
= (L(Yc) + AYc~,.Tc~) =
L(x) + Zflvt(x+. ) + Ax" + )Sc~,x" +Yc~ v=l k
= (L(x) + 2x'~,x ~) + {L(x),Yc~) - Z~p~(x'~,Yc~I) + 2(Yc~,Yc~) v~ 1
= (L(x) + 2x°,x `') + {L(Yq),x) + w(Ycl,xl(b) k -
+
-
Or),
v=l
and hence our claim follows.
[]
4. Proof of the main results
Let )+~ < 2~ < ... denote the eigenvalues of (E*) with corresponding normalized eigenfunctions y~* (i.e., ]IY~*[I= 1). In all the proofs of the results from this section we use the notation from Theorem 5.
K P . Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
163
Theorem 6. If2 E (2~, 2~+1] fq R, then n(2) : k.
Moreover, define a k × k-matrix Z* = (Z~,v) by .
I
y~(t~)
i f t~ is a zero o f y(., 2)
z~, = I w(y,y~)(tu)
i f tu is a node o f T ( . , 2 ) ,
where tl < t2 < • • • < tk are the generalized zeros o f y(., ~). Then Z* is invertible. Proof. Let x = 0, p~ = 2 v, x~ = y ~ , y = y ( - , 2 ) . Then T ( p ( a ) ) = T ( b ) = O. Suppose that y has no generalized zeros in (a, b). Then by T h e o r e m 3 (observe 2(b) = 0 and Yc(p(a)) = 0 if y(p(a)) = 0),
0 <
= v=l
(t)At
< 0,
a)
a contradiction. Hence y has at least one generalized zero in (a, b), say t~ < t2 < ... < _tpare all o f them. Define a p × k-matrix Z* with z~ as above. Let , ilk) ~ K e r Z and x = ~ = l fl~Y~.. Hence ~(tu) = 0 for all zeros t, o f y and w(Yc,y)(tu) = 0 for all nodes t~, o f T . Therefore, by T h e o r e m 5, 0 <~ Z f l ~ ( ~ . - - 2;) = v=l
~)tw2(~,y
(t)At ~<0
a)
so that fll = f12 . . . . . flk = 0 and hence K e r Z* = {0}. Thus p / > k. Further, by L e m m a 5 and T h e o r e m 3 we have p<~k. Therefore p = k and Z* is invertible. [] According to T h e o r e m 6, we have y(t, 2~) > 0 for all t E (a, b) N T. Hence ya(b, 2~) < 0 (note that if b is left-scattered, then it is also right-scattered due to (2), and observe also L e m m a 6 (i)) and
A(2~) = 6y~(b,2~) <<.O,
hence 21 ~<2~
(observe (1) and T h e o r e m 3). Because o f T h e o r e m 6 if c5 ¢ 0, we also have sgn A(2~+1) = sgn 6ya(b, )~+t) = sgnya( b, 2~+1) = ( - 1 ) k+l and hence 2k+1 ~< 2~+1. Theorem 7. I f 2*k < ~ , then /~k+l ~
inf R(x) x(p(a))=x(b)=O xa-!_y~,°, l~
and
/~k+l ~
inf R(x). ~(x)=~(x)=0
a ,o x ±Yv , l~
164
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Proof. Let x E C~d(T ) with x ( p ( a ) ) ~ ( x ) = x ( b ) ~ b ( X ) = O, x # O, xv = y,*,, p~ = 2~, 2 E (2~, 2~+1] n ~, y = y(., 2). Then T ( p ( a ) ) = T(b) = 0. By T h e o r e m 6 we m a y choose i l l , . . . , flk E R such that ~(t,) = 0 for all zeros t u of y and w(Yc, y ) ( t , ) = 0 for all nodes t~, of y (observe that Z* is invertible). Then, by T h e o r e m 5, b
-/L(x)
k
= f
(,),,, +
,Ja
so that - ( L ( x ) , x ~) >>.2(x°,x~).
Theorem 8. The number ~lt(p(a) ) ) - def6.
v=l
[]
o f eigenvalues
of
(E)
is
[[a,b] N T I - def(fl-
Proof. First note that, if I[a, b] n TI = ~ and 22 < oo, there exists an x E C~d(T ) with x ( p ( a ) ) = x(b) = O, x # 0 which is o r t h o g o n a l to all y~,, 1 ~< v <~k. Hence T h e o r e m 7 implies that )~2+1 < oct. Therefore (E*) has infinitely m a n y eigenvalues. Next, suppose [[a, b] N T[ = b - a + 1 < co. Then all points o f [a, b] n T are right-scattered (observe that b is right-scattered since it is left-scattered due to Eq. (2)). Hence y ( a ) = fi - ~p(p(a)),
and the recursion formula y(a2(t)) = {a(t) + )~b(t)}y(a(t)) + c(t)y(t),
t E [p(a), p(b)] n T
with a = l + #~ - q##~, #
b = -##~,
c -
#
can be easily verified so that our claim follows f r o m L e m m a 3.
[]
Proof of Theorem 1. We let x = 0 , x,,=yv, P , , = 2 , , kEt~0, 2=)~k+l if 2k+1 < oo, and y = y(-, ).). Then T ( p ( a ) ) = T(b) = 0, and as in the p r o o f o f T h e o r e m 6 we a p p l y T h e o r e m 5 to find n(2) <~ k. Hence our claim follows as before. Moreover, we observe that the analogously defined k × k-matrix Z is invertible. [] Theorem 7 ( C o m p a r i s o n Theorem). I f k E No with 2*k < oo, then & ~< 2~ < 2k+l.
Proofi This is clear f r o m T h e o r e m 1 and the r e m a r k after T h e o r e m 6.
[]
R.P. Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
165
Proof of Theorem 2. We let x(p(a))~,~(x) = x(b)~b(X) = O, x ~ O, xv = y,,, p~ =)-v, k E t~0, )- : ) - k + : if 2k+l < :XD, and y = y ( . , ) - ) . Then T(p(a))= T(b) = 0 a n d as in the p r o o f o f T h e o r e m 7 we apply T h e o r e m 5 to find 2~+ 1 ~<
inf
x(p(a) ):Ra(X)=x(b)~b(x)=O x~iy~, l <<.v<~k
R(x).
We note that R(yk+l) = 2k+j, and this implies our claim.
[]
We conclude this p a p e r with specializing our results for the contirmous and the discrete case. R e m a r k 1. I f T = ~, then our results are well-known (see e.g. [7], ch. 0), and they m a y be s u m m a r i z e d as follows (we put a = 0 and b = 1): There exist infinitely m a n y eigenvalues 2: < 22 < )`3 < "'" o f j) + ()- + q(t))y = O,
~y(O) + fl))(O) = 7y(1) + 69(1) = O,
a n d an eigenfunction corresponding to )-k+t vanishes exactly k times in (0, 1). We can c o m p u t e )-k+l by taking the m i n i m u m o f
fd (2(t) + q(t)x(t) )x(t) dt x2(t) dt over all C2-functions x that satisfy the b o u n d a r y conditions and l~
~y(0) + flAy(0) = yy(N) + bAy(N) = 0,
and an eigenfunction corresponding to )`k+l has exactly k zeros or changes o f sign in (0, 1) (where an interval (i, i + 1) for i E 2: is said to contain a change of sign o f y whenever y(i)y(i + 1) < 0). We can c o m p u t e 2k+: by taking the m i n i m u m of EN=, (A2x(i) + q(i - 1)x(i))x(i) ZiUl x2(i) N
•
•
over all sequences x that satisfy the b o u n d a r y conditions and E;=0 x(~)yv(~) = 0, 1 <<.v<~k (where yv are the eigenfunctions corresponding to )`v with ~,i'V=oy~(i) = 1).
166
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Acknowledgements MB is grateful to the Alexander von Humboldt Foundation for awarding him a Feodor Lynen Research Fellowship to support this work.
References [1] S. Hilger, Analysis on measure chains - A unified approach to continuous and discrete calculus, Res. Math. 18 (1990) 19-56. [2] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990. [3] V. Lakshmikantham, S. Sivasundaram, B. Kaymak~alan. Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. [4] L. Erbe, S. Hilger, Sturmian theory on measure chains, Differential Equations Dynamical Systems 1 (3) (1993) 223-246. [5] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications (submitted). [6] R.P. Agarwal, M. Bohner, Quadratic functionals for second order matrix equations on time scales, Nonlinear Anal. (to appear). [7] W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995. [8] W.G. Kelley, A.C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991. [9] R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992. [10] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic Publishers, Dordrecht, 1996. [11] M. Bohner, Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199 (1996) 804-826. [12] M. Bohner, On disconjugacy for Sturm-Liouville difference equations, J. Difference Equations 2 (2) (1996) 227-237.
AND
C~TATUON
ELSEVIER
Applied Mathematics and Computation 99 (1999) 153-166
Sturm-Liouville eigenvalue problems on time scales Ravi P. Agarwal a,., Martin Bohner Patricia J.Y. Wong b,2
a,1,
a Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Division of Mathematics, Nanyang Technological University, 469 Bukit Timah Road, Singapore 259756, Singapore
Abstract For Sturm-Liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish Rayleigh's principle. Our results not only unifiy the corresponding theories for differential and difference equations, but are also new in the discrete case. © 1999 Published by Elsevier Science Inc. All rights reserved. AMS classification: 39A10; 39A12; 34B24; 34L15; 93C70 Keywords." Sturm-Liouville eigenvalueproblem; Time scale; Rayleigh's principle; Generalized zero; Oscillation theorem; Comparison theorem
1. Introduction A time scale T is a closed subset o f ~, a n d for a f u n c t i o n f : T ~ • it is possible to define a derivative fA in such a m a n n e r that fA(t)=f'(t)
for all t c ~
if T =
*Corresponding author. E-mail: [email protected]. i E-mail: [email protected]. 2 E-mail: [email protected]. 0096-3003199/$ - see front matter © 1999 Published by Elsevier Science Inc. All rights reserved. PII: S0096-3003(98)00004-6
ILP. Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
154 and
f~(t)=Af(t)=f(t+l)-f(t)
forall tET/ ifT=Z.
Moreover, j u m p operators a, p : T ---, T can be introduced so that
a(t)=p(t)=t
for all t E ~
if T =
tr(t)=t+l,
p(t)=t-1
thrall tE7/
and if T = 7 / .
F u r t h e r details o f time scales will be provided in Section 2; but for reading this introduction it is sufficient to k n o w the particular cases R and 7 / o f T. In this paper we shall consider a Sturm-Liouville eigenvalue problem with separated b o u n d a r y conditions on an arbitrary time scale T o f the f o r m (E)
L(y) + 2y ° = O, ~a(Y) := :~b(Y) = O,
where (we put y~ = y o ~r and yaA = oA)A)
L(y) = yaA + qy, with q : T ~ R continuous, a, b E T with a < b, and
:~,(v) = ~ty(p(a)) + flya(p(a)), ~b(Y) = 7y(b) + 6ya(b) with ( ~ 2 + fl2)(72 + 62) ~ 0. Eigenvalues and eigenfunctions o f (E) are defined in the usual manner. O u r main result is an oscillation theorem for the eigenvalue problem (E), and to state this theorem, we need the concept o f generalized zeros o f solutions o f L ( y ) + ~.y~= 0. F o r this, a solution y o f L(y) + 2y ~ = 0 is said to have a zero at t E T if y(t) = 0, and it has a node at (t + a ( t ) ) / 2 if y(t)y(~r(t)) < 0 (observe that there are no nodes in the case T = ~). A generalized zero o f y is then defined as its zero or its node. O u r main result reads as follows:
Theorem 1 (Oscillation Theorem). The eigenvalues of (E) may be arranged as - c ~ < 21 < 22 < 23 < .... and an eigenfunction corresponding to 2k+1 has exactly k generalized zeros in the open interval (a, b). By introducing a scalar p r o d u c t and the corresponding n o r m
(x,y) = ~(b)x(t)y(t) At
and
Ilxll--
o n the set o f so-called rd-continuous functions (the integral will be defined in
the next section; however, for the m o m e n t we note that it is the " u s u a l " integral f r o m a to b if T = ~ and the sum f r o m a - 1 to b - 1 if T = 7/), we shall also consider the Rayleigh quotient (L(x),x g(x) -
Ilx ll =
,
x
O.
tLP. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
155
Our main result concerning this Rayleigh quotient then reads as the following theorem. Theorem 2 (Rayleigh's Principle). We have 2k+l =
min R(x), ~o(x)=~ (x)=0
where yv are the normalized eigenfunctions corresponding to 2~, 1 <~v <~k. The plan of this paper is as follows: Section 2 gives an introduction of the concept of time scales and provides some preliminaries of the eigenvalue problem (E) such as a characterization of eigenvalues and Lagrange's identity. In Section 3 we state and prove the extended Pieone's identity on time scales. This identity is the essential tool for proving most of the results in this paper. The proofs of Theorems 1 and 2 are contained in the final Section 4. Here, we shall also provide estimates for the Rayleigh quotient in terms of eigenvalues and eigenfunctions of (E*)
L(y) +
= 0,
o(y) = y ( b ) = O.
A comparison result between eigenvalues of (E) and (E*) is also given. Finally, we review the literature on this subject. The concept of time scales has been introduced by Hilger [1] in order to unify continuous and discrete calculus (see also [2]). An up-to-date information of the subject is available in the recently published monograph [3]. Section 4.7 deals with Sturm-Liouville equations on time scales (see also [4]), and we will use some of their results. Further results that we need are contained in [5,6]. The methods we apply are rather similar to those in the recent monograph "Quadratic Functionals in Variational Analysis and Control Theory" [7] ch. 0. However, a great deal of additional effort has to be made which is due to the existence of nodes in the case of general time scales (ofcourse there are no nodes in the case of T = ~). We also wish to add that many of our results are new even for the case T = Y. However, for the discrete case, Theorem 1 can be found (with fl -- 6 = 0) in [8], Theorem 7.6 while [8] Theorem 7.7 gives a result related to Theorem 2 (with fl = 6 = k = 0). Further references for the case T -- 72 are contained in [9-12]. 2. Preliminaries on time scales and eigenvalue problems We let T c ~ be closed so that the jump operators
a(t)=inf{sET:
s>t}
and
p(t)= sup{sCT: s
(supplemented by inf~ := sup T and sup 0 := inf T) are well-defined and map T into T. We call a point t E T left-dense, left-scattered, right-dense,
R.P. Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
156
right-scattered if p(t) = t, p(t) < t, a(t) = t, ~r(t) > t, respectively. The graininess/t : T ~ ~ - is defined by #(t) = ~r(t) - t. W e put T ~ = T if T is u n b o u n d e d a b o v e and T ~ = T \ ( p ( m a x T ) , m a x T ] ) otherwise. A function f : T ~ R is called differentiable at t c T ~ if f a ( t ) := l i m f ( a ( t ) ) - f ( s ) -
s
where s - * t,
s E T\{tT(t)}
exists, and it is called differentiable on T provided it is differentiable at each t C T ~. A function F : T ~ ~ with FA(t) = f ( t ) for all t E T ~ is called an antiderivative o f f on T, and in this case we define
ftf(r)Ar=F(t)-F(s)
for all s , t E T.
Moreover, f is called rd-continuous on T if it is continuous at all right-dense points and has left-sided limits at all left-dense points t c T. T h e following results f r o m [3], ch. 1 (see also [5], L e m m a 1) will be needed. L e m m a 1. Let f , g : T ~ R and t E T ~. Then the following holds:
(i) l f fA(t) exists, then f is continuous at t; (ii) if t is right-scattered and f is continuous at 0c(a(t)) - f(t))/Ft(t); (iii) if f 6 ( t ) exists, then f ( a ( t ) ) = f ( t ) + p(t)fA(t); (iv) if f6(t),gZX(t) exist and (fg)(t) is defined, f(~r(t))gA(t) + fA(t)g(t); (V) if f A exists on T ~ and f is invertible on T, then ( f - l ) a on T~; (vi) if f is rd-continuous on T, then it has an antiderivative
t,
then f A ( t ) =
then
(fg)A(t)=
= _(F)-lf6f-i
on T.
N o w we proceed to give some preliminaries on Sturm-Liouville eigenvalue problems. F o r x , y ~ Crd(T), the set o f twice differentiable functions with rdcontinuous second derivative, we define the Wronskian
w(x, y) = xy A - yx a. L e m m a 2 (Lagrange's Identity, G r e e n ' s F o r m u l a ) . For x , y E C2d(T) we have (i) L(y)x ~ - L ( x ) y ~ = wa(x,y) on [p(a),p(b)] N T ;
(ii) (L(y),x') - (L(x),y ~) = w(x,y)(b) - w(x,y)(p(a)). Proof. W e refer to [4], L e m m a 3.9.
[]
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
157
It is easy to see that w ( x , y ) ( p ( a ) ) = O if ~ a ( X ) = ~ a ( y ) = 0 and w(x,y)(b) = 0 if ~b(X) = ~b(Y) = 0. Hence the problem (E) is self-adjoint and all eigenvalues are real. In what follows we shall always assume without loss of generality that
fl > ~l~(p(a)) if fl ~ ~It(p(a)), 6>0if6¢0,
~ = - 1 if fl = ~kt(p(a)),
7= 1 if6=0.
(1)
We will also assume throughout that if a is right-scattered,
then it is also left-scattered, and (2)
if b is left-scattered,
then it is also right-scattered.
Now we shall provide a characterization for the eigenvalues of (E). For this, we denote the unique solutions (see [4]) of the initial value problems
y(p(a)) = fl,
L(y) + 2y~ = 0,
yA(p(a)) = - ~
by y(., ).), where 2 E ~, and we put A(2) = .~b(Y(', )0). With this notation, we have the following lemmas. Lemma 3. 2 is an eigenvalue o f (E) if and o n l y / f A(2) = 0. Proof. If ~b(Y(', 2)) = 0, then y = y(., 2) satisfies L(y) + & o = 0,
&(y) =
= 0,
i.e., 2 is an eigenvalue of (E). Conversely, let 2 E ~ be an eigenvalue of (E) with corresponding eigenfunction y. Then because of the unique solvability of initial value problems (observe So (y) --- 0),
y=cy(.,2)
with
Hence ~b(Y(', 2)) = 0.
c = fly(p(a) ) - ~ya(p(a) ) []
The proof of Lemma 3 also shows that all eigenvalues of (E) are simple. Furthermore, in view of Lemma 2, eigenfunctions x,y corresponding to different eigenvalues are always orthogonal in the sense
x~ L y ~,
i.e.,
(x°,y~)=0.
Lemma 4. We have for a / / t E [p(a), b] fq T and 2 E R
Y(t'2) O y a ( t ' 2 ) - Y a ( t ' 2 )
Oy(t'2) = -
a)
158
ILP. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Proof. Let v, 2 E ~ with v # 4. Then
[y(t, v){yA(t, 4) - ya(t, v)} - ya(t, v){y(t, 4) - y ( t , v)}] a = [y(t, v)yA(t, 4) - yA(t, v)y(t, 4)] a = y~(t, v)yaa(t, 4) --yAa(t, v)y~(t, 2) = (V -- 2)y~(t, V)y(t, 4).
We divide both sides by 2 - v, let v ~ 2, and integrate from p(a) to t (observe that ( O / 0 2 ) y ( p ( a ) , 2 ) = (O/02)ya(p(a),2) ----0) to obtain our desired equation. [] F r o m Lemma 4 it follows that (yA(t, 2))/(y(t, 2)) is for each t c (a, b] N T strictly decreasing in 2 whenever y(t, 4) # 0. Note that L e m m a 4 implies (O/02)y(b, 2) # 0 if y(b, 2) = 0, and it also implies (0/02){7 + 6(yA(b, 4))/ (y(b, 2 ) ) } # 0 . This t o g e t h e r with L e m m a 3 (observe A ( 2 ) = y ( b , 2) × {7 + 6(Ya( b, 2))/(y(b, 2))}) shows that all eigenvalues of (E) are isolated. Our main concern in this paper is to give the so-called oscillation number for any eigenfunction of (E). We define n(2) to be the number of generalized zeros of y(-, 4) in the open interval (a, b). The following result is needed later, and since its p r o o f is analogous to the p r o o f for the continuous case (in view of Lemma 1 (i), y(t, 4) is continuous in t, whereas its continuity with respect to 2 follows from [3], Section 2.6), it is omitted here. Lemma 5. lim s u p ~
n(v) <<.n(2) + 1 .for all 2 E ~; and if equality holds, then
y(b, 4) = 0.
Lemma 6. Put 4" = - { 1 + maxt~[p(a),b]nTq(t)}. Let 2 ~< 4" and y be a nontrivial solution o f L(y) + 2y ~ = O. Then we have the following: (i) I f y has for some tl E [p(a),b] N T a generalized zero in J = {tl } U(tl, a(tt)), then ya has no generalized zero in J ; (ii) if y has a generalized zero in J , then yyA has a generalized zero in J also; (iii) yya strictly increases on [p(a), b] fq T; ( i v ) / f (yya)(t2) t> Ofor some t2 c [p(a), b] fq T, then y has no generalized zeros in (t2, b]; (v) [p(a), b] contains at most one generalized zero of y; (vi) if n(2) = O, then lim . . . . . n(v) = O. Proof. Assertion (i) follows from (use L e m m a 1 (iii))
ya (tl)yA (a(tl)) = ya (tl) {y~ (tl) + #(tl)yaA (tl) } = (yA(h))Z -- (;o + q(t,))It(tl)y(a(t|))yA(h) = (ya(fi))2 _ (). + q(h))y(tr(tl))(y(¢r(tt)) - y(t,)} = (yA(h))2 -- (). + q(h))(y(~r(h))) 2 + (4 + q(h))y(h)y(tr(tl)) >~ (ya(h))2 + (y(tr(fi))) 2 - y ( t l ) y ( a ( t l ) ) > O.
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
159
For (ii), 0y')(tl) ~< 0 implies (Y~y~°)(tt) > 0 by (i), and these two inequalities of course force (yya(yyA)~)(h) ~<0. Next, 0ya)a(t) = (yA(t))z + )/xA(t)y~(t) = (yA(t))2 -- (2 + q(t))(f(t)) 2 1> (y6(t))2 + (ya(/))2 > 0 proves (iii). The identity (use L e m m a 1 (iii))
y(t)y~(t) = y(t){y(t) + p(t)ya(t)} = y2(t) + p(t)y(t)yA(t) takes care of (iv), while (v) follows from (ii), (iii) and (iv). Finally assume that n(2) = 0 and put ~c = mint~[p(~),~(b)]nTy2(t, 2). Then 1< > 0 and
y(t, v)yA(t, V) = --aft + >
a)
(y6(z, V)) 2 -- (V + q(z))(y'(z, 1:))2 AT
-- a/~ + (~* - v ) ~ ( t - - p ( a ) ) > 0
for all sufficiently small v so that (vi) follows from (iv).
[]
Theorem 3. We have lim~-~_oo n(2) = 0 and lim~_oo A(2) = oo. Proof. Let 2* be as in L e m m a 6 and 2 ~<2*. Then n(2*) ~ {0, 1} according to L e m m a 6 (v). If n(2*) = 0, then lim~,~_~ n(v) = 0 because of L e m m a 6 (vi). Hence we assume n ( 2 * ) = 1. Thus aft > 0 (observe y(p(a), 2*)y~(p(a),2 *) = - a f t and use L e m m a 6 (iv)), and [p(a), a] does not contain a generalized zero of y(., 2*) (observe L e m m a 6 (v)). Therefore there exists some t E (p(a), b] n T such that [p(a), ~ contains no generalized zero ofT(., 2*). Put 5- = (p(a), ~ N T. Hence 5- ¢ 0. Therefore yA(t, 2"______<~) 0 y(t, 2")
for all 6 Y
because of L e m m a 6 (iv). We now assume that
y(t, a---Tc \ y(t, ~ ) ' o
for all t E 3- and all 2 < 2*.
This implies for all t E 5" and all 2 < 2* (use L e m m a 1 (iii)) 1 + p ( t ) ya(t'2) 1> 1 +It(t) yA(t'2*) _y~(t,)~*) I > 0
y(t, 2)
y(t, 2*)
y(t, 2*)
160
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
and hence for all 2 < 2"
y(t, 2)
. p+. ~
/> --fi +
fp(La{ (4" )
. )
. (2+q(t))
\
y(t, 4)
t
-~- txl't)
(YA(t'£)~el2
4) -- (4* + q(t)) - k y(t, 2*)
1} y(t,2) , }
,t.~ y%,;.)
1
At
+ , ;i, y%,,~U At ~,~ ) ~
- y(t, x*) + which tends to infinity as 2 ~ - 2 , a contradiction. Hence and because (ya/y)(t, .) is strictly decreasing according to Lemma 4, there must exist an s ¢ Y- and 2 < 2" such that yA(s, 2) = 0. Then y(s, 2) # 0 and y(., 3) has no generalized zero in (s, b) according to Lemma 6 (iv). If y(., 2-) had a generalized zero in [p(a),s), then (yyA)(6,).)~>0 for some tl E [p(a), s) fq T because of Lemma 6 (ii), and this contradicts Lemma 6 (iii). Hence n(2-) = 0 and lim . . . . . n(v) = 0 due to Lemma 6 (vi). Finally, using what we have shown so far (note also - y A ( b , 2 ) = ~+ f~b(a)(2+q(t))ya(t)At ) and from the proof of Lemma 6 (vi) it follows that ~limy(b, 2) = ~lira .
.
.
.
ya(b, 2) =
lim YA(b' )~) -- oc.
....
y ( b , 4)
Hence lim ~_+_ a(2)-----
.fim_~{y(b,2)(~+6Ya(b'2))}=~
according to our assumption (1).
y(b,2)
[]
F r o m the results obtained so far, we know that the eigenvalues of (E) may be arranged as - 2 < ),1 < 22 < -.. We shall write 2p = cx~ for allp > k if only k eigenvalues exist.
3. The extended Picone's identity
Besides elementary results in Section 2, the p r o o f of our main results requires appropriate applications of the extended Picone's identity on time scales, which we shall derive in this section. Let us introduce some notation which is used below. By Ker M we denote the kernel of the indicated matrix M, and we put d e f M = d i m Ker M. Moreover, we let c t = l/c i f c # 0 and 01 = 0. Finally, for a set S we abbreviate the number of elements of S by iS I.
R.P. Agarwal et aL / Appl. Math. Comput. 99 (1999) 153-166
161
Theorem 4 (Picone's Identity). L e t t E [p(a), p(b)] fq T. S u p p o s e Kery(t) C Kerx(t)
and
Kery*(t) C Kerx*(t),
w h e r e x E C~d(T ) a n d y solves L ( y ) + 2y ~ = O. T h e n at t,
-- {L(x) + 2xa}x ~ = [ x y t w ( x , y ) ] a + ( y y a ) t w Z ( x , y ) .
ProoL First s u p p o s e t h a t (yy~)(t) ¢ 0. T h e n L e m m a s 1 (iv), (v) a n d 2 p r o v i d e at t, Xa
o = y~XV{LCv~x o . - L(x)y ~} = -{L(x)
Next,
suppose
+ ;d}x
w2(x,y) yy~
~°)tw2(x,y).
~ -
t h a t t is right-scattered
with
(yJ)(t)=
0. T h e n
we h a v e
(xx")(t) = 0 a n d hence at t, w(x,y) = xy A - yx ~ = x
y° - y
- y
#
x~ - x #
xy a - yx ~ - - - - 0 . #
I f t is left-scattered, then x y t w ( x , y ) is o b v i o u s l y c o n t i n u o u s at t, a n d if t is leftdense, then it is also c o n t i n u o u s at t, since a c c o r d i n g to l ' H o p i t a l rule (see [5], T h e o r e m 3), lira [xytw(x,y)] (s) = ~ w ( x , y ) ( t )
= o = [xytw(x,y)] (,).
s~t-
H e n c e we m a y a p p l y L e m m a s 1 (ii) a n d 2 (i) to o b t a i n at t, t
[xy w(x,y)]
A
-
[xy t w ( x , y ) ]
a
--xytw(x,y)
[xy t w ( x , y ) ]
/t
= x°(y~)tw~(x,y)
/z =
x°(y~)*{t(y)x o - L(x)y ~}
= xaCv~)ty~{ -- 2x ° -- L(x)} = - x ~ { L ( x ) + 2x ~} = -xa{L(x)
+ 2x a} - Cvy~)twZ(x,y).
Finally, if t is right-dense with y ( t ) = y ( a ( t ) ) = O , x(t) = x ( a ( t ) ) = 0, a n d l ' H o p i t a l rule yields [xy*w(x, y)] a = lim
[xytw(x, Y)I (o-(t)) -
then
we
have
[xytw(x, y)] (t)
a(t) - s
= lim [ x j w ( x , y)] (s) _ lim [xy*] (s)lism w ( x , y ) ( s ) - w ( x , y ) ( t ) s~t s - t ,~t ~ s - t XA(t) -- y a ( t ) W a ( X , y ) ( t ) ----0 = - { L ( x ) + 2x°}(t) - O y " ) t w 2 ( x , y ) ( t ) .
This p r o v e s o u r result.
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ILP. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
162
For a special case of Theorem 4 we refer to [6], Theorem 3. With the aid of Theorem 2 it is now possible to prove the following key result: Theorem 5 (Extended Picone's Identity). Let y be a solution o f L(y) + 2y ~ = 0 and k E ~. Let x~ be solutions o f L(x~) + p~x~ = 0 with p~ E ff~, 1 <~v <~k. Moreover, /et x E Cr2d(T) satisfy x ° I x~, 1 <~v <~k, and suppose x~ ± x ~ k IIx ll=t, Put ~ j = ~ v = l l ~ x v with fl+,ER, l <~v <~k, and Yc = x + Yq. Suppose Kery(t) c K e r x ( t ) holds for all t E [p(a), b] fq T. Then
- { L ( x ) + 2x~,x~) =
[+[
]
(yy~)+w2(~:,y)(t)At
o)
k
+ ~,B~(2 - p,) + T(b) - T(p(a)), ','=1
where r =
+
Proof. We apply Theorem 2 and Lemma 2 (ii) to obtain
= (L(Yc) + AYc~,.Tc~) =
L(x) + Zflvt(x+. ) + Ax" + )Sc~,x" +Yc~ v=l k
= (L(x) + 2x'~,x ~) + {L(x),Yc~) - Z~p~(x'~,Yc~I) + 2(Yc~,Yc~) v~ 1
= (L(x) + 2x°,x `') + {L(Yq),x) + w(Ycl,xl(b) k -
+
-
Or),
v=l
and hence our claim follows.
[]
4. Proof of the main results
Let )+~ < 2~ < ... denote the eigenvalues of (E*) with corresponding normalized eigenfunctions y~* (i.e., ]IY~*[I= 1). In all the proofs of the results from this section we use the notation from Theorem 5.
K P . Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
163
Theorem 6. If2 E (2~, 2~+1] fq R, then n(2) : k.
Moreover, define a k × k-matrix Z* = (Z~,v) by .
I
y~(t~)
i f t~ is a zero o f y(., 2)
z~, = I w(y,y~)(tu)
i f tu is a node o f T ( . , 2 ) ,
where tl < t2 < • • • < tk are the generalized zeros o f y(., ~). Then Z* is invertible. Proof. Let x = 0, p~ = 2 v, x~ = y ~ , y = y ( - , 2 ) . Then T ( p ( a ) ) = T ( b ) = O. Suppose that y has no generalized zeros in (a, b). Then by T h e o r e m 3 (observe 2(b) = 0 and Yc(p(a)) = 0 if y(p(a)) = 0),
0 <
= v=l
(t)At
< 0,
a)
a contradiction. Hence y has at least one generalized zero in (a, b), say t~ < t2 < ... < _tpare all o f them. Define a p × k-matrix Z* with z~ as above. Let , ilk) ~ K e r Z and x = ~ = l fl~Y~.. Hence ~(tu) = 0 for all zeros t, o f y and w(Yc,y)(tu) = 0 for all nodes t~, o f T . Therefore, by T h e o r e m 5, 0 <~ Z f l ~ ( ~ . - - 2;) = v=l
~)tw2(~,y
(t)At ~<0
a)
so that fll = f12 . . . . . flk = 0 and hence K e r Z* = {0}. Thus p / > k. Further, by L e m m a 5 and T h e o r e m 3 we have p<~k. Therefore p = k and Z* is invertible. [] According to T h e o r e m 6, we have y(t, 2~) > 0 for all t E (a, b) N T. Hence ya(b, 2~) < 0 (note that if b is left-scattered, then it is also right-scattered due to (2), and observe also L e m m a 6 (i)) and
A(2~) = 6y~(b,2~) <<.O,
hence 21 ~<2~
(observe (1) and T h e o r e m 3). Because o f T h e o r e m 6 if c5 ¢ 0, we also have sgn A(2~+1) = sgn 6ya(b, )~+t) = sgnya( b, 2~+1) = ( - 1 ) k+l and hence 2k+1 ~< 2~+1. Theorem 7. I f 2*k < ~ , then /~k+l ~
inf R(x) x(p(a))=x(b)=O xa-!_y~,°, l~
and
/~k+l ~
inf R(x). ~(x)=~(x)=0
a ,o x ±Yv , l~
164
R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Proof. Let x E C~d(T ) with x ( p ( a ) ) ~ ( x ) = x ( b ) ~ b ( X ) = O, x # O, xv = y,*,, p~ = 2~, 2 E (2~, 2~+1] n ~, y = y(., 2). Then T ( p ( a ) ) = T(b) = 0. By T h e o r e m 6 we m a y choose i l l , . . . , flk E R such that ~(t,) = 0 for all zeros t u of y and w(Yc, y ) ( t , ) = 0 for all nodes t~, of y (observe that Z* is invertible). Then, by T h e o r e m 5, b
-/L(x)
k
= f
(,),,, +
,Ja
so that - ( L ( x ) , x ~) >>.2(x°,x~).
Theorem 8. The number ~lt(p(a) ) ) - def6.
v=l
[]
o f eigenvalues
of
(E)
is
[[a,b] N T I - def(fl-
Proof. First note that, if I[a, b] n TI = ~ and 22 < oo, there exists an x E C~d(T ) with x ( p ( a ) ) = x(b) = O, x # 0 which is o r t h o g o n a l to all y~,, 1 ~< v <~k. Hence T h e o r e m 7 implies that )~2+1 < oct. Therefore (E*) has infinitely m a n y eigenvalues. Next, suppose [[a, b] N T[ = b - a + 1 < co. Then all points o f [a, b] n T are right-scattered (observe that b is right-scattered since it is left-scattered due to Eq. (2)). Hence y ( a ) = fi - ~p(p(a)),
and the recursion formula y(a2(t)) = {a(t) + )~b(t)}y(a(t)) + c(t)y(t),
t E [p(a), p(b)] n T
with a = l + #~ - q##~, #
b = -##~,
c -
#
can be easily verified so that our claim follows f r o m L e m m a 3.
[]
Proof of Theorem 1. We let x = 0 , x,,=yv, P , , = 2 , , kEt~0, 2=)~k+l if 2k+1 < oo, and y = y(-, ).). Then T ( p ( a ) ) = T(b) = 0, and as in the p r o o f o f T h e o r e m 6 we a p p l y T h e o r e m 5 to find n(2) <~ k. Hence our claim follows as before. Moreover, we observe that the analogously defined k × k-matrix Z is invertible. [] Theorem 7 ( C o m p a r i s o n Theorem). I f k E No with 2*k < oo, then & ~< 2~ < 2k+l.
Proofi This is clear f r o m T h e o r e m 1 and the r e m a r k after T h e o r e m 6.
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R.P. Agarwal et al. / Appl. Math. Comput. 99 (1999) 153-166
165
Proof of Theorem 2. We let x(p(a))~,~(x) = x(b)~b(X) = O, x ~ O, xv = y,,, p~ =)-v, k E t~0, )- : ) - k + : if 2k+l < :XD, and y = y ( . , ) - ) . Then T(p(a))= T(b) = 0 a n d as in the p r o o f o f T h e o r e m 7 we apply T h e o r e m 5 to find 2~+ 1 ~<
inf
x(p(a) ):Ra(X)=x(b)~b(x)=O x~iy~, l <<.v<~k
R(x).
We note that R(yk+l) = 2k+j, and this implies our claim.
[]
We conclude this p a p e r with specializing our results for the contirmous and the discrete case. R e m a r k 1. I f T = ~, then our results are well-known (see e.g. [7], ch. 0), and they m a y be s u m m a r i z e d as follows (we put a = 0 and b = 1): There exist infinitely m a n y eigenvalues 2: < 22 < )`3 < "'" o f j) + ()- + q(t))y = O,
~y(O) + fl))(O) = 7y(1) + 69(1) = O,
a n d an eigenfunction corresponding to )-k+t vanishes exactly k times in (0, 1). We can c o m p u t e )-k+l by taking the m i n i m u m o f
fd (2(t) + q(t)x(t) )x(t) dt x2(t) dt over all C2-functions x that satisfy the b o u n d a r y conditions and l~
~y(0) + flAy(0) = yy(N) + bAy(N) = 0,
and an eigenfunction corresponding to )`k+l has exactly k zeros or changes o f sign in (0, 1) (where an interval (i, i + 1) for i E 2: is said to contain a change of sign o f y whenever y(i)y(i + 1) < 0). We can c o m p u t e 2k+: by taking the m i n i m u m of EN=, (A2x(i) + q(i - 1)x(i))x(i) ZiUl x2(i) N
•
•
over all sequences x that satisfy the b o u n d a r y conditions and E;=0 x(~)yv(~) = 0, 1 <<.v<~k (where yv are the eigenfunctions corresponding to )`v with ~,i'V=oy~(i) = 1).
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R.P. Agarwal et al. I Appl. Math. Comput. 99 (1999) 153-166
Acknowledgements MB is grateful to the Alexander von Humboldt Foundation for awarding him a Feodor Lynen Research Fellowship to support this work.
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