Comparison Theorems for Linear Boundary Value ...

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SIAM J. MATH. ANAL. Voi. 9, No. 5, October 1978

1978 Society for Industrial and Applied Mathematics 0036-1410/78/0905-0019 $01.00/0

COMPARISON THEOREMS FOR LINEAR BOUNDARY VALUE PROBLEMS* JAMES S. MULDOWNEY? Abstract. Conditions on a linear differential operator L are given which guarantee the nonexistence of nontrivial solutions to certain homogeneous boundary value problems. Less restrictive conditions for the nonexistence of nontrivial nonnegative solutions are found and applied to questions of disconjugacy. Some new proofs of known results as well as new disconjugacy criteria are obtained.

1. Introduction. Let L denote the n th order linear differential operator defined

by

Ly

y(n)+aay(n-1)+

+any,

if y ACn-a[a, b], where ak are real valued functions of class C n-k on [a, b]. Also let the boundary form U: ACn-a[a, b] -> R be defined by

(Miiy(i-l)(a)+Nijy(i-)(b)),

Uiy

i= 1,..., n

/=1

where Mj, Nii are real numbers such that the n x 2n matrix [M" N] has rank n; the form U is then said to have rank n. This paper is concerned with boundary value problems of the form

(L; U): Lx f,

Ux

y,

x

AC’-[a, b]

Rn. Conditions on the homogeneous problem Ux O, x C"[a, b] 0(L; U): Lx O,

where fsa[a, b] and 3’

which guarantee the nonexistence of nontrivial solutions to this problem, and more particularly conditions which ensure the nonexistence of nontrivial nonnegative solutions, are found. In the special case

Uiy

y(i-)(a),

i= 1,..., k,

Uy

y0-k-)(b),

k + 1,..., n,

the problems (L; U), 0(L; U) are denoted (L; k, a, b) and 0(L; k, a, b) respectively. Discussions of boundary value problems may be found in the books of Coddington and Levinson [2] and Reid [14]. The main tools in this paper are the Lagrange identity and the boundary-form formula described below. A complete discussion may be found in [2, Chapter 11]. The Lagrange adjoint L* of L is defined by

L*y=(-1)nyO’)+(-1)n-(axy)O’-)+... +any and Lagrange’s identity is

vLu-uL*v [uv]’,

(1.1)

if u and v are n times differentiable functions, where

[uvl

(--1)iu(k)(an-,nV)(i) 2 m=l j+k=m-1

Received by the editors September 2, 1976 and in revised form March 4, 1977.

"

Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. This

research was supported by NRC Grant A-7197.

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JAMES S. MULDOWNEY

and ao 1. For a boundary form U of rank n, there exist boundary forms the 2n x 2n matrix

Uc such that

Mc N has rank 2n; U and Uc are called complementary boundary forms. For each pair of complementary boundary forms U, Uc of rank n, there exists a unique pair of complementary forms U*, U* such that

(1.2)

[xy](b)-[xy](a)= gx U*y + Ux U*y

if the functions x, y are n 1 times differentiable at a and b. This is the boundary-form formula. Note that, in general, if U is specified then adjoint boundary forms U* depend on the coefficients in L, and conversely if U* is specified U depends on L (cf. [2, Theorem 3.1, p. 289]). However, in the case of the boundary form U (k, a, b) it is always possible to choose U* (n- k, a, b). The first comparison result proved in this paper, Theorem 3.1, states that if (L*; U*) has a solution with y 0, f->_ 0 ( 0) then there is no nontrivial nonnegative solution to 0(L; U). This theorem is extended in Theorems 3.2 and 3.3 by the introduction of a class W(U*) of functions H(t, s). It is shown that if [L*sH(’, s)]_ is ’small’ for some H W(U*), then 0(L; U) has no nontrivial nonnegative solution, and if [L*H(., s)l is ’small’, then 0(L; U) has no nontrivial solution.

2. The classes Yg(U*), p(U). DEITO 2.1. A function H: [a, b] [a, b] -> R belongs to the class (U*) if, for each [a, b],

,

(i) H(t, ) C([a, t)[3 (t, b]) C-2[a, b], (ii) (O-/Os-l)H(t, t+)-(o-l/s-)H(t, t-)=( 1) (iii) U*H(t,. 0. Note that if, in addition to (i), (ii), (iii), y- H(t,. satisfies L*y- 0 on [a, t) (t, b] for each [a, b] then H is the Green’s function for (L; U). However, this condition is not required here; indeed if G is any Green’s function corresponding to any differential operator Lo of order n and satisfies U*G(t, .)=0, then G s (U*). However (U*) does not consist entirely of Green’s functions or generalized Green’s functions since, if H (U*) and K: [a, b] x [a, b]-> R is such that

U*K(t, )= 0 K(t, ) C"[a, hi, for each [a, b], then H + K s (U*). DEFINITION 2.2. (U), 1 0, and lxoL#[a,b] if xAC"-[a,b] and Ux-O, where p[a,b] denotes the usual Lebesgue class. DEFINITION 2.3. If Ul, "", such that r0 +" + r, n and to,

b], to," , t,,C"[a, s [a, b], then

U,

Cn(Ul,’’’,

Un)[ ro

",

r, are nonnegative integers

to,’’’,

rm

denotes the determinant of the n x n matrix the kth row of which is

[Uk(to),

u(kr-l)(to),

uk(tm),

ur’--)(t)l.

In particular ?g’,(u,’’’, u,)[] is the Wronskian determinant W(u,..., u,)(t)= det [u/-) (t)].

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LINEAR BOUNDARY VALUE PROBLEMS

Throughout this paper, a+ 1/2(lal + a) and a_ 1/2(lal-a) for any real number a. The following proposition follows from Lagrange’s identity (1.1) and the boundary-form formula (1.2). The suffix s in Ls denotes the variable to which the differentiation pertains.

PROPOSITION 2.1.

If g (U*) and x A C"-[a, b ], then b

b

-Ux U*cH(t, )+

(2.1)

Proof. From (1.1) with u

x, v

Ja

H(t, s)Lx(s) ds

Ja x(s)L*H(t, s)ds.

H(t,. ), it follows that

b

b

(2.2)

fill(t,. )](b)-[xH(t,. )](t+)+ fill(t,. )](t-)-[xH(t,. )](a). But from the conditions (i), (ii) of Definition 2.1 fill(t,. )](t-)-[xH(t,. )](t+) --a (2.3) O "0 =(-)"x(t oSn_n(t, t+)-os,_n(t, t- =x(t),

)[

)]

and from the boundary-form formula (1.2)

fin(t,. )](b)-[xn(t,. )l(a)= Ux un(t,. )+ Ux u*H(t, .) Ux. un(t,. ), from condition (iii) of Definition 2.1. Combining (2.2), (2.3) and (2.4) gives (2.1). Proposition 2.1 shows that any solution of the problem (L; U) must satisfy the

(2.4)

integral equation b

J,

x(t)=-r. n(t, )+ n(t, s)t(s) es

(2.s)

eb

J (s)Ln(t, s)s

if H (U*). Proposition 2.2 gives conditions under which a solution of (2.5) is also a solution of (L; U). PROPOSITION 2.2. (a) If H (U*) is such that 6 R", l[a, b] and b

(2.6)

0

-6.

J

UH(t, )+ H(t, s)(s) ds

O, then any solution of (2.5) such that x [a, b] implies 6 0 and AC"-X[a, b] is a solution of (L; U). (b) If H (U*) is such that the only solution of

for each

b

(2.7)

x(t)=

-J x(s)LH(t,

s) ds

of (2.5) is also a solution of (L; U). Proof of (a). If x satisfies (2.5) and x AC"-[a, b], then

is the zero solution, then a solution

Ux-, Lx-L from Proposition 2.1, satisfy (2.6)so that x satisfies (L; U). The conditions of part (a) are satisfied if H (U*) is a Green’s function for a boundary value problem associated with any differential operator L0 of order n and 6


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JAMES S. MULDOWNEY

the components of the vector valued function U*H(t,. ) are linearly independent on [a, b]. In that case the right-hand side of (2.6) satisfies Lox A, but since x 0 it follows that A 0; therefore t U*cH(t,. )= 0 and hence 8 0. Proof of (b). From Proposition 2.1, any solution of 0(L; U) satisfies (2.7); thus 0(L; U) has only the zero solution. Therefore, given y and f, (L; U) has a unique solution. The conditions of part (b) also imply that (2.5) has a unique solution which must therefore be the unique solution of (L; U). PROPOSITION 2.3. For each boundary form U of rank n the class (U) is nonempty. In particular ( U) contains a Green’s function. bn)= det Proof. If bi Cn[a, hi, 1,. n, are such that W(bl, is defined y)/W(b,..., by Loy W(bl,..., b, ) then 0, det [Uibi] 0 and L0 Green’s function has a G(t, s) 0(L0; U) has only the trivial solutions so (Lo; U) and H (U) if H(t, s)= (-1)’G(s, t). It will be shown that the constants may be chosen so that ci(t)=e ’t, i-1,..., n, satisfy the required conditions. Without loss of generality it may be assumed that a =0, b 1. If the constants A 1," b) O. Also Ubj =/x(hi) where An are distinct then W(b,

,

.,

,

[Mikl k-1 .at_ Nikt

[j,i(/ )__

-leX ].

k=l

An inductive proof is given that the constants A 1, An may be chosen so that the determinants det [/xi(Ai)] 0, i, j 1,. m, 1 --0 follows from Theorem 3.2 (cf. Corollary 3.2.1). The operator L is said to be discon]ugate on an interval I if the only solution of Lx 0 having n zeros or more in I counting multiplicities is the trivial solution. A result of Levin [7] and Sherman [15] shows that L is disconjugate on I if and only if the problem 0(L; k, a, b) has no nontrivial nonnegative solution for each [a, b] c I and k 1,. n 1. Also L is disconjugate on I if and only if L* is disconjugate on I Coppel [3, Chap. 3]). These observations will be used to give examples of (cf. applications for Theorem 3.1 in the following corollaries. COIOLLAR 3.1.1. If L is disconfugate on I and (-1)n-kqn _--> 0, then the problem o(L + qn k, a, b) has no nontrivial nonnegative solution when [a, b c L Proof. If L is disconjugate on I, then so also is L* and the solution of L*O 1, O(k-1)(b)= 0 satisfies (-1)"-k0>0 on (a, b) if 0(-k-1)(a) O(b) O(a) [a, b] c L This follows from a theorem of Pdlya [13, Thm. V] or from the sign of the appropriate Green’s function (cf. [3]). Therefore (L +q,)*O L*O +q,O >-L*O >0 on

,

(a, ). The following comparison principle was first stated by Levin [7]. The first published proof is due to Nehari

[11].

COROLLARY 3.1.2 (Levin, Nehari). Suppose that 1, 2, Liy Ly + qn, iy, where

qn,1 0 on (a, b ). The necessity of this condition follows from P61ya’s Theorem V [13] or from the sign of the Green’s function for (L; k, a, b). COROLLARY 3.1.4. Suppose n > 2, al(t) a,-l(t) 0 and p,(t) 0. The result now follows from Corollary 3.1.3 and the observation that both expressions on the left in (3.3) are increasing functions of b a. To prove the assertion (3.4), first observe that the set of all polynomials g,(t)with [4,(")(t)l 1, having n zeros in [a, b] and at least one zero at each of the points a, b satisfy [4,(t)l 0 when n- k is odd and the second pertains to the case when n- k is even. Thus some improvement is possible in the case of the second inequality in (3.3) when n is even since suitable coefficients for lab.(t)[ in that expression may be obtained by maximizing I’"(n-J)(t)l’gn, for even k only. Thus, for example, when n is even the coefficient of an (t)_ may be replaced by the smaller number

4(n --2) n-2

(---/-i.-; (b and the coefficient of

a )n

max {10,,(t)l" k

2, 4,.

,n

2},

lan-l(t)l may be replaced by

(n -2) (b a (n 1)!(n 1) n-1

> max {16n, (t)[" k 2, 4,

,n

2}.

Corollary 3.1.6 is a disconjugacy criterion of a type introduced for higher order equations by Hartman and Levin [8], [4], [5]. It extends to higher order operators the form of the Sturm comparison theorem which states that a second order operator L is disconjugate on [a, b] if there exists u C2[a, b] such that u >0 and Lu O, f=l,...,n. Also the symbol (Ul,’’’,ti,’’’,Un) denotes the system (ua, ui-1, ui/a, COROLLARY 3.1.6 (Hartman, Levin). Let [q, b] be a compact interval. Suppose there exist functions ua, un- Cn[a, b) such that (i) (-1)n-’Lu. >_- O, j= 1,... ,n-1 on [a,b)

. .,


950

JAMES S. MULDOWNEY

(ii) (Ul, ", Un-1) and (Ul, ai,’’’, Un-1) are Markov systems on [a, b), 1 _-< ]O’

k, n-k ui,...,

un k, n k are all positive

by P61ya’s Theorem V [13]. It now follows from Corollary 3.1.3 that L is disconjugate [a, b). To see that b is not the first conjugate point of a observe that the functions u,,-1) may be extended to an interval [c, b), c < a, and still satisfy conditions (Ul," (i), (ii) on [c, b). Thus L is disconjugate on [c, b) and hence, by Theorem 7 of Sherman [15], L is disconjugate on [a, b]. THEOREM 3.2. Suppose H (U*) and 7 o(U) exist such that

on

,

b

p/q

b

where 1 0. Then, from Proposition 2.1, y r/x is a nontrivial nonnegative solution of b

y(t)=

J, y(s)L*n(t, s)rl(t)/q(s) ds,

y E .p[a, b].

Therefore y(t)--_(--1)"-kH(t,s)>--O,

(t,s)e[a,b]x[a,b],


954

JAMES S. MULDOWNEY

provided there exists a function e AC"-I[a, b] having a zero of multiplicity k (exactly) at a and n-k (exactly) at b, (-1)n-k >0 and (L + q, ) >-0 on (a, b) with strict inequality on a set of positive measure. &(k-l) Proof. Since has a zero of multiplicity k exactly at a (i.e. (a) (a)=0, ()(a)0) and multiplicity n-k exactly at b it follows that x/ is bounded if x(a) x(-)(a) x(b) x("--a)(b) 0 and x e AC"-[a, b]; thus 1/[[ e(k; a, b). The Green’s function H is continuous, H(., s) has a zero of multiplicity k exactly at a and n- k exactly at b and (-1)"-H( s)> 0 on (a, b), a