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Comparison of Green's Functions for a Family of Multipoint Boundary Value Problems
Journal of Mathematical Analysis and Applications 246, 296᎐307 Ž2000. doi:10.1006rjmaa.2000.6811, available online at http:rrwww.idealibrary.com on
Comparison of Green’s Functions for a Family of Multipoint Boundary Value Problems Paul W. Eloe and Lingqin Zhang Department of Mathematics, Uni¨ ersity of Dayton, Dayton, Ohio 45469-2316 Submitted by William F. Ames Received October 27, 1998
1. INTRODUCTION Let n G 2 be an integer and let k g 2, . . . , n4 be given. Let B ) 0, let 0 F x 1 - ⭈⭈⭈ - x ky1 - B be fixed throughout this paper, and let x ky1 x k s b F B be given. Let a i g C w0, B x, i s 1, . . . , n, and define the linear differential operator, L, by n
Ly s y Ž n. q
Ý ai y Ž nyi. .
Ž 1.1.
is1
Let n1 , . . . , n k denote positive integers such that Ý kls1 n l s n. Let W denote the set of nonnegative integers, and let ⍀ n k ; W n k be defined by ⍀ n k s ␣ s Ž ␣ 1 , . . . , ␣ n k . : 0 F ␣ 1 - ⭈⭈⭈ - ␣ n k F n y 1 4 . Define a partial order on ⍀ n k as follows: We say ␣ F  if for each i s 1, . . . , n k , ␣ i F i , and we say ␣ -  if ␣ F  and ␣ /  . For each ␣ g ⍀ n k, consider the homogeneous, k-point boundary conditions of the form y Ž l . Ž x i . s 0,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s 0,
l s ␣1 , . . . , ␣nk .
Ž 1.2.
We shall sometimes denote the boundary conditions Ž1.2. by T Ž k, ␣ , b . y s 0. Note that if ␣ s Ž0, . . . , n k y 1., then T Ž k, ␣ , b . y s 0 represents k-point conjugate boundary conditions w1x. 296 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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297
For each ␣ g ⍀ n k, let GŽ k, ␣ , b; x, s . denote Green’s function, if it exists, of the Boundary Value Problem ŽBVP., Ly s 0, x 1 - x b, T Ž k, ␣ , b . y s 0. For each l s 0, . . . , n, let Gl Ž k, ␣ , b; x, s . s Ž ⭸ lr⭸ x l . G Ž k , ␣ , b; x, s . . Throughout this paper we shall assume sufficient hypotheses such that G exists for each ␣ g ⍀ n k, x ky1 - b F B. The purpose of this paper is to obtain comparison theorems for the family of Green’s functions, GŽ k, ␣ , b; x, s .. For example, let ␣ s Ž0, . . . , n k y 1.. It is known w1x that if L is disconjugate on w x 1 , b x then m
Ž y1. i G Ž k , ␣ , b; x, s . ) 0,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . ,
i s 1, . . . , k y 1, where k
mi s
Ý
nj.
Ž 1.3.
jsiq1
We shall show, for example, that, under the assumption of right disfocality, if ␣ , ␣ ˆ g ⍀ n k, ␣ - ␣ˆ , then m
m
Ž y1. i G Ž k, ␣ˆ , b; x, s . ) Ž y1. i G Ž k, ␣ , b; x, s . ) 0, Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b .. Recall w7x that L, defined by Ž1.1., is right disfocal on w0, B x if the only solution of Ly s 0, 0 F x F B, satisfying y Ž j. Ž t j . s 0, j s 0, . . . , n y 1, where 0 F t 0 F ⭈⭈⭈ F t ny1 F B, is y ' 0. We shall assume throughout this paper that L is right disfocal on w0, B x and thus assume sufficient conditions for the existence of each GŽ k, ␣ , b; x, s .. It is actually the case that right disfocality is too strong an assumption; we assume right disfocality for simplicity. This paper represents an extension of the work by Eloe and Ridenhour w6x, who obtained comparison theorems for Green’s functions for two-point boundary value problems. The work by Eloe and Ridenhour w6x extended some of the results by Peterson w11, 12x and Elias w2, 3x for two-term differential equations where the coefficients have fixed sign, and some of the results by Peterson and Ridenhour w13x for two-term differential equations where the coefficients may change sign. We are not aware of analogous studies for two-term differential equations for multipoint boundary value problems. As this paper is an extension the work by Eloe and Ridenhour, the organization is similar. We shall state the main results in Theorems 2.1 and 2.2 in Section 2. We shall then provide the technical details in Section
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3. As is typical in many of these studies w2, 6, 13x, the primary observation is that the difference of two Green’s functions is a solution of Ly s 0 in x for fixed s. Hence, the technical details in Section 3 focus on determining the sign of solutions of Ly s 0 that satisfy n y 1 of the homogeneous boundary conditions given in Ž1.2..
2. THE MAIN RESULTS Assume L is right disfocal on w0, B x. Let ␣ , ␣ ˆ g ⍀ n k,
THEOREM 2.1. ␣-␣ ˆ. Then
mi
0 - Ž y1 .
G Ž k , ␣ , b; x, s . - Ž y1 .
mi
GŽ k , ␣ ˆ , b; x, s . ,
Ž 2.1.
mi
Gn iŽ k , ␣ ˆ , b; x i , s . ,
Ž 2.2.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . i s 1, . . . , k y 1, 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . - Ž y1 .
s g Ž x 1 , b ., i s 1, . . . , k, where m k s 0, and n
n
0 - Ž y1 . k Gl Ž k , ␣ , b; x, s . - Ž y1 . k Gl Ž k, ␣ ˆ , b; x, s . ,
Ž 2.3.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b ., l s 0, . . . , ␣ 1. THEOREM 2.2. Assume L is right disfocal on w0, B x. Let ␣ , ␣ ˆ g ⍀ n k, ␣ n k - n y 1. Assume ␣ F ␣ ˆ , b1 F b 2 , and assume that one of the inequalities, ␣ F ␣ ˆ , b1 F b 2 , is strict. Then mi
0 - Ž y1 .
G Ž k, ␣ , b1 ; x, s . - Ž y1 .
mi
G Ž k, ␣ ˆ , b 2 ; x, s . ,
Ž 2.4.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b1 ., i s 0, . . . , k y 1, 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b1 ; x i , s . - Ž y1 .
mi
Gn iŽ k , ␣ ˆ , b 2 ; x i , s . , Ž 2.5.
s g Ž x 1 , b1 ., i s 0, . . . , k, and n
n
0 - Ž y1 . k Gl Ž k, ␣ , b1 ; x, s . - Ž y1 . k Gl Ž k , ␣ ˆ , b 2 ; x, s . ,
Ž 2.6.
Ž x, s . g Ž x ky 1 , b1 . = Ž x 1 , b1 ., l s 0, . . . , ␣ 1. Remark. In regard to Theorem 2.1, we say that Žy1. m i GŽ k, ␣ , b; x, s . is monotone increasing as a function of ␣ . Similarly, in regard to Theorem 2.2, where ␣ n k - n y 1 is assumed, we say that Žy1. m i GŽ k, ␣ , b; x, s . is monotone increasing as a function of b.
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299
3. PROOFS OF THE MAIN RESULTS LEMMA 3.1. Assume L is right disfocal on w0, B x. Let x ky 1 - x k s b F B. Let 0 F ␣ 1 - ⭈⭈⭈ - ␣ n ky1 F n y 1. Let y be a nontri¨ ial solution of Ly s 0, x 1 - x - b, satisfying the n y 1 homogeneous boundary conditions y Ž l . Ž x i . s 0,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s 0,
l s ␣ 1 , . . . , ␣ n ky1 .
Ži. There exist x 1 s 1 s ⭈⭈⭈ s n - n q1 - ⭈⭈⭈ - nyn s nyn 1 1 k k ⭈⭈⭈ - n ky 1 s ⭈⭈⭈ s 1 s t ky1 such that y Ž jy1. Ž j . s y Ž jy1. Ž j . s 0, j s 1, . . . , n y n k , and y Ž jy1. Ž x . / 0 on w x 1 , j . j Ž j , x ky1 x. Žii. Let M s max i < i g 0, . . . , n y 14 : y Ž i. Ž b . / 04 . Then M G n y nk . Žiii. y Ž x . / 0,
x g Ž x i , x iq1 . , i s 1, . . . , k y 2,
y Ž l . Ž b . / 0,
l g 0, . . . , n y 1 4 _ ␣ 1 , . . . , ␣ n ky1 4 ,
y Ž l . Ž x . / 0,
x ky 1 - x - b, l s 0, . . . , min n y n k , ␣ 1 4 ,
y Ž n i . Ž x i . / 0,
i s 1, . . . , k y 1.
Živ. if i g 0, . . . , M y 14 , there exists ⑀ ) 0 such that y Ž i. Ž t . y Ž iq1. Ž t . - 0,
b y ⑀ - t - b, y Ž i. Ž b . s 0,
y Ž i. Ž t . y Ž iq1. Ž t . ) 0,
b y ⑀ - t - b, y Ž i. Ž b . / 0.
Proof. To obtain Ži., employ Rolle’s theorem repeatedly to the conditions y Ž l . Ž x i . s 0; l s 0, . . . , n i y 1; and i s 1, . . . , k y 1. y has at least n y n k zeros, counting multiplicity, at x 1 , . . . , x ky1 4 . By Rolle’s theorem, y⬘ has at least n y n k y 1 zeros in w x 1 , x ky1 x. Continue inductively and label as in statement Ži.. Peterson and Ridenhour w13, Lemma 1x obtained statements Žii., Žiii., and Živ. for the two-point n y n k , n k point boundary value problem. The arguments are by contradiction; with repeated applications of Rolle’s theorem one violates the hypothesis of right disfocality. If one replaces the boundary condition of Peterson and Ridenhour, y Ž i. Ž x ky 1 . s 0, i s 0, . . . , n y n k y 1, with the conditions y Ž jy1. Ž j . s 0,
j s 1, . . . , n y n k ,
then the argument of Peterson and Ridenhour applies and Žii. is obtained.
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If one replaces the boundary condition of Peterson and Ridenhour, y Ž i. Ž x ky 1 . s 0,
i s 0, . . . , n y n k y 1,
with the conditions y Ž jy1. Ž j . s 0,
j s 1, . . . , n y n k ,
then the arguments of Peterson and Ridenhour apply and Žiii., Živ., and the statement y Ž jy1. Ž x . / 0 on w x 1 , j . j Ž j , x ky1 x, j s 1, . . . , n y n k , are obtained. Technically, Peterson and Ridenhour do not obtain the two-point analogue of y Ž n i . Ž x i . / 0,
i s 1, . . . , k y 1.
However, precisely the same arguments apply and we do not provide the details. COROLLARY 3.2. Assume that y satisfies the hypotheses of Lemma 3.1. Assume y Ž l . Ž b . / 0 and let j s max i : ␣ i - l 4 . Then
Ž y1.
m iyn k qj
y Ž l . Ž b . y Ž x . ) 0,
j Ž y1. y Ž l . Ž b . y Ž i. Ž x . ) 0,
x g Ž x i , x iq1 . , i s 1, . . . , k y 2, x ky 1 - x - b, i g 0, . . . , min n y n k , ␣ 1 4 4 ,
Ž y1.
m iyn k qj
y Ž l . Ž b . y Ž n i . Ž x i . ) 0,
i s 1, . . . , k y 1.
Before we state and prove the next lemma, we introduce further notation. Let ␣ g ⍀ n k, and let SŽ ␣ . s ␣ 1 q ⭈⭈⭈ q␣ n k denote the sum of the components of ␣ . Thus, n k Ž n k y 1.r2 F SŽ ␣ . F n k Ž2 n y n k y 1.r2. Let i g 0, . . . , n y 14 and let nŽ ␣ , i . denote the number of components of ␣ which exceed i. In particular, nŽ ␣ , i . counts the number of derivatives specified by the boundary conditions, T Ž k, ␣ , b . y s 0, that are greater than i. THEOREM 3.3. Let ␣ g ⍀ n k, and let x ky1 - x k s b - B. If j g 0, . . . , n y 14 _ ␣ 1 , . . . , ␣ n k 4 then
Ž y1. n ␣ , j Gj Ž k, ␣ , b; b, s . ) 0, Ž
.
s g Ž x1 , b . .
Proof. The proof is by triple induction. We first induct on k. Eloe and Ridenhour w6, Lemma 2.4x obtained Theorem 3.3 in the case k s 2. Assume k ) 2 and assume that the truth of the assertion of Theorem 3.3
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301
is valid for all j-point BVPs, Ly s 0, x 1 - x - x j , T Ž j, ␣ , b . s 0, j s 2, . . . , k y 1. We now establish the assertion of Theorem 3.3 in the case j s k. We begin this argument with an induction on n y n k . If n i s 1, i s 1, . . . , k y 1, then n y n k s k y 1 or n k s n y k q 1. Finally, to establish the truth of the assertion of Theorem 3.3 for n y n k s k y 1, we induct on SŽ ␣ .. Set ␣ s Ž0, . . . , n y k . g ⍀ ny kq1 so that S Ž ␣ . s Ž n y k .Ž n y k q 1.r2. Let ␣ ˆ s Ž0, . . . , n y k, j . g ⍀ ny kq2 , j g n y k q 1, . . . , n y 14. We can employ the induction hypothesis on k to GŽ k y 1, ␣ ˆ , b; x, s .. By T Ž k y 1, ␣ ˆ , b . y s 0, the following argument is valid if we ignore the boundary condition at any x i , i g 1, . . . , k y 14 . So, we ignore the boundary condition at x ky 1 and assume that T Ž k y 1, ␣ ˆ , b . y s 0 means y Ž x i . s 0,
i s 1, . . . , k y 2,
y Ž l . Ž b . s 0, l s 0, . . . , n y k , j. Fix s g Ž x 1 , b . and define g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k y 1, ␣ ˆ , b; x, s .. Then, g is n times continuously differentiable is a solution of Ly s 0, x 1 - x - b, and satisfies the boundary conditions, g Ž x i . s 0,
i s 1, . . . , k y 2,
g Ž b . s 0,
l s 0, . . . , n y k,
Žl.
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . . Lemma 3.1 applies to g. By w1x, we know that Gny kq1Ž k, ␣ , b; b, s . ) 0. If j s n y k q 1, then Gny kq1Ž k y 1, ␣ ˆ , b; b, s . s 0 and g Ž nykq1. Ž b . ) 0. If j ) n y k q 1, then nŽ ␣ ˆ , n y k q 1. s 1 and it follows by induction on k that Gny kq1Ž k y 1, ␣ , ˆ b; b, s . - 0 and g Ž nykq1. Ž b . ) 0. Regardless, Ž nykq1. Ž . g b ) 0. By Lemma 3.1, g Ž i. Ž b . ) 0, i s n y k q 1, . . . , n y 1. In particular, g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . ) 0 and this completes the proof in the case SŽ ␣ . s Ž n y k .Ž n y k q 1.r2. Let Ž n y k .Ž n y k q 1.r2 - s F Ž n y k q 1.Ž n q k y 2.r2 and assume that the assertion of Theorem 3.3 is valid for all ␣ satisfying Ž n y k .Ž n y k q 1.r2 F SŽ ␣ . - s. Let ␣ s Ž ␣ 1 , . . . , ␣ nykq1 . g ⍀ nykq1 with SŽ ␣ . s s and let j g 0, . . . , n y 14 _ ␣ 1 , . . . , ␣ nykq1 4 . First, assume j - ␣ ny kq1. Find l such that ␣ ly1 - j - ␣ l Žset l s 1 if j - ␣ 1 . and let ␣ˆ s Ž ␣ 1 , . . . , ␣ ly 1 , j, ␣ lq 1 , . . . , ␣ ny kq 1 . g ⍀ ny kq 1 . Set g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k, ␣ ˆ , b; x, s .. Note that SŽ ␣ˆ . - SŽ ␣ . and so the induction hypothesis applies to GŽ k, ␣ ˆ , b; x, s .. Also note that nŽ ␣ , j . s nŽ ␣ ˆ , ␣ l . q 1. Then g satisfies the BVP, Lg s 0, x 1 - x - b, g Ž x i . s 0, g Ž i. Ž b . s 0,
i s 1, . . . , k y 1, i g ␣ 1 , . . . , ␣ nykq1 4 _ ␣ l 4 ,
g Ž ␣ l . Ž b . s yG␣ lŽ k, ␣ ˆ , b; b, s . .
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Thus,
Ž y1. n ␣ , j g Ž ␣ l . Ž b . s Ž y1. n ␣ , j q1 G␣ lŽ k, ␣ˆ , b; b, s . Ž
.
Ž
s Ž y1 . n
.
Ž␣ ˆ, ␣l.
G␣ lŽ k, ␣ ˆ , b; b, s . ) 0.
Apply Lemma 3.1 and we have that 0 - Ž y1 .
nŽ ␣ , j .
g Ž j. Ž b . s Ž y1 .
nŽ ␣ , j .
Gj Ž k , ␣ , b; b, s . .
The proof is complete for j - ␣ ny kq1. Now, assume j ) ␣ ny kq1. Set ␣ ˆ s Ž ␣ 1 , . . . , ␣ nykq1 , j . g ⍀ nykq2 . Thus, we can employ the induction hypothesis on k for GŽ k y 1, ␣ ˆ , b; x, s .. Set . g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k y 1, ␣ , b; x, s . Then g satisfies the BVP, ˆ Lg s 0, x 1 - x - b, g Ž x i . s 0, g Ž l . Ž b . s 0,
i s 1, . . . , k y 2, l g ␣ 1 , . . . , ␣ nykq1 4 ,
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . . Since S Ž ␣ . ) Ž n y k .Ž n y k q 1.r2, let i g 0, . . . , n y k 4 _ ␣ 1 , . . . , ␣ nykq1 4 . We have verified the truth of Theorem 3.3 in the case j - ␣ ny kq1; in particular, Žy1. nŽ ␣ , i. Gi Ž k, ␣ , b; b, s . ) 0. Moreover, Žy1. nŽ ␣ , i. Gi Ž k y 1, ␣ ˆ , b; b, s . - 0. Thus, Žy1. nŽ ␣ , i. g Ž i. Ž b . ) 0. By Lemma Ž i. Ž . 3.1, the sequence g b , . . . , g Ž j. Ž b . has Žy1. nŽ ␣ , i. sign changes; in particular, g j Ž b . s Gj Ž k, ␣ , b; b, s . ) 0. This completes the proof of Theorem 3.3 in the case n y n k s k y 1. Now assume that l ) k y 1 and assume the validity of Theorem 3.3 for all n y n k - l. Assume n y n k s l. The proof, again, is by induction on SŽ ␣ .. For S Ž ␣ . s n k Ž n k y 1.r2, set ␣ s Ž0, . . . , n k y 1. g ⍀ n k and ␣ ˆs Ž0, . . . , n k y 1, j . g ⍀ n q1 , where j G n k . For some i 0 g 1, . . . , k y 14 , n i k 0 ) 1. Let g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k, ␣ ˆ , b; x, s .. Then Lg s 0, x 1 - x - b, and g Ž l . Ž x j . s 0,
l s 0, . . . , n j y 1, j s 1, . . . , k y 1, i / i 0
g Ž l . Ž x i 0 . s 0,
l s 0, . . . , n i 0 y 2,
g Ž l . Ž b . s 0,
l s 0, . . . , n k y 1,
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . .
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303
By w1x, Gn kŽ k, ␣ , b; b, s . ) 0; by the induction hypothesis, Gn kŽ k, ␣ ˆ , b; b, s . F 0. Thus, g Ž n k . Ž b . ) 0. Apply Lemma 3.1 to obtain that 0 - g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . and the proof is complete for S Ž ␣ . s n k Ž n k y 1.r2. Now assume n k Ž n k y 1.r2 - s F n k Ž2 n y n k y 1.r2 and assume that the assertion of Theorem 3.3 is valid for all n k Ž n k y 1.r2 F SŽ ␣ . - s. Choose ␣ g ⍀ n k with SŽ ␣ . s s. We only outline the details here as they are completely analogous to those employed in the earlier induction arguments. If ␣ iy1 - j - ␣ i , or j - ␣ 1 let ␣ ˆ s Ž ␣ 1 , . . . , ␣ iy1 , j, ␣ iq1 , . . . , ␣ n k . g ⍀ n k. If j ) ␣ n k, let
␣ ˆ s Ž ␣ 1 , . . . , ␣ n k , j . g ⍀ n kq1 . This completes the proof of Theorem 3.3. Let ␣ g ⍀ n k, x ky1 - b - B, and let H Ž k, ␣ , b; x, s . s Ž ⭸r⭸ b . G Ž k, ␣ , b; x, s . . Eloe and Henderson w4x have obtained the following result. LEMMA 3.4. Let ␣ g ⍀ n k, x ky1 - b - B; and x 1 - s - b. Then, as a function of x, H Ž k, ␣ , b; x, s . g C nw x 1 , b x and is the unique solution of the BVP Ly s 0, y Ž l . Ž x i . s 0,
x 1 - x - b,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s yGlq1 Ž k, ␣ , b; b, s . ,
Let ␣ g ⍀ n k and assume that ␣ n k - n y 1. For x ky1
COROLLARY 3.5. - b1 - b 2 F B, 0 - Ž y1 .
l s ␣1 , . . . , ␣nk .
mi
G Ž k, ␣ , b1 ; x, s . - Ž y1 .
mi
G Ž k, ␣ , b 2 ; x, s . ,
mi
Gn iŽ k, ␣ , b 2 ; x i , s . ,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b1 ., i s 0, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b1 ; x i , s . - Ž y1 .
s g Ž x 1 , b1 ., i s 0, . . . , k; and n
n
0 - Ž y1 . k Gl Ž k, ␣ , b1 ; x, s . - Ž y1 . k Gl Ž k , ␣ , b 2 ; x, s . , Ž x, s . g Ž x ky 1 , b1 . = Ž x 1 , b1 ., l s 0, . . . , ␣ 1.
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Proof. Since H satisfies the BVP given in Lemma 3.4, H s Ý kls1 H Ž l ., where H Ž l . satisfies the BVP Ly s 0,
x 1 - x - b,
y Ž j. Ž x i . s 0,
j s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž j. Ž b . s 0,
j g ␣1 , . . . , ␣nk 4 _ ␣l 4 ,
y Ž ␣ l . Ž b . s yG␣ lq1 Ž ␣ , b; b, s . . If ␣ lq1 s ␣ l q 1, then G␣ lq1Ž ␣ , b; b, s . s 0 and H Ž l . ' 0. If ␣ lq1 ) ␣ l q 1, Žy1.Ž n kyl . G␣ lq1Ž ␣ , b; b, s . ) 0 is obtained from Theorem 3.3. Now apply Corollary 3.2 with j s l and y Ž l . Ž b . s Žy1.Ž n k . G␣ lq1Ž ␣ , b; b, s .. Then each Hl , for ␣ lq1 ) ␣ l q 1, and hence H satisfy 0 - Ž y1 .
mi
H Ž k, ␣ , b; x, s . ,
0 - Ž y1 .
mi
Hn iŽ k, ␣ , b; x, s . ,
0 - Ž y1 .
nk
Hl Ž k, ␣ , b; x, s . ,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . , s g Ž x1 , b . ,
Ž x, s . g Ž x ky1 , b . = Ž x 1 , b . ,
l s 0, . . . , ␣ 1. Corollary 3.5 now follows by the mean value theorem. It was pointed out in w6x that without further assumptions on the signs of the coefficients, a i , in Ž1.1., the condition ␣ n k - n y 1 cannot be removed. We refer the reader to the discussion on page 448 of w6x and to w2, 3, 5, 9, 10x. We now complete the proofs of Theorems 2.1 and 2.2. THEOREM 3.6. Assume ␣ 1 - n y n k , and let ␣ s ␣ 1 , . . . , ␣ 1 q n k y 14 . Then, for 0 - b F B, 0 - Ž y1 .
mi
G Ž k , ␣ , b; x, s . ,
Ž 3.1.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b ., i s 1, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . ,
Ž 3.2.
s g Ž x 1 , b ., i s 1, . . . , k; and n
0 - Ž y1 . k Gl Ž k, ␣ , b; x, s . ,
l s 0, . . . , ␣ 1 ,
Ž 3.3.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b .. Proof. We first argue that GŽ k, ␣ , b; x, s ., G␣ 1Ž k, ␣ , b; x, s ., and Gn iŽ k, ␣ , b; x i , s ., i s 1, . . . , k y 1, do not change sign in Ž x i , x iq1 . =
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Ž x 1 , b ., i s 1, . . . , k y 1; Ž x ky1 , b . = Ž x 1 , b .; and Ž x 1 , b ., respectively. The basis for the argument is standard and can be found in Coppel w1, pp. 106, 107x. Suppose for the sake of contradiction that for some c g Ž x ky 1 , b ., G␣ Ž k, ␣ , b; c, s . changes sign for s g Ž x 1 , b .. Find f g C w x 1 , b x 1 such that f Ž x . ) 0, x 1 - x - b, and b
Hx G
Ž k, ␣ , b; c, s . f Ž s . ds s 0.
␣1
1
If hŽ x . s Hxb1 G␣ 1Ž k, ␣ , b; x, s . f Ž s . ds, then h satisfies Lh s f, T Ž k, ␣ , b . h s 0, and hŽ ␣ 1 . Ž c . s 0. In particular, h has at least n y n k zeros, counting multiplicity on w x 1 , x ky1 x. An easy consequence of Rolle’s theorem is that, for j s 1, . . . , n y n k , hŽ j. has at least n y n k y j zeros, counting multiplicity on w x 1 , x ky1 x. If one chooses n y n k y j zeros let t j denote the smallest root of hŽ j. on w x 1 , x ky1 .. Moreover, label n y n k y ␣ 1 roots of hŽ ␣ 1 . in w x 1 , x ky1 x by t␣ 1 s c 0 F c1 F ⭈⭈⭈ F c nyn ky ␣ 1y1 F x ky1. Since hŽ ␣ 1 . also has n k q 1 roots, counting multiplicity, at c, x k , . . . , x k , Muldowney’s Mean Value Theorem w8, Corollary 1, p. 375x applies and Lh s f changes sign in Ž x 1 , x k .. ŽTo apply Muldowney’s result, assume right-Ž1, . . . , 1. invertibility w8, p. 373x and let
Ž t 0 ; . . . , t␣ y1 ; c0 , . . . , c nyn y ␣ y1 , c, x k , . . . x k . 1
k
1
denote an increasing partition of n q 1 points.. Since f ) 0, this is a contradiction and G␣ 1Ž k, ␣ , b; x, s . does not change sign in Ž x 1 , x k .. Similar applications of Muldowney’s Mean Value Theorem give that neither GŽ k, ␣ , b; x, s . nor Gn iŽ k, ␣ , b; x i , s ., i s 1, . . . , k y 1, change sign in Ž x i , x iq1 . = Ž x 1 , b . or Ž x 1 , b ., respectively. Now, suppose for some s g Ž x 1 , b ., that G␣ 1Ž k, ␣ , b; x, s . changes sign for x g Ž x ky 1 , b .. From the preceding paragraph, it follows that there exists c such that G␣ 1Ž k, ␣ , b; c, s . s 0. Muldowney’s Mean Value Theorem is again applied to obtain a contradiction. Similar contradictions are valid for GŽ k, ␣ , b; x, s . and Gn iŽ k, ␣ , b; x, s ., i s 1, . . . , k y 1. Thus, G␣ 1Ž k, ␣ , b; x, s . does not change sign on Ž x ky1 , b . = Ž x 1 , b .. We now verify Ž3.1.. By Theorem 3.3, G␣ 1qn kŽ k, ␣ , b; b, s . ) 0. Repeated applications of Taylor’s theorem give that Žy1. n k G␣ 1Ž k, ␣ , b; x, s . G 0 on Ž x ky 1 , b . = Ž x 1 , b .. Strict inequality follows from Corollary 3.5. To see this, let Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b . and let max x, s4 - b1 - b. Then n
n
0 F Ž y1 . k G␣ 1Ž k, ␣ , b1 ; x, s . - Ž y1 . k G␣ 1Ž k, ␣ , b; x, s . and Ž3.1. is verified. Similarly, Ž3.2. and Ž3.3. are verified.
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THEOREM 3.7.
Let ␣ , ␣ ˆ g ⍀ n k, ␣ - ␣ˆ , and x ky1 - b F B. Then
0 - Ž y1 .
mi
G Ž k , ␣ , b; x, s . - Ž y1 .
mi
GŽ k , ␣ ˆ , b; x, s . ,
Ž 3.4.
mi
Gn iŽ k , ␣ ˆ , b; x i , s . ,
Ž 3.5.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b ., i s 1, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . - Ž y1 .
s g Ž x 1 , b ., i s 1, . . . , k; and n
n
0 - Ž y1 . k Gl Ž k , ␣ , b; x, s . - Ž y1 . k Gl Ž k, ␣ ˆ , b; x, s . ,
Ž 3.6.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b ., l s 0, . . . , ␣ 1. Proof. Let ␣ , ␣ ˆ g ⍀ n k and first consider the case where there exists j g 1, . . . , n k 4 such that ␣ p s ␣ ˆp if p / j and ␣ˆj s ␣ j q 1. If we obtain Theorem 3.7 in this case, then the proof of Theorem 3.7 will follow inductively for any ␣ - ␣ ˆ. Let us return to the special case and note that ␣ 1 F n y n k y 1. Let s g Ž x 1 , b . and set g Ž x . s GŽ k, ␣ ˆ , b; x, s . y GŽ k, ␣ , b; x, s .. Then Lg Ž x . s 0, x 1 - x - b, and g Ž q. Ž x i . s 0,
q s 0, . . . , n i y 1, i s 1, . . . , k y 1,
g Ž q. Ž b . s 0,
q g ␣ ˆ1 , . . . , ␣ˆnyk 4 _ ␣ˆj 4 ,
g Ž ␣ˆ j . Ž b . s yG␣ˆ jŽ k, ␣ , b; b, s . . By Theorem 3.3 Žy1. n kyj g Ž ␣ˆ j . Ž b . - 0. By Lemma 3.1, Žy1. n k g Ž ␣ 1 . Ž b . ) 0, x ky 1 - x - b. In particular, Ž3.5. holds for l s ␣ 1. It follows from Lemma 3.1 and repeated integration from l to x that Ž3.5. holds for l s 0, . . . , ␣ 1. Now g Ž n i . Ž x i . / 0, i s 1, . . . , k y 1, by right disfocality. Thus, g has a root of multiplicity n i at x i and has no other roots. Thus, Ž3.4. follows from Ž3.5. and Ž3.6. follows from Ž3.5. by Taylor’s theorem. The proof of Theorem 3.7 is complete. Theorems 2.1 and 2.2 follow from Corollary 3.5 and Theorems 3.6 and 3.7.
REFERENCES 1. W. Coppel, ‘‘Disconjugacy,’’ Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, New YorkrBerlin, 1971. 2. U. Elias, Green’s functions for a nondisconjugate differential operator, J. Differential Equations 37 Ž1980., 319᎐350.
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3. U. Elias, ‘‘Oscillation Theory for Two-Term Differential Equations,’’ Kluwer, Boston, 1997. 4. P. W. Eloe and J. Henderson, Multipoint boundary value problems for ordinary differential systems, J. Differential Equations 114 Ž1994., 232᎐242. 5. P. W. Eloe and E. R. Kaufmann, A singular boundary value problem for a right disfocal linear differential operator, Dynam. Systems Appl. 5 Ž1996., 174᎐182. 6. P. W. Eloe and J. Ridenhour, Sign properties of Green’s functions for a family of two-point boundary value problems, Proc. Amer. Math. Soc. 120 Ž1994., 443᎐452. 7. J. S. Muldowney, A necessary and sufficient condition for disfocality, Proc. Amer. Math. Soc. 74 Ž1979., 49᎐55. 8. J. S. Muldowney, On invertibility of linear ordinary differential boundary value problems, SIAM J. Math. Anal. 12 Ž1981., 368᎐384. 9. Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 Ž1967., 500᎐516. 10. Z. Nehari, Green’s functions and disconjugacy, Arch. Ration. Mech. Anal. 62 Ž1976., 53᎐76. 11. A. Peterson, Green’s functions for focal type boundary value problems, Rocky Mountain J. Math. 9 Ž1979., 721᎐732. 12. A. Peterson, Focal Green’s functions for fourth-order differential equations, J. Math. Anal. Appl. 75 Ž1980., 602᎐610. 13. A. Peterson and J. Ridenhour, Comparison theorems for Green’s functions for focal boundary value problems, in ‘‘Recent Trends in Differential Equations,’’ World Scientific Series in Applicable Analysis, Vol. 1, pp. 493᎐506, World Scientific, River Edge, NJ, 1992.
Comparison of Green’s Functions for a Family of Multipoint Boundary Value Problems Paul W. Eloe and Lingqin Zhang Department of Mathematics, Uni¨ ersity of Dayton, Dayton, Ohio 45469-2316 Submitted by William F. Ames Received October 27, 1998
1. INTRODUCTION Let n G 2 be an integer and let k g 2, . . . , n4 be given. Let B ) 0, let 0 F x 1 - ⭈⭈⭈ - x ky1 - B be fixed throughout this paper, and let x ky1 x k s b F B be given. Let a i g C w0, B x, i s 1, . . . , n, and define the linear differential operator, L, by n
Ly s y Ž n. q
Ý ai y Ž nyi. .
Ž 1.1.
is1
Let n1 , . . . , n k denote positive integers such that Ý kls1 n l s n. Let W denote the set of nonnegative integers, and let ⍀ n k ; W n k be defined by ⍀ n k s ␣ s Ž ␣ 1 , . . . , ␣ n k . : 0 F ␣ 1 - ⭈⭈⭈ - ␣ n k F n y 1 4 . Define a partial order on ⍀ n k as follows: We say ␣ F  if for each i s 1, . . . , n k , ␣ i F i , and we say ␣ -  if ␣ F  and ␣ /  . For each ␣ g ⍀ n k, consider the homogeneous, k-point boundary conditions of the form y Ž l . Ž x i . s 0,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s 0,
l s ␣1 , . . . , ␣nk .
Ž 1.2.
We shall sometimes denote the boundary conditions Ž1.2. by T Ž k, ␣ , b . y s 0. Note that if ␣ s Ž0, . . . , n k y 1., then T Ž k, ␣ , b . y s 0 represents k-point conjugate boundary conditions w1x. 296 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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297
For each ␣ g ⍀ n k, let GŽ k, ␣ , b; x, s . denote Green’s function, if it exists, of the Boundary Value Problem ŽBVP., Ly s 0, x 1 - x b, T Ž k, ␣ , b . y s 0. For each l s 0, . . . , n, let Gl Ž k, ␣ , b; x, s . s Ž ⭸ lr⭸ x l . G Ž k , ␣ , b; x, s . . Throughout this paper we shall assume sufficient hypotheses such that G exists for each ␣ g ⍀ n k, x ky1 - b F B. The purpose of this paper is to obtain comparison theorems for the family of Green’s functions, GŽ k, ␣ , b; x, s .. For example, let ␣ s Ž0, . . . , n k y 1.. It is known w1x that if L is disconjugate on w x 1 , b x then m
Ž y1. i G Ž k , ␣ , b; x, s . ) 0,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . ,
i s 1, . . . , k y 1, where k
mi s
Ý
nj.
Ž 1.3.
jsiq1
We shall show, for example, that, under the assumption of right disfocality, if ␣ , ␣ ˆ g ⍀ n k, ␣ - ␣ˆ , then m
m
Ž y1. i G Ž k, ␣ˆ , b; x, s . ) Ž y1. i G Ž k, ␣ , b; x, s . ) 0, Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b .. Recall w7x that L, defined by Ž1.1., is right disfocal on w0, B x if the only solution of Ly s 0, 0 F x F B, satisfying y Ž j. Ž t j . s 0, j s 0, . . . , n y 1, where 0 F t 0 F ⭈⭈⭈ F t ny1 F B, is y ' 0. We shall assume throughout this paper that L is right disfocal on w0, B x and thus assume sufficient conditions for the existence of each GŽ k, ␣ , b; x, s .. It is actually the case that right disfocality is too strong an assumption; we assume right disfocality for simplicity. This paper represents an extension of the work by Eloe and Ridenhour w6x, who obtained comparison theorems for Green’s functions for two-point boundary value problems. The work by Eloe and Ridenhour w6x extended some of the results by Peterson w11, 12x and Elias w2, 3x for two-term differential equations where the coefficients have fixed sign, and some of the results by Peterson and Ridenhour w13x for two-term differential equations where the coefficients may change sign. We are not aware of analogous studies for two-term differential equations for multipoint boundary value problems. As this paper is an extension the work by Eloe and Ridenhour, the organization is similar. We shall state the main results in Theorems 2.1 and 2.2 in Section 2. We shall then provide the technical details in Section
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3. As is typical in many of these studies w2, 6, 13x, the primary observation is that the difference of two Green’s functions is a solution of Ly s 0 in x for fixed s. Hence, the technical details in Section 3 focus on determining the sign of solutions of Ly s 0 that satisfy n y 1 of the homogeneous boundary conditions given in Ž1.2..
2. THE MAIN RESULTS Assume L is right disfocal on w0, B x. Let ␣ , ␣ ˆ g ⍀ n k,
THEOREM 2.1. ␣-␣ ˆ. Then
mi
0 - Ž y1 .
G Ž k , ␣ , b; x, s . - Ž y1 .
mi
GŽ k , ␣ ˆ , b; x, s . ,
Ž 2.1.
mi
Gn iŽ k , ␣ ˆ , b; x i , s . ,
Ž 2.2.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . i s 1, . . . , k y 1, 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . - Ž y1 .
s g Ž x 1 , b ., i s 1, . . . , k, where m k s 0, and n
n
0 - Ž y1 . k Gl Ž k , ␣ , b; x, s . - Ž y1 . k Gl Ž k, ␣ ˆ , b; x, s . ,
Ž 2.3.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b ., l s 0, . . . , ␣ 1. THEOREM 2.2. Assume L is right disfocal on w0, B x. Let ␣ , ␣ ˆ g ⍀ n k, ␣ n k - n y 1. Assume ␣ F ␣ ˆ , b1 F b 2 , and assume that one of the inequalities, ␣ F ␣ ˆ , b1 F b 2 , is strict. Then mi
0 - Ž y1 .
G Ž k, ␣ , b1 ; x, s . - Ž y1 .
mi
G Ž k, ␣ ˆ , b 2 ; x, s . ,
Ž 2.4.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b1 ., i s 0, . . . , k y 1, 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b1 ; x i , s . - Ž y1 .
mi
Gn iŽ k , ␣ ˆ , b 2 ; x i , s . , Ž 2.5.
s g Ž x 1 , b1 ., i s 0, . . . , k, and n
n
0 - Ž y1 . k Gl Ž k, ␣ , b1 ; x, s . - Ž y1 . k Gl Ž k , ␣ ˆ , b 2 ; x, s . ,
Ž 2.6.
Ž x, s . g Ž x ky 1 , b1 . = Ž x 1 , b1 ., l s 0, . . . , ␣ 1. Remark. In regard to Theorem 2.1, we say that Žy1. m i GŽ k, ␣ , b; x, s . is monotone increasing as a function of ␣ . Similarly, in regard to Theorem 2.2, where ␣ n k - n y 1 is assumed, we say that Žy1. m i GŽ k, ␣ , b; x, s . is monotone increasing as a function of b.
COMPARISON OF GREEN’S FUNCTIONS
299
3. PROOFS OF THE MAIN RESULTS LEMMA 3.1. Assume L is right disfocal on w0, B x. Let x ky 1 - x k s b F B. Let 0 F ␣ 1 - ⭈⭈⭈ - ␣ n ky1 F n y 1. Let y be a nontri¨ ial solution of Ly s 0, x 1 - x - b, satisfying the n y 1 homogeneous boundary conditions y Ž l . Ž x i . s 0,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s 0,
l s ␣ 1 , . . . , ␣ n ky1 .
Ži. There exist x 1 s 1 s ⭈⭈⭈ s n - n q1 - ⭈⭈⭈ - nyn s nyn 1 1 k k ⭈⭈⭈ - n ky 1 s ⭈⭈⭈ s 1 s t ky1 such that y Ž jy1. Ž j . s y Ž jy1. Ž j . s 0, j s 1, . . . , n y n k , and y Ž jy1. Ž x . / 0 on w x 1 , j . j Ž j , x ky1 x. Žii. Let M s max i < i g 0, . . . , n y 14 : y Ž i. Ž b . / 04 . Then M G n y nk . Žiii. y Ž x . / 0,
x g Ž x i , x iq1 . , i s 1, . . . , k y 2,
y Ž l . Ž b . / 0,
l g 0, . . . , n y 1 4 _ ␣ 1 , . . . , ␣ n ky1 4 ,
y Ž l . Ž x . / 0,
x ky 1 - x - b, l s 0, . . . , min n y n k , ␣ 1 4 ,
y Ž n i . Ž x i . / 0,
i s 1, . . . , k y 1.
Živ. if i g 0, . . . , M y 14 , there exists ⑀ ) 0 such that y Ž i. Ž t . y Ž iq1. Ž t . - 0,
b y ⑀ - t - b, y Ž i. Ž b . s 0,
y Ž i. Ž t . y Ž iq1. Ž t . ) 0,
b y ⑀ - t - b, y Ž i. Ž b . / 0.
Proof. To obtain Ži., employ Rolle’s theorem repeatedly to the conditions y Ž l . Ž x i . s 0; l s 0, . . . , n i y 1; and i s 1, . . . , k y 1. y has at least n y n k zeros, counting multiplicity, at x 1 , . . . , x ky1 4 . By Rolle’s theorem, y⬘ has at least n y n k y 1 zeros in w x 1 , x ky1 x. Continue inductively and label as in statement Ži.. Peterson and Ridenhour w13, Lemma 1x obtained statements Žii., Žiii., and Živ. for the two-point n y n k , n k point boundary value problem. The arguments are by contradiction; with repeated applications of Rolle’s theorem one violates the hypothesis of right disfocality. If one replaces the boundary condition of Peterson and Ridenhour, y Ž i. Ž x ky 1 . s 0, i s 0, . . . , n y n k y 1, with the conditions y Ž jy1. Ž j . s 0,
j s 1, . . . , n y n k ,
then the argument of Peterson and Ridenhour applies and Žii. is obtained.
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ELOE AND ZHANG
If one replaces the boundary condition of Peterson and Ridenhour, y Ž i. Ž x ky 1 . s 0,
i s 0, . . . , n y n k y 1,
with the conditions y Ž jy1. Ž j . s 0,
j s 1, . . . , n y n k ,
then the arguments of Peterson and Ridenhour apply and Žiii., Živ., and the statement y Ž jy1. Ž x . / 0 on w x 1 , j . j Ž j , x ky1 x, j s 1, . . . , n y n k , are obtained. Technically, Peterson and Ridenhour do not obtain the two-point analogue of y Ž n i . Ž x i . / 0,
i s 1, . . . , k y 1.
However, precisely the same arguments apply and we do not provide the details. COROLLARY 3.2. Assume that y satisfies the hypotheses of Lemma 3.1. Assume y Ž l . Ž b . / 0 and let j s max i : ␣ i - l 4 . Then
Ž y1.
m iyn k qj
y Ž l . Ž b . y Ž x . ) 0,
j Ž y1. y Ž l . Ž b . y Ž i. Ž x . ) 0,
x g Ž x i , x iq1 . , i s 1, . . . , k y 2, x ky 1 - x - b, i g 0, . . . , min n y n k , ␣ 1 4 4 ,
Ž y1.
m iyn k qj
y Ž l . Ž b . y Ž n i . Ž x i . ) 0,
i s 1, . . . , k y 1.
Before we state and prove the next lemma, we introduce further notation. Let ␣ g ⍀ n k, and let SŽ ␣ . s ␣ 1 q ⭈⭈⭈ q␣ n k denote the sum of the components of ␣ . Thus, n k Ž n k y 1.r2 F SŽ ␣ . F n k Ž2 n y n k y 1.r2. Let i g 0, . . . , n y 14 and let nŽ ␣ , i . denote the number of components of ␣ which exceed i. In particular, nŽ ␣ , i . counts the number of derivatives specified by the boundary conditions, T Ž k, ␣ , b . y s 0, that are greater than i. THEOREM 3.3. Let ␣ g ⍀ n k, and let x ky1 - x k s b - B. If j g 0, . . . , n y 14 _ ␣ 1 , . . . , ␣ n k 4 then
Ž y1. n ␣ , j Gj Ž k, ␣ , b; b, s . ) 0, Ž
.
s g Ž x1 , b . .
Proof. The proof is by triple induction. We first induct on k. Eloe and Ridenhour w6, Lemma 2.4x obtained Theorem 3.3 in the case k s 2. Assume k ) 2 and assume that the truth of the assertion of Theorem 3.3
COMPARISON OF GREEN’S FUNCTIONS
301
is valid for all j-point BVPs, Ly s 0, x 1 - x - x j , T Ž j, ␣ , b . s 0, j s 2, . . . , k y 1. We now establish the assertion of Theorem 3.3 in the case j s k. We begin this argument with an induction on n y n k . If n i s 1, i s 1, . . . , k y 1, then n y n k s k y 1 or n k s n y k q 1. Finally, to establish the truth of the assertion of Theorem 3.3 for n y n k s k y 1, we induct on SŽ ␣ .. Set ␣ s Ž0, . . . , n y k . g ⍀ ny kq1 so that S Ž ␣ . s Ž n y k .Ž n y k q 1.r2. Let ␣ ˆ s Ž0, . . . , n y k, j . g ⍀ ny kq2 , j g n y k q 1, . . . , n y 14. We can employ the induction hypothesis on k to GŽ k y 1, ␣ ˆ , b; x, s .. By T Ž k y 1, ␣ ˆ , b . y s 0, the following argument is valid if we ignore the boundary condition at any x i , i g 1, . . . , k y 14 . So, we ignore the boundary condition at x ky 1 and assume that T Ž k y 1, ␣ ˆ , b . y s 0 means y Ž x i . s 0,
i s 1, . . . , k y 2,
y Ž l . Ž b . s 0, l s 0, . . . , n y k , j. Fix s g Ž x 1 , b . and define g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k y 1, ␣ ˆ , b; x, s .. Then, g is n times continuously differentiable is a solution of Ly s 0, x 1 - x - b, and satisfies the boundary conditions, g Ž x i . s 0,
i s 1, . . . , k y 2,
g Ž b . s 0,
l s 0, . . . , n y k,
Žl.
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . . Lemma 3.1 applies to g. By w1x, we know that Gny kq1Ž k, ␣ , b; b, s . ) 0. If j s n y k q 1, then Gny kq1Ž k y 1, ␣ ˆ , b; b, s . s 0 and g Ž nykq1. Ž b . ) 0. If j ) n y k q 1, then nŽ ␣ ˆ , n y k q 1. s 1 and it follows by induction on k that Gny kq1Ž k y 1, ␣ , ˆ b; b, s . - 0 and g Ž nykq1. Ž b . ) 0. Regardless, Ž nykq1. Ž . g b ) 0. By Lemma 3.1, g Ž i. Ž b . ) 0, i s n y k q 1, . . . , n y 1. In particular, g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . ) 0 and this completes the proof in the case SŽ ␣ . s Ž n y k .Ž n y k q 1.r2. Let Ž n y k .Ž n y k q 1.r2 - s F Ž n y k q 1.Ž n q k y 2.r2 and assume that the assertion of Theorem 3.3 is valid for all ␣ satisfying Ž n y k .Ž n y k q 1.r2 F SŽ ␣ . - s. Let ␣ s Ž ␣ 1 , . . . , ␣ nykq1 . g ⍀ nykq1 with SŽ ␣ . s s and let j g 0, . . . , n y 14 _ ␣ 1 , . . . , ␣ nykq1 4 . First, assume j - ␣ ny kq1. Find l such that ␣ ly1 - j - ␣ l Žset l s 1 if j - ␣ 1 . and let ␣ˆ s Ž ␣ 1 , . . . , ␣ ly 1 , j, ␣ lq 1 , . . . , ␣ ny kq 1 . g ⍀ ny kq 1 . Set g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k, ␣ ˆ , b; x, s .. Note that SŽ ␣ˆ . - SŽ ␣ . and so the induction hypothesis applies to GŽ k, ␣ ˆ , b; x, s .. Also note that nŽ ␣ , j . s nŽ ␣ ˆ , ␣ l . q 1. Then g satisfies the BVP, Lg s 0, x 1 - x - b, g Ž x i . s 0, g Ž i. Ž b . s 0,
i s 1, . . . , k y 1, i g ␣ 1 , . . . , ␣ nykq1 4 _ ␣ l 4 ,
g Ž ␣ l . Ž b . s yG␣ lŽ k, ␣ ˆ , b; b, s . .
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Thus,
Ž y1. n ␣ , j g Ž ␣ l . Ž b . s Ž y1. n ␣ , j q1 G␣ lŽ k, ␣ˆ , b; b, s . Ž
.
Ž
s Ž y1 . n
.
Ž␣ ˆ, ␣l.
G␣ lŽ k, ␣ ˆ , b; b, s . ) 0.
Apply Lemma 3.1 and we have that 0 - Ž y1 .
nŽ ␣ , j .
g Ž j. Ž b . s Ž y1 .
nŽ ␣ , j .
Gj Ž k , ␣ , b; b, s . .
The proof is complete for j - ␣ ny kq1. Now, assume j ) ␣ ny kq1. Set ␣ ˆ s Ž ␣ 1 , . . . , ␣ nykq1 , j . g ⍀ nykq2 . Thus, we can employ the induction hypothesis on k for GŽ k y 1, ␣ ˆ , b; x, s .. Set . g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k y 1, ␣ , b; x, s . Then g satisfies the BVP, ˆ Lg s 0, x 1 - x - b, g Ž x i . s 0, g Ž l . Ž b . s 0,
i s 1, . . . , k y 2, l g ␣ 1 , . . . , ␣ nykq1 4 ,
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . . Since S Ž ␣ . ) Ž n y k .Ž n y k q 1.r2, let i g 0, . . . , n y k 4 _ ␣ 1 , . . . , ␣ nykq1 4 . We have verified the truth of Theorem 3.3 in the case j - ␣ ny kq1; in particular, Žy1. nŽ ␣ , i. Gi Ž k, ␣ , b; b, s . ) 0. Moreover, Žy1. nŽ ␣ , i. Gi Ž k y 1, ␣ ˆ , b; b, s . - 0. Thus, Žy1. nŽ ␣ , i. g Ž i. Ž b . ) 0. By Lemma Ž i. Ž . 3.1, the sequence g b , . . . , g Ž j. Ž b . has Žy1. nŽ ␣ , i. sign changes; in particular, g j Ž b . s Gj Ž k, ␣ , b; b, s . ) 0. This completes the proof of Theorem 3.3 in the case n y n k s k y 1. Now assume that l ) k y 1 and assume the validity of Theorem 3.3 for all n y n k - l. Assume n y n k s l. The proof, again, is by induction on SŽ ␣ .. For S Ž ␣ . s n k Ž n k y 1.r2, set ␣ s Ž0, . . . , n k y 1. g ⍀ n k and ␣ ˆs Ž0, . . . , n k y 1, j . g ⍀ n q1 , where j G n k . For some i 0 g 1, . . . , k y 14 , n i k 0 ) 1. Let g Ž t . s GŽ k, ␣ , b; x, s . y GŽ k, ␣ ˆ , b; x, s .. Then Lg s 0, x 1 - x - b, and g Ž l . Ž x j . s 0,
l s 0, . . . , n j y 1, j s 1, . . . , k y 1, i / i 0
g Ž l . Ž x i 0 . s 0,
l s 0, . . . , n i 0 y 2,
g Ž l . Ž b . s 0,
l s 0, . . . , n k y 1,
g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . .
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303
By w1x, Gn kŽ k, ␣ , b; b, s . ) 0; by the induction hypothesis, Gn kŽ k, ␣ ˆ , b; b, s . F 0. Thus, g Ž n k . Ž b . ) 0. Apply Lemma 3.1 to obtain that 0 - g Ž j. Ž b . s Gj Ž k, ␣ , b; b, s . and the proof is complete for S Ž ␣ . s n k Ž n k y 1.r2. Now assume n k Ž n k y 1.r2 - s F n k Ž2 n y n k y 1.r2 and assume that the assertion of Theorem 3.3 is valid for all n k Ž n k y 1.r2 F SŽ ␣ . - s. Choose ␣ g ⍀ n k with SŽ ␣ . s s. We only outline the details here as they are completely analogous to those employed in the earlier induction arguments. If ␣ iy1 - j - ␣ i , or j - ␣ 1 let ␣ ˆ s Ž ␣ 1 , . . . , ␣ iy1 , j, ␣ iq1 , . . . , ␣ n k . g ⍀ n k. If j ) ␣ n k, let
␣ ˆ s Ž ␣ 1 , . . . , ␣ n k , j . g ⍀ n kq1 . This completes the proof of Theorem 3.3. Let ␣ g ⍀ n k, x ky1 - b - B, and let H Ž k, ␣ , b; x, s . s Ž ⭸r⭸ b . G Ž k, ␣ , b; x, s . . Eloe and Henderson w4x have obtained the following result. LEMMA 3.4. Let ␣ g ⍀ n k, x ky1 - b - B; and x 1 - s - b. Then, as a function of x, H Ž k, ␣ , b; x, s . g C nw x 1 , b x and is the unique solution of the BVP Ly s 0, y Ž l . Ž x i . s 0,
x 1 - x - b,
l s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž l . Ž b . s yGlq1 Ž k, ␣ , b; b, s . ,
Let ␣ g ⍀ n k and assume that ␣ n k - n y 1. For x ky1
COROLLARY 3.5. - b1 - b 2 F B, 0 - Ž y1 .
l s ␣1 , . . . , ␣nk .
mi
G Ž k, ␣ , b1 ; x, s . - Ž y1 .
mi
G Ž k, ␣ , b 2 ; x, s . ,
mi
Gn iŽ k, ␣ , b 2 ; x i , s . ,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b1 ., i s 0, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b1 ; x i , s . - Ž y1 .
s g Ž x 1 , b1 ., i s 0, . . . , k; and n
n
0 - Ž y1 . k Gl Ž k, ␣ , b1 ; x, s . - Ž y1 . k Gl Ž k , ␣ , b 2 ; x, s . , Ž x, s . g Ž x ky 1 , b1 . = Ž x 1 , b1 ., l s 0, . . . , ␣ 1.
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Proof. Since H satisfies the BVP given in Lemma 3.4, H s Ý kls1 H Ž l ., where H Ž l . satisfies the BVP Ly s 0,
x 1 - x - b,
y Ž j. Ž x i . s 0,
j s 0, . . . , n i y 1, i s 1, . . . , k y 1,
y Ž j. Ž b . s 0,
j g ␣1 , . . . , ␣nk 4 _ ␣l 4 ,
y Ž ␣ l . Ž b . s yG␣ lq1 Ž ␣ , b; b, s . . If ␣ lq1 s ␣ l q 1, then G␣ lq1Ž ␣ , b; b, s . s 0 and H Ž l . ' 0. If ␣ lq1 ) ␣ l q 1, Žy1.Ž n kyl . G␣ lq1Ž ␣ , b; b, s . ) 0 is obtained from Theorem 3.3. Now apply Corollary 3.2 with j s l and y Ž l . Ž b . s Žy1.Ž n k . G␣ lq1Ž ␣ , b; b, s .. Then each Hl , for ␣ lq1 ) ␣ l q 1, and hence H satisfy 0 - Ž y1 .
mi
H Ž k, ␣ , b; x, s . ,
0 - Ž y1 .
mi
Hn iŽ k, ␣ , b; x, s . ,
0 - Ž y1 .
nk
Hl Ž k, ␣ , b; x, s . ,
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b . , s g Ž x1 , b . ,
Ž x, s . g Ž x ky1 , b . = Ž x 1 , b . ,
l s 0, . . . , ␣ 1. Corollary 3.5 now follows by the mean value theorem. It was pointed out in w6x that without further assumptions on the signs of the coefficients, a i , in Ž1.1., the condition ␣ n k - n y 1 cannot be removed. We refer the reader to the discussion on page 448 of w6x and to w2, 3, 5, 9, 10x. We now complete the proofs of Theorems 2.1 and 2.2. THEOREM 3.6. Assume ␣ 1 - n y n k , and let ␣ s ␣ 1 , . . . , ␣ 1 q n k y 14 . Then, for 0 - b F B, 0 - Ž y1 .
mi
G Ž k , ␣ , b; x, s . ,
Ž 3.1.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b ., i s 1, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . ,
Ž 3.2.
s g Ž x 1 , b ., i s 1, . . . , k; and n
0 - Ž y1 . k Gl Ž k, ␣ , b; x, s . ,
l s 0, . . . , ␣ 1 ,
Ž 3.3.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b .. Proof. We first argue that GŽ k, ␣ , b; x, s ., G␣ 1Ž k, ␣ , b; x, s ., and Gn iŽ k, ␣ , b; x i , s ., i s 1, . . . , k y 1, do not change sign in Ž x i , x iq1 . =
COMPARISON OF GREEN’S FUNCTIONS
305
Ž x 1 , b ., i s 1, . . . , k y 1; Ž x ky1 , b . = Ž x 1 , b .; and Ž x 1 , b ., respectively. The basis for the argument is standard and can be found in Coppel w1, pp. 106, 107x. Suppose for the sake of contradiction that for some c g Ž x ky 1 , b ., G␣ Ž k, ␣ , b; c, s . changes sign for s g Ž x 1 , b .. Find f g C w x 1 , b x 1 such that f Ž x . ) 0, x 1 - x - b, and b
Hx G
Ž k, ␣ , b; c, s . f Ž s . ds s 0.
␣1
1
If hŽ x . s Hxb1 G␣ 1Ž k, ␣ , b; x, s . f Ž s . ds, then h satisfies Lh s f, T Ž k, ␣ , b . h s 0, and hŽ ␣ 1 . Ž c . s 0. In particular, h has at least n y n k zeros, counting multiplicity on w x 1 , x ky1 x. An easy consequence of Rolle’s theorem is that, for j s 1, . . . , n y n k , hŽ j. has at least n y n k y j zeros, counting multiplicity on w x 1 , x ky1 x. If one chooses n y n k y j zeros let t j denote the smallest root of hŽ j. on w x 1 , x ky1 .. Moreover, label n y n k y ␣ 1 roots of hŽ ␣ 1 . in w x 1 , x ky1 x by t␣ 1 s c 0 F c1 F ⭈⭈⭈ F c nyn ky ␣ 1y1 F x ky1. Since hŽ ␣ 1 . also has n k q 1 roots, counting multiplicity, at c, x k , . . . , x k , Muldowney’s Mean Value Theorem w8, Corollary 1, p. 375x applies and Lh s f changes sign in Ž x 1 , x k .. ŽTo apply Muldowney’s result, assume right-Ž1, . . . , 1. invertibility w8, p. 373x and let
Ž t 0 ; . . . , t␣ y1 ; c0 , . . . , c nyn y ␣ y1 , c, x k , . . . x k . 1
k
1
denote an increasing partition of n q 1 points.. Since f ) 0, this is a contradiction and G␣ 1Ž k, ␣ , b; x, s . does not change sign in Ž x 1 , x k .. Similar applications of Muldowney’s Mean Value Theorem give that neither GŽ k, ␣ , b; x, s . nor Gn iŽ k, ␣ , b; x i , s ., i s 1, . . . , k y 1, change sign in Ž x i , x iq1 . = Ž x 1 , b . or Ž x 1 , b ., respectively. Now, suppose for some s g Ž x 1 , b ., that G␣ 1Ž k, ␣ , b; x, s . changes sign for x g Ž x ky 1 , b .. From the preceding paragraph, it follows that there exists c such that G␣ 1Ž k, ␣ , b; c, s . s 0. Muldowney’s Mean Value Theorem is again applied to obtain a contradiction. Similar contradictions are valid for GŽ k, ␣ , b; x, s . and Gn iŽ k, ␣ , b; x, s ., i s 1, . . . , k y 1. Thus, G␣ 1Ž k, ␣ , b; x, s . does not change sign on Ž x ky1 , b . = Ž x 1 , b .. We now verify Ž3.1.. By Theorem 3.3, G␣ 1qn kŽ k, ␣ , b; b, s . ) 0. Repeated applications of Taylor’s theorem give that Žy1. n k G␣ 1Ž k, ␣ , b; x, s . G 0 on Ž x ky 1 , b . = Ž x 1 , b .. Strict inequality follows from Corollary 3.5. To see this, let Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b . and let max x, s4 - b1 - b. Then n
n
0 F Ž y1 . k G␣ 1Ž k, ␣ , b1 ; x, s . - Ž y1 . k G␣ 1Ž k, ␣ , b; x, s . and Ž3.1. is verified. Similarly, Ž3.2. and Ž3.3. are verified.
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THEOREM 3.7.
Let ␣ , ␣ ˆ g ⍀ n k, ␣ - ␣ˆ , and x ky1 - b F B. Then
0 - Ž y1 .
mi
G Ž k , ␣ , b; x, s . - Ž y1 .
mi
GŽ k , ␣ ˆ , b; x, s . ,
Ž 3.4.
mi
Gn iŽ k , ␣ ˆ , b; x i , s . ,
Ž 3.5.
Ž x, s . g Ž x i , x iq1 . = Ž x 1 , b ., i s 1, . . . , k y 1; 0 - Ž y1 .
mi
Gn iŽ k, ␣ , b; x i , s . - Ž y1 .
s g Ž x 1 , b ., i s 1, . . . , k; and n
n
0 - Ž y1 . k Gl Ž k , ␣ , b; x, s . - Ž y1 . k Gl Ž k, ␣ ˆ , b; x, s . ,
Ž 3.6.
Ž x, s . g Ž x ky 1 , b . = Ž x 1 , b ., l s 0, . . . , ␣ 1. Proof. Let ␣ , ␣ ˆ g ⍀ n k and first consider the case where there exists j g 1, . . . , n k 4 such that ␣ p s ␣ ˆp if p / j and ␣ˆj s ␣ j q 1. If we obtain Theorem 3.7 in this case, then the proof of Theorem 3.7 will follow inductively for any ␣ - ␣ ˆ. Let us return to the special case and note that ␣ 1 F n y n k y 1. Let s g Ž x 1 , b . and set g Ž x . s GŽ k, ␣ ˆ , b; x, s . y GŽ k, ␣ , b; x, s .. Then Lg Ž x . s 0, x 1 - x - b, and g Ž q. Ž x i . s 0,
q s 0, . . . , n i y 1, i s 1, . . . , k y 1,
g Ž q. Ž b . s 0,
q g ␣ ˆ1 , . . . , ␣ˆnyk 4 _ ␣ˆj 4 ,
g Ž ␣ˆ j . Ž b . s yG␣ˆ jŽ k, ␣ , b; b, s . . By Theorem 3.3 Žy1. n kyj g Ž ␣ˆ j . Ž b . - 0. By Lemma 3.1, Žy1. n k g Ž ␣ 1 . Ž b . ) 0, x ky 1 - x - b. In particular, Ž3.5. holds for l s ␣ 1. It follows from Lemma 3.1 and repeated integration from l to x that Ž3.5. holds for l s 0, . . . , ␣ 1. Now g Ž n i . Ž x i . / 0, i s 1, . . . , k y 1, by right disfocality. Thus, g has a root of multiplicity n i at x i and has no other roots. Thus, Ž3.4. follows from Ž3.5. and Ž3.6. follows from Ž3.5. by Taylor’s theorem. The proof of Theorem 3.7 is complete. Theorems 2.1 and 2.2 follow from Corollary 3.5 and Theorems 3.6 and 3.7.
REFERENCES 1. W. Coppel, ‘‘Disconjugacy,’’ Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, New YorkrBerlin, 1971. 2. U. Elias, Green’s functions for a nondisconjugate differential operator, J. Differential Equations 37 Ž1980., 319᎐350.
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3. U. Elias, ‘‘Oscillation Theory for Two-Term Differential Equations,’’ Kluwer, Boston, 1997. 4. P. W. Eloe and J. Henderson, Multipoint boundary value problems for ordinary differential systems, J. Differential Equations 114 Ž1994., 232᎐242. 5. P. W. Eloe and E. R. Kaufmann, A singular boundary value problem for a right disfocal linear differential operator, Dynam. Systems Appl. 5 Ž1996., 174᎐182. 6. P. W. Eloe and J. Ridenhour, Sign properties of Green’s functions for a family of two-point boundary value problems, Proc. Amer. Math. Soc. 120 Ž1994., 443᎐452. 7. J. S. Muldowney, A necessary and sufficient condition for disfocality, Proc. Amer. Math. Soc. 74 Ž1979., 49᎐55. 8. J. S. Muldowney, On invertibility of linear ordinary differential boundary value problems, SIAM J. Math. Anal. 12 Ž1981., 368᎐384. 9. Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 Ž1967., 500᎐516. 10. Z. Nehari, Green’s functions and disconjugacy, Arch. Ration. Mech. Anal. 62 Ž1976., 53᎐76. 11. A. Peterson, Green’s functions for focal type boundary value problems, Rocky Mountain J. Math. 9 Ž1979., 721᎐732. 12. A. Peterson, Focal Green’s functions for fourth-order differential equations, J. Math. Anal. Appl. 75 Ž1980., 602᎐610. 13. A. Peterson and J. Ridenhour, Comparison theorems for Green’s functions for focal boundary value problems, in ‘‘Recent Trends in Differential Equations,’’ World Scientific Series in Applicable Analysis, Vol. 1, pp. 493᎐506, World Scientific, River Edge, NJ, 1992.