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On an extension of Sturm's comparison theorem to a class of nonselfadjoint second-order systems
Nonlinear
Analysis,
Theory,
Methods
& Applicarrons,
0362.546X/80/0501~97
Vol. 4, No. 3. pp. 497-501
102.00/O
@ Pergamon Press Ltd. 1980. Printed m Great Britain
ON AN
EXTENSION
OF STURM’S
TO A CLASS OF NONSELFADJOINT
COMPARISON SECOND-ORDER
THEOREM SYSTEMS
SHAIR AHMAD* Oklahoma State University, Stillwater, OK 74074, U.S.A
and ALAN C. LAZER? University of Cincinnati, Cincinnati, OH 45221 (Received 1 June 1979) Keywords: Conjugate paints, adjoint system, irreducible matrix
THE PURPOSE of this note is to give an extension
of Sturm’s comparison theorem to a class of second-order linear differential systems considered in [l-5]. In the following theorems, which were proved in [4], x and y are n-dimensional column vectors. THEOREM A. L,et A(t) and B(t) be continuous real n x n matrix-valued functions defined for a < t < b such that if A(t) = (uij(t)) and B(t) = (bij(t)) then a,,(t) 2 bij(t) 2 0 for 1 < i, j d n and for t E [a, b]. Assume that for some f E [a, b], uij(f) > b,i(t), 1 < i, j d n. If there exists a nontrivial solution of
x”(t) + B(t) x(t) = 0 such that x(u) = x(b) = 0 then there exists a nontrivial
solution
(1) of
y”(t) + A@)y(t) = 0 such that y(u) = y(c) = 0
with a < c < b.
B(t) = (bij(t)) is continuous on [a, 2 0 for 1 < i, ,j < n. If there exists a nontrivial solution x(t) of (1) with x(u) if there exists no nontrivial solution x(t) of (1) satisfying x(u) = x(c) = 0 with there exists a nontrivial solution u(t) = co@,(t), . . . , u,(t)) of (1) satisfying u(u) u,(t) > 0 for a < t < b and 1 < k < n.
THEOREM B. Assume that the n x n matrix function
bij(t) and then and
(2)
b] and that = x(b) = 0 u < c -C b,
= u(b) = 0
Actually, these theorems are valid under the seemingly weaker condition that only the offdiagonal elements of B(t) are nonnegative. However, because of certain transformation of variables discussed in [4], it is no less general to assume that all of the elements of B(t) are nonnegative. In order to state the main result of this note we recall two definitions. Let A = (u,i) be a real n x n matrix. We recall that A is reducible if there exist nonempty subsets This research was partially supported by grants (MCS78-01480)* and (MCS77-25147)t. 491
498
SHAM AHMAD~III~ AII\Y C. LAZER
I and J of {1,2,..., n}suchthatInJ=cD,{1,2,...,n}=I~J,andu,~=Oifi~I,j~J. Otherwise A is said to be irreducible. It is easy to see that if A is irreducible the transpose of A is also irreducible. Let C(t) be an n x n matrix-valued function defined and continuous for t, G t < m. If there exists a nontrivial solution z(t) of the vector differential equation z”(f) + C(t)z(t) = 0
(3)
such that z(t,) = z(t,) = 0 with 1, > t, and if t, is the smallest number larger than t, with this property, then t, is called the first point conjugate to t, relative to (3). An elementary compactness argument shows that if there exists a nontrivial solution of (3) all of whose components vanish at t, and also at another point in (t,, co), then the first conjugate point oft, relative to (3) exists. The main result of this note is: THEOREM C. Let the IZ x II matrix function B(t) = (bij(t)) lx continuous and let t, be the first point conjugate to t, relative to (1). Suppose that the matrix B(t) is irreducible at some point in the interval (t,, ti). If A(t) = (uij(t)) IS another n x n continuous matrix function satisfying aij(t) 3 b,,(t) 3 0 for 1 d i, j d n and t, d t 6 f,, and if there exists a pair of integers k and 1 such that u,,(Z) > b$) for some 1 E (to, t,), then 1,, the first point conjugate to f, relative to (2), satisfies f, < t,.
A simple instance of when B(t) will be irreducible at a point of (to, fi) is when all the elements of B(t) are positive at some point of (t,, ti). We shall prove Theorem C with the aid of Theorems A and B and some further preliminary results. PROPOSITION1. Let C(r) be continuous for t 3 t, and suppose that the first conjugate relative to (3) exists. Then, if C’(t) denotes the transpose of C(r), the first conjugate relative to the ‘adjoint-system,’
z”(f) + P(t)z(t) exists and is equal to the first conjugate
point oft,
= 0 relative
point oft, point of r.
(3*) to (3).
Proof: If f, denotes the first point conjugate to t, relative to (3) then there exists a nontrivial solution z(t) of (3) such that z(t,,) = z(ti) = 0. By the uniqueness theorem, z’(t,) # 0. Let w(t) be the n x II matrix function such that
W”(f) + CT(t)W(t) = 0, W(t,) = 0, W’(q)) = I
(4)
where I is the identity matrix. We claim that lV(t,) is a singular matrix. Assuming the contrary, there exists a vector b such that kV(t,) b = z’(tl). If w(t) = w(t) b, then the row vector w(t)’ satisfies the differential equation w”(t)T + w(t)7.C(t) = 0
(5)
On an extension
and the boundary
of Sturm’s comparison
to a class of nonselfadjoint
second-order
systems
499
conditions
w(tJ Multiplying
theorem
the equation
= 0, w(tJT = z’(tJT.
(5) on the right by z(t) and multiplying
(6) (3) on the left by w(t)‘, we obtain
w”(t)Tz(t) = - w(t)TC(t)z(t) = W(t)Tz”(t). Hence, g
(w’(t)‘z(t) - wwz’(t)) = w”(t)Tz(t)- w(t)‘z”(t) = 0.
Therefore, since z(t,) = 0 and wT(t,) = 0, it follows that w’(t)Tz(t) - w(t)%‘(t) = 0 for all t. Therefore, since, z(t,) = 0 and w(t,)T = z’(t,)T, it follows that 0 = - z’(tl)“z’(tl). Hence z’(tl) = 0, which is a contradiction. This contradiction shows that the matrix W(t,) is singular. Thus, there exists a nonzero column vector d such that W(t,)d = 0. If u(t) = W(t)d then it follows from (4) that u”(t) + C(t)Tv(t) = 0, u(t,) = 0, u’(tJ = d # 0, and u(tl) = 0. Therefore, it follows that the first conjugate point oft, relative to the differential equation (3*) exists and if this point is denoted by tt, then t: Q t,. Since (C(t)‘)’ = C(t), it follows from a repetition of the above argument, with C(t) replaced by C(r)T, that t, d t,.* Hence, t, = t: and the result is proved. n x n matrix function which satisfies b,,(t) > 0 PROPOSITION 2. Let B(t) = (bij(t)) be a continuous for t 2 t,. If i 1 is the first conjugate point oft, relative to (1) and if there exists a number s E (t,, tl) such that B(s) is irreducible, then there exists a solution u = col(u,, . . . , u,,) of (1) such that u(t,) = u(tl) = 0 and u,(t) > 0 for k = 1,. . . ,n and t, < t < t,. Proof According to Theorem B, there exists a nontrivial solution u = col(u,, . . , u,,) of (1) such that u(t,) = u(t,) = 0 and u,(t) 2 0 fort, d t d t,. From (1) we see that for k = 1, . . . , n and t, < t < t,, u;(t) = - i bkj@)Uj@)< 0. j=l
Therefore, since uk(t,) = u&t,) = 0 and t+(t) > 0 for t, 6 t < t,, it follows that either u,(t) > 0 for all t E (t,, tl) or u,(t) = 0 for all t E (t,, tl). We define subsets I and .Z of { 1,2,. . . , nl as follows: k EZ if u,(t) = 0 for all t E (t,,, tI), k E J if udr) > 0 for all t E (t,, tl). If i E I, then u:‘(i) = 0 for all t E (to, tJ, so 0 = -u;‘(s)
=
x_bij(S)Uj(S). js.J
Since bij(s) 2 0 and uj(s) > 0 forj E J, it follows that bij(s) = 0 for i E Z,j E J. Since B(s) is irreducible and I n J = Q it follows that either Z = 0 or J = a. If J = 0, u(t) 3 0, which contradicts the fact that u is a nontrivial solution. Thus Z = @‘,and hence u&t) > 0 for k = 1,. . , n and t E (t,, tl). This proves the result. 3. Let the n x n matrix functions A(t) = (aii(t)) and B(t) = (bii(t)) be continuous for t > t,. Let t, be the first conjugate point oft, relative to (1). If aij(t) 3 bij(t) 3 0 for 1 d i,.i < n and to < t < t, then the first conjugate point i, oft, relative to (2) exists and 1, < t,. PROPOSITION
Proof For each positive integer m let A,(t) denote the matrix whose i, jth element is aij(t) + l/m.
S(W)
SHAIR AHMADUI~
ALAN C.LAZER
Since every element of A,(t) is strictly greater than the corresponding element of B(t) on [to, rl], it follows from Theorem A that if ZImdenotes the first conjugate point oft, relative to the differential system x”(t) + A,(t)x(t) = 0 then Zln, < t,. Moreover, if 4 > p then every element of Ap(t) is strictly greater than the corresponding element of A&r); hence, Theorem A implies that I,I, < ilq < I,. Hence, f = Ji_mr i ,m exists and r, < i < r,. For each integer m there exists a nontrivial solution x,(r) of xi(t) + A,(r)x,(t) = 0 such that x,(r,,) = x,(iJ = 0. Since x;(fo) # 0, by multiplying x,(t) by a suitable scalar, we may assume that /)x;(rJ/ = 1 where I/ 11is the usual Euclidean norm. There exists a subsequence {x;,(r,)J;& of {x;(ro)},“=l and a column vector L with //rjl = 1 such that ilmT .$,,(r& = I‘. By standard results from the theory of differential equations it follows that if y(t) denotes the solution of y”(r) + A(r)y(r) = 0 which satisfies the initial conditions y(r,) = 0, y’(t,) = 2’then ,llmX,-y,,(r) = y(r) uniformly on compact subintervals of [r,, a). Thus y(t) = dcrnYx,Jilmk) is also contained
= 0. Since i d r,, it follows that 2, d r,. This proposition
in [S, p. 991.
Proofof Theorem C. Assuming the hypotheses of Theorem C, Proposition 3 implies that if 2, is the first conjugate point of r, relative to (2) then I, d r,. To prove the theorem we shall show that the equality i, = t, is impossible. Suppose, then, that 2, = r,. According to Proposition 2 and the hypotheses of Theorem C, there exists u(t) such that u”(r) + B(r)u(r) = 0,
(7)
u(r,) = u(r,) = 0 and u,(r) > 0 for r E (r,, rl) and 1 d k < n. According is the first conjugate point of r, relative to the differential equation
to Proposition
1, i, = f1
y”(t) + ,4(r)Ty(r) = 0. Since B(r) is irreducible at some point of (r,, r,), the hypotheses of Theorem C imply that A(r) is irreducible at the same point. Therefore, .4(t)r is irreducible at this point and Proposition 2 implies the existence of t>(t) = col(t:,(r), , l.,(r)) such that r”(r) + A(r)‘t!(r) = 0, zl(r,) = [jr,) = 0, and v,(r) > 0 for r E (r,, rl), 1 G k G n. The-row vector z(r)’ satisfies ~“(t)~ + t:(r)TA(r) = 0 and tfr,)7‘ = z:(r,)T = 0. Multiplying and then subtracting we obtain
(8)
(7) on the left by t>(r)“‘,multiplying
(8) on the right by u(r),
Ifr)‘u”(r) - v”(r)“u(t) = Ijr)“[A(r) - B(r)]u(t). Therefore, since rfr)Tu”(r) - r”‘(t)‘.u(r) = d/dt(t(r)Tu’(t) =u(r,)’ = 0, it follows that
- z+(r)Tu(r)) and u(r,) =
rft)T[A(r) - B(r)@(r) dr = 0. The hypotheses of Theorem C imply that the elements of the negative on (r,, r,) and that the matrix is nonzero at some point of the 1 x n matrix z(r)T and the n x 1 matrix u(r) are strictly scalar function zl(r)TIA(r) - B(r)&(r) is nonnegative on (fo, r,)
u(t,) =
0, t:(r,)T
(9)
n x n matrix A(r) - B(r) are nonof this interval. Since the elements positive on (r,, fl), the continuous and nonzero at some point of this
On an extension
of Sturm’s comparison
theorem
to a class of nonselfadjoint
interval. Therefore, (9) is impossible, and we see that the assumption contradiction. Therefore 2, < t, and the theorem is proved. Remark. The assertion of Theorem C is false, in general, without irreducible at some point of the interval (t,, tl). To see this, let
second-order
systems
501
that i, = t, leads to a
the assumption
that B(t) lx
and It is then easy to verify that the first point conjugate
to 0 is 7~relative to both (1) and (2).
REFERENCES properties of second order linear systems, Bull. Am. math. Sot. 82 287-289 1. AHMAD S. and LAZER A. C., Component (1976). ofextremal solutions of second order systems, SIAM J. math. Anal. 8, 2. AHMAD S. & LAZFB A. C. On the components 1623 (1977). of the Sturm Comparison Theorem to selfadjomt systems. Proc. Am. 3. AHMAD S. & LAZER A. C. A new generalization math. Sot. 66, 185-188 (1978). 4. AHMAD S. & LAZER A. C. An h’-dimensional extension of the Sturm separation and comparison theory to a class ot nonselfadjoint systems, SIAM J. math. Anal. 9, 1137-l 150 (1978). bonlinecrr 5. SCHMIDT K. and SMITH H., Positive solutions and conjugate pomts for systems of differential equations, .4na/ysb 2, 93-105 (1978).
Analysis,
Theory,
Methods
& Applicarrons,
0362.546X/80/0501~97
Vol. 4, No. 3. pp. 497-501
102.00/O
@ Pergamon Press Ltd. 1980. Printed m Great Britain
ON AN
EXTENSION
OF STURM’S
TO A CLASS OF NONSELFADJOINT
COMPARISON SECOND-ORDER
THEOREM SYSTEMS
SHAIR AHMAD* Oklahoma State University, Stillwater, OK 74074, U.S.A
and ALAN C. LAZER? University of Cincinnati, Cincinnati, OH 45221 (Received 1 June 1979) Keywords: Conjugate paints, adjoint system, irreducible matrix
THE PURPOSE of this note is to give an extension
of Sturm’s comparison theorem to a class of second-order linear differential systems considered in [l-5]. In the following theorems, which were proved in [4], x and y are n-dimensional column vectors. THEOREM A. L,et A(t) and B(t) be continuous real n x n matrix-valued functions defined for a < t < b such that if A(t) = (uij(t)) and B(t) = (bij(t)) then a,,(t) 2 bij(t) 2 0 for 1 < i, j d n and for t E [a, b]. Assume that for some f E [a, b], uij(f) > b,i(t), 1 < i, j d n. If there exists a nontrivial solution of
x”(t) + B(t) x(t) = 0 such that x(u) = x(b) = 0 then there exists a nontrivial
solution
(1) of
y”(t) + A@)y(t) = 0 such that y(u) = y(c) = 0
with a < c < b.
B(t) = (bij(t)) is continuous on [a, 2 0 for 1 < i, ,j < n. If there exists a nontrivial solution x(t) of (1) with x(u) if there exists no nontrivial solution x(t) of (1) satisfying x(u) = x(c) = 0 with there exists a nontrivial solution u(t) = co@,(t), . . . , u,(t)) of (1) satisfying u(u) u,(t) > 0 for a < t < b and 1 < k < n.
THEOREM B. Assume that the n x n matrix function
bij(t) and then and
(2)
b] and that = x(b) = 0 u < c -C b,
= u(b) = 0
Actually, these theorems are valid under the seemingly weaker condition that only the offdiagonal elements of B(t) are nonnegative. However, because of certain transformation of variables discussed in [4], it is no less general to assume that all of the elements of B(t) are nonnegative. In order to state the main result of this note we recall two definitions. Let A = (u,i) be a real n x n matrix. We recall that A is reducible if there exist nonempty subsets This research was partially supported by grants (MCS78-01480)* and (MCS77-25147)t. 491
498
SHAM AHMAD~III~ AII\Y C. LAZER
I and J of {1,2,..., n}suchthatInJ=cD,{1,2,...,n}=I~J,andu,~=Oifi~I,j~J. Otherwise A is said to be irreducible. It is easy to see that if A is irreducible the transpose of A is also irreducible. Let C(t) be an n x n matrix-valued function defined and continuous for t, G t < m. If there exists a nontrivial solution z(t) of the vector differential equation z”(f) + C(t)z(t) = 0
(3)
such that z(t,) = z(t,) = 0 with 1, > t, and if t, is the smallest number larger than t, with this property, then t, is called the first point conjugate to t, relative to (3). An elementary compactness argument shows that if there exists a nontrivial solution of (3) all of whose components vanish at t, and also at another point in (t,, co), then the first conjugate point oft, relative to (3) exists. The main result of this note is: THEOREM C. Let the IZ x II matrix function B(t) = (bij(t)) lx continuous and let t, be the first point conjugate to t, relative to (1). Suppose that the matrix B(t) is irreducible at some point in the interval (t,, ti). If A(t) = (uij(t)) IS another n x n continuous matrix function satisfying aij(t) 3 b,,(t) 3 0 for 1 d i, j d n and t, d t 6 f,, and if there exists a pair of integers k and 1 such that u,,(Z) > b$) for some 1 E (to, t,), then 1,, the first point conjugate to f, relative to (2), satisfies f, < t,.
A simple instance of when B(t) will be irreducible at a point of (to, fi) is when all the elements of B(t) are positive at some point of (t,, ti). We shall prove Theorem C with the aid of Theorems A and B and some further preliminary results. PROPOSITION1. Let C(r) be continuous for t 3 t, and suppose that the first conjugate relative to (3) exists. Then, if C’(t) denotes the transpose of C(r), the first conjugate relative to the ‘adjoint-system,’
z”(f) + P(t)z(t) exists and is equal to the first conjugate
point oft,
= 0 relative
point oft, point of r.
(3*) to (3).
Proof: If f, denotes the first point conjugate to t, relative to (3) then there exists a nontrivial solution z(t) of (3) such that z(t,,) = z(ti) = 0. By the uniqueness theorem, z’(t,) # 0. Let w(t) be the n x II matrix function such that
W”(f) + CT(t)W(t) = 0, W(t,) = 0, W’(q)) = I
(4)
where I is the identity matrix. We claim that lV(t,) is a singular matrix. Assuming the contrary, there exists a vector b such that kV(t,) b = z’(tl). If w(t) = w(t) b, then the row vector w(t)’ satisfies the differential equation w”(t)T + w(t)7.C(t) = 0
(5)
On an extension
and the boundary
of Sturm’s comparison
to a class of nonselfadjoint
second-order
systems
499
conditions
w(tJ Multiplying
theorem
the equation
= 0, w(tJT = z’(tJT.
(5) on the right by z(t) and multiplying
(6) (3) on the left by w(t)‘, we obtain
w”(t)Tz(t) = - w(t)TC(t)z(t) = W(t)Tz”(t). Hence, g
(w’(t)‘z(t) - wwz’(t)) = w”(t)Tz(t)- w(t)‘z”(t) = 0.
Therefore, since z(t,) = 0 and wT(t,) = 0, it follows that w’(t)Tz(t) - w(t)%‘(t) = 0 for all t. Therefore, since, z(t,) = 0 and w(t,)T = z’(t,)T, it follows that 0 = - z’(tl)“z’(tl). Hence z’(tl) = 0, which is a contradiction. This contradiction shows that the matrix W(t,) is singular. Thus, there exists a nonzero column vector d such that W(t,)d = 0. If u(t) = W(t)d then it follows from (4) that u”(t) + C(t)Tv(t) = 0, u(t,) = 0, u’(tJ = d # 0, and u(tl) = 0. Therefore, it follows that the first conjugate point oft, relative to the differential equation (3*) exists and if this point is denoted by tt, then t: Q t,. Since (C(t)‘)’ = C(t), it follows from a repetition of the above argument, with C(t) replaced by C(r)T, that t, d t,.* Hence, t, = t: and the result is proved. n x n matrix function which satisfies b,,(t) > 0 PROPOSITION 2. Let B(t) = (bij(t)) be a continuous for t 2 t,. If i 1 is the first conjugate point oft, relative to (1) and if there exists a number s E (t,, tl) such that B(s) is irreducible, then there exists a solution u = col(u,, . . . , u,,) of (1) such that u(t,) = u(tl) = 0 and u,(t) > 0 for k = 1,. . . ,n and t, < t < t,. Proof According to Theorem B, there exists a nontrivial solution u = col(u,, . . , u,,) of (1) such that u(t,) = u(t,) = 0 and u,(t) 2 0 fort, d t d t,. From (1) we see that for k = 1, . . . , n and t, < t < t,, u;(t) = - i bkj@)Uj@)< 0. j=l
Therefore, since uk(t,) = u&t,) = 0 and t+(t) > 0 for t, 6 t < t,, it follows that either u,(t) > 0 for all t E (t,, tl) or u,(t) = 0 for all t E (t,, tl). We define subsets I and .Z of { 1,2,. . . , nl as follows: k EZ if u,(t) = 0 for all t E (t,,, tI), k E J if udr) > 0 for all t E (t,, tl). If i E I, then u:‘(i) = 0 for all t E (to, tJ, so 0 = -u;‘(s)
=
x_bij(S)Uj(S). js.J
Since bij(s) 2 0 and uj(s) > 0 forj E J, it follows that bij(s) = 0 for i E Z,j E J. Since B(s) is irreducible and I n J = Q it follows that either Z = 0 or J = a. If J = 0, u(t) 3 0, which contradicts the fact that u is a nontrivial solution. Thus Z = @‘,and hence u&t) > 0 for k = 1,. . , n and t E (t,, tl). This proves the result. 3. Let the n x n matrix functions A(t) = (aii(t)) and B(t) = (bii(t)) be continuous for t > t,. Let t, be the first conjugate point oft, relative to (1). If aij(t) 3 bij(t) 3 0 for 1 d i,.i < n and to < t < t, then the first conjugate point i, oft, relative to (2) exists and 1, < t,. PROPOSITION
Proof For each positive integer m let A,(t) denote the matrix whose i, jth element is aij(t) + l/m.
S(W)
SHAIR AHMADUI~
ALAN C.LAZER
Since every element of A,(t) is strictly greater than the corresponding element of B(t) on [to, rl], it follows from Theorem A that if ZImdenotes the first conjugate point oft, relative to the differential system x”(t) + A,(t)x(t) = 0 then Zln, < t,. Moreover, if 4 > p then every element of Ap(t) is strictly greater than the corresponding element of A&r); hence, Theorem A implies that I,I, < ilq < I,. Hence, f = Ji_mr i ,m exists and r, < i < r,. For each integer m there exists a nontrivial solution x,(r) of xi(t) + A,(r)x,(t) = 0 such that x,(r,,) = x,(iJ = 0. Since x;(fo) # 0, by multiplying x,(t) by a suitable scalar, we may assume that /)x;(rJ/ = 1 where I/ 11is the usual Euclidean norm. There exists a subsequence {x;,(r,)J;& of {x;(ro)},“=l and a column vector L with //rjl = 1 such that ilmT .$,,(r& = I‘. By standard results from the theory of differential equations it follows that if y(t) denotes the solution of y”(r) + A(r)y(r) = 0 which satisfies the initial conditions y(r,) = 0, y’(t,) = 2’then ,llmX,-y,,(r) = y(r) uniformly on compact subintervals of [r,, a). Thus y(t) = dcrnYx,Jilmk) is also contained
= 0. Since i d r,, it follows that 2, d r,. This proposition
in [S, p. 991.
Proofof Theorem C. Assuming the hypotheses of Theorem C, Proposition 3 implies that if 2, is the first conjugate point of r, relative to (2) then I, d r,. To prove the theorem we shall show that the equality i, = t, is impossible. Suppose, then, that 2, = r,. According to Proposition 2 and the hypotheses of Theorem C, there exists u(t) such that u”(r) + B(r)u(r) = 0,
(7)
u(r,) = u(r,) = 0 and u,(r) > 0 for r E (r,, rl) and 1 d k < n. According is the first conjugate point of r, relative to the differential equation
to Proposition
1, i, = f1
y”(t) + ,4(r)Ty(r) = 0. Since B(r) is irreducible at some point of (r,, r,), the hypotheses of Theorem C imply that A(r) is irreducible at the same point. Therefore, .4(t)r is irreducible at this point and Proposition 2 implies the existence of t>(t) = col(t:,(r), , l.,(r)) such that r”(r) + A(r)‘t!(r) = 0, zl(r,) = [jr,) = 0, and v,(r) > 0 for r E (r,, rl), 1 G k G n. The-row vector z(r)’ satisfies ~“(t)~ + t:(r)TA(r) = 0 and tfr,)7‘ = z:(r,)T = 0. Multiplying and then subtracting we obtain
(8)
(7) on the left by t>(r)“‘,multiplying
(8) on the right by u(r),
Ifr)‘u”(r) - v”(r)“u(t) = Ijr)“[A(r) - B(r)]u(t). Therefore, since rfr)Tu”(r) - r”‘(t)‘.u(r) = d/dt(t(r)Tu’(t) =u(r,)’ = 0, it follows that
- z+(r)Tu(r)) and u(r,) =
rft)T[A(r) - B(r)@(r) dr = 0. The hypotheses of Theorem C imply that the elements of the negative on (r,, r,) and that the matrix is nonzero at some point of the 1 x n matrix z(r)T and the n x 1 matrix u(r) are strictly scalar function zl(r)TIA(r) - B(r)&(r) is nonnegative on (fo, r,)
u(t,) =
0, t:(r,)T
(9)
n x n matrix A(r) - B(r) are nonof this interval. Since the elements positive on (r,, fl), the continuous and nonzero at some point of this
On an extension
of Sturm’s comparison
theorem
to a class of nonselfadjoint
interval. Therefore, (9) is impossible, and we see that the assumption contradiction. Therefore 2, < t, and the theorem is proved. Remark. The assertion of Theorem C is false, in general, without irreducible at some point of the interval (t,, tl). To see this, let
second-order
systems
501
that i, = t, leads to a
the assumption
that B(t) lx
and It is then easy to verify that the first point conjugate
to 0 is 7~relative to both (1) and (2).
REFERENCES properties of second order linear systems, Bull. Am. math. Sot. 82 287-289 1. AHMAD S. and LAZER A. C., Component (1976). ofextremal solutions of second order systems, SIAM J. math. Anal. 8, 2. AHMAD S. & LAZFB A. C. On the components 1623 (1977). of the Sturm Comparison Theorem to selfadjomt systems. Proc. Am. 3. AHMAD S. & LAZER A. C. A new generalization math. Sot. 66, 185-188 (1978). 4. AHMAD S. & LAZER A. C. An h’-dimensional extension of the Sturm separation and comparison theory to a class ot nonselfadjoint systems, SIAM J. math. Anal. 9, 1137-l 150 (1978). bonlinecrr 5. SCHMIDT K. and SMITH H., Positive solutions and conjugate pomts for systems of differential equations, .4na/ysb 2, 93-105 (1978).