In this section, we consider general half-linear eigenvalue problem (1), (3). We denote
In the next theorems, we discuss various asymptotic behavior of ratios of the functions C, R, W for , which implies various limit behavior of the first eigenvalue. The behavior of the higher eigenvalues is described at the end of this section.
In the proofs of the next theorems, given , , denotes the Prüfer angle of a solution x of (1) with satisfying (3) at a, i.e., .
We start with the most interesting case in (3).
Theorem 2 Suppose that . Then for any , we have
(12)
Proof We will show that for any if is sufficiently close to a. Since the eigenfunction corresponding to the first eigenvalue has no zero on (see, e.g., [10]), we can use the Riccati equation (8) for instead of equation (7) for φ. Using the mean-value theorem for Lebesgue integrals in computing the integral , integration of (8) gives
where . Hence, for b sufficiently close to a,
i.e., . Thus, since φ was the Prüfer angle corresponding to a solution of (1) with satisfying (3) at , we have for the first eigenvalue when b is in a sufficiently small right neighborhood of a (since we need for ). Therefore, (12) holds. □
Theorem 3
Suppose that
and let
Then we have as for any .
Proof Let be fixed, and take so small that
(13)
Such a positive δ exists according to the definition of the number . Formula (13) implies that for τ sufficiently close to α, we have the inequality
when t is sufficiently close to a. Again, integration of (8), together with the mean value theorem applied to the last integral on the right-hand side of (8), gives
Hence . This implies that for every , and thus, as . □
Theorem 4
Assume that
(14)
Let
(15)
Then we have as for any .
Proof The first formula in (14) implies that for t in a right neighborhood of a. Take , and let be so small that
(16)
Again, such exists according to the definition of . Hence, for τ sufficiently close to α, from (16), we have for t close to a that
Let be arbitrary. Similarly as in the proof of the previous theorem,
if b is sufficiently close to a, i.e., , and hence . Therefore, . □
Theorem 5
Assume that
If
then
Proof Let be arbitrary. Similarly as in the previous theorems,
(17)
(18)
If , the expression in line (17) tends to −∞ as while remaining terms on the right-hand side of the previous formula are bounded. Hence the expression on the right-hand side is negative for b close to a, which means that for these b. However, this implies that in a right neighborhood of a, and since Λ was arbitrary, we have . The same arguments imply that if .
Finally, if , take first . Since the last term in (18) tends to zero as , we obtain using the same argument as in the previous part of the proof that for b sufficiently close to a. Taking , we obtain for b in a right neighborhood of a, and this completes the proof. □
Theorem 6
Suppose that
(19)
and . If
then
(20)
Proof Denote , and let be the Prüfer angle of the solution x of (1) with satisfying (3) with . Then
(21)
(22)
for some . Therefore, from (21), (22), we obtain
Letting and using that
(which means that is bounded in a neighborhood of a), we see that
and (20) is proved. □
Theorem 7 If has a finite value for two different values of and with and , then
exist finite, and for each , we have
(23)
Proof First of all, we have
(24)
Denote for , and let be the Prüfer angle of the solution of (1), (3) with and . Since , we have from (8)
where . Hence, one gets
(25)
Subtracting these two equations and using the fact that
for b sufficiently close to a (this follows from (24)), we have
as , so exists finite, and hence from (25), the same holds for , and the conclusion follows from Theorem 5. □
We finish the paper with a brief treatment of the case in (3). We show that the situation is similar as in the case of a constant coefficients equation treated in Section 3.
Theorem 8 Let , .
-
(i)
If , then
(26)
-
(ii)
If , then
Proof We will prove the part (i) only, the proof of (ii) is analogical. Let be arbitrary, and let be the Prüfer angle of the solution x of (1) with satisfying (3) at . Since φ is a continuous function of t, and , we have if b is sufficiently close to a. Hence, using the same argument as before, we have , which implies (26). □
Remark 2 Until now, we have considered the first eigenvalue only. Concerning the asymptotic behavior of higher eigenvalues , we have
(27)
for any , . This formula follows in the case from the general theory of half-linear eigenvalues problem (see [8, 10]), which says that the eigenvalues form an increasing sequence tending to ∞, and from the fact that . If , then , but for , we have and higher eigenvalues correspond to the situation when the Prüfer angle of a solution of (1) satisfying the first condition in (3) satisfies . Hence, the growth of φ must be unbounded when , and hence (27) holds also for .