In this section, we consider general halflinear eigenvalue problem (1), (3). We denote
R(t)={\int}_{a}^{t}{r}^{1q}(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{2em}{0ex}}C(t)={\int}_{a}^{t}c(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{2em}{0ex}}W(t)={\int}_{a}^{t}w(s)\phantom{\rule{0.2em}{0ex}}ds.
In the next theorems, we discuss various asymptotic behavior of ratios of the functions C, R, W for t\to a+, 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 \mathrm{\Lambda}\in \mathbb{R}, \phi (t)=\phi (t,\mathrm{\Lambda}), denotes the Prüfer angle of a solution x of (1) with \lambda =\mathrm{\Lambda} satisfying (3) at a, i.e., \phi (a,\mathrm{\Lambda})=\alpha.
We start with the most interesting case \alpha =\beta in (3).
Theorem 2 Suppose that {lim}_{t\to a+}\frac{C(t)}{W(t)}=\mathrm{\infty}. Then for any \alpha \in (0,{\pi}_{p}), we have
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}.
(12)
Proof We will show that {\lambda}_{1}(b,\alpha )<\mathrm{\Lambda} for any \mathrm{\Lambda}\in \mathbb{R} if b>a is sufficiently close to a. Since the eigenfunction corresponding to the first eigenvalue {\lambda}_{1} has no zero on (a,b) (see, e.g., [10]), we can use the Riccati equation (8) for v=\mathrm{\Phi}({cot}_{p}\phi ) instead of equation (7) for φ. Using the meanvalue theorem for Lebesgue integrals in computing the integral {\int}_{a}^{b}{r}^{1q}{cot}_{p}\phi {}^{p}, integration of (8) gives
\begin{array}{rl}v(b)v(a)& =\mathrm{\Phi}({cot}_{p}\phi (b))\mathrm{\Phi}({cot}_{p}\alpha )\\ =[\frac{C(b)}{W(b)}\mathrm{\Lambda}(p1)\frac{R(b)}{W(b)}{cot}_{p}\phi ({t}_{b}){}^{p}]W(b),\end{array}
where {t}_{b}\in (a,b). Hence, for b sufficiently close to a,
\mathrm{\Phi}({cot}_{p}\phi (b))\mathrm{\Phi}({cot}_{p}\alpha )\le (\frac{C(b)}{W(b)}\mathrm{\Lambda})W(b)<0,
i.e., \phi (b)>\alpha =\beta. Thus, since φ was the Prüfer angle corresponding to a solution of (1) with \lambda =\mathrm{\Lambda} satisfying (3) at t=a, we have {\lambda}_{1}(b,\alpha )<\mathrm{\Lambda} for the first eigenvalue when b is in a sufficiently small right neighborhood of a (since we need \phi (b)=\alpha for \lambda ={\lambda}_{1}(b,\alpha )). Therefore, (12) holds. □
Theorem 3
Suppose that
\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}<\mathrm{\infty},\phantom{\rule{2em}{0ex}}\underset{t\to a+}{lim}\frac{R(t)}{W(t)}=\mathrm{\infty},
and let
{\alpha}^{\ast}={arccot}_{p}{(max\{0,\frac{1}{p1}\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}\})}^{1/p}\in (0,\frac{{\pi}_{p}}{2}].
Then we have {\lambda}_{1}(b,\alpha )\to \mathrm{\infty} as b\to a+ for any \alpha \in (0,{\pi}_{p})\setminus [{\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast}].
Proof Let \alpha \in (0,{\alpha}^{\ast})\cup ({\pi}_{p}{\alpha}^{\ast},{\pi}_{p}) be fixed, and take \delta >0 so small that
(p1){cot}_{p}\alpha {}^{p}>\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}+2\delta .
(13)
Such a positive δ exists according to the definition of the number {\alpha}^{\ast}. Formula (13) implies that for τ sufficiently close to α, we have the inequality
(p1){cot}_{p}\tau {}^{p}>\frac{C(t)}{R(t)}+\delta
when t is sufficiently close to a. Again, integration of (8), together with the mean value theorem applied to the last integral on the righthand side of (8), gives
\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{R(b)}{W(b)}(\frac{C(b)}{R(b)}(p1){cot}_{p}\phi ({t}_{b}){}^{p})\mathrm{\Lambda}]W(b)<0.\end{array}
Hence \phi (b)>\alpha. This implies that {\lambda}_{1}(b,\alpha )<\mathrm{\Lambda} for every \mathrm{\Lambda}\in \mathbb{R}, and thus, {\lambda}_{1}(b,\alpha )\to \mathrm{\infty} as b\to a+. □
Theorem 4
Assume that
\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{t\to a+}{lim}\frac{C(t)}{W(t)}=\mathrm{\infty}.
(14)
Let
{\alpha}^{\ast}={arccot}_{p}{\left(\frac{1}{p1}\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}\right)}^{1/p}\in [0,\frac{{\pi}_{p}}{2}).
(15)
Then we have {\lambda}_{1}(b,\alpha )\to \mathrm{\infty} as b\to a+ for any \alpha \in ({\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast}).
Proof The first formula in (14) implies that C(t)>0 for t in a right neighborhood of a. Take \alpha \in ({\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast}), and let \delta >0 be so small that
(12\delta )\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}>(p1){cot}_{p}\alpha {}^{p}.
(16)
Again, such \delta >0 exists according to the definition of {\alpha}^{\ast}. Hence, for τ sufficiently close to α, from (16), we have for t close to a that
(1\delta )\frac{C(t)}{R(t)}>(p1){cot}_{p}\tau {}^{p}.
Let \mathrm{\Lambda}\in \mathbb{R} be arbitrary. Similarly as in the proof of the previous theorem,
\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{C(b)}{W(b)}(1(p1)\frac{R(b)}{C(b)}{{cot}_{p}\phi ({t}_{b})}^{p})\mathrm{\Lambda}]W(b)>0\end{array}
if b is sufficiently close to a, i.e., \phi (b)<\alpha, and hence {\lambda}_{1}(b,\alpha )>\mathrm{\Lambda}. Therefore, {lim}_{b\to a+}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}. □
Theorem 5
Assume that
\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{R(t)}{W(t)}<\mathrm{\infty}.
If
\underset{t\to a+}{lim}(\frac{C(t)}{W(t)}(p1)\frac{R(t)}{W(t)}{{cot}_{p}\phi }^{p})=L\in \mathbb{R}\cup \{\pm \mathrm{\infty}\},
then
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L.
Proof Let \mathrm{\Lambda}\in \mathbb{R} be arbitrary. Similarly as in the previous theorems,
\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{C(b)}{W(b)}(p1)\frac{R(b)}{W(b)}{{cot}_{p}\alpha }^{p}\end{array}
(17)
\phantom{\rule{2em}{0ex}}\mathrm{\Lambda}+(p1)\frac{R(b)}{W(b)}({{cot}_{p}\alpha }^{p}{{cot}_{p}\phi ({t}_{b})}^{p})]W(b).
(18)
If L=\mathrm{\infty}, the expression in line (17) tends to −∞ as b\to a+ while remaining terms on the righthand side of the previous formula are bounded. Hence the expression on the righthand side is negative for b close to a, which means that \phi (b)>\alpha for these b. However, this implies that {\lambda}_{1}(b,\alpha )<\mathrm{\Lambda} in a right neighborhood of a, and since Λ was arbitrary, we have {lim}_{b\to a}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}. The same arguments imply that {lim}_{b\to a+}{\lambda}_{1}(b,\alpha )=\mathrm{\infty} if L=+\mathrm{\infty}.
Finally, if L\in \mathbb{R}, take first \mathrm{\Lambda}=L+\epsilon. Since the last term in (18) tends to zero as b\to a+, we obtain using the same argument as in the previous part of the proof that {\lambda}_{1}(b,\alpha )<L+\epsilon for b sufficiently close to a. Taking \mathrm{\Lambda}=L\epsilon, we obtain {\lambda}_{1}(b,\alpha )>L\epsilon for b in a right neighborhood of a, and this completes the proof. □
Theorem 6
Suppose that
\underset{t\to a+}{lim}\frac{R(t)}{W(t)}<\mathrm{\infty}
(19)
and \alpha \in (0,{\pi}_{p}). If
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L\phantom{\rule{1em}{0ex}}\mathit{\text{exists finite}},
then
\underset{t\to a+}{lim}(\frac{C(t)}{W(t)}(p1)\frac{R(t)}{W(t)}{{cot}_{p}\alpha }^{p})=L.
(20)
Proof Denote \lambda (b):={\lambda}_{1}(b,\alpha ), and let \phi (t,\lambda (b)) be the Prüfer angle of the solution x of (1) with \lambda =\lambda (b)={\lambda}_{1}(b,\alpha ) satisfying (3) with \alpha =\beta. Then
\begin{array}{rl}0=& \mathrm{\Phi}({cot}_{p}\beta )\mathrm{\Phi}({cot}_{p}\alpha )\\ =& \mathrm{\Phi}\left({cot}_{p}\phi (b,\lambda (b))\right)\mathrm{\Phi}\left({cot}_{p}\phi (a,\lambda (b))\right)\\ =& C(b)\lambda (b)W(b)(p1)R(b){{cot}_{p}\alpha }^{p}\\ +(p1)R(b)({{cot}_{p}\alpha }^{p}{\left{cot}_{p}\phi ({t}_{b},\lambda (b))\right}^{p})\\ =& [\frac{C(b)}{W(b)}(p1)\frac{R(b)}{W(b)}{{cot}_{p}\alpha }^{p}\end{array}
(21)
\lambda (b)+(p1)\frac{R(b)}{W(b)}({{cot}_{p}\alpha }^{p}{\left{cot}_{p}\phi ({t}_{b},\lambda (b))\right}^{p})]W(b)=0
(22)
for some {t}_{b}\in (a,b). Therefore, from (21), (22), we obtain
\frac{C(b)}{W(b)}(p1)\frac{R(b)}{W(b)}{{cot}_{p}\alpha }^{p}=\lambda (b)(p1)\frac{R(b)}{W(b)}({{cot}_{p}\alpha }^{p}{\left{cot}_{p}\phi ({t}_{b},\lambda (b))\right}^{p}).
Letting b\to a+ and using that
\underset{t\to a+}{lim}\lambda (b)=\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L\in \mathbb{R}
(which means that \lambda (b) is bounded in a neighborhood of a), we see that
\underset{b\to a+}{lim}\phi ({t}_{b},\lambda (b))=\alpha ,
and (20) is proved. □
Theorem 7 If {lim}_{b\to a+}{\lambda}_{1}(b,\alpha ) has a finite value for two different values of {\alpha}_{1} and {\alpha}_{2} with {\alpha}_{1},{\alpha}_{2}\in (0,{\pi}_{p}) and {\alpha}_{1}\ne {\pi}_{p}{\alpha}_{2}, then
\underset{t\to a+}{lim}\frac{R(t)}{W(t)}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{t\to a+}{lim}\frac{C(t)}{W(t)}
exist finite, and for each \alpha \in (0,{\pi}_{p}), we have
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=\underset{t\to a+}{lim}\frac{C(t)}{W(t)}(p1){{cot}_{p}\alpha }^{p}\underset{t\to a+}{lim}\frac{R(t)}{W(t)}.
(23)
Proof First of all, we have
{{cot}_{p}{\alpha}_{1}}^{p}\ne {{cot}_{p}{\alpha}_{2}}^{p}.
(24)
Denote {\lambda}_{1}(b,{\alpha}_{i})=:{\mu}_{i}(b) for i=1,2, and let \phi (t,{\mu}_{i}(b)) be the Prüfer angle of the solution of (1), (3) with \lambda ={\mu}_{i}(b) and \alpha =\beta. Since \phi (b,{\mu}_{i}(b))=\phi (a,{\mu}_{i}(b)), we have from (8)
C(b){\mu}_{i}(b)W(b)(p1)R(b){\left{cot}_{p}\phi ({t}_{i,b},{\mu}_{i}(b))\right}^{p}=0,\phantom{\rule{1em}{0ex}}i=1,2,
where {t}_{i,b}\in (a,b). Hence, one gets
\frac{C(b)}{W(b)}(p1)\frac{R(b)}{W(b)}{\left{cot}_{p}\phi ({t}_{i,b},{\mu}_{i}(b))\right}^{p}={\mu}_{i}(b),\phantom{\rule{1em}{0ex}}i=1,2.
(25)
Subtracting these two equations and using the fact that
\left{cot}_{p}\phi ({t}_{1,b},{\mu}_{1}(b))\right\ne \left{cot}_{p}\phi ({t}_{2,b},{\mu}_{2}(b))\right
for b sufficiently close to a (this follows from (24)), we have
\frac{R(b)}{W(b)}=\frac{{\mu}_{1}(b){\mu}_{2}(b)}{{{cot}_{p}\phi ({t}_{1,b},{\mu}_{1}(b))}^{p}{{cot}_{p}\phi ({t}_{2,b},{\mu}_{2}(b))}^{p}}\to \frac{{\lambda}_{1}(a+,{\alpha}_{1}){\lambda}_{1}(a+,{\alpha}_{2})}{{{cot}_{p}{\alpha}_{1}}^{p}{{cot}_{p}{\alpha}_{2}}^{p}}
as b\to a+, so {lim}_{b\to a+}\frac{R(b)}{W(b)} exists finite, and hence from (25), the same holds for {lim}_{b\to a+}\frac{C(b)}{W(b)}, and the conclusion follows from Theorem 5. □
We finish the paper with a brief treatment of the case \alpha \ne \beta 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 \alpha \in [0,{\pi}_{p}), \beta \in (0,{\pi}_{p}].

(i)
If \alpha <\beta, then
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}.
(26)

(ii)
If \alpha >\beta, then
\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}.
Proof We will prove the part (i) only, the proof of (ii) is analogical. Let \mathrm{\Lambda}\in \mathbb{R} be arbitrary, and let \phi (t,\mathrm{\Lambda}) be the Prüfer angle of the solution x of (1) with \lambda =\mathrm{\Lambda} satisfying (3) at t=a. Since φ is a continuous function of t, and \phi (a,\mathrm{\Lambda})=\alpha <\beta, we have \phi (b,\mathrm{\Lambda})<\beta if b is sufficiently close to a. Hence, using the same argument as before, we have {\lambda}_{1}(b,\alpha ,\beta )>\mathrm{\Lambda}, which implies (26). □
Remark 2 Until now, we have considered the first eigenvalue {\lambda}_{1}(b,\alpha ,\beta ) only. Concerning the asymptotic behavior of higher eigenvalues {\lambda}_{n}(b,\alpha ,\beta ), we have
\underset{b\to a+}{lim}{\lambda}_{n}(b,\alpha ,\beta )=\mathrm{\infty}
(27)
for any \alpha \in [0,{\pi}_{p}), \beta \in (0,{\pi}_{p}]. This formula follows in the case \alpha <\beta from the general theory of halflinear eigenvalues problem (see [8, 10]), which says that the eigenvalues form an increasing sequence tending to ∞, and from the fact that {lim}_{b\to a+}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}. If \alpha >\beta, then {cot}_{p}\alpha <{cot}_{p}\beta, but for n\in \mathbb{N}, we have \alpha <\beta +n{\pi}_{p} and higher eigenvalues correspond to the situation when the Prüfer angle of a solution of (1) satisfying the first condition in (3) satisfies \phi (b)=\beta +n{\pi}_{p}. Hence, the growth of φ must be unbounded when b\to a+, and hence (27) holds also for \alpha >\beta.