Theorem 2.1 (see [, Theorem 2.4])
if and only if
Theorem 2.2 (see [, Theorem 2.7])
(A1) hold and with on any subinterval of
has an infinite sequence of positive eigenvalues
given an arbitrary subinterval of , an eigenfunction that belongs to a sufficiently large eigenvalue changes its sign in that subinterval;
for each , the algebraic multiplicity of is 1.
Theorem 2.3 (see [, Theorem 2.8]) (Maximum principle)
then on .
Let with the norm . Let with its usual norm . By a positive solution of (1.2), we mean x is a solution of (1.2) with (i.e., in and ).
with the inner product
and the norm
. Further, define the linear operator
Then is a closed operator and is completely continuous.
Lemma 2.4 Let be the first eigenfunction of
, for all
, we get
, we have
Integrating by parts, we obtain
be the closure of the set of positive solutions of the problem
We extend the function f
to a continuous function
, let x
be an arbitrary solution of the problem
Since for , we have for . Thus x is a nonnegative solution of (2.11), and the closure of the set of nontrivial solutions of (2.13) in is exactly Σ.
be the Nemytskii operator associated with the function
Then (2.13), with
, is equivalent to the operator equation
In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping
For , let , and let denote the degree of on with respect to 0.
Proof Suppose to the contrary that there exist sequences and in , in E, such that for all , then in .
. Now, from condition (H1), we have the following:
denote the nonnegative eigenfunctions corresponding to
, respectively. Then we have, from the first inequality in (2.19),
From Lemma 2.4, we have
, from (1.6) we have
By the fact that
, we conclude that
Combining this and (2.21) and letting
in (2.20), we get
Similarly, we deduce from the second inequality in (2.19) that
Thus, . This contradicts . □
Corollary 2.6 For and , .
Lemma 2.5, applied to the interval
, guarantees the existence of
such that for
Hence, for any
which ends the proof. □
Lemma 2.7 Suppose
. Then there exists such that with
where is the nonnegative eigenfunction corresponding to .
We assume to the contrary that there exist
and a sequence
, such that
, it follows that
has a unique decomposition
where and . Since on and , we have from (2.32) that .
By (H1), there exists
, there exists
Applying Lemma 2.4 and (2.37), it follows that
This contradicts (2.33). □
Corollary 2.8 For and , .
is the number asserted in Lemma 2.7. As
is bounded in
, there exists
. By Lemma 2.7, one has
Now, using Theorem A, we may prove the following.
Proposition 2.9 is a bifurcation interval from the trivial solution for
(2.15). There exists an unbounded component C of a positive solution of
(2.15), which meets
, let us take that
. It is easy to check that for
, all of the conditions of Theorem A are satisfied. So, there exists a connected component
of solutions of (2.15) containing
, and either
is unbounded, or
By Lemma 2.5, the case (ii) cannot occur. Thus is unbounded bifurcated from in . Furthermore, we have from Lemma 2.5 that for any closed interval , if , then in E is impossible. So, must be bifurcated from in . □