Lemma 3.1 ([27])
Given . Then the following FBVPs:
(3.1)
(3.2)
(3.3)
are equivalent to the operator equation
where
Lemma 3.2 ([27])
Let be given as in the statement of Lemma 3.1. Then we find that:
-
(1)
is a continuous function on the unit square .
-
(2)
for each .
-
(3)
for each .
-
(4)
There is a positive constant
such that
where
Lemma 3.3 Given . Assume that is a solution of problem (3.1)-(3.3), then
Lemma 3.4 Assume that and is a solution of problem (3.1)-(3.3). Then there exists a positive constant such that
where .
Proof From Lemmas 3.1-3.3, we have
Let
We can check that is positive and strictly decreasing on . Furthermore,
Then
Thus,
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Let the space be endowed with the norm
It is well known that X is a Banach space [37]. Define the cone by
Lemma 3.5
Let
be the operator defined by
Then is completely continuous.
Proof The operator is continuous in view of the nonnegativity and continuity of the functions and . Let be bounded. Then there exists a positive constant such that , . Denote
Then for , we have
Hence is bounded. For , , one has
Thus,
By means of the Arzela-Ascoli theorem, we claim that T is completely continuous. Finally, we see that
Thus, we show that is a completely continuous operator. □
Let the nonnegative continuous concave functional φ, the nonnegative continuous convex functionals γ, θ and the nonnegative continuous functional ψ be defined on the cone by
By Lemmas 3.3 and 3.4, the functionals defined above satisfy
where . Therefore condition (2.2) of Lemma 2.4 is satisfied.
Assume that there exist constants with , and
such that
Theorem 3.1 Under assumptions (A1)-(A3), problem (1.1)-(1.3) has at least three positive solutions , , satisfying
Proof Problem (1.1)-(1.3) has a solution if and only if u solves the operator equation
For , we have . From assumption (A1), we obtain
Thus,
Hence, .
The facts that the constant function and imply that . For , we have and for . From assumption (A2), we see
Thus,
which means , . These ensure that condition (S1) of Lemma 2.4 is satisfied. Secondly, for all with ,
Thus, condition (S2) of Lemma 2.4 holds. Finally we show that (S3) also holds. We see that and . Suppose that with . Then by assumption (A3),
Thus, all conditions of Lemma 2.4 are satisfied. Hence problem (1.1)-(1.2) has at least three positive solutions , , satisfying
□