Lemma 3.1 ()
Given . Then the following FBVPs:
are equivalent to the operator equation
Lemma 3.2 ()
Let be given as in the statement of Lemma 3.1. Then we find that:
is a continuous function on the unit square .
for each .
for each .
There is a positive constant
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
Proof From Lemmas 3.1-3.3, we have
We can check that is positive and strictly decreasing on . Furthermore,
Let the space be endowed with the norm
It is well known that X is a Banach space . Define the cone by
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
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
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
The facts that the constant function and imply that . For , we have and for . From assumption (A2), we see
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