In this section, we impose some growth conditions on
which allow us to apply Lemma 2.2 to the operator
defined in Section 2 to establish the existence of three positive solutions of (1.1) and (1.2). We note that, from the nonnegativity of
and
, the solution of (1.1) and (1.2) is nonnegative and concave on
.

First in view of Lemma

in [

27], we know that for

, there is

for

So we get

Let the nonnegative continuous convex functionals

and the nonnegative continuous concave functional

be defined on the cone

by

Then, it is easy to see that
and (2.6), (2.7) hold.

Now, for convenience we introduce the following notations. Let

Theorem 3.1.

Assume
for
. If there are positive numbers
with
, such that the following conditions are satisfied

(i)
for

(ii)
for

(iii)
for

then the problem (1.1), (1.2) has at least three positive solutions

satisfying

Proof.

By the definition of operator
and its properties, it suffices to show that the conditions of Lemma 2.2 hold with respect to the operator

We first show that if the condition (i) is satisfied, then

In fact, if

then

so assumption (i) implies

On the other hand, for

, there is

; then

is concave in

, and

for

, so

Therefore, (3.5) holds.

In the same way, if

, then condition (iii) implies

As in the argument above, we can get that
Thus, condition (
) of Lemma 2.2 holds.

Next we show that condition (

) in Lemma 2.2 holds. We choose

for

It is easy to see that

Therefore, for

there are

Hence in view of hypothesis (ii), we have

So by the definition of the functional

, we see that

Therefore, we get
for
, and condition (
) in Lemma 2.2 is fulfilled.

We finally prove that (

) in Lemma 2.2 holds. In fact, for

with

we have

Thus from Lemma 2.2 and the assumption that

on

, the BVP (1.1) and (1.2) has at least three positive solutions

, and

in

with

The fact that the functionals

and

on

satisfy an additional relation

for

implies that

The proof is complete.

From Theorem 3.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on
we can establish the existence of an arbitrary odd number of positive solutions of (1.1) and (1.2).

Theorem 3.2.

Suppose that there exist constants

such that the following conditions hold:

(i)
for

(ii)
for

Then, BVP (1.1) and (1.2) has at least
positive solutions.

Proof.

When
it is immediate from condition (i) that
which means that
has at least one fixed point
by the Schauder fixed point theorem. When
it is clear that the hypothesis of Theorem 3.1 holds. Then we can obtain at least three positive solutions
and
. Following this way, we finish the proof by induction. The proof is complete.

In the final part of this section, we give an example to illustrate our results.

Example 3.3.

Let

, where

denote nonnegative integer numbers set. If we choose

,

, and

and consider the following BVP on time scale

:

obviously the hypotheses (H1), (H2) hold and

on

. By simple calculations, we have

If we choose

and

, then

satisfies

So all conditions of Theorem 3.1 hold. Thus by Theorem 3.1, the problem (3.20) has at least three positive solutions

such that