In this section, we will apply cone theory to further study the uniqueness of solution for BVP (1) in and the dependence of such a positive solution on the parameter λ. The following hypotheses are needed:
(H7) and for all and , there exists such that ;
(H8) and for , , ;
(H9) for all , and , there exists such that .
Remark 4.1 The inequalities in (H7), (H8), and (H9) are equivalent to the following inequalities, respectively:
Let and define as in Section 1. It is obvious that and if then and , .
Remark 4.2 (H2) and (H7) imply for . Moreover, for .
Remark 4.3 Let be a solution for BVP (1) in . If (H2) and (H7) hold, then . Indeed, from Theorem 2.4 we have . So Remark 4.2 implies . Note that
we conclude .
So, in this section, we only need to consider the unique solution for BVP (1) in .
Lemma 4.1 Assume that (H2) and (H7) hold. Then B has a unique fixed point in , moreover, constructing successively the sequence () for any initial value , we have .
Proof For any , we have , which means . For all , , from (H7) we have . Consequently, the conclusion follows from Lemma 1.2. This completes the proof. □
Lemma 4.2 Assume that (H1), (H2), (H7), and (H8) hold. Then
-
(i)
is an increasing operator;
-
(ii)
for any and , there exists such that ;
-
(iii)
for and , there exists such that
Proof The conclusion (i) follows from (H1), (H2), (H7), and (4).
The proof of (ii). For given , , from (H1) and (4) we have
Let
(10)
then and
The proof of (iii). For any , , from (10) and (11) we have
Moreover, from (H7) and (H8) we have
where . This completes the proof. □
Lemma 4.3 Assume that (H1), (H2), (H7), and (H8) hold. Then has a unique fixed point in iff there exists such that . Moreover, constructing successively the sequence () for any initial value , we have
(12)
Proof ‘⇒’ Let be a fixed point of in , i.e., . Taking , we obtain .
‘⇐’ By virtue of Lemma 4.1, B has a unique fixed point in . Moreover,
Now, we are going to prove
Let , then . Otherwise, , from Lemma 4.2 we have
By the definition of , we get a contradiction . Thus, (14) holds.
Set , , , . From (13) and (14) we have
(15)
Lemma 2.3 implies that and converge to fixed points and of , respectively. From (15), we have
(16)
To prove that has only one fixed point in , let
(17)
then
(18)
From (16)-(18) we infer that , which means that . We assert that . Otherwise, for , then by Lemma 4.2 we deduce that
By (17), we have , moreover, , which is a contradiction. So . Thus, by (16) and (18) we have
which means that is the unique fixed point of in .
Now, we prove that is the unique fixed point of in . By the above proof, we only need to show that does not have any fixed point in . If is a fixed point of in . Let
(19)
It is evident that . If , then . By Lemma 4.2 we have
Thus, from (19) we have , which is a contradiction. So . Moreover, , which implies the contradiction: and .
Finally, the iterative scheme and (12) can be proved in a similar way to the proof of Theorem 3.4 of [21], here it is omitted. The proof is complete. □
Theorem 4.4 Assume that (H1), (H2), (H7), and (H8) hold. Then there exists a such that BVP (1) has a unique solution in for and does not have any solution in for . Moreover, set () for any , then (12) holds.
Proof By Lemma 4.1, B has the unique fixed point in . So , moreover, . Let , we have
(20)
Set . Lemma 4.3 implies that
(21)
Similarly to the proof of Lemma 3.1, we can show that implies .
Now, take and let and , then and . By (H7), (H8), and (20), we have , that is, . Moreover, .
Let , then . We assert that . Indeed, if , from the definition of it is obvious that . Suppose that and . Then by (14) and (21) there exists such that . Similarly to the proof of (20), we have
Denote , then
(22)
Set for given , then . Since , we can choose such that . Therefore, from (22) we have
This means that , which is a contradiction to the definition of . So, . Consequently, an application of Lemma 4.3 completes the proof. □
In what follows, we assume that is the unique fixed point of B in , is the unique fixed point of in and .
Theorem 4.5 Assume that (H1), (H2), (H7), and (H8) hold. Then depends upon the parameter λ as follows:
-
(i)
is nondecreasing with respect to λ for ;
-
(ii)
is continuous with respect to λ for ;
-
(iii)
and .
Proof (i) Let with . Since , from the proof of Lemma 4.3, we find that the unique fixed of belongs to , which means that .
(ii)Let . In order to prove , let sequence satisfy
By virtue of the above conclusion (i) we have
(23)
which implies that is a bounded subset in P. Further, similarly to the proof of the conclusion (ii) in Lemma 3.3 we see that converges to . From (23) we have , which leads to . Note that
By taking the limit we have . Since has only one fixed point in , then . This means that as .
A similar argument can show that for any , as . Thus, the proof of (ii) is complete.
-
(iii)
It is obvious from the above conclusion (ii) that .
In order to finish the proof of , we consider two cases.
Case 1. .
Since , then , which means .
Case 2. .
By the above conclusion (i) we have . Suppose to the contrary that . Similarly to the case 2 in the proof of Lemma 3.3, we conclude that has a fixed point . From Remark 4.3 we have . So , which is a contradiction. This ends the proof. □
Now, we give an estimate for critical value in Theorem 4.4. If (H1) and (H8) hold, then
Moreover, .
Theorem 4.6 Assume that (H1), (H2), (H7), and (H8) hold. Then
(24)
Proof For any , there exists such that
(25)
Note that , we can choose a sufficiently large positive integer number k such that , that is,
(26)
Let , then, from (4), (25), and (26) we have
Moreover, taking , we have
Consequently, from (21) we obtain , that is, , which implies that (24) holds. This completes the proof. □
Remark 4.4 Different from Theorems 3.2 and 3.4, the estimate of in Theorem 4.6 does not take into account effect of . This is valuable, because the conditions (H2) and (H7) cannot ensure as . Certainly, if , then . In particular, if , then .
Corollary 4.7 Assume that (H1), (H2), (H7), and (H9) hold. Then
-
(i)
BVP (1) has a unique positive solution in for . Moreover, for any , set (), then ;
-
(ii)
is nondecreasing with respect to λ for ;
-
(iii)
is continuous with respect to λ for ;
-
(iv)
and .
Proof From (H1), (H2), (H7), and (4), we see that is increasing for any given . Further, for any given we have
where . Thus, the conclusion (i) follows from Lemma 1.2.
From (H9), we have for , and . Therefore, in the same way as in the proof of Theorem 4.5, we can complete the rest of the proof. □
When is a constant function, and . It is evident that B satisfies (H2) and (H7). So we can obtain the following two results.
Corollary 4.8 Assume that (H1) and (H8) hold. If , then
-
(i)
there exists such that BVP (1) with has a unique positive solution in for and does not have any solution in for . Moreover, for any , set (), then ;
-
(ii)
is nondecreasing with respect to λ for ;
-
(iii)
is continuous with respect to λ for ;
-
(iv)
and .
Corollary 4.9 Assume that (H1) and (H8) hold. If , then
-
(i)
for any , BVP (1) with has a unique positive solution in , moreover, for any , set (), then ;
-
(ii)
is nondecreasing in λ for ;
-
(iii)
is continuous with respect to λ for ;
-
(iv)
and .
Corollary 4.10 Assume that (H1) and (H9) hold. Then the conclusions (i), (ii), (iii), and (iv) in Corollary 4.9 hold.
Finally, we give two concrete examples to illustrate those results in the section.
Example 1 In BVP (1), let
it is obvious that the conditions (H1) and (H2) are satisfied. For any ,
as , we have ;
as and , we have ;
as and , we have ,
that is, for and . Similarly, we can obtain for . Therefore, the conditions (H7) and (H8) are satisfied. Note that
By Theorems 4.4, 4.5 and Remarks 4.3, 4.4 we see that there exists such that BVP (1) has a unique positive solution for and does not have any positive solution for . Moreover, for any , set (), then , and such solution satisfies the properties (i), (ii), and (iii) in Theorem 4.5.
Example 2 In BVP (1), let , , , , it is easy to see that (H1) holds. For any , we have
So, (H8) holds. Note that
by Corollary 4.9 we find that BVP (1) has a unique positive solution for . Moreover, for any , set (), then , and such a solution satisfies the properties (ii), (iii), and (iv) in Corollary 4.9 with .
In this example, by using Wolfram Mathematica 9.0, we can plot the graphs of solutions for BVP (1) with , as the Figure 1 shows.