In this section, we shall use the fixed point theorems stated in Section 2 to obtain the existence of at least three positive solutions of the complementary Lidstone boundary value problem (1.1). By a positive solution y of (1.1), we mean a nontrivial satisfying (1.1) and for .
To tackle (1.1), we first consider the initial value problem
(3.1)
whose solution is simply
(3.2)
Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem
(3.3)
If (3.3) has a solution , then by virtue of (3.2), the boundary value problem (1.1) has a solution given by
(3.4)
So the existence of a solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a solution of the Lidstone boundary value problem (3.3). It is clear from (3.4) that ; moreover if is positive, so is . With the tools in Section 2 and a technique to handle the nonlinear term F, we shall study the boundary value problem (1.1) via (3.3).
Let the Banach space be equipped with the norm for . Define the operator by
(3.5)
where is the Green’s function given in (2.4). A fixed point of the operator S is clearly a solution of the boundary value problem (3.3), and as seen earlier is a solution of (1.1).
For easy reference, we shall list the conditions that are needed later. In these conditions the sets K and are defined by
(3.6)
(C1) is an -Carathéodory function.
(C2) We have
(C3) There exist continuous functions f, ν, μ with and such that
(C4) There exists a number such that
If (C2) and (C3) hold, then it follows from (3.5) that for and ,
(3.7)
Let be fixed. We define a cone C in B as
(3.8)
where θ is given in (C4). Clearly, we have .
Lemma 3.1 Let (C1)-(C4) hold. Then the operator S defined in (3.5) is continuous and completely continuous, and S maps C into C.
Proof From (2.4) we have , and the map is continuous from to . This together with is an -Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that S is continuous and completely continuous.
Let . From (3.7) we have for . Next, using (3.7) and Lemma 2.1 gives for ,
(3.9)
Hence, we have
(3.10)
Now, employing (3.7), Lemma 2.2, (C4) and (3.10), we find for ,
This leads to
We have shown that . □
For subsequent results, we define the following constants for fixed and :
(3.11)
Lemma 3.2 Let (C1)-(C4) hold, and assume
(C5) the function on a subset of of positive measure.
Suppose that there exists a number such that for ,
Then
(3.13)
Proof Let . So , which implies immediately that
Then, using (3.9), (C5) and (3.12), we find for ,
This implies . Together with the fact that (Lemma 3.1), we have shown that . Conclusion (3.13) is now immediate. □
Using a similar argument as Lemma 3.2, we have the following lemma.
Lemma 3.3 Let (C1)-(C4) hold. Suppose that there exists a number such that for ,
Then
We are now ready to establish the existence of three positive solutions for the complementary Lidstone boundary value problem (1.1). The first result below uses Leggett-Williams’ fixed point theorem (Theorem 2.1).
Theorem 3.1 Let be fixed. Let (C1)-(C5) hold, and assume
(C6) for each , the function on a subset of of positive measure.
Suppose that there exist numbers , , with
such that the following hold:
-
(P)
for ;
-
(Q)
one of the following holds:
-
(Q1)
or ;
-
(Q2)
there exists a number d () such that for ;
-
(R)
for and .
Then we have the following conclusions:
-
(a)
The Lidstone boundary value problem (3.3) has (at least) three positive solutions (where C is defined in (3.8)) such that
(3.14)
-
(b)
The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions , , such that for ,
(3.15)
(where ’s are those in conclusion (a)). We further have
(3.16)
Proof We shall employ Theorem 2.1 with the cone C defined in (3.8). First, we shall prove that condition (Q) implies the existence of a number , where , such that
(3.17)
Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick . Suppose now that of (Q1) is satisfied. Then there exist and such that
(3.18)
Let
Noting (3.18), it is then clear that for ,
Now, pick the number so that
(3.20)
Let . Using (3.9), (3.19) and (3.20) yields for ,
Hence, and so . Thus, (3.17) follows immediately. Note that the argument is similar if we assume that of (Q1) is satisfied.
Let be defined by
Clearly, ψ is a nonnegative continuous concave functional on C and for all .
We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that
and so . Next, let . Then and which imply
(3.21)
Using (3.7), (3.21), (C6) and (R), it follows that
Therefore, we have shown that for all .
Next, by condition (P) and Lemma 3.2 (with ), we have . Hence, condition (b) of Theorem 2.1 is fulfilled.
Finally, we shall show that condition (c) of Theorem 2.1 holds. Let with . Using (3.7), Lemma 2.2, (C4), (3.10) and the inequality , we find
Hence, we have proved that for all with .
It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions satisfying (2.1). It is easy to see that here (2.1) reduces to (3.14). This completes the proof of conclusion (a).
Finally, it is observed from (3.4) that the complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions , , such that for ,
(3.22)
Moreover, since , we get for ,
(3.23)
Combining (3.22) and (3.23) gives (3.15) immediately.
Further, since for , we have for ,
(3.24)
Hence, noting (3.14), (3.15) and (3.24), we get (3.16). This completes the proof of conclusion (b). □
We shall now employ the five-functional fixed point theorem (Theorem 2.2) to give other existence criteria. In applying Theorem 2.2 it is possible to choose the functionals and constants in different ways, indeed we shall do so and derive two results. Our first result below turns out to be a generalization of Theorem 3.1.
Theorem 3.2 Let be fixed. Let (C1)-(C4) hold. Assume that there exist numbers , , with
such that
(C7) for each , the function on a subset of of positive measure;
(C8) the function on a subset of of positive measure.
Suppose that there exist numbers , , with
such that the following hold:
-
(P)
for and ;
-
(Q)
for ;
-
(R)
for and .
Then we have the following conclusions:
-
(a)
The Lidstone boundary value problem (3.3) has (at least) three positive solutions (where C is defined in (3.8)) such that
(3.25)
-
(b)
The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions , , such that (3.15) holds for . We further have
(3.26)
Proof We shall apply Theorem 2.2 with the cone C defined in (3.8). We define the following five functionals on the cone C:
(3.27)
First, we shall show that the operator S maps into . Note that . By (Q) and Lemma 3.3 (with ), we immediately have .
Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that
since it has an element . Let . Then by definition we have and , which imply
(3.28)
Noting (3.7), (3.28), (C7) and (R), we find
Hence, for all .
We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let be such that . Note that
because it has an element . Let . Then we have and , i.e.,
(3.29)
which lead to the following:
(3.30)
Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find
Therefore, for all .
Next, we shall show that condition (c) of Theorem 2.2 is met. Let . Clearly, we have
(3.31)
Moreover, using the fact that S maps C into C, we find
(3.32)
Combining (3.31) and (3.32) yields
(3.33)
Now, let with . Then it follows from (3.33) and the inequality that
(3.34)
Thus, for all with .
Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let with . Then we have and which give (3.29) and (3.30). As in proving condition (b), we get . Hence, condition (d) of Theorem 2.2 is satisfied.
It now follows from Theorem 2.2 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions satisfying (2.2). Furthermore, (2.2) reduces to (3.25) immediately. This completes the proof of conclusion (a).
Finally, as in the proof of Theorem 3.1, we see that (3.15) holds for the positive solutions , , of the complementary Lidstone boundary value problem (1.1). Moreover, noting that for , we find for ,
Next, noting for , we get for ,
Lastly, using (3.15) and , we find for ,
The proof of conclusion (b) is complete. □
We shall now consider the special case of Theorem 3.2 when
Then, from definitions (3.11), we see that
In this case Theorem 3.2 yields the following corollary.
Corollary 3.1 Let be fixed. Let (C1)-(C4) hold, and assume
(C7)′ for each , the function on a subset of of positive measure;
(C8)′ the function on a subset of of positive measure.
Suppose that there exist numbers , , with
such that the following hold:
-
(P)
for ;
-
(Q)
for ;
-
(R)
for and .
Then we have the following conclusions:
-
(a)
The Lidstone boundary value problem (3.3) has (at least) three positive solutions (where C is defined in (3.8)) such that
(3.35)
-
(b)
The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions , , such that (3.15) holds for . We further have
(3.36)
Remark 3.1 Corollary 3.1 is actually Theorem 3.1. Since Corollary 3.1 is a special case of Theorem 3.2, this shows that Theorem 3.2 is more general than Theorem 3.1.
The next theorem illustrates another application of Theorem 2.2. Compared to the conditions in Theorem 3.2, here the numbers , and have different ranges and condition (P) is also different. Note that in the proof of Theorem 3.3 the functionals ψ and Θ are chosen differently from those in Theorem 3.2.
Theorem 3.3 Let be fixed. Let (C1)-(C4) hold. Assume that there exist numbers , , with
such that (C7) and (C8) hold. Suppose that there exist numbers , , with
such that the following hold:
-
(P)
for and ;
-
(Q)
for ;
-
(R)
for and .
Then we have conclusions (a) and (b) of Theorem 3.2.
Proof To apply Theorem 2.2, we shall define the following functionals on the cone C (see (3.8)):
(3.37)
As in the proof of Theorem 3.2, using (Q) and Lemma 3.3 we can show that .
Next, to see that condition (a) of Theorem 2.2 is fulfilled, we use (R) and a similar argument as in the proof of Theorem 3.2.
We shall now prove that condition (b) of Theorem 2.2 is satisfied. Note that
Let . Then we have , and which imply
(3.38)
and also (3.30). In view of (3.9), (3.38), (3.30), (C8), (P) and (Q), we obtain, as in the proof of Theorem 3.2, that . Therefore, condition (b) of Theorem 2.2 is fulfilled.
Next, using a similar argument as in the proof of Theorem 3.2, we see that condition (c) of Theorem 2.2 is met.
Finally, we shall verify that condition (d) of Theorem 2.2 is fulfilled. Let . It is clear that
(3.39)
Noting that S maps C into C, we find
(3.40)
A combination of (3.39) and (3.40) gives
(3.41)
Let with . Then (3.41) and the inequality lead to
Thus, for all with .
Conclusion (a) now follows from Theorem 2.2 immediately, while conclusion (b) is similarly obtained as in Theorem 3.2. □