Abstract
We consider the following complementary Lidstone boundary value problem:
By using fixed point theorems of LeggettWilliams and Avery, we offer several criteria
for the existence of three positive solutions of the boundary value problem. Examples
are also included to illustrate the results obtained. We note that the nonlinear term
F depends on
MSC: 34B15, 34B18.
Keywords:
positive solutions; complementary Lidstone boundary value problems; derivativedependent nonlinearity; fixed point theorems1 Introduction
In this paper we shall consider the complementary Lidstone boundary value problem
where
The complementary Lidstone interpolation and boundary value problems have been very recently introduced in [1], and drawn on by Agarwal et al. in [2,3] where they consider an
The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [47] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an 2mth order differential equation and the Lidstone boundary conditions
There is a vast literature on Lidstone interpolation and boundary value problems. In fact, the Lidstone interpolation was first introduced by Lidstone [8] in 1929 and further characterized in the work of [916]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in [1,1723]. Meanwhile, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,2437] and the references cited therein. In most of these works the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of papers [30,31,34,35] tackle nonlinear terms that involve even order derivatives. In the present work, our study of the complementary Lidstone boundary value problem (1.1) where F depends on a derivative certainly extends and complements the rich literature on boundary value problems and notably on Lidstone boundary value problems. The literature on complementary Lidstone boundary value problems pales in comparison with that on Lidstone boundary value problems  after the first work [2] on complementary Lidstone boundary value problems, the recent paper [38] discusses the eigenvalue problem, while in [39] the existence of at least one or two positive solutions of the complementary Lidstone boundary value problem is derived by LeraySchauder alternative and Krasnosel’skii’s fixed point theorem in a cone.
In the present work, we shall establish the existence of at least three positive solutions using fixed point theorems of Leggett and Williams [40] as well as of Avery [41]. Estimates on the norms of these solutions will also be provided. Besides achieving
new results, we also compare the results in terms of generality and illustrate the importance
of the results through some examples. As remarked earlier, the presence of the derivative
The paper is organized as follows. Section 2 contains the necessary definitions and fixed point theorems. The existence criteria are developed and discussed in Section 3. Finally, examples are presented in Section 4 to illustrate the importance of the results obtained.
2 Preliminaries
In this section we shall state some necessary definitions, the relevant fixed point
theorems and properties of certain Green’s function. Let B be a Banach space equipped with the norm
Definition 2.1 Let C (⊂B) be a nonempty closed convex set. We say that C is a cone provided the following conditions are satisfied:
(a) If
(b) If
Definition 2.2 Let C (⊂B) be a cone. A map ψ is a nonnegative continuous concave functional on C if the following conditions are satisfied:
(a)
(b)
Definition 2.3 Let C (⊂B) be a cone. A map β is a nonnegative continuous convex functional on C if the following conditions are satisfied:
(a)
(b)
Let γ, β, Θ be nonnegative continuous convex functionals on C and α, ψ be nonnegative continuous concave functionals on C. For nonnegative numbers
The following fixed point theorems are our main tools, the first is usually called LeggettWilliams’ fixed point theorem, and the second is known as the fivefunctional fixed point theorem.
Theorem 2.1[40]
LetC (⊂B) be a cone, and
(a)
(b)
(c)
ThenShas (at least) three fixed points
Theorem 2.2[41]
LetC (⊂B) be a cone. Assume that there exist positive numbers
for all
(a)
(b)
(c)
(d)
ThenShas (at least) three fixed points
We also require the definition of an
Definition 2.4[42]
A function
(a) The map
(b) The map
(c) For any
To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let
The Green’s function
where
Further, it is known that
The following two lemmas give the upper and lower bounds of
Lemma 2.1[38]
For
Lemma 2.2[38]
Let
3 Triple positive solutions
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 solutiony of (1.1), we mean a nontrivial
To tackle (1.1), we first consider the initial value problem
whose solution is simply
Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem
If (3.3) has a solution
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
Let the Banach space
where
For easy reference, we shall list the conditions that are needed later. In these conditions
the sets K and
(C1)
(C2) We have
(C3) There exist continuous functions f, ν, μ with
(C4) There exists a number
If (C2) and (C3) hold, then it follows from (3.5) that for
Let
where θ is given in (C4). Clearly, we have
Lemma 3.1Let (C1)(C4) hold. Then the operatorSdefined in (3.5) is continuous and completely continuous, andSmapsCintoC.
Proof From (2.4) we have
Let
Hence, we have
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
Lemma 3.2Let (C1)(C4) hold, and assume
(C5) the function
Suppose that there exists a number
Then
Proof Let
Then, using (3.9), (C5) and (3.12), we find for
This implies
Using a similar argument as Lemma 3.2, we have the following lemma.
Lemma 3.3Let (C1)(C4) hold. Suppose that there exists a number
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 LeggettWilliams’ fixed point theorem (Theorem 2.1).
Theorem 3.1Let
(C6) for each
Suppose that there exist numbers
such that the following hold:
(P)
(Q) one of the following holds:
(Q1)
(Q2) there exists a numberd (
(R)
Then we have the following conclusions:
(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions
(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions
(where
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
Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick
Let
Noting (3.18), it is then clear that for
Now, pick the number
Let
Hence,
Let
Clearly, ψ is a nonnegative continuous concave functional on C and
We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that
and so
Using (3.7), (3.21), (C6) and (R), it follows that
Therefore, we have shown that
Next, by condition (P) and Lemma 3.2 (with
Finally, we shall show that condition (c) of Theorem 2.1 holds. Let
Hence, we have proved that
It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has
(at least) three positive solutions
Finally, it is observed from (3.4) that the complementary Lidstone boundary value
problem (1.1) has (at least) three positive solutions
Moreover, since
Combining (3.22) and (3.23) gives (3.15) immediately.
Further, since
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 fivefunctional 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 3.2Let
such that
(C7) for each
(C8) the function
Suppose that there exist numbers
such that the following hold:
(P)
(Q)
(R)
Then we have the following conclusions:
(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions
(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions
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:
First, we shall show that the operator S maps
Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that
since it has an element
Noting (3.7), (3.28), (C7) and (R), we find
Hence,
We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let
because it has an element
which lead to the following:
Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find
Therefore,
Next, we shall show that condition (c) of Theorem 2.2 is met. Let
Moreover, using the fact that S maps C into C, we find
Combining (3.31) and (3.32) yields
Now, let
Thus,
Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let
It now follows from Theorem 2.2 that the Lidstone boundary value problem (3.3) has
(at least) three positive solutions
Finally, as in the proof of Theorem 3.1, we see that (3.15) holds for the positive
solutions
Next, noting
Lastly, using (3.15) and
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 3.1Let
(C7)′ for each
(C8)′ the function
Suppose that there exist numbers
such that the following hold:
(P)
(Q)
(R)
Then we have the following conclusions:
(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions
(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions
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
Theorem 3.3Let
such that (C7) and (C8) hold. Suppose that there exist numbers
such that the following hold:
(P)
(Q)
(R)
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)):
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
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
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
Noting that S maps C into C, we find
A combination of (3.39) and (3.40) gives
Let
Thus,
Conclusion (a) now follows from Theorem 2.2 immediately, while conclusion (b) is similarly obtained as in Theorem 3.2. □
4 Examples
In this section, we shall present examples to illustrate the usefulness as well as to compare the generality of the results obtained in Section 3.
Example 4.1 Consider the complementary Lidstone boundary value problem (1.1) with
where
Here,
Let the functions
For convenience, we take
Hence, (4.3) reduces to
and clearly we can easily find numbers
We shall check the conditions of Theorem 3.1. First, condition (P) is obviously satisfied.
Next, from (4.3) we have
Hence, condition (Q2) is met. Finally, (R) is satisfied since for
By Theorem 3.1 (conclusion (b)), the boundary value problem (1.1) with
where
Example 4.2 Consider the complementary Lidstone boundary value problem (1.1) with
where
Here, we fix
and the
Let the functions
By using Lemma 2.2, we get
For convenience, we take
or equivalently (combining the first two inequalities)
It is clear that we can easily find numbers
We shall check the conditions of Theorem 3.2. First, condition (P) is obviously satisfied. Next, since
we find for
Hence, condition (Q) is met. Finally, (R) is satisfied since for
By Theorem 3.2 (conclusion (b)), the boundary value problem (1.1) with
where
Remark 4.1 In Example 4.2, we see that for
Thus, condition (R) of Corollary 3.1 is not satisfied and so Corollary 3.1 cannot be used to establish the existence of triple positive solutions in Example 4.2. Recalling that Corollary 3.1 is actually Theorem 3.1, this illustrates the case when Theorem 3.2 is applicable but not Theorem 3.1. Hence, this example shows that Theorem 3.2 is indeed more general than Theorem 3.1.
Competing interests
The author declares that she has no competing interests.
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