Triple solutions of complementary Lidstone boundary value problems via fixed point theorems

We consider the following complementary Lidstone boundary value problem:

By using fixed point theorems of Leggett-Williams 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 y^{\prime } and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem.

MSC:34B15, 34B18.

In this paper we shall consider the complementary Lidstone boundary value problem

where m\ge 1 and F is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term F involves y^{\prime }, a derivative of the dependent variable. Most research papers on boundary value problems consider nonlinear terms that involve y only, and derivative-dependent nonlinearities are seldom tackled as special techniques are required.

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 (2m+1)th order differential equation together with boundary data at the odd order derivatives

The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4–7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an 2m th 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 [9–16]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in [1, 17–23]. Meanwhile, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4, 18, 24–37] 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 Leray-Schauder 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 y^{\prime } in the nonlinear term F requires a special technique to tackle the problem.

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.

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 \parallel \cdot \parallel.

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:

1. (a)

   If x\in C and \alpha \ge 0, then \alpha x\in C;

2. (b)

   If x\in C and -x\in C, then x=0.

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:

1. (a)

   \psi :C\to [0,\mathrm{\infty }) is continuous;

2. (b)

   \psi (ty+(1-t)z)\ge t\psi (y)+(1-t)\psi (z) for all y,z\in C and 0\le t\le 1.

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:

1. (a)

   \beta :C\to [0,\mathrm{\infty }) is continuous;

2. (b)

   \beta (ty+(1-t)z)\le t\beta (y)+(1-t)\beta (z) for all y,z\in C and 0\le t\le 1.

Let γ, β, Θ be nonnegative continuous convex functionals on C and α, ψ be nonnegative continuous concave functionals on C. For nonnegative numbers w_i, 1\le i\le 3, we shall introduce the following notations:

The following fixed point theorems are our main tools, the first is usually called Leggett-Williams’ fixed point theorem, and the second is known as the five-functional fixed point theorem.

Theorem 2.1 [40]

Let C (⊂B) be a cone, and w_4>0 be given. Assume that ψ is a nonnegative continuous concave functional on C such that \psi (x)\le \parallel x\parallel for all x\in \overline{C}(w_4), and let S:\overline{C}(w_4)\to \overline{C}(w_4) be a continuous and completely continuous operator. Suppose that there exist numbers w_1, w_2, w_3, where , such that

1. (a)

   \{x\in C(\psi ,w_2,w_3)\mid \psi (x)>w_2\}\ne \mathrm{\varnothing }, and \psi (Sx)>w_2 for all x\in C(\psi ,w_2,w_3);

2. (b)

   for all x\in \overline{C}(w_1);

3. (c)

   \psi (Sx)>w_2 for all x\in C(\psi ,w_2,w_4) with \parallel Sx\parallel >w_3.

Then S has (at least) three fixed points x_1, x_2 and x_3 in \overline{C}(w_4). Furthermore, we have

Theorem 2.2 [41]

Let C (⊂B) be a cone. Assume that there exist positive numbers w_5, M, nonnegative continuous convex functionals γ, β, Θ on C, and nonnegative continuous concave functionals α, ψ on C, with

for all x\in \overline{P}(\gamma ,w_5). Let S:\overline{P}(\gamma ,w_5)\to \overline{P}(\gamma ,w_5) be a continuous and completely continuous operator. Suppose that there exist nonnegative numbers w_i, 1\le i\le 4, with such that

1. (a)

   \{x\in P(\gamma ,\mathrm{\Theta },\alpha ,w_3,w_4,w_5)\mid \alpha (x)>w_3\}\ne \mathrm{\varnothing }, and \alpha (Sx)>w_3 for all x\in P(\gamma ,\mathrm{\Theta },\alpha ,w_3,w_4,w_5);

2. (b)

   , and for all x\in Q(\gamma ,\beta ,\psi ,w_1,w_2,w_5);

3. (c)

   \alpha (Sx)>w_3 for all x\in P(\gamma ,\alpha ,w_3,w_5) with \mathrm{\Theta }(Sx)>w_4;

4. (d)

   for all x\in Q(\gamma ,\beta ,w_2,w_5) with .

Then S has (at least) three fixed points x_1, x_2 and x_3 in \overline{P}(\gamma ,w_5). Furthermore, we have

We also require the definition of an L^1-Carathéodory function.

Definition 2.4 [42]

A function P:[0,1]\times {\mathbb{R}}^2\to \mathbb{R} is an L^1-Carathéodory function if the following conditions hold:

1. (a)

   The map t\to P(t,u) is measurable for all u\in {\mathbb{R}}^2.

2. (b)

   The map u\to P(t,u) is continuous for almost all t\in [0,1].

3. (c)

   For any r>0, there exists {\mu }_r\in L^1[0,1] such that |u|\le r implies that |P(t,u)|\le {\mu }_r(t) for almost all t\in [0,1].

To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let g_m(t,s) be the Green’s function of the Lidstone boundary value problem

The Green’s function g_m(t,s) can be expressed as [4, 5]

where

Further, it is known that

The following two lemmas give the upper and lower bounds of |g_m(t,s)|, they play an important role in subsequent development. We remark that the bounds in the two lemmas are sharper than those given in the literature [4, 5, 35, 37].

Lemma 2.1 [38]

For (t,s)\in [0,1]\times [0,1], we have

Lemma 2.2 [38]

Let \delta \in (0,\frac{1}{2}) be given. For (t,s)\in [\delta ,1-\delta ]\times [0,1], we have

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 y\in C[0,1] satisfying (1.1) and y(t)\ge 0 for t\in [0,1].

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 x^{\ast }, then by virtue of (3.2), the boundary value problem (1.1) has a solution given by

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 \parallel y^{\ast }\parallel \le \parallel x^{\ast }\parallel; moreover if x^{\ast } is positive, so is y^{\ast }. 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 B=C[0,1] be equipped with the norm \parallel x\parallel ={sup}_{t\in [0,1]}|x(t)| for x\in B. Define the operator S:C[0,1]\to C[0,1] by

where g_m(t,s) is the Green’s function given in (2.4). A fixed point x^{\ast } of the operator S is clearly a solution of the boundary value problem (3.3), and as seen earlier y^{\ast }(t)={\int }_0^t{x}^{\ast }(s)ds 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 \tilde{K} are defined by

(C1) F:[0,1]\times {\mathbb{R}}^2\to \mathbb{R} is an L^1-Carathéodory function.

(C2) We have

(C3) There exist continuous functions f, ν, μ with f:[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to [0,\mathrm{\infty }) and \nu ,\mu :(0,1)\to [0,\mathrm{\infty }) such that

(C4) There exists a number such that

If (C2) and (C3) hold, then it follows from (3.5) that for x\in \tilde{K} and t\in [0,1],

Let \delta \in (0,\frac{1}{2}) be fixed. We define a cone C in B as

where θ is given in (C4). Clearly, we have C\subseteq \tilde{K}.

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 g_m(t,s)\in C[0,1]\subseteq L^{\mathrm{\infty }}[0,1], t\in [0,1] and the map t\to g_m(t,s) is continuous from [0,1] to C[0,1]. This together with F:[0,1]\times {\mathbb{R}}^2\to \mathbb{R} is an L^1-Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that S is continuous and completely continuous.

Let x\in C. From (3.7) we have Sx(t)\ge 0 for t\in [0,1]. Next, using (3.7) and Lemma 2.1 gives for t\in [0,1],

Hence, we have

Now, employing (3.7), Lemma 2.2, (C4) and (3.10), we find for t\in [\delta ,1-\delta ],

This leads to

We have shown that Sx\in C. □

For subsequent results, we define the following constants for fixed \delta \in (0,\frac{1}{2}) and {\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4\in [0,1]:

Lemma 3.2 Let (C1)-(C4) hold, and assume

(C5) the function \nu (s)sin\pi s>0 on a subset of [0,1] of positive measure.

Suppose that there exists a number d>0 such that for u,v\in [0,d],

Then

Proof Let x\in \overline{C}(d). So \parallel x\parallel \le d, which implies immediately that

Then, using (3.9), (C5) and (3.12), we find for t\in [0,1],

This implies . Together with the fact that Sx\in C (Lemma 3.1), we have shown that Sx\in C(d). 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 d>0 such that for u,v\in [0,d],

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 \delta \in (0,\frac{1}{2}) be fixed. Let (C1)-(C5) hold, and assume

(C6) for each t\in [\delta ,1-\delta ], the function |g_m(t,s)|\mu (s)>0 on a subset of [\frac{1}{2},1-\delta ] of positive measure.

Suppose that there exist numbers w_1, w_2, w_3 with

such that the following hold:

1. (P)

   for u,v\in [0,w_1];

2. (Q)

   one of the following holds:

   1. (Q1)

      or ;

   2. (Q2)

      there exists a number d (\ge w_3) such that f(u,v)\le \frac{d}{q} for u,v\in [0,d];

3. (R)

   f(u,v)>\frac{w_2}{r} for u\in [w_2(\frac{1}{2}-\delta ),w_3(\frac{1}{2}-\delta )] and v\in [w_2,w_3].

Then we have the following conclusions:

1. (a)

   The Lidstone boundary value problem (3.3) has (at least) three positive solutions x_1,x_2,x_3\in C (where C is defined in (3.8)) such that

   (3.14)
2. (b)

   The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y_1, y_2, y_3 such that for i=1,2,3,

   \{\begin{array}{l}{y}_i(t)={\int }_0^t{x}_i(s)ds,t\in [0,1];\\ \parallel y_i\parallel \le \parallel x_i\parallel ;y_i(t)\ge \frac{2\delta \theta }{\pi }\parallel x_i\parallel (t-\delta ),t\in [\delta ,1-\delta ]\end{array}

   (3.15)

(where x_i’s are those in conclusion (a)). We further have

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 w_4, where w_4\ge w_3, such that

Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick w_4=d. Suppose now that of (Q1) is satisfied. Then there exist N>0 and such that

Let

Noting (3.18), it is then clear that for u,v\ge 0,

Now, pick the number w_4 so that

Let x\in \overline{C}(w_4). Using (3.9), (3.19) and (3.20) yields for t\in [0,1],

Hence, and so Sx\in C(w_4)\subset \overline{C}(w_4). Thus, (3.17) follows immediately. Note that the argument is similar if we assume that of (Q1) is satisfied.

Let \psi :C\to [0,\mathrm{\infty }) be defined by

Clearly, ψ is a nonnegative continuous concave functional on C and \psi (x)\le \parallel x\parallel for all x\in C.

We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that

and so \{x\in C(\psi ,w_2,w_3)\mid \psi (x)>w_2\}\ne \mathrm{\varnothing }. Next, let x\in C(\psi ,w_2,w_3). Then \psi (x)\ge w_2 and \parallel x\parallel \le w_3 which imply

Using (3.7), (3.21), (C6) and (R), it follows that

Therefore, we have shown that \psi (Sx)>w_2 for all x\in C(\psi ,w_2,w_3).

Next, by condition (P) and Lemma 3.2 (with d=w_1), we have S(\overline{C}(w_1))\subseteq C(w_1). Hence, condition (b) of Theorem 2.1 is fulfilled.

Finally, we shall show that condition (c) of Theorem 2.1 holds. Let x\in C(\psi ,w_2,w_4) with \parallel Sx\parallel >w_3. Using (3.7), Lemma 2.2, (C4), (3.10) and the inequality \frac{\pi w_2}{2\delta \theta }\le w_3, we find

Hence, we have proved that \psi (Sx)>w_2 for all x\in C(\psi ,w_2,w_4) with \parallel Sx\parallel >w_3.

It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions x_1,x_2,x_3\in \overline{C}(w_4) 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 y_1, y_2, y_3 such that for i=1,2,3,

Moreover, since x_i\in C, we get for t\in [\delta ,1-\delta ],

Combining (3.22) and (3.23) gives (3.15) immediately.

Further, since x_2(t)>w_2 for t\in [\delta ,1-\delta ], we have for t\in [\delta ,1-\delta ],

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 \delta \in (0,\frac{1}{2}) be fixed. Let (C1)-(C4) hold. Assume that there exist numbers {\tau }_j, 1\le j\le 4, with

such that

(C7) for each t\in [{\tau }_2,{\tau }_3], the function |g_m(t,s)|\mu (s)>0 on a subset of [\frac{1}{2},{\tau }_3] of positive measure;

(C8) the function \nu (s)sin\pi s>0 on a subset of [{\tau }_1,{\tau }_4] of positive measure.

Suppose that there exist numbers w_i, 2\le i\le 5, with

such that the following hold:

1. (P)

   for u\in [0,{\tau }_1{w}_5+({\tau }_4-{\tau }_1)w_2] and v\in [0,w_2];

2. (Q)

   f(u,v)\le \frac{w_5}{q} for u,v\in [0,w_5];

3. (R)

   f(u,v)>\frac{w_3}{p_1} for u\in [w_3(\frac{1}{2}-{\tau }_2),w_4(\frac{1}{2}-{\tau }_2)] and v\in [w_3,w_4].

Then we have the following conclusions:

1. (a)

   The Lidstone boundary value problem (3.3) has (at least) three positive solutions x_1,x_2,x_3\in \overline{C}(w_5) (where C is defined in (3.8)) such that

   (3.25)
2. (b)

   The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y_1, y_2, y_3 such that (3.15) holds for i=1,2,3. 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:

First, we shall show that the operator S maps \overline{P}(\gamma ,w_5) into \overline{P}(\gamma ,w_5). Note that \overline{P}(\gamma ,w_5)=\overline{C}(w_5). By (Q) and Lemma 3.3 (with d=w_5), we immediately have S(\overline{C}(w_5))\subseteq \overline{C}(w_5).

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that

since it has an element x(t)=\frac{1}{2}(w_3+w_4). Let x\in P(\gamma ,\mathrm{\Theta },\alpha ,w_3,w_4,w_5). Then by definition we have \alpha (x)\ge w_3 and \mathrm{\Theta }(x)\le w_4, which imply

Noting (3.7), (3.28), (C7) and (R), we find

Hence, \alpha (Sx)>w_3 for all x\in P(\gamma ,\mathrm{\Theta },\alpha ,w_3,w_4,w_5).

We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let w_1 be such that . Note that

because it has an element x(t)=\frac{1}{2}(w_1+w_2). Let x\in Q(\gamma ,\beta ,\psi ,w_1,w_2,w_5). Then we have \beta (x)\le w_2 and \gamma (x)\le w_5, i.e.,

which lead to the following:

Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find