We consider the following complementary Lidstone boundary value problem

where λ > 0. The values of λ are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of λ such that for any λ in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y' and this derivative dependence is seldom investigated in the literature.

AMS Subject Classification: 34B15.

In this article, we shall consider the complementary Lidstone boundary value problem

where m ≥ 1, λ > 0, and F is continuous at least in the interior of the domain of interest. Note that the nonlinear term F involves a derivative of the dependent variable--this is seldom studied in the literature and most research articles on boundary value problems consider nonlinear terms that involve y only.

We are interested in the existence of a positive solution of (1.1). By a positive solution y of (1.1), we mean a nontrivial y ∈ C^(2m+1)(0, 1) satisfying (1.1) and y(t) ≥ 0 for t ∈ (0, 1). If, for a particular λ the boundary value problem (1.1) has a positive solution y, then λ is called an eigenvalue and y is a corresponding eigenfunction of (1.1). We shall denote the set of eigenvalues of (1.1) by E, i.e.,

The focus of this article is eigenvalue problem, as such we shall characterize the values of λ so that the boundary value problem (1.1) has a positive solution. To be specific, we shall establish criteria for E to contain an interval, and for E to be an interval (which may either be bounded or unbounded). In addition explicit subintervals of E are derived.

The complementary Lidstone interpolation and boundary value problems are very recently introduced in [1], and studied by Agarwal et. al. [2,3] where they consider an odd order ((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 even order (2mth order) differential equation and the Lidstone boundary conditions

There is a vast literature on Lidstone interpolation and boundary value problems. The Lidstone interpolation has a long history from 1929 when Lidstone [8] introduced a generalization of Taylor's series that approximates a given function in the neighborhood of two points instead of one. Further characterization can be found in the study of [9-16]. More research on Lidstone interpolation as well as Lidstone spline is seen in [1,17-23]. On the other hand, 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. It is noted that in most of these studies the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of articles [30,31,34,35] tackle nonlinear terms that involve even order derivatives. In the present study, 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 in particular on Lidstone boundary value problems.

The plan of the article is as follows. In Section 2, we shall state a fixed point theorem due to Krasnosel'skii [38], and develop some inequalities for certain Green's function which are needed later. The characterization of the set E is presented in Section 3. Finally, in Section 4, we establish explicit subintervals of E.

Theorem 2.1. [38] Let B be a Banach space, and let C(⊂ B) be a cone. Assume Ω₁, Ω₂ are open subsets of B with , and let be a completely continuous operator such that, either

(a) ∥Sy∥ ≤ ∥y∥, y ∈ C∩∂Ω₁, and ∥Sy∥ ≥ ∥y∥, y ∈ C ∩ ∂Ω₂, or

(b) ∥Sy∥ ≥ ∥y∥, y ∈ C∩∂Ω₁, and ∥Sy∥ ≤ ∥y∥, y ∈ C ∩ ∂Ω₂.

Then, S has a fixed point in .

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

We also have the inequality

The following two lemmas give the upper and lower bounds of |g_m(t, s)|, they play an important role in subsequent development.

Lemma 2.1. For (t, s) ∈ [0, 1] × [0, 1], we have

Proof. For (t, s) ∈ [0, 1] × [0, 1], it is clear from (2.3) that

Using (2.7), (2.4), and (2.5) in (2.2) yields for (t, s) ∈ [0, 1] × [0, 1],

By induction, we can show that

Now (2.6) is immediate by applying (2.9) to (2.8).

Lemma 2.2. Let be given. For (t, s) ∈ [δ, 1-δ] × [0, 1], we have

Proof. For (t, s) ∈ [δ, 1-δ] × [0, 1], from (2.3) we find

Then, using (2.11), (2.4), and (2.5) in (2.2), we get for (t, s) ∈ [δ, 1 - δ ] × [0, 1],

which, in view of (2.9), gives (2.10) immediately.

Remark 2.1. The bounds in Lemmas 2.1 and 2.2 are sharper than those given in the literature [4,5,35,37].

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 positive solution x*, then by virtue of (3.2), is a positive solution of (1.1). Hence, the existence of a positive solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a positive solution of the Lidstone boundary value problem (3.3). It is clear that an eigenvalue of (3.3) is also an eigenvalue of (1.1), thus

With the lemmas developed in Section 2 and a technique to handle the nonlinear term F, we shall study the eigenvalue problem (1.1) via (3.3).

For easy reference, we list below the conditions that are used later. In these conditions, f, α, and β are continuous functions with f : (0, ∞) × (0, ∞) → (0, ∞) and α, β : (0, 1) → [0, ∞).

(A1) f is nondecreasing in each of its arguments, i.e., for u, u₁, u₂, v, v₁, v₂ ∈ (0, ∞) with u₁ ≤ u₂ and v₁ ≤ v₂, we have

(A2) for t ∈ (0, 1) and u, v ∈ (0, ∞),

(A3) α(t) is not identically zero on any nondegenerate subinterval of (0, 1) and there exists a₀ ∈ (0, 1] such that α(t) ≥ a₀β(t) for all t ∈ (0, 1);

(A4) ;

(A5) for t ∈ (0, 1) and u, u₁, u₂, v, v₁, v₂ ∈ (0, ∞) with u₁ ≤ u₂ and v₁ ≤ v₂, we have

We shall consider the Banach space B = C[0, 1] equipped with the norm

For a given , let the cone C_δ be defined by

where (a₀ is defined in (A3)). Further, let

Let the operator S : C_δ → B be defined by

To obtain a positive solution of (3.3), we shall seek a fixed point of the operator S in the cone C_δ.

Further, we define the operators U, V : C_δ → B by

and

If (A2) holds, then

Lemma 3.1. Let (A1)-(A4) hold. Then, the operator S is compact on the cone C_δ.

Proof. Let us consider the case when α(t) is unbounded in a deleted right neighborhood of 0 and also in a deleted left neighborhood of 1. Clearly, β(t) is also unbounded near 0 and 1.

For n ∈ {1, 2, 3, ...}, let α_n, β_n : [0, 1] → [0, ∞) be defined by

and

Also, we define the operators U_n, V_n : C_δ → B by

and

It is standard that for each n, both U_n and V_n are compact operators on C_δ. Let M > 0 and x ∈ C_δ(M). For t ∈ [0, 1], we get

By the monotonicity of f (see (A1)), we have

Coupling with Lemma 2.1, it follows that

The integrability of β(t) sin πt (see (A4)) ensures that V_n converges uniformly to V on C_δ(M). Hence, V is compact on C_δ. By a similar argument, we see that U_n converges uniformly to U on C_δ(M) and therefore U is also compact on C_δ. It follows immediately from inequality (3.5) that the operator S is compact on C_δ.

Remark 3.1. From the proof of Lemma 3.1, we see that if the functions α and β are continuous on the close interval [0, 1], then the conditions (A1) and (A4) are not needed in Lemma 3.1.

The first result shows that E contains an interval.

Theorem 3.1. Let (A1)-(A4) hold. Then, there exists c > 0 such that the interval (0, c] ⊆ E.

Proof. Let M > 0 be given. Define

Let λ ∈ (0, c]. We shall prove that S(C_δ(M)) ⊆ C_δ(M). Let x ∈ C_δ (M). First, we shall show that Sx ∈ C_δ. It is clear from (3.5) that

Further, from (3.5) and Lemma 2.1 we get

which leads to

Now, applying (3.5), Lemma 2.2, (A3) and (3.9) successively, we find for t ∈ [δ, 1-δ],

Therefore,

Inequalities (3.8) and (3.10) imply that Sx ∈ C_δ.

Next, we shall verify that ∥Sx∥ ≤ M. For this, an application of (3.5), Lemma 2.1, (3.6) and (3.7) provides

or equivalently

Hence, S(C_δ(M)) ⊆ C_δ(M). Also, the standard arguments yield that S is completely continuous. By Schauder fixed point theorem, S has a fixed point in C_δ(M). Clearly, this fixed point is a positive solution of (3.3) and therefore λ is an eigenvalue of (3.3). Since λ ∈ (0, c] is arbitrary, it follows immediately that the interval (0, c] ⊆ E.

Remark 3.2. From the proof of Theorem 3.1, we see that (A2) and (A3) lead to S : C_δ → C_δ.

Theorem 3.2. Let (A1)-(A5) hold. Suppose that λ* ∈ E, for any λ ∈ (0, λ*), we have λ ∈ E, i.e., (0, λ*] ⊆ E.

Proof. Let x* be the eigenfunction corresponding to the eigenvalue λ*. Thus, we have

Define

Let λ ∈ (0, λ*) and x ∈ K*. Using (A5), we get

where the last equality follows from (3.11). This immediately implies that the operator S maps K* into K*. Moreover, the operator S is continuous and completely continuous. Schauder's fixed point theorem guarantees that S has a fixed point in K*, which is a positive solution of (3.3). Hence, λ is an eigenvalue, i.e., λ ∈ E.

The following result shows that E is an interval.

Corollary 3.1. Let (A1)-(A5) hold. If , then E is an interval.

Proof. Suppose E is not an interval. Then, there exist and with τ ∉ E. However, this is not possible as Theorem 3.2 guarantees that τ ∈ E. Hence, E is an interval.

The following two results give the upper and lower bounds of an eigenvalue in terms of some parameters of the corresponding eigenfunction.

Theorem 3.3. Let (A1) and (A2) hold. Assume that m is odd. Let λ be an eigenvalue of (3.3) and x ∈ C_δ be a corresponding eigenfunction. If x⁽ⁱ⁾ (0) = b_i, i = 1, 3, ..., 2m - 1, where b_2m-1 > 0, then λ satisfies

where

and

Proof. For n ∈ {1, 2, 3, ...}, we define f_n = f * ω_n, where ω_n is a standard mollifier [25] such that f_n is Lipschitz and converges uniformly to f.

For a fixed n, let λ_n be an eigenvalue and x_n(t), with be a corresponding eigenfunction of the following boundary value problem

where F_n converges uniformly to F, and for u, v ∈ (0, ∞),

(see the proof of Lemma 3.1 for the definitions of α_n(t) and β_n(t)).

It is clear that x_n(t) is the unique solution of the initial value problem (3.13),

First, we shall establish an upper bound for x_n. Since

we have is nonincreasing and hence

In view of the initial conditions (3.16) and also (3.17), we find

Next, an application of (3.18) gives

By repeating the process, we get

By the monotonicity of f_n, we have

and

Coupling with (3.13) and (3.15), it follows that

Once again, using the initial conditions (3.16), repeated integration of (3.20) from 0 to t provides

where

and

In order to satisfy the boundary conditions , from inequality (3.21) it is necessary that

This readily implies

where

and

From (3.20) it is observed (by using the initial conditions (3.16) and repeated integration) that is a uniformly bounded sequence on [0, 1]. Thus, there exists a subsequence, which can be relabeled as , that converges uniformly (in fact, in C^(2m-1)-norm) to some x on [0, 1]. We note that each x_n(t) can be expressed as

Since is a bounded sequence (from (3.22)), there is a subsequence, which can be relabeled as , that converges to some λ. Then, letting n → ∞ in (3.23) yields

This means that x(t) is an eigenfunction of (3.3) corresponding to the eigenvalue λ. Further, x⁽ⁱ⁾(0) = b_i, i = 1, 3, ..., 2m - 1 and inequality (3.12) follows from (3.22) immediately.

Theorem 3.4. Let (A1)-(A4) hold. Let λ be an eigenvalue of (3.3) and x ∈ C_δ be a corresponding eigenfunction. Further, let ∥x∥ = p. Then,

and

where t₁ is any number in (0, 1) such that x(t₁) ≠ 0.

Proof. Let t₀ ∈ [0, 1] be such that

Then, using (3.5), Lemma 2.1 and the monotonicity of f, we find

which gives (3.24) readily.

Next, we employ (3.5), the monotonicity of f and the fact that min_{t∈[δ, 1-δ]} x(t) ≥ γp to get

from which (3.25) is immediate.

The following result gives the criteria for E to be a bounded/unbounded interval.

Theorem 3.5. Define

(a) Let (A1)-(A5) hold. If f ∈ W_B, then E = (0, c) or (0, c] for some c ∈ (0, ∞).

(b) Let (A1)-(A5) hold. If f ∈ W₀, then E = (0, c] for some c ∈ (0, ∞).

(c) Let (A1)-(A4) hold. If f ∈ W_∞, then E = (0, ∞).

Proof. (a) This is immediate from (3.25) and Corollary 3.1.

(b) Since W₀ ⊆ W_B, it follows from Case (a) that E = (0, c) or (0, c] for some c ∈ (0, ∞).

In particular,

Let be a monotonically increasing sequence in E which converges to c, and let be a corresponding sequence of eigenfunctions in the context of (3.3). Further, let p_n = ∥x_n∥. Then, (3.25) together with f ∈ W₀ implies that no subsequence of can diverge to infinity. Thus, there exists R > 0 such that p_n ≤ R for all n. So is uniformly bounded. This implies that there is a subsequence of , relabeled as the original sequence, which converges uniformly to some x, where x(t) ≥ 0 for t ∈ [0, 1]. Clearly, we have Sx_n = x_n, i.e.,

Since x_n converges to x and λ_n converges to c, letting n → ∞ in (3.26) yields

Hence, c is an eigenvalue with corresponding eigenfunction x, i.e., c = sup E ∈ E. This completes the proof for Case (b).

(c) Let λ > 0 be fixed. Choose ε > 0 so that

By definition, if f ∈ W_∞, then there exists M = M(ε) > 0 such that

We shall prove that S(C_δ(M)) ⊆ C_δ(M). Let x ∈ C_δ (M). As in the proof of Theorem 3.1, we have (3.8) and (3.10) and so Sx ∈ C_δ. Thus, it remains to show that ∥Sx∥ ≤ M. Using (3.5), Lemma 2.1, (3.6), (3.28), and (3.27), we find for t ∈ [0, 1],

It follows that ∥Sx∥ ≤ M and hence S(C_δ(M)) ⊆ C_δ(M). Also, S is continuous and completely continuous. Schauder's fixed point theorem guarantees that S has a fixed point in C_δ(M). Clearly, this fixed point is a positive solution of (3.3) and therefore λ is an eigenvalue of (3.3). Since λ > 0 is arbitrary, we have proved that E = (0, ∞).

Example 3.1. Consider the complementary Lidstone boundary value problem

where λ > 0 and q ≥ 0.

Here, m = 2 and

Clearly, F(t, u, v) is nondecreasing in u and v, thus (A5) is satisfied.

Choose

and

We see that (A1)-(A4) are satisfied.

Case 1. 0 ≤ q < 1. Clearly f ∈ W_∞. It follows from Theorem 3.5(c) that the set E = (0, ∞). As an example, when λ = 24, the boundary value problem (3.29) has a positive solution given by .

Case 2. q = 1. Here f ∈ W_B. By Theorem 3.5(a) the set E is an open or a half-closed interval. Further, from Case 1 and Theorem 3.2 we note that E contains the interval (0, 24].

Case 3. q > 1. Clearly f ∈ W₀. By Theorem 3.5(b) the set E is a half-closed interval. Again, as in Case 2 we note that (0, 24] ⊆ E.

In this section, we shall establish explicit subintervals of E. Here, the functions α and β in (A2)-(A4) are assumed to be continuous on the closed interval [0, 1]. Hence, noting Remark 3.1, we shall not require conditions (A1) and (A4) to show the compactness of the operator S. For the function f in (A2), we define

Let be given. Define by

Theorem 4.1. Let (A2)-(A4) hold. Then, λ ∈ E if λ satisfies

Proof. We shall use Theorem 2.1. Let λ satisfy (4.2) and let ε > 0 be such that

First, we pick p > 0 so that

Let x ∈ C_δ be such that ∥x∥ = p. Note that for s ∈ [0, 1],

Then, using (3.5), Lemma 2.1, (4.4) and (4.3) successively, we find for t ∈ [0, 1],

Hence,

If we set Ω₁ = {x ∈ B |∥x∥ < p}, then (4.5) holds for x ∈ C_δ ∩ ∂ Ω₁.

Next, let q > 0 be such that

Let x ∈ C_δ be such that

It is clear that

Then, an application of (3.5), (4.7), and (4.6) gives for t ∈ [0, 1],

Taking supremum both sides and using (4.3) then provides (see (4.1) for the definition of t*)

Therefore, if we set Ω₂ = {x ∈ B| ∥x∥ < q₀}, then for x ∈ C_δ ∩ ∂ Ω₂ we have

Now that we have obtained (4.5) and (4.8), it follows from Remark 3.2 and Theorem 2.1 that S has a fixed point such that p ≤ ∥x∥ ≤ q₀. Obviously, this x is a positive solution of (3.3) and hence λ ∈ E.   □

Theorem 4.2. Let (A2)-(A4) hold. Then, λ ∈ E if λ satisfies

Proof. We shall apply Theorem 2.1 again. Let λ satisfy (4.9) and let ε > 0 be such that

First, we choose r > 0 so that

Let x ∈ C_δ be such that ∥x∥ = r. Then, on using (3.5), (4.11), and (4.10) successively, we have for t ∈ [0, 1],

Taking supremum both sides and using (4.10) then yields (see (4.1) for the definition of )

Hence, if we set Ω₁ = {y ∈ B| ∥x∥ < r}, then (4.8) holds for x ∈ C_δ ∩ ∂ Ω₁.

Next, pick w > 0 such that

We shall consider two cases - when f is bounded and when f is unbounded.

Case 1. Suppose that f is bounded. Then, there exists some M > 0 such that

Let x ∈ C_δ be such that

From (3.5), Lemma 2.1 and (4.13), it is clear for t ∈ [0, 1] that

Hence, (4.5) holds.

Case 2. Suppose that f is unbounded. Then, there exists w₀ > max {r + 1, w} such that

Let x ∈ C_δ be such that ∥x∥ = w₀. Then, applying (3.5), Lemma 2.1, (4.14), (4.12), and (4.10) successively gives for t ∈ [0, 1],

Thus, (4.5) follows immediately.

In both Cases 1 and 2, if we set Ω₂ = {x ∈ B| ∥x∥ < w₀}, then (4.5) holds for x ∈ C_δ ∩ ∂Ω₂.

Now that we have obtained (4.8) and (4.5), it follows from Remark 3.2 and Theorem 2.1 that S has a fixed point such that r ≤ ∥x∥ ≤ w₀. It is clear that this x is a positive solution of (3.3) and hence λ ∈ E.

Remark 4.1. In (4.2) and (4.9), although t* and can be computed from (4.1), we can circumvent the computation by giving further bounds. Indeed, applying Lemma 2.2 we find

and

The following corollary is immediate from Theorems 4.1, 4.2 and Remark 4.1.

Corollary 4.1. Let (A2)-(A4) hold. Then,

and

Remark 4.2. If f is superlinear (i.e., and ) or sublinear (i.e., and ), then we conclude from Corollary 4.1 that E = (0, ∞), i.e., the boundary value problem (3.3) (or (1.1)) has a positive solution for any λ > 0.

Example 4.1. Consider the complementary Lidstone boundary value problem

where λ, a, b, c > 0 and r ≤ 1.

Here, m = 2. It is clear that (A2)-(A4) are satisfied with

Case 1. r < 1. It is clear that f is sublinear. Therefore, by Remark 4.2 the boundary value problem (4.15) has a positive solution for any λ > 0. In fact, we note that when λ = 120, (4.15) has a positive solution given by .

Case 2. r = 1, a = b = 0.5, c = 10. Here, and . It follows from Corollary 4.1 that

Once again we note that when λ = 120 ∈ (0,528.99), the corresponding eigenfunction is given by .

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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