### Abstract

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.

##### Keywords:

eigenvalues; positive solutions; complementary Lidstone boundary value problems### 1 Introduction

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 ((2*m*+ 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 (2*m*th 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*.

### 2 Preliminaries

**Theorem 2.1**. [38] Let *B *be a Banach space, and let *C*(⊂ *B*) be a cone. Assume Ω_{1}, Ω_{2 }are open subsets of *B *with

(a) ∥*Sy*∥ ≤ ∥*y*∥, *y *∈ *C*∩*∂*Ω_{1}, *and *∥*Sy*∥ ≥ ∥*y*∥*, y *∈ *C *∩ *∂*Ω_{2}, or

(b) ∥*Sy*∥ ≥ ∥*y*∥, *y *∈ *C*∩*∂*Ω_{1}, *and *∥*Sy*∥ ≤ ∥*y*∥, *y *∈ *C *∩ *∂*Ω_{2}.

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
*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].

### 3 Eigenvalues of (1.1)

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 positive solution *x**, then by virtue of (3.2),
*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*, *u*_{1}, *u*_{2}, *v*, *v*_{1}, *v*_{2 }∈ (0, ∞) with *u*_{1 }≤ *u*_{2 }and *v*_{1 }≤ *v*_{2}, 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
*a*_{0 }∈ (0, 1] such that *α*(*t*) ≥ *a*_{0}*β*(*t*) for all *t *∈ (0, 1);

(A4)

(A5) for *t *∈ (0, 1) and *u*, *u*_{1}, *u*_{2}, *v*, *v*_{1}, *v*_{2 }∈ (0, ∞) with *u*_{1 }≤ *u*_{2 }and *v*_{1 }≤ *v*_{2}, we have

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

For a given
*C*_{δ }be defined by

where
*a*_{0 }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
*E *is an interval.

**Proof**. Suppose *E *is not an interval. Then, there exist
*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, ..., 2*m *- 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

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

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

This readily implies

where

and

From (3.20) it is observed (by using the initial conditions (3.16) and repeated integration)
that
*C*^{(2m-1)}-norm) to some *x *on [0, 1]. We note that each *x*_{n}(*t*) can be expressed as

Since
*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, ..., 2*m *- 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 *t*_{1 }is any number in (0, 1) such that *x*(*t*_{1}) ≠ 0.

**Proof**. Let *t*_{0 }∈ [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 *∈ *W*_{0}, 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 *W*_{0 }⊆ *W*_{B}, it follows from Case (a) that *E *= (0, *c*) or (0, *c*] for some *c *∈ (0, ∞).

In particular,

Let
*E *which converges to *c*, and let
*p*_{n }= ∥*x*_{n}∥. Then, (3.25) together with *f *∈ *W*_{0 }implies that no subsequence of
*R *> 0 such that *p*_{n }≤ *R *for all *n*. So
*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 *∈ *W*_{0}. By Theorem 3.5(b) the set *E *is a half-closed interval. Again, as in Case 2 we note that (0, 24] ⊆ *E*.

### 4 Eigenvalue intervals

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

**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 Ω_{1 }= {*x *∈ *B *|∥*x*∥ < *p*}, then (4.5) holds for *x *∈ *C*_{δ }∩ *∂ *Ω_{1}.

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 Ω_{2 }= {*x *∈ *B*| ∥*x*∥ < *q*_{0}}, then for *x *∈ *C*_{δ }∩ *∂ *Ω_{2 }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
*p *≤ ∥*x*∥ ≤ *q*_{0}. 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 Ω_{1 }= {*y *∈ *B*| ∥*x*∥ < *r*}, then (4.8) holds for *x *∈ *C*_{δ }∩ *∂ *Ω_{1}.

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 *w*_{0 }> max {*r *+ 1, *w*} such that

Let *x *∈ *C*_{δ }be such that ∥*x*∥ = *w*_{0}. 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 Ω_{2 }= {*x *∈ *B*| ∥*x*∥ < *w*_{0}}, then (4.5) holds for *x *∈ *C*_{δ }∩ *∂*Ω_{2}.

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
*r *≤ ∥*x*∥ ≤ *w*_{0}. 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

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.,
*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