The Sturm-Liouville Problem (SLP) is about finding all values of λ ∈ ℂ and corresponding nonzero functions u which satisfy the differential equation

and the separated boundary conditions

defined on an interval ( a , b ) , − ∞ ≤ a < b ≤ ∞ . Here, α 1 , α 2 , β 1 , β 2 ∈ ℝ and are such that α 1 and α 2 do not vanish at the same time, similarly, for β 1 and β 2 (see ([1]) chapter 12). The parameter λ is called the eigenvalue and the corresponding function u is called an eigenfunction of problem (1)-(2). The set comprising all eigenvalues of problem (1)-(2) is called the spectrum, [2]-[4]. Each eigenvalue may have at least one eigenfunction. An eigenvalue with only one eigenfunction is called a simple eigenvalue, otherwise it is non-simple. The functions p ′ , p , q , and r are continuous on ( a , b ) and p − 1 , q , r ∈ L l o c ( a , b ) . The solution of the problem, is a generally complex-valued function u ( x ) of the real variable x such that u and p u ′ are absolutely continuous on ( a , b ) that satisfies equation (1) and the boundary conditions (2). If the endpoints a and b are finite and p ( x ) > 0 for all x ∈ ( a , b ) the problem is called regular, otherwise it is singular if any of the two regularity conditions (or both) are not satisfied, [4][5].

We consider the general weighted regular Sturm-Liouville problem (GWRSLP) in which the coefficient function q ( x ) and the weight function r ( x ) have no sign restrictions. The paper focuses on estimating the lower bound on the eigenvalue of the smallest modulus of the problem:

It should be noted that problem (3)-(4) is in the form of problem (1)-(2), with p ( x ) ≡ 1 , α 1 = 0 = β 2 . Our work extends the findings of Kikonko and Mingarelli [6] and Mingarelli [7] who examined similar bounds under Dirichlet boundary conditions. Mingarelli [7] obtained estimates with the assumption that q ∈ L ∞ ( a , b ) , while Kikonko and Mingarelli [6] extended these results to q ∈ L 1 ( a , b ) . In our study, we replace Dirichlet boundary conditions with mixed boundary conditions while maintaining the assumptions on q ( x ) and r ( x ) . In the next section, we classify the GWRSLP problems in order to show the nature of the spectrum and of the smallest eignvalue, in particular.

In this section, we find the integral representations of the two Cauchy problems:

We convert problem (13) to an integral equation:

Integrating we get:

Repeating the process yields

Changing the order of integration yields:

Similarly, the integral representation of (14) is

which simplifies to

We now consider the two spaces for q ( x ) to obtain the bounds, in the next section.

Consider the integral Equation (15) given by

where q ∈ L 1 ( a , b ) . The solution can be expressed as a Neumann series:

where the terms are defined iteratively by

For all n ≥ 1 , we have;

where

Proof. We prove this by mathematical induction. For n = 1 :

Hence it is true for n = 1 .

Assume (18) holds for n = k , that is

we show that (18) holds for n = k + 1 . From (17) we have

Using the inductive hypothesis (19), we have

Thus, the result holds for n = k + 1 , and by induction it holds for all n ≥ 1 , n ∈ ℤ . □

Let c = ‖ q ‖ 1 ,

By using a similar approach as above and using the integral Equation (16) given by

we have:

where

and

which gives a general expression as:

Let c = ‖ q ‖ 1 , we have:

Using a similar approach in the previous section and using the integral equation in (15), i.e.

we have:

Then

Similarly, using the integral equation

we have

then

We prove (23) by induction.

Proof. For n = 1 and from the definition,

Taking absolute values,

Thus,

Hence, the result holds for n = 1 .

Assume that (24) holds for n = k , i.e.,

Then we prove that it is true for n = k + 1 , as well. From the recursive definition,

Thus,

Thus, the result holds for k + 1 .

Using a similar approach for B n ( x ) will give us

□

Lemma 6.1.Let y ( x ) denote the solution of the Cauchy problem

satisfying

Then for fixed x ∈ [ a , b ] we have,

Proof. To prove (28), we use the inequality (21), to get:

□

Since the series

converges, the Neumann series converges, too. This yields the following.

which establishes the bound in (28).

Lemma 6.2.Let y denote a linearly independent solution of (25) satisfying(26) then

Proof. We prove the above result by calculating the ℋ -norms of y and use the bounds in Lemma 6.1 as shown below.

Taking the supremum on ‖ y ( x ) ‖ in the inequality (33) yields the bounds in (32). □

Lemma 6.3.Let z ( x ) denotes the solution of the Cauchy problem

then for fixed x ∈ [ a , b ] we have,

Proof. To prove (36), we use the inequality (22) as shown below;

which yields

and the bounds in (36) is established.

Lemma 6.4.Let z denote the linearly independent solutions of(34) satisfying(35) then

Proof. We prove the above result by calculating the ℋ -norms of z and use the bounds in Lemma 6.3 as shown below.

Taking the supremum on ‖ z ( x ) ‖ in the inequality (40) yields the bounds in (39). □

Lemma 7.1.Let y ( x ) denote the solution of the Cauchy problem:

Then for fixed x in [a, b]

with equality in the case where q ( x ) = c a.e.on(a, b)

Similarlyif z ( x , q ) denotes the solution of (41) and q as defined in(43) satisfying,

then for fixed x in [ a , b ] ,

Proof. To prove (44), we use the Neumann series of (15) and inequality (23). Therefore we have that,

And the bounds in (44) are established.

To prove (46) we use the Neumann series of (16) and inequality (24), thus;

which yields that

and the bounds in (46) is established.

Combining the pointwise estimations on y , z in Lemma 7.1 we may obtain (sharp) estimates on the ℋ -norms of y , z which leads to the following results.

Lemma 7.2.Let y and z denote the two linearly independent solutions of(41) satisfying(42) and(45), respectively, then,

1.

2.

Proof. We prove the above results by calculating the ℋ -norms of y and z by integrating the pointwise bounds given by (44) and (46) and use the bounds in (7.1) for a given q ∈ B ( c ) . □

Applying the results in Lemma 6.2 and Lemma 6.4 to the lower bounds in (12), we get the main result when q ∈ L 1 , which is a variant of Theorem 1 in [7].

Theorem 2.Let ‖ q ‖ 1 = c . Then for problem (3)-(4), an eigenvalue λ 0 of the smallest modulus may be estimated by:

where x ∈ ( a , b ) and y ( b ) is a solution of the Cauchy problem(25) and(26) evaluated at x = b .

Applying the results in lemma (7.2) to the lower bounds in (12), gives our main result when q ∈ L ∞ .

Theorem 3.Let ‖ q ‖ ∞ = c . Then for problem (3)-(4),an eigenvalue λ 0 of thesmallest modulus may be estimatedby;

x ∈ ( a , b ) where y ( b ) is a solution of the Cauchy problem (41) and (42) evaluated at x = b.

Here we give examples to verify if the theorems 2 and 3 hold. Without loss of generality, we consider the case where q ( x ) = q ∈ ℝ on the interval [−1,1]. The eigenvalues are found using the Maple C package RootFinding [Analytic].

Example 1. (the case q = − 4 π 2 and r ( x ) changes sign).This is an example of the non-definite SLP.

we take,

The solution to the problem (54) is

and

Substituting(57), (58) and (59) in inequality(52) gives us

The spectrum of(55)when r is given in (56)is

−477.4, −343.8, −229.7, −134.8, −57.1, −39.5, −30.8, −7.315 ± 14.59i, 24.18 ± 9.767i, 36.45, 94.27, 179.9, 284.3, 408.2,… and so | − 7.315 ± 14.59 i | = 16.3 is thesmallest modulus and we see that

Thus theorem2 is satisfied.

Next we verify theorem3