1 Introduction

Consider the following one-dimensional Sturm–Liouville equation:

where the real-valued function is a complex spectral parameter, and .

Remark 1. In this article,, where , , and . Then a map is the bijection between and [21].

In this study, we will investigate nonlocal eigenvalue problems, which consist of equation (1) on [0, 1] with one classical (local) Boundary Condition (BC)

another integral-type Nonlocal Boundary Condition (NBC)

or

For. where . Both these BC we can write as

where in Case 1, and in Case 2. We note that . The first work on Boundary Value Problems (BVPs) with nonlocal integral-type BCs belongs to J.R. Cannon [2]. These kinds of BCs together with a parabolic equation arise, for example, in the study of the process of heat transmission in a thin heated rod when the part of the rod adjoining one of its ends [7]. Parabolic equations with NBCs are also encountered in the study of the heat conduction within linear thermo elasticity [3, 4].

Eigenvalues and eigenfunctions of BVPs with integral-type NBCs and discrete case have been investigated in [1, 6, 8, 10, 11, 16, 19]. Structure of eigenvalues of multi-point BVPs were presented in [5,13,14]. The spectrum structure of one-dimensional differential operator with nonlocal conditions and of the difference operator, corresponding to it, has been exhaustively investigated in [15]. A more comprehensive list can be found in the survey article [20].

Spectral asymptotics of eigenvalues and eigenfunctions of SLPs with Bitsadze– Samarskii-type NBC

where , have been investigated recently [17, 18]. In [17], for sufficiently large k and , it is derived that the asymptotic expansions

are valid for eigenvalues and eigenfunctions, respectively, for the SLP (1)–(2), (6), where , are the positive roots of sin . Under the condition , it is obtained that the asymptotic formulas

are valid for eigenvalues and eigenfunctions, respectively, for the SLP (1)–(2), (6). Here

In [18] the authors consider the equation with retarded argument

together with the BCs (2), (6), where the real-valued function ; the real- valued function is continuous on [ is a complex spectral parameter. They calculate the asymptotics of eigenvalues and eigenfunctions. To speak more precise, under the conditions exist and bounded in [0, 1], in [0, 1], and they find the asymptotic formulas

for eigenvalues and eigenfunctions, respectively, for the SLP (11), (2), (6), where

In both of these studies, it is proven that . Furthermore, in these articles the authors prove the simplicity and countability of eigenvalues and show that all eigenvalues are real.

The article is organized as follows. The statement of the problem and a literature review are given in Section 1. In Sections 1–3, notation and definitions used in the paper are stated. In Section 2, some results about the case are presented. In Section 3, we write the fundamental solutions of the Initial Value Problem (IVP) and find formulas for their asymptotics. In Section 4, we analyze the characteristic equation of the BVP (1)–(2), (5). In Section 5, we investigate the distribution of eigenvalues and obtain asymptotic for- mulas for eigenvalues and eigenfunctions. Also, we calculate normalized eigenfunctions.

2 Properties of a spectrum in the case q ≡ 0

In the case the spectrum of problems (1)–(3) and (1)–(2), (4) have countably many eigenvalues [11, 16]. A unique negative eigenvalue exists for in Case 1

and in Case 2. Also, is eigenvalue if and only if in Case 1 and in Case 2.

Let us define a Constant Eigenvalue (CE) as the eigenvalue , which does not depend on the parameter for fixed . In [11] the spectrum and eigenfunctions with BC (2) and integral-type BCs (3) and (4) were investigated for the case . Constant eigenvalues exist only for rational numbers , and those eigenvalues , are given by: for and for in Case 1; for and for in Case 2. So, all CE are positive.

ll nonconstant (that depend on the parameter ) eigenvalues , are γ-points of the Characteristic Function [21]

So, for fixed , the roots of this meromorphic function describe nonconstant eigenvalues. The graphs of CF on are presented in Fig. 1(a) in Case 1 and Fig. 1(c) in Case 2.

In Case 2, all nonconstant eigenvalues are real and simple [11, 12]. All poles of CF belong to one of the families of the first order poles:

If , then .

Lemma 1. (See [11]) If , then in Case 2, points , where or for are the first-order poles of CF and CE points. A set of these points is intersection of and .

So, all poles of CF are of the first order. We can enumerate all poles in nondecreasing order: . If a pole is CE point, then we write it twice . Additionaly, we denote . Then in Case 2, we can enumerate positive eigenvalues , where for nonconstant eigenvalues, and for CE. Note that , in the case and for all k.

In Case 1, nonconstant eigenvalues can be complex [11, 12]. All poles of CF belong to the family

If , then all poles are of the second order.

Lemma 2.(See [11].) If , then in Case 1, points , where for or for are the first-order poles of CF and CE points, else we have the second-order poles.

Lemma 3.Let , be eigenvalues of problem (1)–(3) in the case . Then exists such that for fixed , all eigenvalues , are positive, simple and , i.e. for all .

Proof. For not simple positive eigenvalues we have

From this system we get

Then we estimate

So, all eigenvalues in the angle for are positive and simple. CE points are the first-order poles of CF. Eigenvalues corresponding to these points are positive and simple. Since CF has zeros at points , we have .

3. Solutions of initial value problem and their asymtotics

Let , and be a solution of equation (1) satisfying the initial conditions

According to [9, Chap. I, Thm. 1.1], this IVP determine a unique solution of (1) on [0, 1]. The function is an analytic (holomorphic) function of s. We will use notation for derivatives , .

Lemma 4. (See [17].)Let be a solution of IVP (1),(14). Then the following integral equation holds

We will use notation for integrals :

We note that .

Then we rewrite equation (15) as

Taking derivative with respect to t and s in (15), we get

For derivatives of the second order, we get formulas

Remark 2. The following formulas

are valid, where ,

Lemma 5.Let . Then there exists such that for , the asymptotic formulas

are valid. These formulas hold uniformly for .

Proof. Put and . Then from (16)–(20) we obtain

Let and Since

and , from (26)–(28) we have

If , then

It follows from (29), (30) that

So, we prove asymptotic formulas

Now, substituting formulas (31)–(32) into the integrals of (26)–(30), we obtain

Lemma is proved.

Remark 3. .

Remark 4. The asymptotic formulas (21)–(24) were proved in [17].

Remark 5. The asymptotic formulas

are valid for ,. The proof is the same as in Lemma 5.

Corollary 1. Let , then we have the following formulas:

Proof. We substitute expressions (21)–(23) into formula

and get

Analogously, we prove the second formula

Lemma 6. Let . Then the following asymptotic formulas

are valid.

Proof. We use integration by parts formula

The other four formulas follow from formula (34).

For real , we can find formulas (35) in [9, 22].

Let . Then we generalize the last two asymptotic formulas in Lemma 6:

For proof, we use integration by parts formula and Lemma 6.

Under the condition that , the more exact asymptotic formulas may be obtained

where .

Now we derive formulas for . We can use the mathematical induction.

Let us substitute

into integral in right-hand side of (16):

Then we rewrite the sum

and apply (36)–(37) for :

We look for terms near ,

or

So, we prove recursive formula

for . This formula shows that . So, the application of formulas (36)–(37) was correct. We note that for all j.

For example, . It is obvious that the function is bounded for .

Let us substitute (41) into integral in right-hand side of (17). Then we get formula

for If we add (42) and (43), we get

So, more simple formula may be used for calculation (we note that ). This formula can be proved directly, but formula (43) is useful independently. We use notation, too.

If we substitute (38) and (40) into integrals , then from (18) we get recursive formula

For example, . We see that . Using the mathematical induction and formulas (44) and (42), we can prove simple formula .

Lemma 7.Let and . Then for , the asymptotic formula

are valid for is calculated by (42).

Proof. We prove (45) formula in the case l=0, 1 and (46) formula in the case l=0. The other cases we can prove by mathematical induction by l.

For example, (see Remark 5, too).

Corollary 2.Let , are bounded functions. If ,

then we have the following formulas:

From previous results we have . Now we derive general formula for . We substitute expressions (45), (47) into formula

and get the following expression for :

From binomial formula we have

Collecting terms near , we get

For , we get formula

We write explicit formulas in the case :

We formulate these results in the next two statements.

Lemma 8. Let and . Then for , the asymptotic formulas

are valid.

Corollary 3.Let is bounded function. If , , then we have the following formulas:

4 Characteristic equation for problem with integral condition

Substituting into (5), we get the characteristic equation

The set of eigenvalues of the BVP (1), (2), (5) coincides with the set . The function h is analytic function of parameter , and

Substituting (45) into (51), we get

where .

Remark 6. In the case r=0 the last term is .

If , then we use integration by parts formula and have

So, we derive

We look for terms near ,

Thus, the next lemma immediately follows from results in the above.

Lemma 9. Let and . Then for , the asymptotic formula

is valid.

Corollary 4.If , then we have (see Remark 6, too)

where

Corollary 5.If , then we have

where

For example, .

Analytic functions have the same nonzero roots as function h.

Corollary 6. If , then we have

Remark 7.The asymptotic formula (53) for the integral condition(5) are simpler than in the case Bitsadze–Samarskii NBC (6) (see [17]). For Bitsadze–Samarskii NBC,

Substituting (45) into these expressions, we get

We look for terms near

So, Remark 9 is valid in the case Bitsadze–Samarskii NBC with above defined ,, and we get two corollaries.

Corollary 7 [Bitsadze–Samarskii NBC]. If , then we have formula (53), where . For example, .

Corollary 8 [Bitsadze–Samarskii NBC]. If , then we have formula (54), where . For example, .

Let us consider real eigenvalues. In this case, (52) is valid with , and functions ,are bounded. We investigate equation , with additional condition .

Lemma 10. If and δ=o (1), then the following asymptotic formula is valid:

Proof. From formula (57) we have

Since we have . We multiply this equality by (using condition and get (59).

Corollary 9. If , then .

Lemma 11. If and , then asymptotic formula

is valid, where , are bounded functions.

Proof. Formula (60) is valid for r=0. So, . If r>0, then substituting (54) into equality

we have

i.e. , where

We derive formulas for . We can use the mathematical induction. Suppose that . Substituting (52) expression in the casey y=0 into equality

we get

or

where ,

For , we have expressions

So,

Collecting terms near , we get

Lemma is proved.

Corollary 10.If , then

Corollary 11 [Integral NBC]. If , then

Corollary 12 [Bitsadze–Samarskii NBC]. If , then

Formula (66) was proved in [17].

5 Spectral asymptotics for eigenvalues and eigenfunctions for problem with integral condition

In this section, we investigate eigenvalues for SLP (1), (2), (5).

Lemma 12.The real eigenvalues of the SLP (1), (2), (5) are bounded from below.

Proof. Set . Then

It is clear that . Then there exists a y₀ > 0 such that for negative

Corollary 13.The number of negative eigenvalues of problem (1), (2), (5) are finite (maybe zero).

Lemma 13.The function has at least one positive root in the interval for large k.

Proof. If , then y=0. In this case, formulas (55) is

We have for large x. The function sin x takes its local maximum points at, and its local minimum points at ,. Thus, from Intermediate value theorem at least one root of the function lies in each interval , for large K. So, we have infinite (countable) number positive roots of equation.

Corollary 14.The SLPs (1), (2), (5) have infinitely many (countable) positive eigenvalues.

Remark 8. The function sin s has the same property, but only one root is in the interval .

Let us denote domain , , and a contour .

Remark 9. The corresponding contour in the plane will be the boundary of the domain .

Lemma 14.There exists such that all eigenvalues of problems (1)–(2), (5) in the domains are positive and, more precisely, there exists only one positive root of function in each interval for sufficiently large k.

Proof. On the vertical part of contour . We estimate

On the remaining part of contour , we estimate

So, we have sin for sufficiently large k.

From formula (55) . Hence, we have on the contours for sufficiently large k. Therefore, by Rouché theorem it follows that the number of zeros of and sin s are the same inside for sufficiently large k.

In the domain between contours and , there is exactly one positive root of the function sin s (see Remark 8). The function H has one root in this domain for sufficiently large k. But interval belongs to this domain. So, the single root of H in this domain is positive.

This lemma clarifies Lemma 13.

Corollary 15. The function has one positive root in the interval for large k.

We can enumerate the zeros of H as s_k, . The first zeros can be complex numbers or not simple. From Corollary 15 we have that sk are positive for sufficiently large k. Now we will investigate the distribution of these positive eigenvalues of problem (1)–(3), and we leave out the note about sufficiently large k. Now we consider only real positive . Since , we have and .

Let us denote . The functions H and sin s are analytic. So, from (67) we have

Theorem 1. Let . For eigenvalues and eigenfunctions u_k of problem (1)–(2), (5), the asymptotic formulas

are valid for sufficiently large k.

Proof. For our problem, and . We have . So, all conditions of Lemma 10 are valid, and from Corollary 9 it follows .

Then we apply Corollary 1 and get

Remark 10. Normalized eigenfunctions are

Theorem 2. Let . For eigenvalues and eigenfunctions uk of problem (1)–(2), (5), the asymptotic formulas

are valid for sufficiently large k.

Proof. We have (see Theorem 1). So, all conditions of Lemma 11 are valid, and it follows

Then we apply Corollary 2 and get

Corollary 16.If , then the asymptotic formulas

are valid for sufficiently large k, where

Remark 11. In this case, normalized eigenfunctions are

6 Conclusion

In this paper the spectrum, existence of solutions and spectral properties of eigenfunctions for a SLP with one integral-type NBC was investigated. The considered problem differs from the classical (local) one-dimensional SLP with BCs in that it contains a NBC in two cases. Therefore, it is not obvious how to apply the classical methods of theory to such type BVPs. Therefore, suggesting own approach and modifying the techniques of classical Sturm theory, we obtained asymptotic formulas for eigenvalues and normalized eigenfunctions. The results obtained in this work can be extended to two- or higher- dimensional SLPs and to higher-order differential equations. Furthermore, asymptotics of eigenvalues and eigenfunctions of the same differential equation but with different NBCs such as eigenvalue-parameter dependent NBCs can be also investigated.

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