`International Journal of Mathematics and Mathematical SciencesVolumeΒ 2012, Article IDΒ 456517, 28 pageshttp://dx.doi.org/10.1155/2012/456517`
Research Article

Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

1Department of Mathematics, Baku State University, Baku AZ 1148, Azerbaijan
2Department of Mathematics, Mersin University, 33343 Mersin, Turkey

Received 15 August 2011; Accepted 12 November 2011

Copyright Β© 2012 Ziyatkhan S. Aliyev and Nazim B. Kerimov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the fourth-order spectral problem with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations (in complex plane) and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in , , of the system of root functions of this problem.

1. Introduction

The following boundary value problem is considered:βwhere is a spectral parameter, , is absolutely continuous function on , , , , , , and are real constants such that , and . Moreover, we assume that the equation is disfocal in , that is, there is no solution of (1.3) such that for any . Note that the sign of which satisfies the disfocal condition may change in .

Problems of this type occur in mechanics. If , , , and in the boundary conditions, then the problem (1.1), (1.2a)β(1.2d) arises when variables are separated in the dynamical boundary value problem describing small oscillations of a homogeneous rod whose left end is fixed rigidly and on whose right end a servocontrol force in acting. In particular, the case when corresponds to the situation where this is a particle of mass at the right end of the rod. For more complete information about the physical meaning of this type of problem see [1β3].

Boundary value problems for ordinary differential operators with spectral parameter in the boundary conditions have been considered in various formulations by many authors (see, e.g., [1, 4β25]). In [14β16, 20, 22] the authors studied the basis property in various function spaces of the eigen- and associated function system of the Sturm-Liouville spectral problem with spectral parameter in the boundary conditions. The existence of eigenvalues, estimates of eigenvalues and eigenfunctions, oscillation properties of eigenfunctions, and expansion theorems were considered in [4, 7, 9, 12, 17, 18, 21, 24] for fourth-order ordinary differential operators with a spectral parameter in a boundary condition. The locations, multiplicities of the eigenvalues, the oscillation properties of eigenfunctions, the basis properties in , of the system of root functions of the boundary value problem (1.1), (1.2a)β(1.2d) with , , are considered in [18] and, with , , , are considered in [4, 5].

The subject of the present paper is the study of the general characteristics of eigenvalue locations on a complex plane, the structure of root subspaces, the oscillation properties of eigenfunctions, the asymptotic behaviour of the eigenvalues and eigenfunctions, and the basis properties in , , of the system of root functions of the problem (1.1), (1.2a)β(1.2d).

Note that the sign of plays an essential role. In the case we associate with problem (1.1), (1.2a)β(1.2d) a selfadjoint operator in the Hilbert space with an appropriate inner product. Using this fact and extending analytic methods to fourth-order problems, we show that all the eigenvalues are real and simple and the system of eigenfunctions, with arbitrary function removed, forms a basis in the space , . For problem (1.1), (1.2a)β(1.2d) can be interpreted as a spectral problem for a selfadjoint operator in a Pontryagin space . It is proved below that nonreal and nonsimple (multiple) eigenvalues are possible and the system of root functions, with arbitrary function removed, forms a basis in the space , , except some cases where the system is neither completed nor minimal.

2. The Operator Interpretation of the Problem (1.1), (1.2a)β(1.2d)

Let be a Hilbert space equipped with the inner product where .

We define in the operator with domain that is dense in [23, 25], where denotes the set of separated boundary conditions (1.2a)β(1.2c).

Obviously, the operator is well defined. By immediate verification we conclude that problem (1.1), (1.2a)β(1.2d) is equivalent to the following spectral problem: that is, the eigenvalue of problem (1.1), (1.2a)β(1.2d) and those of problem (2.4) coincide; moreover, there exists a correspondence between the eigenfunctions and the adjoint functions of the two problems: Problem (1.1), (1.2a)β(1.2d) has regular boundary conditions in the sense of [23, 25]; in particular, it has a discrete spectrum.

If , then is a selfadjoint discrete lower-semibounded operator in and hence has a system of eigenvectors , that forms an orthogonal basis in .

In the case the operator is closed and non-selfadjoint and has compact resolvent in . In we now introduce the operator by . is a unitary, symmetric operator in . Its spectrum consists of two eigenvalues: β1 with multiplicity 1, and +1 with infinite multiplicity. Hence, this operator generates the Pontryagin space by means of the inner products (-metric) [26]:

Lemma 2.1. is a -selfadjoint operator in .

Proof. is selfadjoint in by virtue of Theorem 2.2 [11]. Then, J-selfadjointness of on follows from [27, Section 3, Proposition 30].

Lemma 2.2 (see [27, Section 3, Proposition 50]). Let be an operator adjoined to the operator in . Then, .

Let be an eigenvalue of operator of algebraic multiplicity . Let us suppose that is equal to if and equal to whole part if .

Theorem 2.3 (see [28]). The eigenvalues of operator arrange symmetrically with regard to the real axis. for any system of eigenvalues with nonnegative parts.

From Theorem 2.3 it follows that either all the eigenvalues of boundary value problem (1.1), (1.2a)β(1.2d) are simple (all the eigenvalues are real or all, except a conjugate pair of nonreal, are real) or all the eigenvalues are real and all, except one double or triple, are simple.

3. Some Auxiliary Results

As in [17, 19, 29, 30] forthe analysis of the oscillation properties of eigenfunctions of the problem (1.1), (1.2a)β(1.2d) we will use a PrΓΌfer-type transformation of the following form:

Consider the boundary conditions (see [29, 30]) where , .

Alongside the spectral problem (1.1), (1.2a)β(1.2d) we will consider the spectral problem (1.1), (1.2a)β(1.2c), and (). In [30], Banks and Kurowski developed an extension of the PrΓΌfer transformation (3.1) to study the oscillation of the eigenfunctions and their derivatives of problem (1.1), (), (1.2b), (1.2c), and () with , and in some cases when (1.3) is disfocal and , . In [19], the authors used the PrΓΌfer transformation (3.1) to study the oscillations of the eigenfunctions of the problem (1.1), (), (1.2b), (1.2c), and () with and . In this work it is proved that problem (1.1), (), (1.2b), (1.2c), and () may have at most one negative and simple eigenvalue and sequence of positive and simple eigenvalues tending to infinity, the number of zeros of the eigenfunctions corresponding to positive eigenvalues behaves in that usual way (it is equal to the serial number of an eigenvalue increasing by 1); the function associated with the lowest eigenvalue has no zeros in (however in reality, this eigenfunction has no zeros in if the least eigenvalue is positive; the number of zeros can by arbitrary if the least eigenvalue is negative). In [31], Ben Amara developed an extension of the classical Sturm theory [32] to study the oscillation properties for the eigenfunctions of the problem (1.1), (1.2a)β(1.2c), and () with , in particular, given an asymptotic estimate of the number of zeros in of the first eigenfunction in terms of the variation of parameters in the boundary conditions.

Let be a solution of (1.3) which satisfies the initial conditions , . Then the disfocal condition of (1.3) implies that in . Therefore, if denotes the solution of (1.3) satisfying the initial conditions , , where is a sufficiently small constant, then we have also on . Thus, in , and hence the following substitutions [33, Theorem 12.1]: transform into the interval and (1.1) into where , ; , are taken as functions of and . Furthermore, the following relations are useful in the sequel: It is clear from the second relation (3.4) that the sign of is not necessarily preserved after the transformation (3.2). For this reason this transformation cannot be used in any straightforward way. The following lemma of Leighton and Nehari [33] will be needed throughout our discussion. In [30, Lemma 2.1], Banks and Kurowski gave a new proof of this lemma for . However, in the case when (1.3) is disfosal on , they partially proved it [30, Lemma 7.1], and therefore they were able to study problem (1.1), (1.2a)β(1.2c), and () with , . In [31], Ben Amara shows how Lemma 3.1 together with the transformation (3.2) can be applicable to investigate boundary conditions (1.2a)β(1.2c), and () with .

Lemma 3.1 (see [33, Lemma 2.1]). Let , and let be a nontrivial solution of (3.3). If , , , and are nonnegative at (but not all zero), they are positive for all . If , , , and are nonnegative at (but not all zero), they are positive for all .

We also need the following results which are basic in the sequel.

Lemma 3.2. All the eigenvalues of problem (1.1), (1.2a)β(1.2c), and () for or , are positive.

Proof. In this case, the transformed problem is determined by (3.3) and the boundary conditionswhere .
It is known that the eigenvalues of (3.3), (3.5a)β(3.5d) are given by the max-min principle [13, Page 405] using the Rayleigh quotient where . It follows by inspection of the numerator in (3.6) that zero is an eigenvalue only in the case . Hence, all the eigenvalues of problem (3.3), (3.5a)β(3.5d) for or , , are positive. Lemma 3.2 is proved.

Lemma 3.3. Let be the space of solution of the problem (1.1), (1.2a)β(1.2c). Then, .

The proof is similar to that of [19, Lemma 2] using transformation (3.2), Lemmas 3.1 and 3.2 (see also [31, Lemma 2.2]). However, it is not true if (see, e.g., [31, Page 9]). Therefore, Lemma 3.1 together with the transformation (3.2) cannot be applicable to investigate more general boundary conditions, for example, (), (1.2b), and (1.2c) for .

Lemma 3.4 (see [29, Lemma 2.2]). Let and be a solution of (3.3) which satisfies the boundary conditions (3.5a)β(3.5c). If is a zero of and in the interval , then in a neighborhood of . If is a zero of or in , then in a neighborhood of .

Theorem 3.5. Let be a nontrivial solution of the problem (1.1), (1.2a) and (1.2c) for . Then the Jacobian of the transformation (3.1) does not vanish in .

Proof. Let be a nontrivial solution of (1.1) which satisfies the boundary conditions (1.2a) and (1.2c). Assume first that the corresponding angle satisfies for some integer and for some . Then, the transformation (3.1) implies that . Using the transformation (3.2), the solution of (3.3) also satisfies , where . However, it is incompatible with the conclusion of Lemma 3.4.
The proof of the inequality , , proceeds in the same fashion as in the previous case. The proof of Theorem 3.5 is complete.

Let be a nontrivial solution of the problem (1.1), (1.2a)β(1.2c) for and , the corresponding functions in (3.1). Without loss of generality, we can define the initial values of these functions as follows (see [30, Theorem 3.3]):

With obvious modifications, the results stated in [30, Sections 3β5] are true for the solution of the problem (1.1), (1.2a)β(1.2c), and () for . In particular, we have the following results.

Theorem 3.6. is a strictly increasing continuous function on .

Theorem 3.7. Problem (1.1), (1.2a)β(1.2c), and () for (except the case ) has a sequence of positive and simple eigenvalues Moreover, , ; the corresponding eigenfunctions have simple zeros in .

Remark 3.8. In the case the first eigenvalue of boundary value problem (1.1), (1.2a)β(1.2c), and () is equal to zero and the corresponding eigenfunction is constant; the statement of Theorem 3.7 is true for .

Obviously, the eigenvalues , , of the problem (1.1), (1.2a)β(1.2c), and () are zeros of the entire function . Note that the function is defined for , where .

Lemma 3.9 (see [19, Lemma 5]). Let . Then, the following relation holds:

In (1.1) we set . As is known (see [34, Chapter II, Section 4.5, Theorem 1]) in each subdomain of the complex -plane equation (1.1) has four linearly independent solutions , , regular in (for sufficiently large ) and satisfying the relations where , , are the distinct fourth roots of unity, .

For brevity, we introduce the notation . Using relation (3.10) and taking into account boundary conditions (1.2a)β(1.2c), we obtain

Remark 3.10. As an immediate consequence of (3.11), we obtain that the number of zeros in the interval of function tends to as .

Taking into account relations (3.11) and (3.12), we obtain the asymptotic formulas Furthermore, we have

We define numbers , , , , , , , and a function , , , as follows: By virtue of [18, Theorem 3.1], one has the asymptotic formulas where relation (3.17) holds uniformly for .

By (3.14), we have From Property 1 in [30] and formulas (3.9), one has the relations

Remark 3.11. It follows by Theorem 3.7, Lemma 3.9, and relations (3.18) and (3.19) that if or , , then ; besides, if and , then .

Let be the number of zeros of the function in the interval .

Lemma 3.12. If and , , then .

The proof is similar to that of [19, Lemma 10] using Theorems 3.6 and 3.7 and Remark 3.11.

Theorem 3.13. The problem (1.1), (1.2a)β(1.2c), and () for has a sequence of real and simple eigenvalues including at most one negative eigenvalue. Moreover, (a) if , then for ; for for , where ; (b) if , then ; (c) the eigenfunction , corresponding to the eigenvalue , has exactly simple zeros in .

The proof parallels the proof of [19, Theorem 4] using Theorems 3.5β3.7 and Lemmas 3.9 and 3.12.

Lemma 3.14. The following non-selfadjoint boundary value problem: has an infinite set of nonpositive eigenvalues tending to and satisfying the asymptote

Setting in (3.12), we obtain (3.22).

Remark 3.15. By Remark 3.10 the number of zeros of the eigenfunction corresponding to an eigenvalue can by arbitrary. In views of [31, Corollary 2.5], as varies, new zeros of the corresponding eigenfunction enter the interval only through the end point (since ), and hence the number of its zeros, in the case , is asymptotically equivalent to the number of eigenvalues of the problem (3.21) which are higher than . In the case see [31, Theorem 5.3].

We consider the following boundary conditions:

Note that since . The boundary condition () coincides the boundary condition () for (resp., ) in the case (resp., ), and the boundary condition () coincides the boundary condition () for (resp., ) in the case ( resp., ).

Let . The eigenvalues of the problem (1.1), (1.2a)β(1.2c), and () (resp., (1.1) (1.2a)β(1.2c), and ()) are the roots of the equation (resp., ). By (3.9), this equation has only simple roots; hence all the eigenvalues of the problems (1.1), (1.2a)β(1.2c), and () and (1.1), (1.2a)β(1.2c), and () are simple. On the base of (3.9), (3.18), and (3.19) in each interval , , the equation (resp., ) has a unique solution (resp., ); moreover, if and if . Besides, if and if and .

Taking into account (), (), (3.23), and (3.24) and using the corresponding reasoning [18, Theorem 3.1] we have where relation (3.26) holds uniformly for and eigenfunctions and , , correspond to the eigenvalues and , respectively.

Let us denote .

Remark 3.16. Note that if is the eigenvalue of problem (1.1), (1.2a)β(1.2d), then since .

It is easy to see that the eigenvalues of problem (1.1), (1.2a)β(1.2d) are roots of the equation

By virtue of Remark 3.16 and formula (3.27), a simple calculation yields that the eigenvalues of the problem (1.1), (1.2a)β(1.2d) can be realized at the solution of the equation

Denote , , where .

We observe that the function is well defined for and is a finite-order meromorphic function and the eigenvalues and , , of boundary value problems (1.1), (1.2a)β(1.2c), and () and (1.1), (1.2a)β(1.2c), and () are zeros and poles of this function, respectively.

Let . Using formula (3.9), we get

Lemma 3.17. The expansion holds, where , , are some negative numbers.

Proof. It is known (see [35, Chapter 6, Section 5]) that the meromorphic function with simple poles allows the representation where is an entire function, and integers , , are chosen so that series (3.31) are uniformly convergent in any finite circle (after truncation of terms having poles in this circle).
We consider the case . By virtue of relation (3.18), we have . Hence, for and for . Without loss of generality, we can assume for . Then, for . Since the eigenvalues and , , are simple zeros of functions and , respectively, then by (3.29) the relations are true.
Taking into account (3.33), in (3.32) we get , . The cases , , and can be treated along similar lines.
Denote where is some small number. From the asymptotic formula (3.25), it follows that for the domains asymptotically do not intersect and contain only one pole of the function .
By (3.11), (3.12), (3.23), (3.24), and (3.25), we see that outside of domains the asymptotic formulae are true: where
Following the corresponding reasoning (see [36, Chapter VII, Sectionββ2, formula (27)]), we see that outside of domains the estimation holds; using it in (3.32) we get where , , , , , are some positive constants. By (3.37) and asymptotic formula (3.25) the series converges. Then, according to Theorem 2 in [35, Chapter 6, Section 5], in formula (3.31) we can assume , .
Let be a sequence of the expanding circles which are not crossing domains . Then, according to Formula (9) in [37, Chapter V, Section 13], we have By (3.38), we get
From (3.36), the right side of (3.39) tends to zero as . Then, passing to the limit as in (3.39), we obtain which implies .
Differentiating the right side of the least equality, we have Note that the function has the following expansion: where
Now let , that is, . has the following expansion: Again, according to Formula (9) in [37, Chapter V, Section 13], we have By (2.6) [18] and (3.9), we get Using (3.42), (3.41), and (3.46), we obtain Passing to the limit as in (3.47), we get Using (3.48) in (3.45), we have
In view of (3.49) and (3.45), we get Passing to the limit as in (3.50), we obtain
Lemma 3.17 is proved.

4. The Structure of Root Subspaces, Location of Eigenvalues on a Complex Plane, and Oscillation Properties of Eigenfunctions of the Problem (1.1), (1.2a)β(1.2d)

For , we find a positive integer from the inequality .

Theorem 4.1. The problem (1.1), (1.2a)β(1.2d) for has a sequence of real and simple eigenvalues including at most number of negative ones. The corresponding eigenfunctions have the following oscillation properties.(a)If , then the eigenfunction , , has exactly zeros in , the eigenfunction has no zeros in in the case , and the number of zeros of can be arbitrary in the case .(b)If , then the eigenfunction corresponding to the eigenvalue has exactly simple zeros for and exactly simple zeros for in and the eigenfunctions associated with the negative eigenvalues may have an arbitrary number of simple zeros in .

The proof of this theorem is similar to that of [18, Theorem 2.2] using Remark 3.15.

Throughout the following, we assume that .

Let be the eigenvalue of the operator . The eigenvectors and corresponding to the eigenvalues and , respectively, are orthogonal in , since the operator is -selfadjoint in . Hence, by (2.4), we have

Lemma 4.2. Let be an eigenvalue of boundary value problem (1.1), (1.2a)β(1.2d) and , where . Then, problem (1.1), (1.2a)β(1.2d) has no nonreal eigenvalues.

Proof. Let be an eigenvalue of problem (1.1), (1.2a)β(1.2d). Then, from Remark 3.16 and equality (4.2), we obtain
In view of formula (3.29), the inequality is true.
By (4.3), From (4.4)β(4.6), we get
From the second relation it follows that , which by (1.1) contradicts the condition . The obtained contradictions prove Lemma 4.2.

Lemma 4.3. Let , be eigenvalues of problem (1.1), (1.2a)β(1.2d) and . Then, .

Proof. Let . By (3.29) and (4.2), we have
Hence, we get
From (4.9), it follows that for . Therefore, .
Since , then by (1.1) . The obtained contradictions prove Lemma 4.3.

By Lemmas 4.2 and 4.3 problem (1.1), (1.2a)β(1.2d) can have only one multiple real eigenvalue. From (3.41), we get , , whence it follows that the multiplicity of real eigenvalue of problem (1.1), (1.2a)β(1.2d) does not exceed three.

Theorem 4.4. The boundary value problem (1.1), (1.2a)β(1.2d) for has only point spectrum, which is countable infinite and accumulates at and can thus be listed as with eigenvalues repeated according to algebraic multiplicity and ordered so as to have increasing real parts. Moreover, one of the following occurs.(1)All eigenvalues are real, at that contains algebraically two (either two simple or one double) eigenvalues, and , , contain precisely one simple eigenvalues.(2)All eigenvalues are real, at that contains no eigenvalues but, for some , contains algebraically three (either three simple, or one simple and one double, or one triple) eigenvalues, and , , contain precisely one simple eigenvalue.(3)There are two nonreal eigenvalues appearing as a conjugate pair, at that contains no eigenvalues, and , , contain precisely one simple eigenvalue.

Proof. Remember that the eigenvalues of problem (1.1), (1.2a)β(1.2d) are the roots of the equation , where , (see (3.28)). From (3.41), it follows that for ; therefore, the function is convex on the interval . By virtue of (3.18) and (3.30), we have
That is why for each fixed number there exists number such that the lines , , touch the graph of function at some point . Hence, in the interval , (3.28) has two simple roots if , one double root if , and no roots if .
By (3.29) and (3.30) we have , , . Therefore, (3.28) has at least one solution in the interval , .
Let . If , then , ; if then . By (3.29), (3.28) has only simple root for for in the interval , .
Let . By Lemma 4.3 either for any , or there exists such that and , . Assume that . Obviously, . Choose natural number such that the inequalities are fulfilled; where , is sufficiently small positive number, and .
We have where (see [37, ChapterββIV, Sectionββ10]). By (4.11) hence, the point does not go out of circle . Therefore, vector cannot turn around the point , and the second summand in (4.13) equals zero. Thus, By the argument principle (see [37, Chapter IV, Section 10, Theorem 1]) we have where and are multiplicity of zero and pole of the function , respectively . Obviously, and . Then, by (4.18) we obtain From (4.17) and (4.19) follows the validity of the equality
Using the argument principle again, by (4.20) we get whence it follows that where , , are roots of the equation . From the above-mentioned reasoning, by (4.22) we have and, therefore, problem (1.1), (1.2a)β(1.2d) in the interval for , , has only one simple eigenvalue.
Consider the following two cases.
Case 1. For all real eigenvalues of problem (1.1), (1.2a)β(1.2d) the inequalities , , are fulfilled. The problem (1.1), (1.2a)β(1.2d) in every interval , , has one simple eigenvalue. Hence, problem (1.1), (1.2a)β(1.2d) in the interval , , has simple eigenvalues, and hence, by (4.23), this problem in the circle has one pair of simple nonreal eigenvalues. In this case, the location of the eigenvalues will be in the following form: , , , , .Case 2. Let , , and , . By Lemma 4.2 problem (1.1), (1.2a)β(1.2d) has no nonreal eigenvalues. From the above-mentioned reasoning it follows that in each interval , , , problem (1.1), (1.2a)β(1.2d) has one simple eigenvalue.Subcase 1. Let ,, that is, the eigenvalue is a double one (by this ). Then, from (4.23) it follows that the interval besides the eigenvalue contains one more simple eigenvalue: at that it is either (by this ) or (by this ). Hence, , , (by this either or ), , .Subcase 2. Let , . By (3.41), . Hence, is a triple eigenvalue of the problem (1.1), (1.2a)β(1.2d) (by this ). Then, from (4.23) it follows that in the interval problem (1.1), (1.2a)β(1.2d) has unique triple eigenvalue , and therefore, . At this , , , ,.Subcase 3. Let , that is, the eigenvalue is simple. Then, by (4.23), in the interval problem (1.1), (1.2) has an eigenvalue as well as two more simple eigenvalues, which, by Lemma 4.3, are and (and hence ). In this case, we have , , (), , .
Theorem 4.4 is proved.

By Theorem 4.4 we have , that is, if or ; , that is, if (If assertion (2) in Theorem 4.4 holds, then we set ).

Let be a system of eigen- and associated functions corresponding to the eigenvalue system of problem (1.1), (1.2a)β(1.2d), where if ; , , , is an arbitrary constant, if , , , , , are arbitrary constants, if . Here, is an eigenfunction for and when ; , when are the associated functions (see [34, Pages 16β20] for more details).

We turn now to the oscillation theorem of the eigenfunctions corresponding to the positive eigenvalues of problem (1.1), (1.2a)β(1.2d) since the eigenfunctions associated with the negative eigenvalues may have an arbitrary number of simple zeros in .

Theorem 4.5. For each (resp., ), has (resp., ) zeros in the interval . Similarly , both have (resp., ) zeros if (resp., ). Finally, if , then each of (and , if ) has ββorββ zeros according to , , or , and if and , then , both have zeros.

The proof of this theorem is similar to that of [11, Theorem 4.4] using Lemma 3.12.

5. Asymptotic Formulae for Eigenvalues and Eigenfunctions of the Boundary Value Problem (1.1), (1.2a)β(1.2d)

For , let be an integer such that (interpreting ).

Lemma 5.1. The following relations hold for sufficiently large , :