Indefinite Eigenvalue Problem with Eigenparameter in the Two Boundary Conditions

The object of this paper is to establish an expansion theorem for a regular indefinite eigenvalue problem of second order differential equation with an eigenvalue parameter ,k in the two boundary conditions We associated with this problem a J-selfadjoint operator with compact resolvent defined in a suitable Krein space and then we develop an associated eigenfunction expansion theorem KEY WORDS AND PHRASES: An expansion theorem, a regular indefinite eigenvalue problem, eigenvalue parameter in the boundary conditions, a J-selfadjoint operator, Krein space formulation 1991 AMS SUBJECT CLASSIFICATION CODES: 65NXX, 65N25.


INTRODUCTION
The regular fight-definite eigenvalue problem with eigenparameter in the two boundary conditions where r(x) and q(z) are positive functions on [a, b] has been studied by Zayed and Ibrahim [1] Daho and Langer [2] have made an extension of Everitt's paper [3] and have replaced the Hilbert space in some cases by a Pontragin space with index one.Everitt [3] has shown that for a 6 [0, ] the singular Sturm-Liouville with indefinite weight function r(x) can be represented by a selfadjoint operator in a suitable Hilbert space Recently, Fleckinger and Mingarelli [4] have studied an indefinite problem with the usual homogeneous Dirichlet or Neumann boundary conditions.
(ii) Both the weight function r(x) and the potential function q(x) change sign on [a, b] in the sense that the problem (1 1)-(1.3) is an indefinite.

(4)
The parameter A is a complex number.
(ii) The decomposable, non-degenerate inner product space H is a Krein space [5] if and only if for every fundamental symmetric operator J, the J-inner product turns H into a Hilbert space, that is, we have If, g] (Jr, g), f,g _ H. ( DEFINITION 2.2.We define a closed linear operator A D(A) H by A/:= (r/1,Mo(A), Ma(A)), Vy e D(A) (2.11) such that M,(f) cif (a) a2(pf)(a), MO(A) xA () where the domain D(A) of the closed linear operator A is defined as the set of all f (fl, f2, f3) H which satisfy the following conditions: (i) f, f{ are absolutely cominuous functions on [a, b] with /., (, b) and (a) h (A), (iii) f3 REMARK 2.2 (i) The domain D(A) is dense in H with respect to the indefinite inner product (2.2).
(ii) A is an eigenvalue and f is a corresponding eigenfunction of problem (1.1)-(1.3)if and only it" f (fl,h, f3) D(A) and Af Af Therefore, the eigenvalues and the eigenfunctions of problem (1 1)-(1.3)are equivalent to the eigenvalues and the eigenfunctions of the operator A.
(2.13) Integrating the first term of (3.1) by parts twice, we get 3. THE J-SELFADJOINTNESS OF OPERATOR A DEFINITION 3.1.In the Krein space H, a symmetric operator and a selfadjoint operator with respect to indefinite scalar product are called ,/-symmetric and J-selfadjoint respectively (see [5]).
Let J be a conjugation operator on H; this means that J is a conjugate-linear involution with As in Knowles [6] we can define an inner product on the domain D(JA'J) by Since J is a conjugation operator on H, we find that (3.6 With this indefinite inner product, D(JA'J) becomes a Krein space (see Dunford and Schwartz [7, p. 1225]).LEMMA 3.2.If A is a J-symmetric operator in H, then D(JA'J) D(A).
PROOF.Let g D(JA" J) (9 then If, g] 0, for f .D(A). ( Making use of(3.7),(3.9) and the fact that JA'Jf Af, for f e D(A) we get On using the definition of an adjoint operator, (3.1 O) implies JA'JgED(A').
This gives g e D(A" JA'J).
(3 I) From (3.8) and (3.11), we can conclude that the vector function 9 is the zero vector function This implies that D(JA'J) D(A).
(3 12) REMARK 3.1.Our closed linear operator A is J-symmetric in Krein space H, the domain D(A) is dense in H and A JA* J. Therefore, the operator A is J-selfadjoint operator in H (see Race [8]).

THE BOUNDEDNESS OF THE OPERATOR A
In this section we shall show that the J-selfadjoint operator A in H is unbounded from above and bounded from below.To this end we need the following lemma for all signs of r(x) and q(x).
On using the same ments ofLena 4.2, we c show that where c is a constt.Letting m , we get lim [Ax, X]n . ( This proves that A in H is unbounded from above.

THE EIGENVALUES OF OPERATOR A
The problem (1 1)-(1.3) in the indefinite case gives us positive and negative eigenvalues Thus we consider the infinite sequence ofthe eigenvalues of A: where A,, < 0 < A=+I.For brevity, the eigenvalues and eigenfunctions are together called eigenpairs DEFINITION 5.1.The J-selfadjoint operator A is called J-non-negative if JAr, f] n > O, f E D(A).

TIIE RESOLVENT OPERATOR AND TIIE EXPANSION THEOREM
Suppose that @l(X;,X), @2(x;)), are the fundamental solutions of (I.I) on [a,b] which satisfy the initial conditions: (6 l) where A E C is not an eigenvalue ofthe operator A, and put which is independent of x 6 [a, b].Putting x a, we therefore have: Mo((a; a)) + a((; )). ( Similarly, putting z b, we therefore have It is also from (6. l) that P((a;A)) p and R(2 where p, p2 are given by (1.4).From (6.1) it is clear that for all A 6 C, M(q1 (a; ,)) ,X/(@I (a; R)), which gives that q1 (x; A) satisfies the first boundary condition (1.2) at z a and M(q;2(b; A)) RR(@2(b; R)), which gives that @2(z;R) satisfies the second boundary condition (1.3) at x b.Employing the same type of argument as in the regular Sturm-Liouville problem [9, Sec.1.8] it follows that the zeros of o(R) are real and that if A,, n 1, 2, 3, denotes these zeros, the three-component vectors are eigenfunctions of operator A satisfies the onhogonality relation [,,,,]H=0 for n#rn, (67) where the indefinite inner product is given by (2.2).The initial conditions (6.1) also serve to guarantee that /'a (z; A) and 2 (z; A) are entire in A for fixed z, and so it follows that wx (A) is an entire function of A.
(iii) On using Theorem 3 in Hellwig [10, p. 30], we deduce that the density of D(A) in Krein space H gives the completeness of the orthonormal system ofthe real eigenfunctions of A.
The results of our investigations are summarized in the following expansion theorem THEOREM 6.1.The closed linear operator A in Krein space H has an unbounded set of real eigenvalues of finite multiplicity without accumulation points in (oo, oo), and they can be ordered according to size A,+,oo as s--oa with If the corresponding real eigenfunctions <I)l, 2,-.., n, n+l,-.. form a complete onhonormal system, then for any function f(x) E H, we have the expansion in the sense of strong convergence in H f If, :I:'n]HO, (6 2:5)

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning ) 3.ql (12) c4(]9.q[)(a),Ra(gl) 193g] (b) 4 (pg' )(b) H := P+(Ll(a,b 9C 9C)

LEMMA 4 .
1. Let f, f' be continuous functions on[a, b] with Tfl E Lrl(a,b Since p(x) is continuous on [a, b] as well as p(x) > 0 on [a, b], then there exists a positive constant co with p(x) >_ co such that bp(:)IfI(:r.)12dx>_ (b c a) I.fl (b) fl (a)]2.

LE 4 . 3 .
The J-selfadjoint operator A H is unbounded from above.PROOF.Let X(Z) be a test nction in the ein space H th compact suppo on [a, b] d define a suence of ts test nction by X(Z) :=X(mZ) for z [a,b], m 1,2,3,...,

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation