WRONSKIANS AND SUBSPACES OF CERTAIN FOURTH ORDER DIFFERENTIAL EQUATIONS

The objectives of the paper are to study the behavior of Wronskians of solutions of the fourth order differential equations and to relate this behavior with the oscillations of these equations, as well as to the structure of the sub- spaces of the solution spaces of the equations.


i. INTRODUCTION.
This paper is concerned with the fourth order differential equation (p(x)y")" q(x)y" r(x)y 0 (L)   and its adjoint [p(x)y" q(x)y]" r(x)y 0 (L) where p, q and r are assumed to be continuous real-valued functions on the interval R + [0, ).In addition, it is assumed throughout that p > 0, q >-0 and r > 0 on R+, with r not identically zero on any subinterval.It is clear that if q is a constant, then (L) is selfadjoint; otherwise (L) is non-selfadjoint.
The objectives of the paper are to study the behavior of Wronskians of solutions of (L) and (L*), and to relate this behavior with the oscillation of (L) and (L*), as well as to the structure of the subspaces of the solution spaces of (L) and (L*).
A nontrivial solution y of (L) {(L*)} is oscillatory if the set of zeros of y is not bounded above.If the set of zeros of y is bounded above, implying that y has only finitely many zeros on R+, then y is nonoscillatory.Equation (L) (L*)} is oscill- atory if it has at least one nontrivial oscillatory solution.For convenience here- after, the term "solution" shall be interpreted to mean "nontrivlal solution." Various special cases of (L) have been studied in detail.In particular, we refer to the fundamental work of W. Leighton and Z. Nehari [10,Part I] on the self- adj oint equation (p(x)y") r(x)y 0 (I.I) M. Keener [7,Part I] continued the investigation of (i.i), concentrating on the oscillatory behavior of solutions.S. Ahmad [1] considered the selfadjoint equation (4)   y r(x)y 0 (1.2) and gave a necessary and sufficient condition for the existence of a linearly inde- pendent pair of oscillatory solutions.In [12] and [13] V. Pudei investigated the behavior of solutions of the equation y(4) q(x)y" r(x)y 0 (1.3) Finally, we refer to the authors' work in 5 where sufficient conditions for the oscillation of (L) and (L*) are given, and where the behavior of both oscillatory and nonoscillatory solutions is studied.
As a notational convenience in treating the solutions of equations (L) and (L*), we introduce the following differential operators.
The next theorem provides the existence of a bounded nonoscillatory solution.
Pudei [13, Theorem 55] has shown that equation (1.3) is oscillatory if and only if its adjoint is oscillatory.With obvious modifications, his proof can be extended to the case of (L) and (L*).An alternative proof of this fact can th be accomplished by showing that the n--conjugate point of a with respect to (L) th coincides with the n---conjugate point of a with respect to (L*), and then applying Leighton and Nehari's result [i0, Theorem 3.8].The final theorem in this section gives a necessary and sufficient condition for the existence of oscillatory solutions of (L) and (L*).The authors established this result for (L) (see [5, Theorem 4.1]) using the approach developed by Ahmad in [i].This approach can also be used to establish the result for (L*).
THEOREM 2.3.The following two statements are equivalent: (a) Equation (L) (L*) is oscillatory (b) If y is a nonoscillatory, eventually positive solution of (L) (L*)}, then y is either strongly increasing or strongly decreasing.
Let $ and * denote the space of solutions of (L) and (L*) respectively.
The theorems of the previous section suggest the identification of the following subsets of $, and of $*: I {y e $ either y or -y is strongly increasing} {w e $ either w or-w is strongly decreasing} O= {z S z,S uD} Let I*, D* and 0* be the corresponding subsets of S*.Theorems 2.1, 2.2 and 2.3 specify that none of these subsets is empty.Moreover, according to Theorem 2.3, (L) {(L*)} is oscillatory if and only if every solution in 0 (*) is oscillatory.
In this section we study the Wronskians of solutions of (L) and (L*), we give some basic identities satisfied by these Wronskians, and we give a neces- sary and sufficient condition for the nonoscillation of (L) and (L*) in terms of these Wronskians.
Let u, v, y, z S Then W2 (u,v) u v Dlu Dlv W3(u,v, Y) The Wronskians W2(u,v), W3(u, v, y) and W4(u, v, y, z) of solutions u, v, y, z , of (L) are defined in a similar manner.Of course W4(u v, y, z)--k (constant) {W(u, v, y, z) =-k} on R+, and k # 0 if and only if {u, v, y, z} is a solution basis for (L) {(L )}.Also, it is well-known (and easy to verify by direct calculation) that if u, v, y are any three linearly independent solutions of (L) }.
(L) {(L*)} then W 3(u, v, y) {W(u v y)} is a solution of (L* , The linear operators L and L associated with equations (L) and (L), respectively, are defined by A function y defined on R + is said to be admissable for /_ {*} if each of y and R/" py" {py' qy} is twice differentiable on If y is admissable for /_ and z is admissable for L then they satisfy the Lagrange identity.

z/_[y]
yL [z] {y;z}', where {y;z} z(py") z (py") + (pz"-qz)y (pz"-qz) y , If follows from (3.1) that if y S and z S then {y;z} 0, which implies {y;z} k (constant) on Thus the relation (3.2) determines a function }: S x / R (the reals), and it is easy to verify that this function is linear in each of its arguments.This function can also be used to express a relationship between certain Wronskians of orders 3 and 4. In particular, let u, v, y S be linearly independent, and let z S. Then W3(u, v, y) S and {z; W3(u v, y)} W4(u, v, y, z) Similarly, if v, y, z E S'are linearly independent and u S then W3(v y, z) e S and {W3(v, y, z); u} W4(u, v, y, z) ( These identities are an extension of the ideas introduced by J. M. Dolan in [4]. , They can be verified by expanding W 4 along its last column and W 4 along its first column.It is clear from identities (3.3) and (3.4) that {u, v, y, z} forms a solution basis for S (S) if and only if {z;W3(u, v, y)} k # 0 ({W3(v, y, z); u} k # 0).
Our first two results establish a connection between two and three dimensional subspaces of S (S*) and certain second and third order Wronskians.
These results are related to the Wronskian identities established by W. J. Kim in [8].
, , , THEOREM 3.1.Let y ( S (S) and let Sy (Sy) be the subset of S (S) (iii) Let u, v, z be a basis for S (Sy).Then W3(u v, z) my Y (W3(u, v, z) my) for some nonzero constant m.In fact, the basis u, v, z may be chosen such that W3(u, v, z) y (W3(u v, z) y) NOTE.The proofs in this section will be given in terms of elements of S only.It will be clear that the same arguments apply equally as well for elements of S PROOF.Part (i) follows from the fact that {y;z} 0 essentially defines a third order, linear, homogeneous differential equation.
Consider part (ii).Since {. ,.} is linear in each of its arguments, it is easy to see that if y ku, then S Su).To show the converse y u S assume that y, u S are linearly independent Since is a three dimensional Y subspace of S it follows that if a is any point in R + and i, j are any two distinct integers, 0 < i, j < 3, then there exists a solution z S* such that y Diz(a) Djz(a) 0. Fix any a R+.Suppose y(a) 0, u(a) # 0. Then it is easy to verify, using the Lagrange identity, that S contains the solution z Y satisfying D0z(a) Dlz(a) D2z(a) 0 D3z(a) i and z S Thus S # S u y u The same argument applies if u(a) 0, y(a) # 0. Now suppose y(a) # 0 and u(a) # 0. From part (i) we can assume that y(a) u(a).Let i, i _< i <_ 3 be the least integer such that DiY(a) # Diu(a).Such an integer exists since y and S u are linearly independent Now choose z Y such that D.z(a)3 DkZ(a) 0, j # k, 0 < j, k < 2 and j # 3 i, k # 3 i.Then from the Lagrange identify ztS.
U , , For (iii), let {u, v, z} be a basis for S and let W--W3(u v, z}.Then, Y , , from (3.4), {W;u} {W;v} {W;z} 0, which implies S W Sy, and so W my from (ii).The last part of (iii) follows from the fact that i* , THEOREM 3.2.Let y,z S(S be linearly independent.Then: (i) S* * * N S (Sy N S z) is a two dimensional subspace of S (S).W 2(y, z) kW 2(u, v) {W 2(y, z) kW 2(u, v)} for some nonzero constant k.
PROOF.Part (i) follows from the fact that the intersection of two distinct three dimensional subspaces of a four dimensional vector space has dimension two.
To prove part (ii), assume that y, z S are linearly independent and {u,v} Now D2 p q so that this equation can be written.W2(Y, z) W2(Y z) + [y D3z z D3Y q] 0 P Since u and v satisfy this equation, we have, by Abel's identity, , W 2(y, z) k W 2 (u, v), for some nonzero constant k, on any interval on which W2(Y z) # 0. The continuity of W2, W 2 and their derivatives imply that this equation actually , , holds on R+.Finally, since k W2(u v) W2(ku, v), we have (ii) in the case y, z We conclude this section with a necessary and sufficient condition that , each of (L) and (L) be nonoscillatory.This condition is stated in terms of the nonoscillation of second and third order Wronskians of linearly independent solutions.In particular, the Wronskian W i (Wi) i 2 or 3, of linearly * + independent solutions of ..q (S) is nonoscillator if there is a number b R such that W i (W i) is nonzero on [b, ); otherwise W i (Wi) is oscillatory.
THEOREM 3.3.Equations (L) and (L) are nonoscillatory if and only if every second and third order Wronskian W2, W 3 {W2, W 3} of linearly independent solutions of S {S*} is nonoscillatory.
PROOF.Assume that all second and third order Wronskians of linearly independent solutions of (L) are nonoscillatory.Then, in particular, all third order Wronskians W 3 are nonoscillatory.By Theorem 3.1 (iii) every solution of , ..q is the Wronskian of three linearly independent solutions of S. Thus we can conclude that (L), and hence (L), are both nonoscillatory.Now assume that (L) is nonoscillatory.Then Theorem 3.1 (iii) and the fact , that (L) is also nonoscillatory implies that all third order Wronskians of linearly independent solutions of (L) are nonoscillatory.Thus, it remains to examine the second order Wronskians.Let y, z S be linearly independent and R+.
assume y > O, z > 0 on [b, =) for some b e It was shown in [5, Lemma 3.1] that i u is any nonoscillatory solution of (L), then 2 D.u # 0 on [c, =) for some c R + i=O Therefore, by taking b large enough, we may assume y > O, z > O, DiY O, D.zl # 0, i i, 2, on [b, ).Suppose W2(Y z) is oscillatory.Recall the subsets I, D and 0 of S. We show first that we cannot have y I, z D u 0 (or vice versa).Since W2(Y z) yz zy it is clear that y e , z is impossible.Suppose, therefore that y e and z 0. There are three possible cases for the signs of z and its "derivatives".
(I) z > 0, z > 0, D2z < 0 on [b, ), Since y > 0 and y is strongly increasing, we may assume that DiY > 0, i i, 2, 3, on [b, =).If z satisfies (II), then W2(Y z) < 0 on [c, =).If z satisfies (I), then W2(Y z) yz zy < 0 on [b, =) which implies that W 2 has constant sign on [d, ) for some d > b.Finally, suppose z satisfies (III).and as a result of the signs of y, z and their "derivatives", we find that the first term on the right hand side is monotone decreasing and the second term has limit -.Thus lim pW 2 -,.This implies that W 2 < 0 on [d, ,) for some d >-b and so W 2 is eventually of one sign.We can now conclude that if W2(Y z) oscillates, then either y, z I, or y, z D u 0. Assume y, z and W2(Y z) oscillates.Choose u 0 such that y, z, u are linearly independent.Then W3(y, z, u) # 0 on [a, ) for some a R+, and {y, z, u} is a solution basis for the third order differential equation which can be written in the form WB(Y z, u)(p WB(Y z, u)(p + f(x) + g(x) O. (3.5) Now, the fact that u and W2(Y, u) are nonoscillatory implies that there exists b > a such that u 0 and W2(Y, u) 0 on [b, ).Therefore, according to Ahmad [2, p292] (see also, G. Polya .F12]),( 8) is disconjugate on Fb, ).Thls implies that the adjoint of equation (3.5) is disconjugate on [b, ) (see J. H. Barrett [3]).But (x)= W2(Y, z)/W3(Y, z, u) is an oscillatory solution of the adjolnt equation, and we have a contradiction.The same method of proof can be used if we assume that y, z D u 0 and W 2(y, z) oscillates.

SUBSPACES OF S AND S:
In this section we consider the structure of the three dimensional subspaces of S and S*, and as a corollary, we also identify certain two dimensional sbspaces.We will be making use of the subsets , D and 0 of * * O* * S and I and of defined in Section 2, and we shall also be concerned with the "complimentary" subspaces S (Sy) determined by the solutions y of Y S (S).In this regard recall that if y S (S) then S (Sy) is the three on [a, =) if any solution with a double zero at b >a, is THEOREM 4.1.Let y e S. W2(u v) # 0 on ['a, ) for some a > 0, and the third order equation ( 9) is in class CII on [a, The corresponding statements hold if y E S PROOF.As in Section 3, we will prove the theorem only for the case y E S.
(1).Suppose y E [, and assume, without loss of generality, that y is eventually positive.Then there exists a > 0 such that Diy > 0, i 0, i, 2, 3, on [a, ).Choose any b > a and let z be a solution of (9), i.e. let z be an , element of S such that z has a double zero at b. Then it is easy to verify Y , , from (9) that_ D2z(b) and D3z(b) cannot have opposite sign.We may assume, , therefore, that D2z(b) > 0, D3z(b) >_ 0 with at least one inequality being strict, , and so, by Theorem 2.1, z E [ and equation (4.1) is in class CII on [a, )., Y (ll).Suppose y D, and assume, without loss of generality, that i (-i) DiY > 0"on [0, ).Choose any b > 0 and let z be a solution of (4.1) such , that z has a double zero at b. Then it is easy to see from (4.1) that D 2 z( z(b) must have opposite sign, say D 2 z(b) > 0, D 3 z(b) < 0. Thus, by i * Theorem 2.1, we have (-i) D i z > 0, i 0, i, 2, 3, on [0, b) and equation (4.1) is in class C I on [0, ).Now, for each positive integer n, let z be a solution n of (4.1) which satisfies Zn(n) D I Zn(n) 0, D2Zn(n) > 0, D3zn(n) < 0. Then by a standard "sequence argument" (see A. C. Lazer [9, Theorem 1.17 or Ahmad [17), we can construct a solution w of (4.1) such that sg w sgn D2w # sgn DlW sgn D3w on [0, =), i.e. such that w e D Now choose any y e I and assume D iY > 0 i 0, i, 2, 3 on [a, =), a e 0. Then by using the same argument as in part (i), we can conclude that S [w, u, v], Y where w u, v 0 and W2(u,v) # 0 on [a, ).
(iii).Suppose y e 0. Choose any z e I and any w e .Then S 0 S n S Now, Y , suppose that y is oscillatory, and let {z I, z 2, z 3} be a basis for S Then, by Y Theorem 3.1 (iii), W3(zI, z2, z 3) my for some nonzero constant m.Let b be a , zero of y.Then W3(zI, z 2, z3)(b) 0 which implies that there exists a nontrivial linear combination z of the solutions z I, z 2, z 3 such that z has a triple zero at b.We may assume that D3z(b) > 0 and conclude, by Theorem 2.1, * that D.z > 0, i 0, I, 2, 3, on (b,=) so that z e Let {b be the sequence 1 n of zeros of y.Then for each positive integer n there is a solution z of (4.1) n such that Diz (b n) 0 i 0 i 2 and D3Zn(bn) < 0. Thus by Theorem 2.1 n i * (-i) D.z > 0, i 0, i, 2, 3 on [0, b ), and so, by using the "sequence i n n * argument" cited above we can construct a solution w in S such that w Y Finally, it is easy to verify that S [z, w, u].Y Now suppose that y is nonoscillatory.As observed in the proof of Theorem 3.3, y and its derivatives must satisfy one of the following three sets of inequalities on an interval [a, =).
(I) y > 0, y > 0, D2Y < 0 (II) y > 0, y < 0, D2Y > 0, D3Y > 0 D2y (III) y > 0, y > 0, > 0, D3Y < 0 In each case y has two consecutive derivatives D iY, D i+lY, 0 -< i _< 2 which have the same sign on [a, ,), and two consecutive derivatives Djy, Dj+lY, 0 -< j -< 2 which have opposite sign on [a, ).Choose any b > a.Let z be a solution of (4.1) such that D z(b) 0 for the two m's with the property that D z(b) is not m m a coefficient of either DiY(b) or Di+lY(b).It then follows that the remaining two "derivatives" of z at b cannot have opposite sign.Assutng that at least one of these "derivatives" is positive, we have z e I Now let {b be an n increasing sequence in [a, ,) with limb ,.For each positive integer n, let n--t n z be a solution of (4.1) such that D z _(b n) 0 for the two m's with the n m n property that DmZn(bn) is not a coefficient of either Djy(bn) or Dj+lY(bn).
Then we can conclude from equation (9) that the retning o "derivatives" of z at b must have opposite sign.Thus the consecutive "derivatives" of z n n n must have opposite sign on [0, n), and the "sequence argument" allows us to construct a solution w of (4.1) such that w Finally, it is easy to verify that the three solutions, u, z, nd w are linearly independent, and this completes the proof of the theorem.
There are a variety of consequences of Theorem 4.1 which describe the structure of the two and three dimensional subspaces of S and S*.We list these results in the following corollaries.
COROLLARY: (i) Every three dimensional subspace of S($ has a non- oscillatory solution. (2) Equation (L) (L)} is oscillatory if and only if every three dimensional subspace of S (S) has an oscillatory solution.
(3) Every three dimensional subspace of S (S) contains a pair of solutions whose Wronskian is nonzero on [a, ) for some a >-0.
PROOF.Let T be any three dimensional subspace of .q (q), and let {Yl' Y2' Y3 } be a basis for T. Then y W(Yl, Y2' Y3 {y W (Yl' Y2' Y3 )} is S* * an element of (S) and T S (T y).y , We now consider the two dimensional subspaces of and S We note that corresponding to any two dimensional subspace T of there is a unique two T* * * * dimensional subspace S n S of S where {y, z} is any basis for T. y z , , Similarly, a two dimensional subspace T of S determines a unique two dimensional subspace T of S. PROOF.The four cases follow from the fact that if T is any two dimensional subspace of S, then T o (0 u D) @ and T 0 (0 u I) # @.This fact can be established by considering the intersections T n S and T n S where y E I* y w' w E D and using Theorem 4.1.
The structure of T specified in cases (i) We conclude by noting that if the equations (L) and (L*) are oscillatory, then the structure of a two dimensional subspace can be determined by

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 If z is a solution of (L) {(L*)} such that (-l)iD* z(b) > 0 {(-i) i Diz(b > 0} i O, I, 2,3, for some b R+, b > 0, with strict inequality for at least one i, ) is a three dimensional subspace of (S).(ii) Let u S (S*).Then S* S* (Sy S u) if and only if y ku for Y U some nonzero constant k.
DlZ D2c + D2z D 1-D3 z D0q: 0 Multiplying these two equations by y and z, respectively, and subtracting, yields the second order equation W2(Y, z) D 2 p(yz zy )D I+ [y D3z z D3Y]= 0 D2 + D2Y DI D3Y DO 0}In[6], M. Hanan defined two classes of third order linear differential equations.In particular, a third order equation is in class C I on [a, ) if amy soltion,wlth D double zero t some point b > a is nonzero on [a, b), and it is in class CII nonzero on (b, ).
u, v] (= the space spanned by z, u, v), where Y 0* * z E I u, v E v], where u, v Finally, from Theorem 3.2 (ll), there is a y , nonzero constant k such that W 2 (u, v) k W2(Y w) 0 on [a, ).Therefore , S [z, u, v] and the proof of Dart (i) is complete.
S contains an element u in

COROLLARY 2 .I
Let  T be a two dimensional subspace of S. Then T satisfies exactly one of the following:(i) T [y,u], y E I, u E 0, and [z, vl, z T [y,w], y I, w D, and [u, v u E v 0* and every combination of u and v is in

( 3 )P
is easy to verify.The 0* structure specified in(4) follows from the fact that if y E then any two dimensional subspace of S must contain either an element of I, or an element of Y

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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