doi:10.1155/2009/389624
Research Article
On the Nonexistence and Existence of Solutions for
a Fourth-Order Discrete Boundary Value Problem
Shenghuai Huang and Zhan Zhou
School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China
Correspondence should be addressed to Zhan Zhou,[email protected]
Received 16 July 2009; Accepted 16 October 2009
Recommended by Patricia J. Y. Wong
By using the critical point theory, we establish various sets of sufficient conditions on the nonexistence and existence of solutions for the boundary value problems of a class of fourth-order difference equations.
Copyrightq2009 S. Huang and Z. Zhou. 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.
1. Introduction
In this paper, we denote byN,Z,Rthe set of all natural numbers, integers, and real numbers, respectively. Fora, b∈Z, defineZa {a, a1, . . .},Za, b {a, a1, . . . , b}whena≤b.
Consider the following boundary value problemBVP:
Δ2pn−1Δ2un−2 ΔqnΔun−1fn, un, n∈Z1, k,
u−1 u0 0uk1 uk2.
1.1
Here,k ∈N,pnis nonzero and real valued for eachn∈Z0, k1,qnis real valued for eachn∈Z1, k1.fn, uis real-valued for eachn, u∈Z1, k×Rand continuous in the second variableu.Δis the forward difference operator defined byΔun un1−un, andΔ2un ΔΔun.
We may think of1.1as being a discrete analogue of the following boundary value problem:
ptxtqtxtft, xt, t∈ a, b,
which are used to describe the bending of an elastic beam; see, for example, 1–10 and references therein. Owing to its importance in physics, many methods are applied to study fourth-order boundary value problems by many authors. For example, fixed point theory 1, 3, 5–7, the method of upper and lower solutions 8, and critical point theory 9, 10 are widely used to deal with the existence of solutions for the boundary value problems of fourth-order differential equations.
Because of applications in many areas for difference equations, in recent years, there has been an increased interest in studying of fourth-order difference equation, which include results on periodic solutions 11, results on oscillation 12–14, and results on boundary value problems and other topics 15, 16. Recently, a few authors have gradually paid attention to applying critical point theory to deal with problems on discrete systems; for example, Yu and Guo in 17considered the existence of solutions for the following BVP:
ΔpnΔun−1qnun fn, un,
ua αua1 A, ub2 βub1 B. 1.3
The papers 17–20show that the critical point theory is an effective approach to the study of the boundary value problems of difference equations. In this paper, we will use critical point theory to establish some sufficient conditions on the nonexistence and existence of solutions for the BVP1.1.
Let
an qn1−2pn pn1,
bn pn−1 4pn pn1−qn−qn1. 1.4
Then the BVP1.1becomes
Lun fn, un, n∈Z1, k,
u−1 u0 0uk1 uk2, 1.5
where
Lun pn1un2 anun1 bnun an−1un−1
pn−1un−2. 1.6
2. Preliminaries
In order to apply the critical point theory, we are going to establish the corresponding variational framework of BVP1.5. First we give some notations.
LetRkbe the real Euclidean space with dimensionk. Define the inner product onRk as follows:
For BVP1.5, consider the functionalJdefined onRkas follows:
Ju 1
After a careful computation, we find that the Fr´echet derivative ofJis
Ju Mu−fu, 2.6
Expanding out Ju, one can easily see that there is an one-to-one correspondence between the critical point of functional J and the solution of BVP 1.5. Furthermore, u u1, u2, . . . , ukT is a critical point of J if and only if {ut}kt−21 u−1, u0, u1, . . . , uk, uk1, uk2T is a solution of BVP 1.5, where u−1
u0 0uk1 uk2.
Therefore, we have reduced the problem of finding a solution of1.5to that of seeking a critical point of the functionalJdefined onRk.
In order to obtain the existence of critical points ofJ onRk, for the convenience of readers, we cite some basic notations and some known results from critical point theory.
Let H be a real Banach space, J ∈ C1H,R, that is, J is a continuously Fr´echet
differentiable functional defined onH, andJ is said to satisfy the Palais-Smale condition P-S condition, if any sequence{xn} ⊂ Hfor whichJxnis bounded andJxn → 0 as
n → ∞possesses a convergent subsequence inH.
LetBr denote the open ball inHabout 0 of radiusrand let∂Br denote its boundary.
The following lemmas are taken from 23,24and will play an important role in the proofs of our main results.
Lemma 2.1Linking theorem. LetHbe a real Banach space,HH1
H2,whereH1is a
finite-dimensional subspace ofH. Assume thatJ∈C1H,Rsatisfies the P-S condition and the following.
F1There exist constantsσ,ρ >0such thatJ|∂Bρ∩H2 ≥σ.
F2There is ane ∈ ∂B1∩H2 and a constantR0 > ρsuch thatJ|∂Q ≤ 0 andQ BR0 ∩
H1{re|0< r < R0}.
ThenJpossesses a critical valuec≥σ, where
cinf
h∈Γmaxu∈Q Jhu, 2.7
andΓ {h∈CQ, H|h|∂Qid}, whereiddenotes the identity operator.
Lemma 2.2Saddle point theorem. LetHbe a real Banach space,HH1
H2,whereH1/{0}
and is finite-dimensional. Suppose thatJ∈C1H,Rsatisfies the P-S condition and the following.
F3There exist constantsσ,ρ >0such thatJ|∂Bρ∩H1 ≤σ.
F4There ise∈Bρ∩H1and a constantω > σsuch thatJ|eH2≥ω.
ThenJpossesses a critical valuec≥ω, where
cinf
h∈Γu∈maxBρ∩H1
Jhu, 2.8
andΓ {h∈CBρ∩H1, H|h|∂Bρ∩H1id}, whereiddenotes the identity operator.
Lemma 2.3Clark theorem. LetH be a real Banach space,J ∈ C1H,R,with J being even,
bounded from below, and satisfying P-S condition. SupposeJθ 0, there is a setK⊂Hsuch that
Kis homeomorphic toSj−1(j−1dimension unit sphere) by an odd map, andsup
KJ <0. ThenJhas
3. Nonexistence of Nontrivial Solutions
In this section, we give a result of nonexistence of nontrivial solutions to BVP1.5.
Theorem 3.1. Suppose that matrixMis negative semidefinite and forn1,2, . . . , k,
zfn, z>0, forz /0. 3.1
Then BVP1.5has no nontrivial solutions.
Proof. Assume, for the sake of contradiction, that BVP1.5has a nontrivial solution. ThenJ
has a nonzero critical pointu∗. Since
Ju Mu−fu, 3.2
we get
fu∗, u∗ Mu∗, u∗≤0. 3.3
On the other hand, it follows from3.1that
fu∗, u∗k
n1
u∗nfn, u∗n>0. 3.4
This contradicts with3.3and hence the proof is complete.
In the existing literature, results on the nonexistence of solutions of discrete boundary value problems are scarce. HenceTheorem 3.1complements existing ones.
4. Existence of Solutions
Theorem 3.1 gives a set of sufficient conditions on the nonexistence of solutions of BVP 1.5. In this section, with part of the conditions being violated, we establish the existence of solutions of BVP1.5by distinguishing three cases:f is superlinear,fis sublinear, andf is Lipschitzian.
4.1. The Superlinear Case
In this subsection, we need the following conditions. P1For anyn, z∈Z1, k×R,
z
0fn, sds≥0, and
z
0fn, sdso|z|2, asz → 0.
P2There exist constantsa1>0, a2>0 andβ >2 such that
z
0
P3MatrixMexists at least one positive eigenvalue.
P4fn, zis odd for the second variablez, namely,
fn,−z −fn, z, ∀n, z∈Z1, k×R. 4.2
Theorem 4.1. Suppose thatfn, zsatisfiesP2. Then BVP1.5possesses at least one solution.
Proof. For anyu u1, u2, . . . , ukT ∈Rk, we have
Fu k
j1
uj
0
fj, sds≥a1
⎛ ⎝k
j1
ujβ
⎞ ⎠−a2k
≥a1
⎛
⎝k2−β/βk
j1
uj2
⎞ ⎠
β/2
−a2ka1k2−β/2uβ−a2k.
4.3
LetA1a1k2−β/2, A2a2k. We have, for anyu u1, u2, . . . , ukT ∈Rk,
Fu≥A1uβ−A2. 4.4
Since matrixMis symmetric, its all eigenvalues are real. We denote byλ1, λ2, . . . , λkits
eigenvalues. Setλmaxmax{|λ1|,|λ2|, . . . ,|λk|}. Thus for anyu u1, u2, . . . , ukT ∈Rk,
Ju 1
2Mu, u−Fu
≤ 1
2λmaxu
2−A
1uβA2
−→ −∞ asu −→ ∞.
4.5
The above inequality means that −Ju is coercive. By the continuity of Ju, J attains its maximum at some point, and we denote it by u, that is, Ju cmax, where cmax
supu∈RkJu. Clearly,uis a critical point ofJ. This completes the proof ofTheorem 4.1. Theorem 4.2. Suppose thatfn, zsatisfies the assumptionsP1,P2,andP3. Then BVP1.5
possesses at least two nontrivial solutions.
To proveTheorem 4.2, we need the following lemma.
Proof. Assume that{un} ⊂Rkis a P-S sequence. Then there exists a constantc1such that for
Due toβ >2, the above inequality means{un}is bounded. SinceRkis a finite-dimensional Hilbert space, there must exist a subsequence of{un}which is convergent inRk. Therefore, P-S condition is satisfied.
2 E, thenRkhas the following decomposition of direct sum:
RkX
1
By assumptionP1, there exists a constantρ >0, such that for anyn∈Z1, k,z∈Bρ,
That is to say,Jsatisfies assumptionF1of Linking theorem.
Letu∈Rkbe a critical point corresponding to the critical valuecofJ, that is,Ju c. Clearly,u /0 sincec >0. On the other hand, byTheorem 4.1,Jhas a critical pointusatisfying
Ju supu∈RkJu ≥ c. Ifu /u, thenTheorem 4.2holds. Otherwise,u u. Then cmax Ju Ju c, which is the same as supu∈RkJu infh∈Γsupu∈QJhu.
Choosinghid, we have supu∈QJu cmax. Because the choice ofe∈∂B1∩X2∈Q
BR0∩X1
{re|
0< r < R0}is arbitrary, we can take−e∈∂B1∩X2. Similarly, there exists a
positive numberR1> ρ, for anyu∈∂Q1, Ju≤0, whereQ1 BR1∩X1
{−re|0< r < R1}.
Again, by the Linking theorem,Jpossesses a critical valuec0≥σ >0, where
c0 inf
h∈Γ1
max
u∈Q1
Jhu, Γ1
h∈CQ1,Rk|h| ∂Q1id
. 4.16
If c0/cmax, then the proof is complete. Otherwise c0 cmax,supu∈Q1Ju cmax. Because J|∂Q≤0 andJ|∂Q1 ≤0, thenJattains its maximum at some point in the interior of setsQand Q1. ButQ∩Q1⊂X1, andJu≤0 foru∈X1. Thus there is a critical pointu∈Rksatisfying
u /u,Ju c0cmax.
The proof ofTheorem 4.2is now complete.
Theorem 4.4. Suppose thatfn, zsatisfies the assumptionsP1,P2,P3,andP4. Then BVP
1.5possesses at leastl distinct pairs of nontrivial solutions, wherel is the dimension of the space spanned by the eigenvectors corresponding to the positive eigenvalues ofM.
Proof. From the proof ofTheorem 4.2, it is easy to know thatJ is bounded from above and
satisfies the P-S condition. It is clear thatJ is even andJ0 0, and we should find a setK and an odd map such thatKis homeomorphic toSl−1by an odd map.
We takeK ∂Bρ∩X2,whereρandX2 are defined as in the proof ofTheorem 4.2. It
is clear thatKis homeomorphic toSl−1l−1 dimension unit sphereby an odd map. With
4.13, we get supK−J < 0. Thus all the conditions ofLemma 2.3are satisfied, andJ has at leastldistinct pairs of nonzero critical points. Consequently, BVP1.5possesses at leastl distinct pairs nontrivial solutions. The proof ofTheorem 4.4is complete.
4.2. The Sublinear Case
In this subsection, we will consider the case where f is sublinear. First, we assume the following.
P5There exist constantsa1>0, a2>0, R >0 and 1< α <2 such that
Fu≤a1uαa2, ∀u u1, u2, . . . , ukT ∈Rk, u ≥R. 4.17
The first result is as follows.
Theorem 4.5. Suppose thatP5is satisfied and that matrix M is positive definite. Then BVP1.5
Proof. The proof will be finished when the existence of one critical point of functional J defined as in2.3is proved.
Assume that matrixMis positive definite. We denote byλ1, λ2, . . . , λkits eigenvalues,
where 0< λ1≤λ2≤ · · · ≤λk. Then for anyu u1, u2, . . . , ukT ∈Rk,u ≥R, followed
byP5we have
Ju 1
2Mu, u−Fu
≤λ1u2−a1uα−a2
−→∞ asu −→ ∞.
4.18
By the continuity ofJ on Rk, the above inequality means that there exists a lower bound of values of functionalJ. Classical calculus shows thatJ attains its minimal value at some point, and then there existusuch thatJu min{Ju|u∈Rk}. Clearly,uis a critical point of the functionalJ.
Corollary 4.6. Suppose that matrix M is positive definite, and fn, z satisfies that there exist constantsa1>0,a2>0and1< α <2such that
z
0
fn, sds≤a1|z|αa2, ∀n, z∈Z1, k×R. 4.19
Then BVP1.5possesses at least one solution.
Corollary 4.7. Suppose that matrix M is positive definite, andfn, zsatisfies the following.
P6There exists a constantt0 >0such that for anyn, z∈Z1, k×R,|fn, z|≤t0.
Then BVP1.5possesses at least one solution.
Proof. Assume that matrix M is positive definite. In this case, for any u
u1, u2, . . . , ukT ∈Rk,
|Fu| ≤k j1
uj
0
fj, sds ≤
k
j1 t0u
j≤t0 ku. 4.20
Since the rest of the proof is similar toTheorem 4.5, we do not repeat them here.
WhenMis neither positive definite nor negative definite, we now assume thatMis nonsingular, and we have the following result.
Theorem 4.8. Suppose thatMis nonsingular,fn, zsatisfiesP6. Then BVP1.5possesses at
least one solution.
Proof. We may assume that M is neither positive definite nor negative definite. Let
0< λ1 ≤λ2 ≤ · · · ≤λmandlmk. For anyj ∈Z−l,−1∪Z1, m, letξjbe an eigenvector
ofMcorresponding to the eigenvalueλj,j −l, −l1, . . . ,−1,1,2, . . . , m, such that
ξi, ξj
⎧ ⎨ ⎩
0, asi /j,
1, asij, 4.21
LetX1andX2be subspaces ofRkdefined as follows:
X1span{ξi, i∈Z1, m}, X2span
ξj, j ∈Z−l,−1. 4.22
ThenRkhas the following decomposition of direct sum:
RkX
1
X2. 4.23
LetJube defined as in2.3. ThenJ ∈C1Rk,R. By4.20,
|Fu| ≤t0 ku, ∀u∈Rk. 4.24
Suppose that{un} ⊂Rkis a P-S sequence. Then there exists a constantt1such that for
anyn∈Z1,|Jun| ≤t1andJun → 0 asn → ∞. Thus, for sufficiently largenand
for anyu∈Rk, we have|Jun, u| ≤ u.
Letunxnyn∈X1X2. We have, by2.6, for anyu u1, u2, . . . , ukT ∈
Rk,
Jun, uMun, u−k
j1
fj, unj·uj. 4.25
Thus for sufficiently largen, we get
Mun, xn≤k
j1
fj, unj·xnjxn
≤t0
k
j1
xnjxn
≤t0 k1xn,
Mun, xnMxn, xn≥λ
1xn 2
.
Thus,
λ1xn 2
≤t0 k1xn, 4.27
which implies that{xn}is bounded.
Now we are going to prove that{yn}is also bounded. By4.25,
Mun, yn≥k
j1
fj, unj·ynj−yn
≥ −t0 k1yn,
Mun, ynMyn, yn≤λ
−1yn 2
.
4.28
Thus we have
λ−1yn 2
≥ −t0 k1yn. 4.29
And so
−λ−1yn 2
−t0 k1yn≤0. 4.30
Due to λ−1 < 0,{yn} is bounded. Then {un} is bounded. Since Rk is a
finite-dimensional Hilbert space, there must exist a subsequence of{un}which is convergent in
Rk. Therefore, P-S condition is satisfied.
In order to apply the saddle point theorem to prove the conclusion, we consider the functional−Jand verify the conditions ofLemma 2.2.
For anyy∈X2,y y1, y2, . . . , ykT, we have
−Jy−1
2
My, yFy
≥ −1
2λ−1y
2−t
0 ky
≥ 1
2λ−1
t0 k
2 .
4.31
For anyx∈X1,x x1, x2, . . . , xkT,
So far we have verified all the assumptions ofLemma 2.2and hence−Jhas at least a critical point inRk. This completes the proof.
Consider the following special case
Δ4un−2 fn, un, n∈Z1, k,
It can be verified thatMis positive definite, then we have the following corollaries. Corollary 4.9. Suppose that there exist constantsa1>0, a2>0and1< α <2such that
z
0
fn, sds≤a1|z|αa2, ∀n, z∈Z1, k×R. 4.35
Then BVP4.33possesses at least one solution.
4.3. The Lipschitz Case
In this subsection, we suppose the following.
P7Assume that there exist positive constantsL,Ksuch that for anyn, z∈Z1, k×
R,
fn, z≤L|z|K. 4.36
Whenfn, zis Lipschitzian in the second variablez, namely, there exists a constant
L >0 such that for anyn∈Z1, k,z1,z2∈R,
fn, z1−fn, z2≤L|z1−z2|, 4.37
then condition4.36is satisfied.
Theorem 4.11. Suppose thatP7is satisfied and M is nonsingular. IfL < λmin min{λ1,−λ−1},
where λ1 and λ−1 are the minimal positive eigenvalue and maximal negative eigenvalue of M,
respectively, then BVP1.5possesses at least one solution.
Proof. Assume that{un} ⊂ Rkis a P-S sequence. ThenJun → 0 asn → ∞. Thus for
sufficiently largen, we getJun ≤1. SinceJun Mun−fun, then for sufficiently largen,
Mun≤fun1. 4.38
In view of4.36, we have
fun2k
j1
f2j, unj≤k
j1
LunjK2
L2k
j1
unj22LKk
j1
unjK2k.
4.39
It follows, by using the inequality√abc≤√a√b√cfora≥0, b≥0, c≥0 and H ¨older’s inequality, that
fun≤Lun 2LK1/2k1/4un1/2Kk1/2. 4.40
By a similar argument to the proof ofTheorem 4.8, we can decompose Rk into the following form of direct sum:
RkX
1
whereX1andX2can be referred to4.22. Thusuncan be expressed by
the P-S condition is verified.
Now we are going to check conditionsF3andF4for functional−J. In fact, by4.36,
for some positive constantω.
For anyx∈X1,x x1, x2, . . . , xkT, we have
Finally, we consider the special case thatfn, zis independent of the second variable
z; that is,fn, z≡gnfor anyn, z∈Z1, k×R, the BVP1.1becomes
Δ2pn−1Δ2un−2 ΔqnΔun−1gn, n∈Z1, k,
u−1 u0 0uk1 uk2.
4.48
As inSection 2, we reduce the existence of solutions of BVP4.48to the existence of critical points of a functionalJ1defined onRkas follows:
J1u 1
2Mu, u−G, u, ∀u u1, u2, . . . , uk
T∈Rk, 4.49
where Mis defined as in2.4, andG g1, g2, . . . , gkT. Then we can see that the critical point ofJ1is just the solution to the following system of linear algebraic equations:
Mu−G0. 4.50
By using the theory of linear algebra, we have the next necessary and sufficient conditions.
Theorem 4.12. i BVP4.48has at least one solution if and only ifrM rM, G, where
rMdenotes the rank of matrixMandM, Gis the augmented matrix defined as follows:
Acknowledgment
This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction no. 20071078001.
References
1 A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundary value problems,”
Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986.
2 C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,”
Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988.
3 R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,”Differential and Integral Equations, vol. 2, no. 1, pp. 91–110, 1989.
4 C. P. Gupta, “Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems,”Applicable Analysis, vol. 36, no. 3-4, pp. 157–169, 1990.
5 R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,”Applicable Analysis, vol. 59, no. 1–4, pp. 225–231, 1995.
6 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,”Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357–368, 2002.
7 Q. Yao, “Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 5-6, pp. 1570– 1580, 2008.
8 M. Ruyun, Z. Jihui, and F. Shengmao, “The method of lower and upper solutions for fourth-order two-point boundary value problems,”Journal of Mathematical Analysis and Applications, vol. 215, no. 2, pp. 415–422, 1997.
9 F. Li, Q. Zhang, and Z. Liang, “Existence and multiplicity of solutions of a kind of fourth-order boundary value problem,”Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 5, pp. 803– 816, 2005.
10 G. Han and Z. Xu, “Multiple solutions of some nonlinear fourth-order beam equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3646–3656, 2008.
11 X. Cai, J. Yu, and Z. Guo, “Existence of periodic solutions for fourth-order difference equations,”
Computers & Mathematics with Applications, vol. 50, no. 1-2, pp. 49–55, 2005.
12 R. P. Agarwal, S. R. Grace, and J. V. Manojlovic, “On the oscillatory properties of certain fourth order nonlinear difference equations,”Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 930–956, 2006.
13 E. Thandapani and I. M. Arockiasamy, “Fourth-order nonlinear oscillations of difference equations,”
Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 357–368, 2001.
14 E. Schmeidel and B. Szmanda, “Oscillatory and asymptotic behavior of certain difference equation,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4731–4742, 2001.
15 Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,”Applied Mathematics and Computation, vol. 161, no. 1, pp. 139–148, 2005.
16 B. Zhang, L. Kong, Y. Sun, and X. Deng, “Existence of positive solutions for BVPs of fourth-order difference equations,”Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 583–591, 2002. 17 J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,”
Journal of Differential Equations, vol. 231, no. 1, pp. 18–31, 2006.
18 L. Jiang and Z. Zhou, “Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations,”Advances in Difference Equations, vol. 2008, Article ID 345916, 10 pages, 2008. 19 L. Jiang and Z. Zhou, “Multiple nontrivial solutions for a class of higher dimensional discrete
boundary value problems,”Applied Mathematics and Computation, vol. 203, no. 1, pp. 30–38, 2008. 20 D. Bai and Y. Xu, “Nontrivial solutions of boundary value problems of second-order difference
equations,”Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 297–302, 2007. 21 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
23 P. H. Rabinowitz,Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 ofCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
