Discrete
Oscillation
Theory
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c
2005 Hindawi Publishing Corporation
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Preface xi
1. Oscillation theory for second-order linear difference equations 1
1.1. Introduction 1
1.2. Basic concepts 3
1.3. Riccati-type transformations 11
1.4. Reid’s roundabout theorem and Sturmian theory 15
1.4.1. The disconjugacy characterization theorem 15
1.4.2. Sturmian theory 16
1.5. Disconjugacy and disfocality 17
1.5.1. Dominant and recessive solutions 23
1.5.2. More on disconjugacy characterization 26
1.5.3. Disfocality 31
1.6. Conjugacy criteria 33
1.7. Methods of linear discrete oscillation theory 40
1.7.1. Riccati technique 40
1.7.2. Variational principle 40
1.8. Oscillation and nonoscillation criteria 41
1.9. Comparison theorems 69
1.10. Oscillation, nonoscillation, and monotone solutions 71
1.11. Oscillation criteria by Riccati technique 75
1.12. Oscillation criteria by averaging techniques 87
1.13. Further criteria by averaging techniques 102
1.14. Oscillation criteria for linear damped difference equations 117 1.14.1. Oscillation of a reducible damped equation 118 1.14.2. Oscillation criteria of summation averaging type 121 1.14.3. Oscillation criteria of nonsummation type 127 1.15. Oscillation and nonoscillation criteria for forced equations 131
1.15.1. Implicit-type results 133
1.15.2. Some explicit criteria 137
1.16. Some qualitative properties of solutions 141
1.16.1. Some properties of dominant and recessive solutions 141
1.16.2. Reciprocity principle 148
1.16.3. Behavioral properties of dominant and recessive solutions148
1.16.4. Further classification of solutions 156
1.16.5. Comparison theorems 161
1.17. More on nonoscillation criteria 163
1.18. Limit point results 168
1.19.1. Growth of solutions 173
1.19.2. Oscillation criteria 175
1.20. Mixed difference equations 176
1.21. Notes and general discussions 180
2. Oscillation theory for systems
of difference equations 183
2.1. Introduction 183
2.2. Discrete variational problems 185
2.2.1. Problem and notation 185
2.2.2. Inhomogeneous discrete quadratic functionals 186 2.2.3. Solution of discrete variational problems 189
2.3. Linear Hamiltonian difference systems 191
2.3.1. Notation and assumptions 191
2.3.2. Special solutions of the Hamiltonian system 194
2.3.3. Riccati matrix difference equations 198
2.4. The discrete Picone formula 203
2.4.1. Discrete quadratic functionals 203
2.4.2. The proof of the discrete Picone identity 207
2.5. Disconjugacy and controllability 212
2.5.1. Focal points 212
2.5.2. Generalized zeros 215
2.5.3. Controllability 220
2.6. Positivity of discrete quadratic functionals 226
2.6.1. The Reid roundabout theorem 226
2.6.2. Sturm’s separation and comparison theorems 227
2.6.3. The extended Reid roundabout theorem 231
2.6.4. Separated boundary conditions 235
2.6.5. Solvability of discrete Riccati equations 239
2.7. Sturm-Liouville difference equations 244
2.7.1. Disconjugacy 244
2.7.2. The Reid roundabout theorem 252
2.8. Moore-Penrose inverses 257
2.9. Notes and general discussions 263
3. Oscillation theory for half-linear
difference equations 265
3.1. Introduction 265
3.2. Preliminaries and basic concepts 266
3.3. Reid’s roundabout theorem and Sturmian theory 273
3.3.1. The disconjugacy characterization theorem 273
3.3.2. Sturmian theory 282
3.4. Nonoscillation and conjugacy criteria 283
3.4.1. Nonoscillation theorems based on the Riccati technique 283 3.4.2. Nonoscillation theorems based on the variational principle294
3.5. Oscillation criteria 304 3.5.1. Oscillation criteria based on the Riccati technique 304 3.5.2. Further oscillation criteria based on the Riccati technique 319 3.5.3. Oscillation criteria based on the variational principle 336
3.6. Comparison theorems 342
3.6.1. Some results from algebra 342
3.6.2. Some comparison theorems 343
3.6.3. Comparison theorem for generalized Riccati difference
equations 347
3.7. More on oscillation, nonoscillation, and positive solutions 347
3.7.1. Preparatory lemmas 348
3.7.2. Sufficient conditions 350
3.7.3. Necessary conditions 353
3.8. Strong oscillation 361
3.8.1. Strong oscillation, nonoscillation, and conditional
oscillation 361
3.8.2. Generalized Euler difference equation 364
3.9. Half-linear difference equations with damping term 365 3.10. Half-linear difference equations with forcing term 372
3.11. Notes and general discussions 375
4. Oscillation theory for nonlinear
difference equations I 377
4.1. Oscillation criteria 377
4.2. Superlinear oscillation 391
4.3. Sublinear oscillation 399
4.4. Oscillation characterizations 412
4.5. Oscillation of damped nonlinear difference equations 426 4.6. Asymptotic behavior for nonlinear difference equations 435 4.7. Oscillation criteria via Liapunov’s second method 443
4.8. Notes and general discussions 451
5. Oscillation theory for nonlinear
difference equations II 453
5.1. Oscillation criteria 453
5.1.1. Oscillation criteria—(I1) 454
5.1.2. Superlinear oscillation 463
5.1.3. Sublinear oscillation 474
5.1.4. Oscillation criteria—(I2) 479
5.1.5. Oscillation criteria for damped equations 486 5.2. Monotone solutions of nonlinear difference equations 488 5.2.1. Existence of positive monotone solutions of (5.2.2) 488 5.2.2. Existence of monotone solutions of (5.2.1) 491
5.3. Bounded, unbounded, and monotone properties 492
5.3.1. Positive decreasing solutions 492
5.4. Notes and general discussions 514
6. Oscillation theory for difference equations with deviating arguments 515
6.1. Oscillation criteria (I) 516
6.1.1. Oscillation of equation (6.1.1) 516
6.1.2. Comparison theorems 541
6.1.3. Oscillation of equation (6.1.2) 545
6.2. Oscillation criteria (II) 553
6.2.1. Oscillation of equation (6.2.1) 554
6.2.2. Oscillation of equation (6.2.2) 565
6.2.3. Asymptotic behavior 572
6.2.4. Decaying nonoscillatory solutions of equation (6.2.130) 577
6.3. Oscillation via characteristic equations 584
6.3.1. Positive solutions of equation (6.3.1) 585
6.3.2. Oscillation of unbounded solutions of equation (6.3.2) 589
6.4. Oscillation of damped difference equations 594
6.4.1. Almost oscillation 600
6.5. Oscillation of forced difference equations 615
6.6. Notes and general discussions 624
7. Oscillation theory for neutral
difference equations 627
7.1. Oscillation criteria via comparison 627
7.1.1. Linearization theorems 635
7.1.2. Comparison results for forced neutral equations 643
7.2. Nonoscillation criteria 648
7.3. Existence of nonoscillatory solutions 656
7.4. Classification of nonoscillatory solutions 667
7.4.1. Nonoscillatory solutions of equation (7.4.1+) 668
7.4.2. Nonoscillatory solutions of equation (7.4.1−) 674
7.4.3. Oscillation and asymptotic behavior 678
7.5. Further oscillation criteria 680
7.6. Oscillation criteria for mixed neutral difference equations 692 7.6.1. Oscillation of equations with constant coefficients 693 7.6.2. Oscillation of equations with periodic coefficients 707
7.6.3. Oscillation via characteristic equations 717
7.7. Notes and general discussions 726
8. Stability and oscillation theory for differential equations with piecewise
constant arguments 727
8.1. Oscillation of linear first-order delay equations 727
8.1.1. Oscillation criteria 728
8.1.2. First-order equations of alternately retarded and advanced
type 743
8.1.3. Characteristic equations 748
8.2.1. Existence of solutions 752
8.2.2. The caseω=0 756
8.2.3. The caseω=0 760
8.2.4. Second-order equations of alternately retarded and
advanced type 766
8.2.5. Neutral differential equations with piecewise constant
argument 770
8.3. Systems of alternately retarded and advanced type 776
8.3.1. The caseA=0 777
8.3.2. The caseAnonsingular 781
8.4. Applications 784
8.5. Notes and general discussions 787
9. Miscellaneous topics 791
9.1. Generalized characteristic equations 791
9.2. Difference equations with periodic coefficients 804
9.2.1. Oscillation of equation (9.2.1) 805
9.2.2. Delay difference equations with positive and negative
coefficients 820
9.2.3. Oscillation of equation (9.2.3) 825
9.2.4. Oscillation of equation (9.2.4) 832
9.3. Linearized oscillations for difference equations 840 9.3.1. Linearized oscillations for equation (9.3.1) 840 9.3.2. Linearized oscillations for equation (9.3.2) 843
9.4. Oscillation of recursive sequences 849
9.4.1. Oscillation ofx(k+ 1)=ax(k)/[1 +bx(k−τ)] 850 9.4.2. Oscillation ofx(k+ 1)=[a+bx(k)]/[c+x(k−τ)] 854 9.4.3. Oscillation of the discrete delay logistic equation 863 9.4.4. Oscillation of certain recursive sequences 865 9.5. Global behavior of certain difference equations 871
9.5.1. Global attractivity in a nonlinear second-order difference
equation 874
9.5.2. Global asymptotic stability of a second-order difference
equation 880
9.5.3. A rational recursive sequence 883
9.5.4. On the recursive sequencex(k+ 1)=α+ (x(k−1)/x(k)) 889 9.5.5. Global stability of a certain recursive sequence 894 9.5.6. Global attractivity in a nonlinear delay difference equation 899 9.5.7. On the recursive sequencex(k+ 1)=ax(k) + (b+cx(k−
1))e−x(k) 905
9.5.8. Global attractivity in a differential equation with piecewise
constant arguments 914
9.9. Notes and general discussions 942
Bibliography 947
The theory of difference equations, the methods used in their solutions, and their wide applications have advanced beyond their adolescent stage to occupy a cen-tral position in applicable analysis. In fact, in the last ten years, the proliferation of the subject is witnessed by numerous research articles and several monographs, annual international conferences, and new journals devoted to the study of dif-ference equations and their applications. Now even those experts who believe in the universality of differential equations are discovering the sometimes striking divergence between the continuous and the discrete.
This book is devoted to a rapidly developing branch of the qualitative theory of difference equations with or without delays. It presents the theory of oscillation of difference equations, exhibiting classical as well as very recent results in that area. Mainly second-order difference equations are treated, but results on first-order and higher-first-order equations and systems of difference equations are included as well.
There are nine chapters in this book, among which the first one is by far the longest one.Chapter 1features an in-depth presentation of oscillation theory for second-order linear difference equations. Generalized zeros and the concepts of disconjugacy, disfocality, and oscillation are introduced, and a characterization of disconjugacy in terms of a discrete quadratic functional and a discrete Ric-cati equation is given. We present oscillation and nonoscillation criteria as well as comparison theorems, and discuss the main tools used in deriving those crite-ria, namely, the variational principle and the Riccati technique. A discrete version of the well-known Sturmian theory is presented, and the concepts of recessive and dominant solutions are discussed.
Another way of extending the theory of linear second-order difference equa-tions is presented inChapter 3, namely, a generalization to second-order half-linear difference equations (the equations in Chapter 1are special cases of the equations inChapter 3). Again the concepts of generalized zeros and oscillation are introduced and many analogs of the results fromChapter 1, among them nu-merous oscillation and nonoscillation criteria, are offered inChapter 3. Most of the results on discrete half-linear equations are due to Pavel ˇReh´ak.
Next, Chapters4and5feature an oscillation theory for second-order non-linear difference equations. Many necessary and sufficient conditions for oscilla-tion of superlinear and sublinear difference equations are provided. We also derive some oscillation criteria for nonlinear difference equations via Liapunov’s second method. Damped nonlinear difference equations are considered as well.Chapter 5 also presents a discussion and classification of positive, bounded, unbounded, in-creasing, and decreasing solutions of discrete nonlinear equations of second order. An oscillation theory for difference equations with deviating arguments is presented inChapter 6. We establish oscillation criteria for certain linear, non-linear, and half-linear difference equations with deviating (both advanced and re-tarded) arguments. Oscillation criteria for linear difference equations with devi-ating arguments via their characteristic equations are given, and many criteria for the oscillation and almost oscillation of linear and nonlinear damped and forced difference equations with deviating arguments are offered.
Chapter 7consists of an oscillation theory for neutral difference equations. We establish many oscillation criteria for discrete linear and nonlinear neutral second-order equations with and without forcing term. Nonoscillation results for neutral equations with positive and negative coefficients are presented. We also introduce a classification scheme for nonoscillatory solutions of neutral difference equations. Oscillation criteria for neutral equations of mixed type with constant coefficients as well as periodic coefficients are offered, and some of those criteria are derived via associated characteristic equations.
Next,Chapter 8features an oscillation theory for differential equations with piecewise constant arguments. Such equations have applications in control theory and certain biomedical models. Many criteria for oscillation, nonoscillation, and stability of first- and second-order differential equations with piecewise constant arguments are developed. We also discuss second-order equations and systems of alternately advanced and retarded type as well as neutral differential equations with piecewise constant arguments. As an application to the techniques presented, necessary and sufficient conditions for all positive solutions of the logistic equation with quadratic nonlinearity and piecewise constant arguments to oscillate about its positive equilibrium are obtained.
global asymptotic stability of various types of recursive sequences is presented. Results on oscillation of second-order nonlinear difference equations with con-tinuous variables are given. Oscillation for systems of delay difference equations is studied. Finally, oscillatory behavior of linear functional equations of second order is discussed.
This book is addressed to a wide audience of specialists such as mathemati-cians, physicists, engineers, and biologists. It can be used as a textbook at the grad-uate level and as a reference book for several disciplines. The presented theory is illustrated with 121 examples throughout the book. Each chapter concludes with a section that is devoted to notes and bibliographical and historical remarks.
Finally, we wish to express our thanks to the staffofHindawi Publishing Cor-porationfor their excellent cooperation during the preparation of this book for publication.
Ravi P. Agarwal, Martin Bohner, Said R. Grace, and Donal O’Regan
1
Oscillation theory for second-order
linear difference equations
1.1. Introduction
In this chapter, we discuss systematically oscillatory and nonoscillatory and vari-ous other qualitative properties of solutions of second-order linear difference equations. Since the classical paper of Sturm [258] in which he discussed some qualitative properties of second-order linear differential equations of the type
c(t)x(t)+q(t)x(t)=0, (1.1.1)
wherec,q∈C([t0,∞),R) andc(t)>0 fort≥t0, and from that time, thousands
of papers have been published concerning this equation. It is a well-known fact that there is a striking similarity between the qualitative theories of differential equations and difference equations. Moreover, it turns out that it makes sense to study qualitative properties of difference equations (inspite of the fact that they are actually recurrence relations). Therefore, in the last years, a considerable ef-fort has been made to investigate qualitative properties (including very important oscillation theory) of the discrete counterpart of the above equation, namely, the second-order linear difference equation
∆c(k)∆x(k)+q(k)x(k+ 1)=0 (1.1.2)
or
∆c(k−1)∆x(k−1)+q(k)x(k)=0, (1.1.3)
of the three-term recurrence relation
c(k+ 1)x(k+ 2) +p(k)x(k+ 1) +c(k)x(k)=0, (1.1.4)
wherep(k)=q(k)−c(k+ 1)−c(k) (or
c(k)x(k+ 1) +b(k)x(k) +c(k−1)x(k−1)=0, (1.1.5)
whereb(k)=q(k)−c(k)−c(k−1)). We also note that the behavioral properties of equations (1.1.2)–(1.1.5) are the same and hence we will have occasions to discuss each of these equations in appropriate places in the text.
Section 1.2reviews relevant material and basic concepts of second-order lin-ear difference equations. InSection 1.3the important concepts of Riccati transfor-mation and Riccati equation are introduced.Section 1.4contains the statements of some important theorems, namely, the Picone identity, the disconjugacy char-acterization theorem, and Sturm-type comparison and separation theorems. In Section 1.5, some criteria for disconjugacy as well as necessary and sufficient con-ditions for disconjugacy and disfocality of second-order linear difference equa-tions are obtained. Several sufficient conditions for second-order linear difference equations to be conjugate onZare given inSection 1.6. In Sections1.7–1.11, we present several types of oscillation and nonoscillation results for second-order lin-ear difference equations. Sections1.12–1.14are devoted to the study of the oscilla-tion and nonoscillaoscilla-tion of second-order linear difference equations by employing weighted average techniques.Section 1.15contains several results on oscillation and nonoscillation of second-order forced linear difference equations. InSection 1.16, we present some qualitative properties, namely, boundedness, monotonic-ity, and zero convergent solutions of second-order difference equations. InSection 1.16we will also provide some necessary and sufficient conditions for the nonoscil-lation as well as some comparison results. Sufficiency criteria for the nonoscilla-tion are given inSection 1.17. Limit point results are discussed inSection 1.18. In Section 1.19we will investigate the growth of solutions as well as some oscillation criteria, and inSection 1.20we will consider mixed-type difference equations and relate them with usual difference equations used in the literature.
Now we list some notation used in this book:∆is the usualforward difference operator, that is,
∆x(k)=x(k+ 1)−x(k), ∆2x(k)=∆∆x(k), (1.1.6)
the most frequently applied rules:
∆u(k) +v(k)=∆u(k) +∆v(k), ∆αu(k)=α∆u(k) forα∈R, ∆u(k)v(k)=v(k+ 1)∆u(k) +u(k)∆v(k)=v(k)∆u(k) +u(k+ 1)∆v(k),
∆u(k)
v(k)
=v(k)∆u(k)−u(k)∆v(k)
v(k)v(k+ 1) ,
u(k) +v(k)=u(k) +v(k), βu(k)=βu(k) forβ∈R,
u(k)∆v(k)=u(k)v(k)−v(k+ 1)∆u(k),
k−1
j=m
u(j)=v(k)−v(m)
(1.1.7)
forv(k) such that∆v(k)=u(k).
1.2. Basic concepts
In this section we will introduce the second-order self-adjoint difference equation, which is the main topic of this chapter. We will show which second-order linear difference equations can be put in self-adjoint form and establish a number of useful identities.
Consider the second-order linear difference equation
∆c(k−1)∆x(k−1)+q(k)x(k)=0, k∈N0= {0, 1, 2, 3,. . .}, (1.2.1)
where ∆is the forward difference operator, that is, ∆x(k) = x(k+ 1)−x(k), ∆2x(k)=∆(∆x(k)), andc(k),q(k) are real numbers withc(k)>0 for allk∈N
0.
Equation (1.2.1) can be viewed as a discrete version of the second-order linear self-adjoint differential equation (1.1.1), wherec(t)>0 in [a,b] andc,qare con-tinuous on [a,b].
First, we will describe the process of discretization of equation (1.1.1). For smallh=(b−a)/n,n∈N= {1, 2, 3,. . .},
x(t) x(t)−x(t−h)
h ,
c(t)x(t)1
h
c(t+h)x(t+h)−x(t)
h −
c(t)x(t)−x(t−h)
h ,
thus
c(t)x(t) 1
h2
c(t+h)x(t+h)−c(t+h) +c(t)x(t) +c(t)x(t−h). (1.2.3)
Lett=a+kh, wherekis a discrete variable taking on the integer values 0≤k≤n. Ifxis a solution of equation (1.1.1), then we have
ca+ (k+ 1)hxa+ (k+ 1)h−ca+ (k+ 1)h+c(a+kh)x(a+kh) +c(a+kh)xa+ (k−1)h+h2q(a+kh)x(a+kh)0.
(1.2.4)
If we setc(k−1)=c(a+kh),q(k)=h2q(a+kh) for 1≤k≤nand 1≤k≤n−1,
respectively, then
c(k)x(k+ 1)−c(k) +c(k−1)x(k) +c(k−1)x(k−1) +q(k)x(k)0.
(1.2.5)
Finally, we can write this equation in the form (1.2.1).
The linear second-order self-adjoint difference equation is then defined to be the equation (1.2.1), where we assume thatc(k) is defined and positive on the set of integers [a,b+ 1]= {a,a+ 1,. . .,b+ 1}andq(k) is defined on the set of integers [a+ 1,b+ 1].
We can also write equation (1.2.1) in the form
c(k)x(k+ 1) +p(k)x(k) +c(k−1)x(k−1)=0, (1.2.6)
where
p(k)=q(k)−c(k)−c(k−1) (1.2.7)
fork∈[a+ 1,b+ 1].
Since equation (1.2.1) is in fact a recurrence relation, given real initial values
x(a) andx(a+ 1) for somea ∈ N, it is clear that we can inductively obtain the valuesx(a+ 2),x(a+ 3),. . . .Hereby, the existence and uniqueness of a solution of the initial value problem (1.2.1),x(m)=A,x(m+ 1) =Bis guaranteed. Thus, a sequence of real numbersx= {x(k)}∞
k=1is called a solution of equation (1.2.1), if
Next, we will show that any equation of the form
u(k)x(k+ 1) +v(k)x(k) +w(k)x(k−1)=0, (1.2.8)
whereu(k)>0 on [a,b+ 1] andw(k)>0 on [a+ 1,b+ 1] can be written in the self-adjoint form of equation (1.2.1). To see this, multiply both sides of equation (1.2.8) by a positive functiong(k) (to be chosen later) to obtain
u(k)g(k)x(k+ 1) +v(k)g(k)x(k) +w(k)g(k)x(k−1)=0. (1.2.9)
This would be of the form of equation (1.2.6), which we know we can write in self-adjoint form, providedu(k)g(k)=c(k) andw(k)g(k)=c(k−1). Consequently, we select a functiong(k)>0 so thatu(k)g(k)=w(k+ 1)g(k+ 1), or
g(k+ 1)= u(k)
w(k+ 1)g(k) (1.2.10)
fork∈[a,b]. Then,
g(k)=A
k−1
n=a
u(n)
w(n+ 1), (1.2.11)
whereAis a positive constant. If we choose
c(k)=Au(k)
k−1
n=a
u(n)
w(n+ 1) (1.2.12)
and takeq(k)=v(k)g(k) +c(k) +c(k−1), then we have that equation (1.2.8) is equivalent to equation (1.2.1).
Letx(k) andy(k) be solutions of equation (1.2.1) in [a,b+ 2]. We define the
CasoratianorWronskianofxandyby
W(k)=Wx(k),y(k)=det
x(k) y(k)
x(k+ 1) y(k+ 1)
=det
x(k) y(k) ∆x(k) ∆y(k)
.
Define a linear operatorLon the set of functionsxdefined on [a,b+ 2] by
Lx(k)=∆c(k−1)∆x(k−1)+q(k)x(k) fork∈[a+ 1,b+ 1]. (1.2.14)
Theorem 1.2.1 (Lagrange’s identity). Ifxandyare defined on[a,b+ 2], then for
k∈[a+ 1,b+ 1],
y(k)Lx(k)−x(k)Ly(k)=∆c(k−1)Wy(k−1),x(k−1). (1.2.15)
Proof. We have
y(k)Lx(k)=y(k)∆c(k−1)∆x(k−1)+y(k)q(k)x(k)
=∆y(k−1)c(k−1)∆x(k−1)
−∆y(k−1)c(k−1)∆x(k−1) +y(k)q(k)x(k)
=∆y(k−1)c(k−1)∆x(k−1)−x(k−1)c(k−1)∆y(k−1) +x(k)∆c(k−1)∆y(k−1)+x(k)q(k)y(k)
=∆c(k−1)Wy(k−1),x(k−1)+x(k)Ly(k)
(1.2.16)
fork∈[a+ 1,b+ 1].
By summing both sides of Lagrange’s identity froma+ 1 tob+ 1, we obtain the following corollary.
Corollary 1.2.2 (Green’s theorem). Assume thatxandyare defined on[a,b+ 2]. Then
b+1
k=a+1
y(k)Lx(k)−
b+1
k=a+1
x(k)Ly(k)=c(k)Wy(k),x(k)ba+1. (1.2.17)
Corollary 1.2.3 (Liouville’s formula). Ifxandyare solutions of equation (1.2.1), thenW[x(k),y(k)]=d/c(k)fork∈[a,b+ 1], wheredis a constant.
Theorem 1.2.4 (P ´olya’s factorization). Assumeyis a solution of equation (1.2.1) with y(k) > 0on[a,b+ 2]. Then there exist functionsg andhwithg(k) > 0on
[a,b+ 2]andh(k) > 0on[a+ 1,b+ 2]such that for any functionx defined on
[a,b+ 2], fork∈[a+ 1,b+ 1],
Lx(k)=g(k)∆h(k)∆g(k−1)x(k−1). (1.2.18)
Proof. Since yis a positive solution of equation (1.2.1), we have by the La-grange identity that
Lx(k)= 1
y(k)∆
c(k−1)Wy(k−1),x(k−1) fork∈[a+ 1,b+ 1].
(1.2.19)
Now we have
∆x(k−1)
y(k−1)
= y(k−1)∆x(k−1)−x(k−1)∆y(k−1)
y(k−1)y(k)
=W
y(k−1),x(k−1)
y(k−1)y(k) .
(1.2.20)
Thus, it follows that
Lx(k)= 1
y(k)∆
c(k−1)y(k−1)y(k)∆
x(k−1)
y(k−1)
. (1.2.21)
Finally, to complete the proof, in (1.2.21) we letg(k)=1/ y(k)>0 fork∈[a,b+2] andh(k)=c(k−1)y(k−1)y(k)>0 fork∈[a+ 1,b+ 2].
Definition 1.2.5. TheCauchy functionX: [a,b+ 2]×[a+ 1,b+ 1]→Ris defined as the function that for each fixedm∈[a+ 1,b+ 1] is the solution of the initial value problem (1.2.1),X(m,m)=0,X(m+ 1,m)=1/c(m).
Theorem 1.2.6. If yand zare linearly independent solutions of (1.2.1), then the Cauchy function for (1.2.1) is given by
X(k,m)= det
y(m)z(m)
y(k) z(k)
dety(ym(m+1)) z(zm(m+1)) , k∈[a,b+ 2], m∈[a+ 1,b+ 1]. (1.2.22)
Proof. Since yandzare linearly independent solutions of equation (1.2.1),
W[y(k),z(k)] = 0 fork ∈ [a,b+ 1]. Hence, equation (1.2.22) is well defined. Note that by expandingX(k,m) in (1.2.22) by the second row in the numerator we have that for each fixedm∈[a+ 1,b+ 1],X(k,m) is a linear combination of
y(k) andz(k) and so it is a solution of equation (1.2.1). Clearly,X(m,m)=0 and
The next theorem shows how the Cauchy function is used to solve initial value problems.
Theorem 1.2.7 (variation of constants formula). The solution of the initial value problem
Lx(k)=e(k) fork∈[a+ 1,b+ 1],
x(a)=0, x(a+ 1)=0 (1.2.23)
is given by
x(k)=
k
m=a+1
X(k,m)e(m) fork∈[a,b+ 2], (1.2.24)
whereX(k,m)is the Cauchy function forLx(k)= 0. (Here, ifk =b+ 2, then the termX(b+ 2,b+ 2)e(b+ 2)is understood to be zero.)
Proof. Letxbe given by equation (1.2.24). By conventionx(a)=0. Also,
x(a+ 1)=X(a+ 1,a+ 1)e(a+ 1)=0,
x(a+ 2)=X(a+ 2,a+ 1)e(a+ 1) +X(a+ 2,a+ 2)e(a+ 2)
=e(a+ 1)
c(a+ 1).
(1.2.25)
It follows thatxsatisfiesLx(k)=e(k) fork=a+ 1. Now, assumea+ 2≤k≤b+ 1. Then, we have
Lx(k)=c(k−1)x(k−1) +p(k)x(k) +c(k)x(k+ 1)
=
k−1
m=a+1
c(k−1)X(k−1,m)e(m) +
k
m=a+1
p(k)X(k,m)e(m)
+
k+1
m=a+1
c(k)X(k+ 1,m)e(m)
=
k−1
m=a+1
LX(k,m)e(m) +p(k)X(k,k)e(k)
+c(k)X(k+ 1,k)e(k) +c(k)X(k+ 1,k+ 1)e(k+ 1)
=e(k).
(1.2.26)
Corollary 1.2.8. The solution of the initial value problem
Lx(k)=e(k) fork∈[a+ 1,b+ 1],
x(a)=A, x(a+ 1)=B (1.2.27)
is given by
x(k)=y(k) +
k
m=a+1
X(k,m)e(m), (1.2.28)
whereX(k,m)is the Cauchy function forLx(k) = 0, and yis the solution of the initial value problemLy(k)=0,y(a)=A,y(a+ 1)=B.
Proof. Sinceyis a solution ofLy(k)=0 andkm=a+1X(k,m)e(m) is a solu-tion ofLx(k)=e(k), we have that
x(k)=y(k) +
k
m=a+1
X(k,m)e(m) (1.2.29)
solvesLx(k)=e(k). Also,x(a)=y(a)=Aandx(a+ 1)=y(a+ 1)=B. Next, instead of the zero of a solution, known for the continuous case, it is necessary to use the following more complicated concept in the corresponding discrete case.
First, we present the following simple lemma which shows that there is no nontrivial solution of equation (1.2.1) withx(m)=0 andx(m−1)x(m+ 1)≥0,
m > a. In some sense this lemma says that nontrivial solutions of equation (1.2.1) can have only “simple” zeros.
Lemma 1.2.9. Ifxis a nontrivial solution of (1.2.1) withx(m)=0,a < m < b+ 2, thenx(m−1)x(m+ 1)<0.
Proof. Sincexis a solution of (1.2.1) withx(m)=0,a < m < b+2, we obtain from (1.2.6),c(m)x(m+ 1)= −c(m−1)x(m−1). Sincex(m+ 1),x(m−1)=0 andc(k)>0, it follows thatx(m−1)x(m+ 1)<0. Because ofLemma 1.2.9we can define a generalized zero of a solution of equa-tion (1.2.1) as follows
Definition 1.2.10. A solutionx of (1.2.1) is said to have ageneralized zeroatm
providedx(m)=0 ifm=a, and ifm > aeitherx(m)=0 orx(m−1)x(m)<0.
Definition 1.2.11. Equation (1.2.1) is calleddisconjugate on [a,b+ 1] provided no nontrivial solution of equation (1.2.1) has two or more generalized zeros on [a,b+ 2]. Otherwise, equation (1.2.1) is said to be conjugate on [a,b+ 2].
Example 1.2.12. The difference equation
x(k+ 1) + 2x(k) + 2x(k−1)=0, (1.2.30) which can be put in self-adjoint form, has
x(k)=2k/2sin
3πk
4
, y(k)=2k/2cos
3πk
4
(1.2.31)
as linearly independent solutions. Note that both of these solutions have a gener-alized zero atk=2.
Example 1.2.13. The difference equation
x(k+ 1)−√3x(k) +x(k−1)=0 (1.2.32) is disconjugate on any interval of length less than six. This follows from the fact that any solution of this equation is of the formx(k)=Asin((πk/6) +B), where
AandBare appropriate constants.
We also see that the difference equation
x(k+ 2)−7x(k+ 1) + 12x(k)=0 (1.2.33) is disconjugate on any interval.
Next, we introduce a discrete quadratic functional.
Definition 1.2.14. (i) Define a classU=U(a,b) of so-called admissible sequences by
U(a,b)=ξ: [a,b+ 2] →R:ξ(a)=ξ(b+ 1)=0. (1.2.34) (ii) Define thequadratic functionalF onU(a,b) by
F(ξ;a,b)=
b
k=a
c(k)∆ξ(k)2−q(k)ξ(k+ 1)2. (1.2.35)
(iii) Say thatFispositive definiteonUprovidedF(ξ)≥0 for allξ∈U(a,b), andF(ξ)=0 if and only ifξ=0.
The concept of oscillation and nonoscillation of equation (1.2.1) is defined in the following way.
Oscillation of equation (1.2.1) may be equivalently defined as follows. A non-trivial solution of equation (1.2.1) is called oscillatory if it has infinitely many gen-eralized zeros. In view of the fact that the Sturm-type separation theorem holds (see Theorem 1.4.4below), we have the following equivalence: any solution of equation (1.2.1) is oscillatory if and only if every solution of equation (1.2.1) is oscillatory. Hence, we can speak about oscillation and nonoscillation of equation (1.2.1).
1.3. Riccati-type transformations
One of the approaches to oscillation and nonoscillation for equation (1.2.1) will be based largely on a discrete version of theRiccati equation. Ifxis a solution of equation (1.2.1) withx(k)x(k+ 1)>0 fork≥m≥a, we let
w(k)=c(k−1)∆x(k−1)
x(k−1) =c(k−1)
x(k)
x(k−1)−1
fork≥a+ 1. (1.3.1)
Then since
w(k) +c(k−1)=c(k−1)
x(k)
x(k−1)
>0, (1.3.2)
we have
∆w(k)= 1
x(k)∆
c(k−1)∆x(k−1)+c(k−1)∆x(k−1)∆ 1
x(k−1)
= −q(k) +w(k)x(k−1)
1
x(k)− 1
x(k−1)
= −q(k) +w(k)
x(k−1)
x(k) −1
= −q(k) +w(k)
c(k−1)
w(k) +c(k−1)−1
= −q(k)− w
2(k)
w(k) +c(k−1).
(1.3.3)
Hence,
Rw(k)=∆w(k) + w
2(k)
w(k) +c(k−1)+q(k)=0 fork≥a+ 1. (1.3.4) Similarly, we have
Rw(k)=∆w(k) + w(k)
c(k−1)
It should be noted that there are several other discrete versions of the Riccati tion corresponding to equation (1.2.6). Of particular interest, we consider equa-tion (1.2.6) withb(k)= −p(k), that is,
c(k)x(k+ 1) +c(k−1)x(k−1)=b(k)x(k) fork≥a+ 1≥1, (1.3.6) withc(k) > 0,k ≥ a. In fact, equation (1.3.6) is equivalent to the self-adjoint equation
−∆c(k−1)∆x(k−1)+q(k)x(k)=0 fork≥a+ 1, (1.3.7)
whereq(k)=b(k)−c(k)−c(k−1), that is,q(k) in equation (1.2.1) is replaced by
−q(k).
Next, we present the following properties for the solutions of equation (1.3.7).
Lemma 1.3.1. If equation (1.3.7) is such that any nontrivial solutionxcan have at most one valuex()=0with > a+ 1, then any two valuesx(m),x(n)withm=n, uniquely determine the solutionx.
Proof. Letuandvbe the solutions defined byu(a)=0,u(a+ 1)=1 and
v(a)=1,v(a+ 1)=0. Thenuandvare linearly independent. Consider the system of equations
x(m)=α1u(m) +α2v(m),
x(n)=α1u(n) +α2v(n). (1.3.8)
If this system does not have a unique solution in terms ofα1andα2, then there
exist values ofα1andα2, not both zero, such that
0=α1u(m) +α2v(m)=α1u(n) +α2v(n). (1.3.9)
However, this implies that the solutionα1u(k) +α2v(k) assumes the value 0 twice,
which is a contradiction.
Lemma 1.3.2. Suppose|b(k)| ≥ c(k−1) +c(k). Ifvis a solution of (1.3.6) such that|v(N+ 1)| ≥ |v(N)|for some integerN≥a+ 1, then|v(k+ 1)| ≥ |v(k)|for all
k≥N. If there exists a sequence{η(k)}of nonnegative numbers such that
b(k)≥
1 +η(k)c(k) +c(k−1),
k
η(k)= ∞, (1.3.10)
Proof. Use induction and assume that it is true for somek≥N+ 1, that is,
|v(k)| ≥ |v(k−1)|. Then for allk≥N,
v(k+ 1)= 1
c(k)b(k)v(k)−c(k−1)v(k−1)
≥ 1
c(k)b(k)−c(k−1)v(k).
(1.3.11)
If in addition we have (1.3.10), then (1.3.11) becomes|v(k+1)| ≥[1+η(k)]|v(k)|. However, assumingk≥N, the above inequality implies that
v(k+ 1)≥v(N)
j
1 +η(j) forN≤ j≤k, (1.3.12)
which means|v(k)| → ∞ask→ ∞. This completes the proof.
Remark 1.3.3. (i) We always assumec(k)>0,k ≥a ≥0. If we have in addition
b(k) ≥ c(k) +c(k−1), then the solutionv of (1.3.6) satisfying v(a) = 1 and
v(a+ 1) =1 must havev(k+ 1) ≥v(k)≥1. Clearly,vis nonoscillatory, so that the conditionb(k)−c(k)−c(k−1)≥0 is sufficient for nonoscillation of equation (1.3.6).
(ii)Lemma 1.3.2 implies that the hypothesis of Lemma 1.3.1is satisfied if
|b(k)| ≥c(k) +c(k−1).
(iii)Lemma 1.3.2is sharp in case of constant coefficients. Assumeb(k)≡ b
andc(k) ≡ 1 for allk. By solving such a difference equation with constant co-efficients, we see that all solutions of equation (1.3.6) are bounded if and only if
|b|<2.
The following result is elementary but useful.
Lemma 1.3.4. If there exists a subsequenceb(kj)≤0, wherekj→ ∞asj→ ∞, then
equation (1.3.6) is oscillatory.
Proof. Suppose not. Then, we may assume the existence of a solutionxsuch thatx(k)>0 for all sufficiently largek. However, the left-hand side of equation (1.3.6) will always be positive, while the right-hand side will be nonpositive for all
values ofkj, which is a contradiction.
Because ofLemma 1.3.4, in addition to the assumption thatc(k)> 0 for all
k≥a≥0, we assume the following. In equation (1.3.6),b(k)>0 fork≥a+ 1. It would suffice to assumeb(k)>0 for all sufficiently largek, but we assume the condition for allk, for simplicity in the results we will present.
Suppose thatx(k),k ≥ a ≥ 0, is a solution of equation (1.3.6) such that
x(k)=0 fork≥Nfor someN≥a. The substitutionr(k)=x(k+ 1)/x(k),k≥N, leads to the nonlinear difference equation
c(k)r(k) +c(k−1)
Similarly, if we letz(k)=c(k)x(k+ 1)/x(k) forn≥N, thenzsatisfies
z(k) +c
2(k−1)
z(k−1) =b(k) forn > N. (1.3.14) If we lets(k)=b(k+ 1)x(k+ 1)/(c(k)x(k)) forn≥N, thenssatisfies
g(k)s(k) + 1
s(k−1)=1 forn > N, (1.3.15) whereg(k)=c2(k)/(b(k)b(k+ 1)).
We note that equation (1.3.14) may be written in the alternative form
∆z(k−1) + 1
b(k)z(k)z(k−1)−z(k) +
c2(k−1)
b(k) =0 forn > N. (1.3.16) The transformationz(k)=c(k)x(k−1)/x(k), which leads to equation (1.3.14), is perhaps the nearest analogue for difference equations to the classical Riccati transformationz(t)=c(t)x(t)/x(t), which transforms the self-adjoint differential equation (1.1.1) into the Riccati equation
z(t) + 1
c(t)z
2(t) +q(t)=0. (1.3.17)
Finally, we prove the following theorem.
Theorem 1.3.5. The following conditions are equivalent.
(i) Equation (1.3.6) is nonoscillatory.
(ii) Equation (1.3.13) has a positive solutionr(k),k≥Nfor someN≥a >0.
(iii) Equation (1.3.14) has a positive solutionz(k),k≥Nfor someN≥a >0.
(iv) Equation (1.3.15) has a positive solutions(k),k≥Nfor someN≥a >0. Proof. If equation (1.3.6) is nonoscillatory andx(k),k≥0 is any solution of equation (1.3.6), then there existsN ≥a ≥0 such thatx(k)x(k−1)>0 for all
k≥N. The necessity conditions (ii), (iii), and (iv) then follow immediately from the transformation which leads to equations (1.3.13), (1.3.14), and (1.3.15).
Conversely, ifr(k),k ≥ N is a positive solution of equation (1.3.13), then we may let x(N) = 1 and x(k−1) = r(k)x(k) for all n ≥ N. This defines a positive solution of equation (1.3.6) forn≥N. Givenx(N) andx(N+1), the terms
1.4. Reid’s roundabout theorem and Sturmian theory
We will consider second-order difference operators of the form
L1x(k)=∆
c1(k)∆x(k)
+q1(k)x(k+ 1), (1.4.1)
L2y(k)=∆
c2(k)∆y(k)
+q2(k)y(k+ 1), (1.4.2)
wherek ∈ [a,b] witha,b ∈ Z,a ≤ b, andqi(k), i ∈ {1, 2}, are real-valued
sequences defined on [a,b]. The sequencesci(k),i∈ {1, 2}, are positive real valued
and defined on [a,b+ 1].
1.4.1. The disconjugacy characterization theorem
Now, we will formulate the discrete version of the so-called Picone identity [235]. Also, we will state some important results, for example, the disconjugacy charac-terization theorem, the Sturm-type separation theorem, and the Sturm-type com-parison theorem. The proofs of these results are postponed to Chapters2and3, where more general results for systems of difference equations and for half-linear difference equations (which include second-order linear difference equations as a special case) are established.
Lemma 1.4.1 (Picone’s identity). Letxandybe defined on[a,b+ 2]and assume
y(k)=0fork∈[a,b+ 1]. Then fork∈[a,b], the following equality holds:
∆x(k)
z(k)
y(k)c1(k)∆x(k)−x(k)c2(k)∆y(k)
=q2(k)−q1(k)
x2(k+ 1) +c
1(k)−c2(k)
∆x(k)2 +x(k+ 1)
y(k+ 1)
y(k+ 1)L1x(k)−x(k+ 1)L2y(k)
+
c2(k)
∆x(k)2−c2(k)∆y(k)
y(k+ 1) x
2(k+ 1) +c2(k)∆y(k)
y(k) x
2(k)
.
(1.4.3)
Next, we consider equation (1.2.1), which we may write in the form
∆c(k)∆x(k)+q(k)x(k+ 1)=0 fork∈[a,b], (1.4.4)
Now, we state the following disconjugacy characterization theorem, also known as Reid’s roundabout theorem.
Theorem 1.4.2 (Reid’s roundabout theorem). All of the following statements are equivalent.
(i) Equation (1.4.4) is disconjugate on[a,b].
(ii) Equation (1.4.4) has a solutionx(k)without generalized zeros in the in-terval[a,b+ 1].
(iii) The Riccati difference equation associated with (1.4.4), namely, Rw(k)=∆w(k) + w
2(k)
w(k) +c(k)+q(k)=0, (1.4.5)
wherew(k) =c(k)∆x(k)/x(k), has a solutionw(k)on[a,b]satisfying
c(k) +w(k)>0on[a,b].
(iv) The functionalFdefined by (1.2.35) is positive definite onU(a,b).
1.4.2. Sturmian theory
This subsection is devoted to Sturmian theory. Consider two equations
L1x(k)=0, L2y(k)=0, (1.4.6)
where the operatorsL1andL2are defined by (1.4.1) and (1.4.2), respectively.
Now, we state the following Sturmian theorems.
Theorem 1.4.3 (Sturm’s comparison theorem). Suppose thatc2(k)≥c1(k)and
q1(k)≥q2(k)fork∈[a,b]. IfL1x(k)=0is disconjugate on[a,b], thenL2y(k)=0 is also disconjugate on[a,b].
Theorem 1.4.4 (Sturm’s separation theorem). Two linearly independent solutions of equation (1.2.1) cannot have a common zero. If a nontrivial solution of equation (1.2.1) has a zero att1and a generalized zero att2> t1, then any second linearly inde-pendent solution has a generalized zero in(t1,t2]. If a nontrivial solution of equation (1.2.1) has a generalized zero att1and a generalized zero att2> t1, then any second linearly independent solution has a generalized zero in[t1,t2].
The next example shows that, under the definition of a generalized zero in the sense ofDefinition 1.2.10, the Sturm separation theorem does not hold for all second-order linear homogeneous difference equations.
Example 1.4.5. Consider theFibonacci difference equation
x(k+ 1)−x(k)−x(k−1)=0. (1.4.7) The characteristic equation isλ2−λ−1=0. Hence the characteristic values are
(1±√5)/2. Takex(k)=[(1−√5)/2]kandy(k)=[(1 +√5)/2]k. Note thatx(k)
Theorem 1.4.4do not hold for equation (1.4.7), at least not with our current def-inition of generalized zeros as given inDefinition 1.2.10. This of course does not contradictTheorem 1.4.4because the self-adjoint form of equation (1.4.7) is
∆(−1)k−1∆x(k−1)+ (−1)k−1x(k)=0, (1.4.8) wherec(k)=(−1)k>0 for allk≥a≥0.
However, in Chapters2and3we extend the definition of generalized zeros to equations of the form (1.2.1) withc(k)=0 for allk∈Z, that is,cwill be allowed to change sign. With such an extended definition, a situation as in the case of the Fibonacci sequence may not occur. Note thatcin (1.4.8) is in fact changing sign.
Remark 1.4.6. InTheorem 1.4.4it was noted that two linearly independent solu-tions cannot have a common zero. In fact,Example 1.2.13shows that this is not true for generalized zeros as defined inDefinition 1.2.10.
1.5. Disconjugacy and disfocality
In this section we will show that disconjugacy is important in obtaining compari-son results for solutions of initial value problems and an existence and uniqueness result for solutions of boundary value problems. Also, we will establish some nec-essary and sufficient conditions for disconjugacy and disfocality of second-order difference equations.
UsingTheorem 1.2.7, we can prove the following comparison theorem.
Theorem 1.5.1. Assume that
Lx(k)=∆c(k−1)∆x(k−1)+q(k)x(k)=0 (1.5.1)
is disconjugate on[a,b+ 2]and thatu,vsatisfyLu(k)≥Lv(k)fork∈[a+ 1,b+ 1],
u(a)=v(a), andu(a+ 1)=v(a+ 1). Thenu(k)≥v(k)on[a,b+ 2].
Proof. Setw(k)= u(k)−v(k). Thenh(k)=Lw(k)= Lu(k)−Lv(k) ≥0 fork ∈ [a+ 1,b+ 1]. Hencew solves the initial value problemLw(k) = h(k),
w(a)=0,w(a+ 1) =0. By the variation of constants formula, that is,Theorem 1.2.7, we have
w(k)=
k
m=a+1
X(k,m)h(m), (1.5.2)
whereX(k,m) is the Cauchy function for equation (1.5.1). Since equation (1.5.1) is disconjugate andX(m,m)=0,X(m+ 1,m)=1/c(m)>0, we haveX(k,m)>0 form+ 1≤k ≤b+ 2. It follows thatw(k)≥0 on [a,b+ 2], which gives us the
Consider the boundary value problem
∆2x(k−1) + 2x(k)=0,
x(0)=A, x(2)=B, (1.5.3)
whereAandBare real numbers. IfA=B=0, then this boundary value problem has infinitely many solutions. IfA = 0 andB = 0, then it has no solution. We show in the following theorem that with the assumption of disconjugacy this type of boundary value problem has a unique solution.
Theorem 1.5.2. If equation (1.5.1) is disconjugate on[a,b+ 2], then the boundary value problem
Lx(k)=h(k),
xm1
=A, xm2
=B, (1.5.4)
wherea≤m1< m2≤b+ 2andA,Bare constants, has a unique solution.
Proof. Letx1 andx2 be linearly independent solutions of equation (1.5.1)
andxp be a particular solution of equation (1.5.4). Then a general solution of
equation (1.5.4) is x(k) = α1x1(k) +α2x2(k) +xp(k), whereαi,i ∈ {1, 2}, are
arbitrary constants. Now, the boundary conditions lead to the system of equations
α1x1
m1
+α2x2
m1
=A−xp
m1
,
α1x1
m2
+α2x2
m2
=B−xp
m2
. (1.5.5)
This system has a unique solution if and only if
D=det
x1
m1
x2
m1
x1
m2
x2
m2
=0. (1.5.6)
IfD=0, then there are constantsβi,i∈ {1, 2}, not both zero such that the
non-trivial solutionx(k)=β1x1(k) +β2x2(k) satisfiesx(m1)=x(m2)=0. This
con-tradicts the disconjugacy of equation (1.5.1) on [a,b+ 2] and the proof is
com-plete.
The following results characterize the disconjugacy on a certain interval.
Theorem 1.5.3. Equation (1.5.1) is disconjugate on[a,b+ 2]if and only if there is a positive solution of equation (1.5.1) on[a,b+ 2].
Proof. Assume that equation (1.5.1) is disconjugate on [a,b+ 2]. Letuandv
be solutions of equation (1.5.1) satisfyingu(a)=0,u(a+ 1)=1 andv(b+ 1)=1,
v(b+ 2)=0. By disconjugacy,u(k)>0 on [a+ 1,b+ 2] andv(k)>0 on [a,b+ 1]. It follows thatx=u+vis a positive solution of equation (1.5.1).
Conversely, assume that equation (1.5.1) has a positive solution on [a,b+ 2]. It follows from the Sturm separation theorem that no nontrivial solution has two
Corollary 1.5.4. Equation (1.5.1) is disconjugate on[a,b+ 2]if and only if it has a P´olya factorization on[a,b+ 2].
Proof. If equation (1.5.1) is disconjugate, then byTheorem 1.5.3it has a positive solution. ByTheorem 1.2.4, equation (1.5.1) has a P ´olya factorization. Conversely, assume that equation (1.5.1) has the P ´olya factorization
g(k)∆h(k)∆h(k−1)x(k−1)=0, (1.5.7) whereg(k)>0 on [a,b+2] andh(k)>0 on [a+1,b+2]. Thusx=1/gis a positive solution. ByTheorem 1.5.3, equation (1.5.1) is disconjugate on [a,b+ 2].
Now, we denote the determinant of them×m-tridiagonal matrix
p(k) c(k) 0 0 0 · · · 0
c(k) p(k+ 1) c(k+ 1) 0 0 · · · 0 0 c(k+ 1) p(k+ 2) c(k+ 2) 0 · · · 0
..
. ... ... ... ... ... ...
0 0 · · · 0 c(k+m−3) p(k+m−2) c(k+m−2) 0 0 · · · · · · 0 c(k+m−2) p(k+m−1)
(1.5.8)
byMm(k),a+ 1≤k≤b+ 1, 1≤m≤b−k+ 2, wherep(k) is given by equation
(1.2.7).
Theorem 1.5.5. If
(−1)mMm(a+ 1)>0 for1≤m≤b−a+ 1, (1.5.9)
then equation (1.5.1) is disconjugate on[a,b+ 2].
Proof. Letybe the solution of (1.5.1) satisfyingy(a)=0 and y(a+ 1)=1. ByTheorem 1.4.4it suffices to show thaty(k)>0 on [a+ 1,b+ 2]. We will show thaty(a+m)>0 for 1≤m≤b−a+ 2 by induction onm. Form=1, we have
y(a+ 1)=1>0. Assume 1≤m≤b−a+ 2 and thaty(a+m−1)>0. Using
Ly(k)=0,a+ 1≤k≤a+m−1, andy(a)=0, we getm−1 equations
p(a+ 1)y(a+ 1) +c(a+ 1)y(a+ 2)=0,
c(a+ 1)y(a+ 1) +p(a+ 2)y(a+ 2) +c(a+ 2)y(a+ 3)=0, ..
.
c(a+m−3)y(a+m−3)+p(a+m−2)y(a+m−2)+c(a+m−2)y(a+m−1)=0,
c(a+m−2)y(a+m−2)+p(a+m−1)y(a+m−1)+c(a+m−1)y(a+m)=0.
By Cramer’s rule (hereM0(a+ 1)=1),
y(a+m−1)= −c(a+m−1)y(a+m)
Mm−2(a+ 1)
Mm−1(a+ 1)
. (1.5.11)
It follows thaty(a+m)>0, so by inductiony(k)>0 in [a,b+ 2]. Hence, equation
(1.5.1) is disconjugate on [a,b+ 2].
Next, we have the following disconjugacy characterization.
Theorem 1.5.6. Equation (1.5.1) is disconjugate on[a,b+ 2]if and only if the coef-ficients of equation (1.5.1) satisfy
(−1)mMm(k)>0 (1.5.12)
fora+ 1≤k≤b+ 1,1≤m≤b−k+ 2.
Proof. Assume that equation (1.5.1) is disconjugate on [a,b+ 2]. We will show that (1.5.12) holds for 1≤m≤b−a+ 1,a+ 1≤k≤b−m+ 2 by induction onm. Form=1, we now show that
−M1(k)= −p(k)>0 fora+ 1≤k≤b+ 1. (1.5.13)
To this end fixk0 ∈ [a+ 1,b+ 1] and letx be the solution of equation (1.5.1)
satisfyingx(k0−1)=0 andx(k0)=1. SinceLx(k0)=0, we have from equation
(1.2.6) thatc(k0)x(k0+ 1) +p(k0)x(k0)=0, that is,p(k0)= −c(k0)x(k0+ 1). By
the disconjugacy,x(k0+ 1)>0, sop(k0)<0. Sincek0∈[a+ 1,b+ 1] is arbitrary,
p(k)<0 fora+ 1≤k≤b+ 1, and the first step of the induction is complete. Now assume 1≤m≤b−a+ 1 and
(−1)m−1Mm−1(k)>0 (1.5.14)
fora+1≤k≤b−m+3. We will use this induction hypothesis to show that (1.5.12) holds. Fixk1 ∈ [a+ 1,b−m+ 2] and letybe the solution of equation (1.5.1)
satisfyingy(k1−1)=0 andy(k1+m)=1. Using these boundary conditions and
equation (1.5.1) fork1≤k≤k1+m−1, we arrive at the equations
pk1
yk1
+ck1
yk1+ 1
=0,
ck1
yk1
+pk1+ 1
yk1+ 1
+ck1+ 1
yk1+ 2
=0, ..
.
(cy)k1+m−3
+ (py)k1+m−2
+ck1+m−2
yk1+m−1
=0,
ck1+m−2
yk1+m−2
+pk1+m−1
yk1+m−1
= −ck1+m−1
.
