KALHORN, REBECCA ISABEL BURTON On the Solvability of Nonlinear
Boundary Value Problems on Time Scales. (Under the direction of Dr. Jes´us Rodr´ıguez.)
In this manuscript we study boundary value problems on time scales. First we will examine weakly nonlinear boundary value problems of the form
x∆(t) =A(t)x(t) +h(t) +f(t, x(t)), t∈[a, b]T
subject to
B1x(a) +B2x(b) = 0,
and analyze problems at resonance; that is, problems where the homogeneous linear boundary value problem has a nontrivial solution space. We establish conditions for the existence of solutions and discuss the dependence of solutions on parameters.
Next we consider the existence and properties of solutions to nonlinear dynamic equations of the form
x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[a, b]T
on Time Scales
by
Rebecca Isabel Burton Kalhorn
A dissertation submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the Degree of
Doctor of Philosophy
Mathematics
Raleigh, North Carolina 2009
APPROVED BY:
Dr. Jes´us Rodr´ıguez Dr. James Selgrade Chair, Advisory Committee Committee Member
Dr. Kailash Misra Dr. Ernest Stitzinger
Rebecca Isabel Burton Kalhorn, the second eldest of six children, was born in St. Louis, Missouri to physicians Drs. Frank and Mary Burton. While attending Nerinx Hall High School she was active in theater and choir most notably trying on a Scottish brogue as Lundi in Brigadoon and qualifying for state choir auditions twice. During her senior year she met her future husband, Tim Kalhorn. After graduating, in 1999, she went on to Marquette University, where she majored in mathematics and minored in computer science and business. Her initial goal was to pursue a career in actuarial science, but after three actuarial internships and passing one actuarial exam, she decided she wanted to pursue an advanced degree in mathematics instead.
During the summer of 2003 she graduated from Marquette, married Tim, and started her graduate work at North Carolina State University. She earned her Master of Science degree in applied mathematics in 2005. In 2007 she and her husband welcomed a son, Frank, to their family.
I would like to thank Dr. Jes´us Rodr´ıguez for his generous investment of time and mathematical guidance during the last four years. He was always patient, no matter how slow I was that day; gracious, no matter how nonsensical I sounded at the time; and calming, no matter how stressful the week. He has influenced me not only mathematically, but also in the importance of communicating clearly and precisely. I have also enjoyed our many conversations about academia, life, and the adventures of parenting.
I am grateful for the help of Miriam Ansley, Seyma Bennett-Shabbir, Di Bucklad, Brenda Currin, Carolyn Gunton, Nicole Newkirk Dahlke, Marilyn McCollum, Denise Seabrooks, and Charlene Wallace during my time at NC State. I have especially appreciated their friendly faces and supportive nature. I would also like to thank Dr. Kailash Misra, Dr. James Selgrade, Dr. Ernie Stitzinger, and Dr. Heather Cheshire for taking the time to be on my advisory committee. I also appreciate the effort of Dr. John Griggs, Ms. Denise Seabrooks, Dr. Ernie Stitzinger, and Dr. Jes´us Rodr´ıguez in preparing letters of recommendation to be sent out for job applications.
to take the time to share it.
I would like to thank my officemates April Alston and Jordan Bostic for their companionship through this process as well as my officemates from the past, Drew Pasteur, Rebecca Wills, and Laurie Zack. I would also like to thank my mathemat-ical comrades Kristen Abernathy, Zack Abernathy, Mike Allocca, John Brown, Ji-mena Davis, Kelly Dickson, Karen Dillard, Rizwana Rehman, Johnny Samuels, Ryan Siskind, Monique Taylor, Paddy Taylor, and Ryan Therkelsen for their friendship. Whether it was studying for qualifiers, sharing teaching strategies, fielding LaTex questions, venting graduate school frustration, or enjoying some down time you have all been gracious with your time and knowledge.
Finally I would like to thank my family. They have supported me, challenged me, and loved me. I am thankful to my parents for stressing the importance of education and providing me with excellent academic opportunities. I would like to thank my siblings, Ben, Anastasia, Elizabeth, Sam, and Isaac for helping me hone my ability to argue effectively, which was essential in the writing of this thesis.
Chapter 1 Introduction . . . 1
1.1 Time Scales . . . 1
1.2 Boundary Value Problems of Interest . . . 2
Chapter 2 Weakly Nonlinear Two-Point Boundary Value Problems at Resonance 5 2.1 Preliminary . . . 7
2.2 Main Results . . . 17
2.3 Application . . . 20
Chapter 3 Global Nonlinear Boundary Value Problems . . . 24
3.1 Preliminaries . . . 25
3.2 Main Results . . . 28
3.3 Applications . . . 35
Chapter 4 Nonlinear Scalar Two-Point Boundary Value Problems . . . 43
4.1 Preliminaries . . . 46
4.2 Main Results . . . 48
4.3 Periodic Boundary Conditions . . . 56
4.4 Example . . . 59
Introduction
1.1
Time Scales
In 1988 Stefan Hilger introduced the idea of time scales as a way to unify continuous and discrete analysis. A time scale, denoted T, is defined to be any closed subset of the real line. Clearly both R and Z are time scales. Hence examining dynamic equations on time scales allows for the simultaneous study of differential and difference equations, but perhaps more importantly it provides a means of analyzing problems which are inherently different from either differential or difference equations. Please consult the references for basic concepts regarding time scale calculus [3, 5, 6].
An example of a time scale application is a model for the population growth of a particular kind of plant. Suppose that the plant population exhibits exponential growth during the months of April through September, and at the beginning of Octo-ber all the plants die while the seeds remain in the ground. Further, suppose that the seeds begin to grow at the beginning of April the following year at which point there are twice as many plants. We can use the time scale
∞
S
n=0
by t = 2k and t = 2k + 1 respectively. Therefore x0(t) = x(t) for t ∈ [2k,2k + 1) and x(t+ 1) = 2x(t) (which implies that x(t + 1)− x(t) = x(t)) for t = 2k + 1. Since the delta derivative on the integers is just the forward difference, we can write x∆(t) = x(t) for allt in the time scale
∞
S
n=0
[2n,2n+ 1] [3]. There are similar examples concerning insect populations, where the insects die out while the eggs incubate for some length of time before hatching [3, 5].
1.2
Boundary Value Problems of Interest
In this manuscript we study the solvability of several general families of nonlinear boundary value problems on time scales. We begin, in Chapter 2, by studying prob-lems of the form
x∆(t) =A(t)x(t) +h(t) +f(t, x(t)), t∈[a, b]T
subject to
B1x(a) +B2x(b) = 0
where the boundary value problem
x∆(t) = A(t)x(t), t∈[a, b]T
subject to
B1x(a) +B2x(b) = 0
has nontrivial solutions. We establish conditions for the existence of solutions, and, through the use of the Implicit Function Theorem, demonstrate the relationship be-tween these solutions and that of the corresponding linear problem.
In Chapter 3 we study nonlinear dynamic equations subject to generalized global boundary conditions, where the corresponding linear homogeneous boundary value problem has a trivial solution space. We begin by considering systems of the form
x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[a, b]T
subject to the above mentioned global boundary conditions. For problems of this type we present sufficient conditions for the existence of solutions both when the nonlinearity exhibits sublinear growth and when the nonlinearity is Lipschitz. Subse-quently we discuss problems where the nonlinearity is of the perturbation type; that is, problems of the form
x∆(t) =A(t)x(t) +q(t) +λf(t, x(t)), t ∈[a, b]T
For this type of problem we establish conditions for the existence of solutions as well as provide a qualitative analysis of the behavior of solutions as a function of the parameter.
Chapter 4 is devoted to the study of nonlinear scalar two-point boundary value problems where the corresponding linear homogeneous boundary value problem has a one dimensional solution space. Through use of the Lyapunov–Schmidt Procedure conditions are established to guarantee the existence of solutions to the these bound-ary value problems. We pay particular attention to second order equations subject to periodic boundary conditions. Results are obtained which significantly extend previ-ous work by Etheridge and Rodr´ıguez concerning the periodic behavior of nonlinear discrete dynamical systems [10].
Weakly Nonlinear Two-Point Boundary Value
Problems at Resonance
This chapter is devoted to the study of nonlinear boundary value problems on time scales. We consider problems of the form
(2.1) x∆(t) =A(t)x(t) +h(t) +f(t, x(t)), t∈[a, b]T
subject to
(2.2) B1x(a) +B2x(b) = 0
where B1 and B2 are constant n ×n matrices and is a “small” real parameter. Throughout this chapter we assume that T is a time scale where [a, b]T ⊂T; h is an rd-continuous function from T intoRn; A is a regressive rd-continuous n×n matrix
valued function onT; andf is a continuously differentiable function fromT×Rninto
Rn. Crd[a, b]T will denote the space of rd-continuous Rn-valued maps on [a, b]T, and
C[a, b]T will denote the subspace of Crd[a, b]T where the maps are continuous.
those where the boundary value problem
(2.3) x∆(t) = A(t)x(t), t∈[a, b]T
subject to
(2.4) B1x(a) +B2x(b) = 0
has nontrivial solutions. We establish conditions for the existence of solutions, and we provide a qualitative analysis of the dependence of the solution on the parameter .
The literature on time scales is vast. Bohner and Peterson provide a reference for the general theory of dynamic equations on time scales [5]. Comprehensive survey articles, and the references therein, may be helpful for the reader interested in bound-ary value problems on time scales [3, 5]. We provide references for functional analytic methods for nonresonant boundary value problems [14] and for results regarding the existence of periodic solutions [27].
2.1
Preliminary
In order to analyze (2.1)–(2.2) we will use operators defined on the following spaces.
X ={x∈C[a, b]T:B1x(a) +B2x(b) = 0} and Y =Crd[a, b]T.
We will use the supremum norm on the spaces X and Y; that is, for x∈X∪Y
||x||= sup
t∈[a,b]T
|x(t)|
where | · | denotes the Euclidean norm on Rn. With this norm it is clear that X and Y are Banach spaces. We will use the operator norm on matrices, and for v = (v1, v2,· · · , vm) an element of the product space V1×V2× · · · ×Vm, we will use
the norm given by ||v||=
m
P
i=1
We define the linear operator L:D(L)→Y where D(L) =X∩Crd1 [a, b]T by
(Lx)(t) = x∆(t)−A(t)x(t), t ∈[a, b]T,
and the operator F :X →Y by
(F x)(t) = f(t, x(t)), t ∈[a, b]T.
It is evident that x is a solution to (2.1)–(2.2) if and only if Lx = h+F x. Φ will denote the fundamental matrix solution forx∆(t) =A(t)x(t),t∈[a, b]
T that satisfies
Φ(a) =I.
Proposition 2.1. The solution space for the boundary value problem (2.3)–(2.4)and the kernel of B1+B2Φ(b) have the same dimension.
Proof.
x∈ ker(L) ⇐⇒ x∆(t) =A(t)x(t) for all t ∈[a, b]T and B1x(a) +B2x(b) = 0
⇐⇒ There exists a c∈Rn such thatx(t) = Φ(t)c and B1c+B2Φ(b)c= 0
⇐⇒ x(t) = Φ(t)c, where c∈ker(B1+B2Φ(b)).
Throughout the remainder of this chapter we will assume that the dimension of the kernel of L, and hence the dimension of the kernel of B1+B2Φ(b), is m.
Proposition 2.2. The map F : X → Y is continuously Fr´echet differentiable and
DF(φ) :X →Y is given by
(DF(φ))(y)(t) =
∂f
∂x(t, φ(t))
(y(t)).
Proof. First we will show that F is differentiable. Letφ ∈X, and let Γ :X →Y be defined by
(Γy)(t) = ∂f
∂x(t, φ(t))y(t).
Let >0. By the Mean Value Theorem, for each t∈[a, b]T, we have
|f(t, φ(t) +y(t))−f(t, φ(t))− ∂f
∂x(t, φ(t))y(t)|
≤ |y(t)|sup
v∈vt
∂f
∂x(t, v)− ∂f
∂x(t, φ(t))
,
where vt is the line segment between φ(t) and φ(t) + y(t). Since ∂f∂x is uniformly
continuous on compact sets, there exists a δ such that if ||y||< δ, then
|f(t, φ(t) +y(t))−f(t, φ(t))− ∂f
∂x(t, φ(t))y(t)| ≤ ||y||. Therefore
||F(φ+y)−F(φ)−Γy||
and thus F is differentiable with
(DF(φ))(y)(t) =
∂f
∂x(t, φ(t))
y(t).
Now we will show that F is continuously differentiable. Suppose ||φ−ψ||< δ.
||DF(φ)−DF(ψ)||= sup
||h||=1
||DF(φ)h−DF(ψ)h||
= sup
||h||=1
(
sup
t∈[a,b]T
∂f
∂x(t, φ(t))(h(t))− ∂f
∂x(t, ψ(t))(h(t))
)
≤ sup
t∈[a,b]T
∂f
∂x(t, φ(t))− ∂f
∂x(t, ψ(t))
.
The continuity of DF follows from the continuity of ∂f∂x.
Definition 2.1. Define S : [a, b]T →Rn×m by
S(t) = Φ(t)D,
where the columns of D∈Rn×m make up a basis for the kernel of B
1+B2Φ(b).
Observation 1. From Proposition 2.1 we see that x is an element of the kernel of L
if and only if x(t) = S(t)α for some α∈Rm and for all t∈[a, b] T.
In order to use the Lyapunov–Schmidt procedure we now construct projections onto the kernel and image of L.
Definition 2.2. Define P :X →X by
Proposition 2.3. P is a projection onto the kernel of L.
Proof. First we will show thatP is well defined by proving thatS(a)TS(a) is
invert-ible. Assume S(a)TS(a)c= 0 for some c∈
Rm.
cTS(a)TS(a)c= 0 ⇒ (S(a)c)T(S(a)c) = 0
⇒ |S(a)c|2 = 0 ⇒ S(a)c= 0
⇒ Φ(a)(d1c1+· · ·+dmcm) = 0,
where c= (c1, . . . , cm)T and D= (d1,· · · , dm); ⇒ d1c1+· · ·+dmcm = 0
⇒ ci = 0 for alli= 1,· · ·, m.
Therefore S(a)TS(a) is invertible.
Now we show that P2 =P.
(P(P x))(t) =S(t)(S(a)TS(a))−1S(a)T(P x)(a)
=S(t)(S(a)TS(a))−1S(a)TS(a)(S(a)TS(a))−1S(a)Tx(a) = (P x)(t), for all t∈[a, b]T.
Hence P2 =P.
Finally we will show that Im(P) = ker(L). Let x∈X; then
where α= (S(a)TS(a))−1S(a)Tx(a) Therefore Im(P)⊂ ker(L).
Let x∈ ker(L); then there exists aβ ∈Rn that satisfies x(t) = S(t)β, and
(P x)(t) =S(t)(S(a)TS(a))−1S(a)Tx(a) = S(t)(S(a)TS(a))−1S(a)TS(a)β =S(t)β =x(t).
Hence ker(L)⊂ Im(P), and therefore Im(P) = ker(L). Since it is clear that P is linear and bounded we have shown that P is a projection onto the kernel of L.
Definition 2.3. Let K ∈Rn×m such that the columns of K span the kernel of (B
1+ B2Φ(b))T. Define Ψ : [a, b]T→R
n×m by
Ψ(t) = [B2Φ(b)Φ−1(σ(t))]TK, t∈[a, b]T.
Proposition 2.4. y is in the image of L if and only if
b
R
a
yT(τ)Ψ(τ)∆τ = 0.
Proof. By the variation of constant formula [5] and the boundary conditions we have that y∈ Im(L) if and only if there exists x∈X such that
B1x(a) +B2
Φ(b)x(a) +
b
Z
a
Φ(b)Φ−1(σ(τ))y(τ)∆τ
= 0
⇐⇒ B2
b
Z
a
Φ(b)Φ−1(σ(τ))y(τ)∆τ ∈Im(B1+B2Φ(b))
⇐⇒
b
Z
a
B2Φ(b)Φ−1(σ(τ))y(τ)∆τ
T
where the columns of K span ker((B1+B2Φ(b))T)
⇐⇒ b
Z
a
yT(τ)[B2Φ(b)Φ−1(σ(τ))]TK∆τ = 0.
Proposition 2.5. If the n×2n matrix [B1|B2] has rank n, then the columns of Ψ
are linearly independent.
Proof. Assume Ψ(t)c = 0 for some c ∈ Rm. Then if Ψ
i is the i-th column of Ψ
it must be true that
m
P
i=1
Ψi(t)ci = 0, for all t ∈ [a, b]T. But, this equivalent to m
P
i=1
[B2Φ(b)Φ−1(σ(t))]TKici = 0, where Ki is the i-th column of K. From this it is
clear that (Φ−1(σ(t)))TΦ(b)T Pm
i=1 BT
2Kici = 0 which implies m
P
i=1 BT
2Kici = 0.
Suppose Ki ∈ ker(B2T) for some i = 1,· · · , m. By the definition of K, (B1T + ΦT(b)B2T)Ki = 0. This implies that Ki ∈ ker(B1T) since Ki ∈ ker (B2T). Therefore [B1|B2]KiT = 0, and since [B1|B2] has full rank Ki = 0. This is a contradiction since
the m columns of K span an m dimensional space. Hence ci = 0 for i = 1,· · · , m
and the columns of Ψ are linearly independent.
Throughout the rest of this chapter we will assume that [B1|B2] has rank n.
Definition 2.4. Define W :Y →Y by
(W y)(t) = Ψ(t)
b
Z
a
ΨT(τ)Ψ(τ)∆τ
−1 b
Z
a
ΨT(τ)y(τ)∆τ, t∈[a, b]T.
Proposition 2.6. E =I−W is a projection onto the image of L.
b
R
a
Ψ(τ)TΨ(τ)∆τ is invertible. Letα∈
Rm, and assume
b R
a
Ψ(τ)TΨ(τ)∆τ
α= 0. We can write αT
b R
a
Ψ(τ)TΨ(τ)∆τ
α = 0 which implies
b
R
a
|Ψ(τ)α|2∆τ = 0. Therefore Ψ(t)α= 0 for all t∈ [a, b]T. α = 0 since the columns of Ψ are linearly independent, and thus
b
R
a
Ψ(τ)TΨ(τ)∆τ is invertible.
Clearly W is a bounded linear map which implies E is also. If we showW2 =W, then we have shown the same for E.
(W(W y))(t) = W
Ψ(t) b Z a
Ψ(τ)TΨ(τ)∆τ
−1 b
Z
a
ΨT(τ)y(τ)∆τ
= Ψ(t) b Z a
Ψ(τ)TΨ(τ)∆τ
−1 b
Z
a
ΨT(τ)Ψ(τ)∆τ
b
Z
a
|Ψ(η)|2∆η
−1 b
Z
a
ΨT(η)y(η)∆η
= Ψ(t) b Z a
|Ψ(η)|2 ∆η
−1 b
Z
a
ΨT(η)y(η)∆η = (W y)(t),
for all t ∈[a, b]T.
Finally we will show that Im(E) = Im(L). Let y ∈Y. ClearlyEy∈ Im(E).
b
Z
a
ΨT(τ)(Ey)(τ)∆τ =
b
Z
a
ΨT(τ)(y−W y)(τ)∆τ
=
b
Z
a
ΨT(τ)y(τ)∆τ − b
Z
a
ΨT(τ)(W y)(τ)∆τ
=
b
Z
a
ΨT(τ)y(τ)∆τ − b
Z
a
ΨT(τ)Ψ(τ)∆τ
b
Z
a
|Ψ(η)|2∆η
−1 b
Z
a
ΨT(η)y(η)∆η
Hence Ey∈ Im(L) and Im(E)⊂ Im(L). Now let y∈ Im(L). Since
b
R
a
ΨT(τ)y(τ)∆τ = 0 we can write
(Ey)(t) = y(t)−Ψ(t)
b
Z
a
Ψ(τ)TΨ(τ)∆τ
−1 b
Z
a
ΨT(τ)y(τ)∆τ =y(t),
for t∈[a, b]T. Therefore y∈ Im(E) and Im(L)⊂ Im(E).
The characterization of the kernel of L and the image of L used in projections is essentially that which appears in [10, 13, 19, 25]. We provide details for the readers convenience. Utilizing the fact that P and E are projections, we write
X = Im(P)⊕Im(I−P)
and
Y = Im(I−E)⊕Im(E).
Proposition 2.7. The dimension of the image of P is the same as the dimension of the image of I−E.
Proof. Lety∈Y. We write (W y)(t) = Ψ(t)(a1, a2,· · · , am)T whereai ∈Ris the i-th
element of the vector
b R
a
ΨT(τ)Ψ(τ)∆τ
−1 b R
a
ΨT(τ)y(τ)∆τ. This implies that all the
y(t) = Ψ(t)(b1, b2,· · · , bm)T, where bi ∈R for i= 1,· · · , m.
(W y)(t) = Ψ(t)
b
Z
a
Ψ(τ)TΨ(τ)∆τ
−1 b
Z
a
ΨT(τ)y(τ)∆τ
= Ψ(t)(b1, b2,· · · , bm)T =y(t), t ∈[a, b]T.
Thereforey∈Im(W), and hence the dimension of the image ofP equals the dimension of the image of I−E.
The following is clear.
Proposition 2.8. L: Im(I−P)∩D(L)→ Im(L) is a bijection.
Recall forx∈X there exists au∈ker(L) and av ∈Im(I−P) such thatx=u+v. Since L : Im(I −P)∩D(L) → Im(L) is a bijection, there exists a bounded linear map M :Im(L)→ Im(I−P)∩D(L) such that
LM y =y for all y ∈ Im(L) and M Lx=v for all x∈X
Definition 2.5. Define H :R× Im(P)× Im(I−P)→ Im(I −E)× Im(I−P) by
H(, u, v) =
W F(u+v)
v−M h−M EF(u+v)
.
Proposition 2.9. Suppose there exists a solution to the linear boundary value problem
Proof.
Lx=h+F x ⇔ E(Lx−h−F x) = 0 and (I−E)(Lx−h−F x) = 0
⇔ Lx−h−EF x= 0 and W F x= 0
⇔ v−M h−M EF(u+v) = 0 and W F(u+v) = 0
⇔ H(, u, v) = 0.
Proposition 2.10. H is a continuously Fr´echet differentiable map fromR× Im(P)×
Im(I−P)into Im(I−E)× Im(I−P). If (, u, v)∈R×Im(P)× Im(I−P) then for each (α, p, q)∈R× Im(P)× Im(I−P),
DH(, u, v)(α, p, q) =
W DF(u+v)(p+q)
q−αM EF(u+v)−M EDF(u+v)(p+q)
.
The proof of this proposition is standard [9, 25] and will be omitted.
2.2
Main Results
differential and difference equations [19, 22, 25, 26]. In this section we will see how our analysis of boundary value problems on time scales allows us to establish the exis-tence of solutions to problems which are neither completely continuous nor completely discrete.
Theorem 2.11. Assume f is continuously differentiable, [B1|B2] has full rank, and
there exists a solution to (2.1)–(2.2) when = 0. If there exists an αˆ ∈Rm such that b
Z
a
[f(τ, S(τ) ˆα+M h(τ))]TΨ(τ)∆τ = 0
and
b
Z
a
Ψ(τ)T∂f
∂x(τ, S(τ) ˆα+M h(τ))S(τ)∆τ is invertible,
then for eachsmall enough, there exists a solution, x, to the boundary value problem
(2.1)–(2.2). Furthermore, lim
→0||x−S(·) ˆα||= 0.
Proof. F and H are continuously Fr´echet differentiable as a result of f being con-tinuously differentiable. Let ˆu ∈ Im(P) be given by ˆu(t) = S(t) ˆα.
b
R
a
[f(τ,u(τ) +ˆ M h(τ))]TΨ(τ)∆τ = 0 implies that F(ˆu+M h) is in the image of L and therefore H(0,u, M h) = 0.ˆ
∂H
∂(u, v)(0,u, M h) : Im(Pˆ )× Im(I−P)→ Im(I−E)× Im(I −P) is given by
∂H
∂(u, v)(0,u, M h)(zˆ 1, z2) =
W DF(ˆu+M h)(z1+z2) z2
since the mapDH(, u, v) :R×Im(P)×Im(I−P)→Im(I−E)×Im(I−P) is given by
DH(, u, v)(α, p, q) =
W DF(u+v)(p+q)
q−αM EF(u+v)−M EDF(u+v)(p+q)
.
Now we will prove that W DF(ˆu+M h) : Im(P)→ Im(I−E) is a bijection. Let z be an element of the kernel of W DF(ˆu+M h). Therefore,
Ψ(t) b Z a
Ψ(τ)TΨ(τ)∆τ
−1 b
Z
a
ΨT(τ)∂f
∂x(τ,u(τˆ ) +M h(τ))z(τ)∆τ = 0
for all t ∈ [a, b]T. Since the columns of Ψ are linearly independent and since z(t) = S(t)α for some α∈Rm, by virtue of z being an element of Im(P),
b
Z
a
ΨT(τ)∂f
∂x(τ,u(τ) +ˆ M h(τ))S(τ)α∆τ = 0.
Since
b
R
a
Ψ(τ)T ∂f
∂x(τ, S(τ) ˆα+M h(τ))S(τ)∆τ is invertible α must be zero and hence
z = 0.
Let (z1, z2) be an element of the kernel of ∂(∂Hu,v)(0,u, M h). This implies thatˆ
W DF(ˆu+M h)(z1+z2) z2 = 0 0
. Therefore z2 = 0. W DF(ˆu+M h)(z1 +
SinceH(0,u, M h) = 0,ˆ ∂(∂Hu,v)(0,u, M h) is a bijection, andˆ His continuously differ-entiable, by the Implicit Function Theorem for small enough there exists a (u, v)
such thatH(, u, v) = 0. Therefore L(u+v) = h+F(u+v) and thus u+v is
a solution of (2.1)–(2.2). Furthermore, lim
→0||u+v−uˆ−M h||= 0.
Our discussion leading to and including Theorem 1 has the same structure to analogous problems for both discrete and continuous dynamic systems [9,10,26]. Our presentation provides a framework for the analysis of a more general class of problems. In the next section we apply this analysis to the study of a boundary value problem on a time scale which is neither discrete nor continuous.
2.3
Application
We present the following discussion as preparation for the next theorem. We consider the following time scale
T=
1− 1
22n,1−
1 22n+1
:n= 0,1,2,· · ·
∪ {1}.
It is clear that
σ(t) =
1
2(1 +t) = 1− 1
22n+2 when t= 1− 22n1+1, n= 0,1,2,· · ·
and µ(t) = 1
2(1−t) = 1
22n+2 when t= 1−
1
22n+1, n= 0,1,2,· · ·
0 otherwise
.
Now we consider the nonlinear second order scalar equation
(2.5) au∆∆+bu∆+cu=f(u(t))¯
subject to
(2.6)
b1x(0) +b2x0(0) +d1x(1) +d2x0(1) = 0 and
b3x(0) +b4x0(0) +d3x(1) +d4x0(1) = 0
wherebi, di ∈Rfori= 1,2. In order to carry out our analysis we write the boundary
value problem in system form. Let A=
0 1 −c a − b a ,
B1 =
b1 b2 b3 b4
,B2 =
d1 d2 d3 d4
, andx=
x1 x2 = u u∆
, whereAis regressive
and [B1|B2] has full rank. We also letf(x) =
0 ¯ f(x1)
for allt∈T. Now we consider the following system:
subject to
(2.8) B1x(0) +B2x(1) = 0.
Let Φ be the fundamental solution of x∆=Ax that satisfies Φ(0) =I. Suppose
ker((B1+B2Φ(b))T) = span
β1 β2
, and ker(B1+B2Φ(b)) = span
α1 α2 .
So now we have,
Ψ(t) = Ψ1 Ψ2
= [B2Φ(1)Φ−1(σ(t))]T
β1 β2 and S(t) = S1 S2 = Φ(t) α1 α2 .
There will be a solution to (2.7)-(2.8) (and thus (2.5)-(2.6)) if there exists an ˆα ∈R
such that the following are true.
1
Z
0
fT(t, S(t) ˆα)Ψ(t)∆t=
∞
X
i=0 1− 1
22i+1
Z
1− 1 22i
¯
f(S1(t) ˆα)Ψ2(t)∆t
+
∞
X
i=0 1 22i+2f¯
S1
1− 1
22i+1
ˆ α Ψ2
1− 1
22i+1
and
1
Z
0
ΨT(t)∂f
∂x(t, S(t) ˆα)S(t)∆t=
∞
X
i=0 1− 1
22i+1
Z
1− 1 22i
Ψ2(t) df¯ dx1
(S1(t) ˆα)S1(t)∆t
+
∞
X
i=0 1 22i+2Ψ2
1− 1
22i+1
df¯ dx1
S1
1− 1
22i+1
ˆ α S1
1− 1
22i+1
is invertible.
Theorem 2.12. Suppose neither Ψ2 nor S1 change sign on the time scale, f¯is C1
and strictly monotonic, and there exists anM > 0such thatf(s)f(−s)<0whenever
s > M. Then, there exists a solution u0 of (2.5)-(2.6) when = 0 and 0 > 0 such
that for each such that||< 0 (2.5)-(2.6)has a solution u and furthermoreu →u
uniformly as →0.
Proof. It is clear that for there exists α0 such that for α > α0, 1
R
0
fT(t, S(t)α)Ψ(t)∆t
and 1
R
0
fT(t, S(t)(−α))Ψ(t)∆t have different signs. So by continuity there must exist
an ˆα such that 1
R
0
fT(t, S(t) ˆα)Ψ(t)∆t= 0. Since ¯f is strictly monotonic,
∞
X
i=0 1− 1
22i+1
Z
1− 1 22i
Ψ2(t) df¯ dx1
(S1(t) ˆα)S1(t)∆t
+
∞
X
i=0 1 22i+2Ψ2
1− 1
22i+1
df¯ dx1
S1
1− 1
22i+1
ˆ α S1
1− 1
22i+1
6
= 0.
Global Nonlinear Boundary Value Problems
This chapter is devoted to the study of nonlinear boundary value problems on time scales. We consider systems of nonlinear dynamic equations subject to generalized global boundary conditions.
First we consider systems of the form
x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[a, b]T
subject to the above mentioned global boundary conditions. For problems of this type we present sufficient conditions for the existence of solutions. These conditions depend on the rate of growth of the nonlinearity as well as on properties of the corresponding linear homogeneous boundary value problem.
Subsequently we discuss problems where the nonlinearity in the dynamic equation is of the perturbation type; that is, problems of the form
x∆(t) =A(t)x(t) +q(t) +λf(t, x(t)), t ∈[a, b]T
the existence of solutions, and we provide a qualitative analysis of the behavior of solutions as a function of the parameter.
The results presented in this chapter allow a unified treatment of differential and difference equations subject to very general boundary conditions. Much more impor-tant, however, is the fact that these results provide a framework for the analysis of boundary value problems where the dynamic equations are inherently different from either differential or difference equations. We provide relevant references for readers interested in global boundary value problems [1, 2, 7, 20–26].
3.1
Preliminaries
For the basic terminology of time scales, as well as for standard results in the field, the reader may consult the references [3, 5, 6, 14, 27]. Throughout this chapter T will denote an arbitrary time scale for which [a, b]T ⊂Tκ.
If v is an element of Rn, |v| will denote the Euclidean norm of v. If B is a
bounded linear map,||B||will represent the operator norm. C[a, b]T will be the space of continuous,Rn-valued maps defined on [a, b]T, andCrd[a, b]Twill represent the class
of rd-continuous functions from [a, b]T intoRn. The norm on each of these spaces will
be the supremum norm; that is ||x||= sup
t∈[a,b]T
|x(t)|.
map; q : T → Rn is an rd-continuous function; and A :
T → Rn×n is a regressive,
rd-continuous matrix valued function. We will require that f : R×Rn →
Rn be continuous. For each such f we define
F : C[a, b]T → C[a, b]T by (F x)(t) = f(t, x(t)). Verifying that F is continuous is straightforward. Throughout this chapter we will assume that the only solution of
(3.1) x∆(t) = A(t)x(t), t∈[a, b]T
subject to
(3.2) Γx= 0
is the trivial one. Φ will denote the matrix solution for x∆(t) = A(t)x(t), t ∈ [a, b]
T
which satisfies Φ(a) = I. We will write ΓΦ to denote the n ×n matrix whose jth column is ΓΦj, where Φj is the jth column of Φ, that is,
ΓΦ = [ΓΦ1|ΓΦ2| · · · |ΓΦn].
The following lemma is a direct consequence of the fact that the general solution of x∆(t) = A(t)x(t), t∈[a, b]
T is given by x(t) = Φ(t)k wherek is in Rn.
Lemma 3.1. ΓΦ is invertible if and only if the only solution of (3.1)–(3.2) is the trivial one.
and only one x that satisfies
(3.3) x∆(t) =A(t)x(t) +h(t), t∈[a, b]T
subject to
(3.4) Γx= 0.
Moreover x is given by
x(t) = Φ(t)x0+ Φ(t)
Z t
a
Φ−1(σ(s))h(s)∆s
where
x0 =−(ΓΦ)−1Γ[Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s].
Proof. Suppose h is in Crd[a, b]T. Let x(t) = Φ(t)x0+ Φ(t)
Z t
a
Φ−1(σ(s))h(s)∆s. By the variation of constants formula [5], x satisfies x∆(t) = A(t)x(t) +h(t),t ∈[a, b]
T.
Γx= 0 ⇐⇒ Γ(Φ(·)x0) =−Γ
Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s
⇐⇒ (ΓΦ)x0 =−Γ
Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s
⇐⇒ x0 =−(ΓΦ)−1Γ
Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s
Definition 3.1. K:Crd[a, b]T →C[a, b]T is defined by
K(h)(t) = Φ(t)x0+ Φ(t)
Z t
a
Φ−1(σ(s))h(s)∆s
where
x0 =−(ΓΦ)−1Γ[Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s].
Clearly, Kis a bounded linear map, and Proposition 3.2 establishes the fact that for eachhinCrd[a, b]T,Khis the unique solution ofx
∆(t) =A(t)x(t) +h(t),t∈[a, b]
T
where Γx= 0.
3.2
Main Results
We now consider the solvability of the boundary value problem
(3.5) x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[a, b]T
subject to
(3.6) Γx= 0.
It is obvious that solving (3.5)–(3.6) is equivalent to finding a fixed point of the map T :C[a, b]T→C[a, b]T defined by (T x)(t) =K(q+F x)(t),t ∈[a, b]T. It is clear that T is continuous.
that there are constants M1, M2, and a number α∈(0,1)such that
|f(t, x)| ≤M1|x|α+M2 for all (t, x) ∈ T×Rn. Then, the boundary value problem (3.5)–(3.6) has at least one solution.
Proof. We will use Schauder’s Fixed Point Theorem to establish the existence of a fixed point of the operator T. Let Br be the closed ball in C[a, b]T having radius r
and centered at the origin. Suppose h∈Br then
||T h|| ≤ ||K|| · ||q+F h|| ≤ ||K|| ·(||q||+||F h||),
and
||F h||= sup
t∈[a,b]T
|f(t, h(t))| ≤ sup
t∈[a,b]T
{M1|h(t)|α+M2} ≤M1||h||α+M2 ≤M1rα+M2.
Therefore ||T h|| ≤ ||K||(||q||+M1rα+M2). This implies that
||T h||
r ≤ ||K||
M1 r1−α +
M2+||q|| r
. Since lim
r→∞
M1 r1−α +
M2+||q|| r
= 0, it follows that for r sufficiently large T(Br)⊂Br.
In order to prove that T(Br) is relatively compact we now establish that it is
equicontinuous. Let s1, s2 ∈[a, b]T such that s1 < s2.
|(T h)(s2)−(T h)(s1)|=|(KF)(h)(s2)−(KF)(h)(s1)|
≤
Φ(s2)x0+ Φ(s2)
Z s2
a
Φ−1(σ(τ))f(τ, h(τ))∆τ
−Φ(s1)x0−Φ(s1)
Z s1
a
Φ−1(σ(τ))f(τ, h(τ))∆τ
Therefore,
|(T h)(s2)−(T h)(s1)| ≤ ||Φ(s2)−Φ(s1)|||x0| +
Φ(s2)
Z s2
a
Φ−1(σ(τ))f(τ, h(τ))∆τ −Φ(s1)
Z s1
a
Φ−1(σ(τ))f(τ, h(τ))∆τ
. It is clear that there exist real numbers B1 and B2 such that fors ∈[a, b]T,
|Φ−1(σ(s))f(s, h(s))| ≤ sup
t∈[a,b]T
||Φ−1(σ(t))||(M1rα+M1)< B1, and
|x0|=
−(ΓΦ)−1Γ[Φ(·)
Z ·
a
Φ−1(σ(s))h(s)∆s]
≤ ||(ΓΦ)−1||||Γ|| sup
t∈[a,b]T
||Φ(t)||(b−a) sup
t∈[a,b]T
||Φ−1(σ(t))||r ≤B2. It is also evident that
Φ(s2)
Z s2
a
Φ−1(σ(τ))f(τ, h(τ))∆τ −Φ(s1)
Z s1
a
Φ−1(σ(τ))f(τ, h(τ))∆τ
≤ kΦ(s2)−Φ(s1)k
Z s1
a
Φ−1(σ(τ))f(τ, h(τ))∆τ
+||Φ(s2)||
Z s2
s1
Φ−1(σ(τ))f(τ, h(τ))∆τ
≤ |s2−s1| sup
t∈[a,b]T
||Φ∆(t)||
Z s1
a
B1∆τ+||Φ(s2)|||s2−s1|B1
≤ |s2−s1|B1 sup
t∈[a,b]T
||Φ∆(t)||(b−a) + sup
t∈[a,b]T
||Φ(t)||
!
.
Therefore,
|(T h)(s2)−(T h)(s1)|
≤ |s2−s1|
"
sup
t∈[a,b]T
||Φ∆(t)||B2+B1 sup
t∈[a,b]T
||Φ∆(t)||(b−a) + sup
t∈[a,b]T
||Φ(t)||
!#
.
The existence of a fixed point of T is a direct consequence of the Schauder Fixed Point Theorem.
The use of Schauder’s Theorem is well documented. We provide references for readers interested in applications in differential equations as well as other aspects of nonlinear analysis [9, 15]
Theorem 3.4. Suppose that the only solution to (3.1)–(3.2) is the trivial one, f is Lipschitz continuous with Lipschitz constant L and ||K||L < 1 then there exists a unique solution to the boundary value problem (3.5)–(3.6).
Proof. We will use the Contraction Mapping Principle to establish the existence of a unique solution. Let h, w∈C[a, b]T. Then,
|F h(t)−F w(t)|=|f(t, h(t))−f(t, w(t))| ≤L|h(t)−w(t)| ⇒ ||F h−F w|| ≤L||h−w||,
and
||T h−T w||=||K(F h−F w)|| ≤ ||K|| · ||F h−F w|| ≤ ||K||L||h−w||.
Since T clearly mapsC[a, b]T into itself, by the Contraction Mapping Principle there is a unique solution to (3.5)–(3.6).
We now consider boundary value problems of the form
(3.7) x∆(t) =A(t)x(t) +q(t) +λf(t, x(t)), t ∈[a, b]T
subject to
(3.8) Γx= 0
where λ is a “small” parameter and everything else is defined as before.
Define the map G : R×C[a, b]T → C[a, b]T by G(λ, x) = x− K(q+λF x). It is clear that x is a solution to (3.7)–(3.8) if and only if G(λ, x) = 0. We provide references for readers interested in Calculus on Banach Spaces [13, 16, 17].
Proposition 3.5. Suppose f is continuously differentiable. Then the map F is con-tinuously Fr´echet differentiable and DF(φ) :C[a, b]T→C[a, b]T is given by:
(DF(φ))(h)(t) =
∂f
∂x(t, φ(t))
(h(t)).
Proof. Letφ ∈C[a, b]T and >0. Let r be positive, and define the set D to be
D:={(t, x(t)) :t∈[a, b]T,||x−φ|| ≤r}.
∂f
∂x is uniformly continuous onDsince ∂f
∂x is continuous and thus uniformly continuous
on compact sets. Therefore there exists a δ >0 such that if (t1, z1) and (t2, z2) are in D and |(t1, z1)−(t2, z2)|< δ then ∂f∂x(t1, z1)− ∂f∂x(t2, z2)
< .
t∈[a, b]T,
f(t, φ(t) +h(t))−f(t, φ(t))− ∂f
∂x(t, φ(t))h(t) k
≤sup
v∈vt
∂f
∂x(t, v)h(t)− ∂f
∂x(t, φ(t))h(t)
where vt is the line segment between φ(t) +h(t) and φ(t). Let ||h|| < δ. vt ⊂ D for
allt ∈[a, b]T, and
f(t, φ(t) +h(t))−f(t, φ(t))− ∂f
∂x(t, φ(t))h(t)
≤ ||h||.
DefineW :C[a, b]T →C[a, b]T to be
(W h)(t) = ∂f
∂x(t, φ(t))h(t). Then,
||F(φ+h)−F(φ)−W h||= sup
t∈[a,b]T
f(t, φ(t) +h(t))−f(t, φ(t))− ∂f
∂x(t, φ(t))h(t)
≤ ||h||.
Since ||F(φ+h)−F(φ)−W h||
||h|| ≤,F is differentiable, with
(DF(φ))(h)(t) =
∂f
∂x(t, φ(t))
Now we will show that F is continuously differentiable. φ, , andD are as before.
||DF(φ)−DF(ψ)||= sup
||h||=1
||DF(φ)h−DF(ψ)h||
≤ sup
||h||=1
sup
t∈[a,b]T
∂f
∂x(t, φ(t))(h(t))− ∂f
∂x(t, ψ(t))(h(t))
!
≤ sup
t∈[a,b]T
∂f
∂x(t, φ(t))− ∂f
∂x(t, ψ(t))
If||φ−ψ||< δthen||DF(φ)−DF(ψ)|| ≤, and henceF is continuously differentiable.
Theorem 3.6. Suppose the only solution to (3.1)–(3.2) is the trivial one. If f is continuously differentiable then there exists a λ0 >0 such that if |λ|< λ0, there is a solution to the boundary value problem (3.7)–(3.8), xλ. Furthermore ||xλ−x|| → 0
as λ →0, where x is the unique solution of the linear boundary value problem
x∆(t) =A(t)x(t) +q(t), t ∈[a, b]T
subject to
Γx= 0.
xλ is a solution of (3.7)–(3.8). Also, as a result of the Implicit Function Theorem, ||xλ− Kq|| →0 as λ →0.
3.3
Applications
Throughout this section we consider boundary value problems of the form
x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[0,1]T
subject to
Γx=
∞
X
k=0
Bkx(tk) = 0
where eachBk is a constant matrix and ∞
P
k=0
||Bk||is finite. Of course, this implies that
the time scales we discuss must contain infinitely many points. We will develop criteria for the existence of solutions to boundary value problems on time scales of this form using theorems that appear in this chapter. We will examine the relation between multipoint boundary value problems with constraints of the form
∞
X
k=0
Bkx(tk) = 0 and
those with boundary conditions of the form
N
X
k=0
Bkx(tk) = 0.
be thought of as pairs (l, r) such that the open interval (l, r) is a removed “middle third”. Therefore the delta derivative of TC is given by
x∆(t) =
x(r)−x(t) r−t
for t=l where (l, r) is any one of the removed “middle thirds”,
lim β→t−
β∈TC
x(β)−x(t)
β−t for t = 1,
lim β→t+
β∈TC
x(β)−x(t)
β−t otherwise. The second example is the time scale
T1 =
1− 1
22n,1−
1 22n+1
:n= 0,1,2,· · ·
∪ {1}
whose delta derivative is given by
x∆(t) =
x0(t) for t∈ 1− 1
22n,1− 1 22n+1
n= 0,1,· · · ,
x0+(t) for t= 1− 1
22n n = 0,1,· · · , (x(t+ 2−2n−2)−x(t))22n+2 for t= 1− 1
22n+1 n = 0,1,· · ·,
lim β→t−
β∈T1
x(β)−x(t)
β−t for t= 1.
.
Consider the boundary value problem
(3.9) x∆(t) =A(t)x(t) +q(t), t∈[0,1]T
subject to
(3.10) Γx=
∞
X
k=0
Bkx(tk) = 0.
It is clear that Γ is linear. The boundedness of Γ follows from the fact that for any x ∈ C[a, b]T, ||Γx|| ≤ ||x||
∞
P
k=0
||Bk||. If ΓΦ is invertible let K be given by Definition
3.1; that is, for each q ∈ Crd[a, b]T, Kq is the unique solution of (3.9)–(3.10).
Cor-responding to the boundary value problem (3.9)–(3.10) we associate the boundary value problem
(3.11) x∆(t) =A(t)x(t) +q(t), t∈[0,1]T
subject to
(3.12) ΓNx=
N
X
k=0
Bkx(tk) = 0.
If ΓNΦ is invertible let KN be given by Definition 3.1; that is, for eachq ∈Crd[a, b]T, KNq is the unique solution of (3.11)–(3.12).
Proposition 3.7. Suppose there exists a positive integer J such that
J
P
k=0
invertible and J P k=0
BkΦ(tk)
−1 < " sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=J+1
||Bk||
#−1
then (3.9)–(3.10)
has a unique solution for each q∈Crd[a, b]T.
Proof.
ΓΦ =
∞
X
k=0
BkΦ(tk) = J
X
k=0
BkΦ(tk) + ∞
X
k=J+1
BkΦ(tk)
=
J
X
k=0
BkΦ(tk)
I+
J
X
k=0
BkΦ(tk)
!−1 ∞ X
k=J+1
BkΦ(tk)
. Since J P k=0
BkΦ(tk)
−1 < " sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=J+1
||Bk||
#−1
it follows that
J P k=0
BkΦ(tk)
−1 ∞ P
k=J+1
BkΦ(tk)
<1. Therefore ΓΦ is invertible [15, 17], and thus by Proposition 3.2, (3.9)–(3.10) has a unique solution for eachq, x=Kq.
Proposition 3.8. Suppose that for each q ∈ Crd[a, b]T (3.9)–(3.10) has exactly one
solution, x = Kq, then there exists a positive integer Nˆ such that for each N > Nˆ (3.11)–(3.12) has a unique solution, xN =KNq. FurthermorexN →x as N → ∞.
Proof. There exists a positive integer ˆN such that for each N >Nˆ
∞ P k=0
BkΦ(tk)
−1 < " sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=N+1
||Bk||
#−1
. Therefore, for each N >Nˆ
ΓNΦ = ∞
X
k=0
BkΦ(tk)
I−
∞
X
k=0
BkΦ(tk)
!−1 ∞ X
k=N+1
BkΦ(tk)
is invertible, and by Proposition 3.2 (3.11)–(3.12) has a unique solution, xN =KNq.
δ1 =
(b−a) sup
t∈[a,b]T
||Φ−1(σ(t))|| then
∞
X
k=N+1
kBkΦ(tk)k< δ1 implies that for any h in Crd[a, b]T,
∞ X
k=N+1
BkΦ(tk)
Z tk
a
Φ−1(σ(s))h(s)∆s
< ||h||.
By continuity there exists aδ2 >0 such that
∞
X
k=N+1
||BkΦ(tk)||< δ2 implies that
" ∞ X k=0
BkΦ(tk)
#−1
−
" N X
k=0
BkΦ(tk)
#−1 < .
There also exists a positive number N1 such that if N > N1,
∞
X
k=N+1
kBkΦ(tk)k<min (δ1, δ2). Let N >max( ˆN , N1).
KNh− Kh= Φ(·)
" ∞ X
k=0
BkΦ(tk)
#−1 ∞ X
k=0
BkΦ(tk)
Z tk
a
Φ−1(σ(s))h(s)∆s
−Φ(·)
" N X
k=0
BkΦ(tk)
#−1 N X
k=0
BkΦ(tk)
Z tk
a
Φ−1(σ(s))h(s)∆s
= Φ(·)
" ∞ X
k=0
BkΦ(tk)
#−1 ∞ X
k=N+1
BkΦ(tk)
Z tk
a
Φ−1(σ(s))h(s)∆s
+ Φ(·) " ∞ X k=0
BkΦ(tk)
#−1
−
" N X
k=0
BkΦ(tk)
#−1
N
X
k=0
BkΦ(tk)
Z tk
a
Φ−1(σ(s))h(s)∆s.
Therefore,
||KN − K||= sup ||h||=1
||KNh− Kh||
≤ sup
t∈[a,b]T
||Φ(t)||
" ∞ X k=0
BkΦ(tk)
#−1
+ (b−a) sup
t∈[a,b]T
||Φ−1(σ(t))|| ∞
X
k=0
||BkΦ(tk)||
and ||KN − K|| →0 as N → ∞. Clearly||x−xN||=||Kq− KNq|| →0 as
N → ∞.
Next we provide the examination of some nonlinear cases starting with
(3.13) x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[0,1]T
subject to
(3.14) Γx=
∞
X
k=0
Bkx(tk) = 0
and the finite multipoint boundary value problem associated with (3.13)–(3.14), namely,
(3.15) x∆(t) =A(t)x(t) +q(t) +f(t, x(t)), t∈[0,1]T
subject to
(3.16) ΓNx=
N
X
k=0
Bkx(tk) = 0.
The following is a corollary to Theorem 3.3.
Corollary 3.9. Suppose there are constants M1, M2, and a number α ∈ (0,1) such that|f(t, x)| ≤M1|x|α+M2 for all (t, x)∈T×Rn and there exists a positive integer J such that
J
P
k=0
J P k=0
BkΦ(tk)
−1 < " sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=J+1
||Bk||
#−1
then there exists at least one
so-lution to (3.13)–(3.14).
Proof. This is a direct result of Proposition 3.7 and Theorem 3.3.
Theorem 3.10. Suppose f is Lipschitz with Lipschitz constant L and there exists a positive integer J such that
J
P
k=0
BkΦ(tk) is invertible and
J P k=0
BkΦ(tk)
−1 < " sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=J+1
||Bk||
#−1
. If L is sufficiently small then there exists a unique solution, x, to (3.13)–(3.14). There also exists a positive integer
ˆ
N such that for each N > Nˆ (3.15)–(3.16) has a unique solution, xN. Further,
xN →x as N → ∞.
Proof. From Proposition 3.7 we see that ΓΦ is invertible and thus the operator K
is well defined. The fact that there exists a unique solution, x = K(q +F x), to (3.13)–(3.14) is a direct result of Theorem 3.4.
Suppose ||K||L < 1. Using Proposition 3.8 we see that there exists a positive integer ˆN such that if N >Nˆ, KN is well defined and sup
N >Nˆ
(||KN||L)<1.
||x−xN||=||K(q+F x)− KN(q+F xN)||
≤ ||K − KN||||q||+||KF x− KNF x+KNF x− KNF xN|| ≤ ||K − KN||(||q||+||F x||) +||KN||L||x−xN||.
Thus||x−xN|| ≤
||K − KN||
(1− ||KN||L)
Now we consider the weakly nonlinear boundary value problem
(3.17) x∆(t) = A(t)x(t) +q(t) +λf(t, x(t)), t∈[0,1]T
subject to
(3.18) Γx=
∞
X
k=0
Bkx(tk) = 0
and the finite multipoint boundary value problem associated with (3.17)–(3.18), namely,
(3.19) x∆(t) = A(t)x(t) +q(t) +λf(t, x(t)), t∈[0,1]T
subject to
(3.20) ΓNx=
N
X
k=0
Bkx(tk) = 0.
The following is a corollary to Theorem 3.6.
Corollary 3.11. Suppose f is continuously differentiable and there exists a positive integer J such that
J
P
k=0
BkΦ(tk) is invertible and
J P
k=0
BkΦ(tk)
−1
<
"
sup
t∈[a,b]T
||Φ(t)|| ∞
P
k=J+1
||Bk||
#−1
then for λ small enough there is a solution to (3.17)–(3.18).
Nonlinear Scalar Two-Point Boundary Value
Problems
This chapter is devoted to the study of scalar nonlinear boundary value problems on time scales. We examine problems of the form
(4.1) u∆n(t) +an−1(t)u∆ n−1
(t) +· · ·+a0(t)u(t) =q(t) +g(u(t)), t ∈[a, b]T
subject to
(4.2)
n
X
j=1 biju∆
j−1
(a) +
n
X
j=1 diju∆
j−1
(b) = 0,
for i = 1,2,· · ·, n. Throughout this chapter T will denote an arbitrary time scale, and we will assume that [a, b]T ⊂Tκn
problem, namely,
(4.3) u∆n(t) +an−1(t)u∆ n−1
(t) +· · ·+a0(t)u(t) = 0, t ∈[a, b]T
subject to
(4.4)
n
X
j=1 biju∆
j−1
(a) +
n
X
j=1 diju∆
j−1
(b) = 0, for i= 1,2,· · · , n,
has dimension one.
Let A(t) be the n×n matrix-valued function given by
A(t) =
0 1 0 · · · 0
0 0 1 0
..
. ... . .. ...
0 0 0 · · · 1
−a0(t) −a1(t) −a2(t) · · · −an−1(t)
.
Clearly A is rd-continuous. We assume that A is also regressive. Let the matrices B and D be defined by B = (bij) and D = (dij). It should be observed that linear
equivalentn×n system,
(4.5) x∆(t) = A(t)x(t) +h(t) +f(x(t)), t∈[a, b]T
subject to
(4.6) Bx(a) +Dx(b) = 0
where
[f(x)]i =
0 for i= 1,2,· · ·n−1 g([x]1) for i=n
and
[h(t)]i =
0 for i= 1,2,· · ·n−1 q(t) for i=n
.
Notice that the solution space of
(4.7) x∆(t) = A(t)x(t), t∈[a, b]T
subject to
(4.8) Bx(a) +Dx(b) = 0
We will pay particular attention to second order equations subject to periodic boundary conditions. We obtain results which significantly extend previous work by Etheridge and Rodr´ıguez concerning the periodic behavior of nonlinear discrete dynamical systems [10].
4.1
Preliminaries
The notation and preliminary results presented here are a straightforward generaliza-tion of previous work in differential equageneraliza-tions and discrete time systems [10–12, 19, 24–26]. We provide references concerning general information on time scales [3, 5, 6] as well as boundary value problems [14, 27].
Let
X ={x∈C[a, b]T :Bx(a) +Dx(b) = 0},
and
Y =Crd[a, b]T
where Crd[a, b]T denotes the space of rd-continuous R
n-valued maps on [a, b] T, and
C[a, b]T denotes the subspace of Crd[a, b]T where the maps are continuous. | · | will
denote the Euclidean norm on Rn. The operator norm will be used for matrices, and the supremum norm will be used for x∈Y ∪X, that is,
||x||= sup
t∈[a,b]
It is clear that X and Y are Banach spaces with this norm. We define the norm of a product space, V1×V2× · · · ×Vm, by
||(v1, v2,· · · , vm)||= m
X
i=1
||vi||i
where || · ||i denotes the norm on Vi.
We define the operator L:D(L)→Y whereD(L) = X∩C1
rd[a, b]T by
(Lx)(t) =x∆(t)−A(t)x(t), t∈[a, b]T
and the operator F :X →Y by
(F x)(t) = f(x(t)), t ∈[a, b]T.
Clearly x is a solution to (4.5)–(4.6) if and only if Lx = h+F x. Φ will denote the fundamental matrix solution for x∆(t) = A(t)x(t), t∈[a, b]
T where Φ(a) = I.
Proposition 4.1. The solution space for the homogeneous boundary value problem
(4.7)–(4.8) and the kernel of (B+DΦ(b)) have the same dimension.
Proof. The the solution space of (4.7)–(4.8) and kernel ofLhave the same dimension. x ∈ ker(L) if and only if x∆(t) = A(t)x(t), t ∈ [a, b]T and x satisfies the boundary conditions. This is true if and only if there is a c in Rn such that x(t) = Φ(t)c for
(B+DΦ(b)) have the same dimension.
Definition 4.1. Let dbe a unit vector which spans the kernel of(B+DΦ(b)). Define
S : [a, b]T →Rn by
S(t) = Φ(t)d.
The following result is obvious.
Corollary 4.2. The kernel of L consists of x such that x(t) = S(t)α for some real number α.
4.2
Main Results
We will now construct projections onto the kernel and image of Lin order to use the Lyapunov–Schmidt Procedure [9, 10].
Definition 4.2. Define P :X →X by
(P x)(t) = S(t)dTx(a), t ∈[a, b]T.
Proposition 4.3. P is a projection onto the kernel of L.
Im(P) = ker(L). Let x ∈ X. (P x)(t) = S(t)dTx(a) = S(t)α where α = dTx(a). Therefore Im(P)⊂ ker(L).
Let x∈ ker(L). There exists aβ ∈R such thatx(t) =S(t)β.
(P x)(t) = S(t)dTx(a) = S(t)dTS(a)β = S(t)β = x(t). Therefore ker(L) ⊂ Im(P).
Definition 4.3. Let k be a vector that spans the kernel of ((B +DΦ(b))T). Define the map Ψ : [a, b]T →Rn by
Ψ(t) = [DΦ(b)Φ−1(σ(t))]Tk, t∈[a, b]T.
Proposition 4.4. y is in the image of L if and only if
b
R
a
yT(τ)Ψ(τ)∆τ = 0.
Proof. Using the variation of constants formula [5] and the boundary conditions it is clear that y∈ Im(L) if and only if there exists x∈X such that
(B+DΦ(b))x(a) +D
b
R
a
Φ(b)Φ−1(σ(τ))y(τ)∆τ = 0, which is equivalent to
−xT(a)(B + DΦ(b))T =
b R
a
DΦ(b)Φ−1(σ(τ))y(τ)
T
∆τ. This holds if and only if
b
R
a
[DΦ(b)Φ−1(σ(τ))y(τ)]T ∆τ β= 0 where β is an element of the the kernel of (B+DΦ(b))T and therefore must be a multiple of k. Therefore,
b
R
a
yT(τ)Ψ(τ)∆τ = 0.
Definition 4.4. Define the operator W from Y into Y by
(W y)(t) = Ψ(t)
b
Z
a
|Ψ(τ)|2∆τ
−1 b
Z
a
Proposition 4.5. E, defined by E =I−W, is a projection onto the image of L.
Proof. First we will show that E is a projection. Since W is a bounded linear map E is also. To prove E2 =E it will be sufficient to show that W2 =W. Let y∈Y.
(W(W y))(t) = W
Ψ(·) b Z a
|Ψ(τ)|2∆τ
−1 b
Z
a
ΨT(τ)y(τ)∆τ
(t), t∈[a, b]T
= Ψ(t) b Z a
|Ψ(τ)|2∆τ
−1 b
Z
a
ΨT(τ)Ψ(τ)∆τ
b
Z
a
|Ψ(ν)|2∆ν
−1 b
Z
a
ΨT(ν)y(ν)∆ν
= Ψ(t) b Z a
|Ψ(ν)|2∆ν
−1 b
Z
a
ΨT(ν)y(ν)∆ν
= (W y)(t).
Finally we will prove that Im(E) = Im(L). It is clear that Ey ∈Im(E).
b
Z
a
ΨT(τ)(Ey)(τ)∆τ =
b
Z
a
ΨT(τ)(y−W y)(τ)∆τ
=
b
Z
a
ΨT(τ)y(τ)∆τ − b
Z
a
ΨT(τ)Ψ(τ)∆τ
b
Z
a
|Ψ(ν)|2∆ν
−1 b
Z
a
ΨT(ν)y(ν)∆ν
= 0.
Therefore Ey∈ Im(L), and Im(E) ⊂Im(L). Now suppose y ∈Im(L).
(Ey)(t) = y(t)−Ψ(t)
b
Z
a
|Ψ(τ)|2∆τ
−1 b
Z
a
Therefore y∈Im(E), and Im(L)⊂ Im(E).
By constructing the projectionsP andE we are now able to analyze the existence of solutions to (4.5)–(4.6) using the classic Lyapunov–Schmidt Procedure. We provide a self-contained presentation of our approach, but offer references [9,13,19,21,23] for a more general formulation and for applications to differential and difference equations. We can utilize the fact that P and E are projections and write
X = Im(P)⊕Im(I−P) and Y = Im(I−E)⊕Im(E).
For all x∈ X there exists u ∈ ker(L) and v ∈ Im(I −P) such that x =u+v. It is clear that L: Im(I−P)∩D(L)→ Im(L) is a bijection, and therefore there exists a bounded linear mapM : Im(L)→Im(I−P)∩D(L) such that
LM y =y, for all y ∈ Im(L) and M Lx=v, for all x∈X.
Definition 4.5. Define the map H1 :R×Im(I−P)→R by H1(α, v) =α−
b
Z
a
g([αS(τ) +M h(τ) +M EF(Sα+v)(τ)]1)[Ψ(τ)]n∆τ,
H2 :R×Im(I−P)→ Im(I−P) by
and H :R×Im(I−P)→R×Im(I−P) by
H(α, v) = (H1(α, v), H2(α, v)).
Proposition 4.6. Lx=h+F xif and only if there exists(α, v)∈R×Im(I−P)such that H(α, v) = (α, v).
Proof. Letx∈X. There exist α ∈R and v ∈Im(I−P) such that x=Sα+v and,
Lx=h+F x ⇐⇒
E[Lx−h−F x] = 0 (I−E)[Lx−h−F x] = 0
⇐⇒
Lv−h−EF(x) = 0 (I −E)F(x) = 0
⇐⇒
v =M h+M EF(Sα+v)
b
R
a
g([αS(τ) +M h(τ) +M EF(Sα+v)(τ)]1)[Ψ(τ)]n∆τ = 0
⇐⇒ H(α, v) = (α, v).
Definition 4.6. Define g(±∞) as follows, provided the corresponding limits exist,
lim
x→±∞g(x) =g(±∞).
sign, and g(∞)g(−∞)
b
R
a
[Ψ(τ)]n∆τ 6= 0. Then
b
Z
a
g([±αS(τ) +M h(τ) +M EF x(τ)]1)[Ψ(τ)]n∆τ −→g(±∞) b
Z
a
[Ψ(τ)]n∆τ
as α → ∞.
Proof. We will show that
b
R
a
g([αS(τ) +M h(τ) +M EF x(τ)]1)[Ψ(τ)]n∆τ →g(∞) b
R
a
[Ψ(τ)]n∆τ as α → ∞. The
proof for the corresponding result with the opposite sign follows an analogous argu-ment.
Let >0. SinceM handM EF are bounded on [a, b]TandSachieves it’s minimum on the set, there exists α0 >0 such that for all α > α0
|g(∞)−g([αS(t) +M h(t) +M EF x(t)]1)|< .
Letα > α0.
g(∞)
b
Z
a
[Ψ(τ)]n∆τ − b
Z
a
g([αS(τ) +M h(τ) +M EF x(τ)]1)[Ψ(τ)]n∆τ
≤ b Z a
|g(∞)−g([αS(τ) +M h(τ) +M EF x(τ)]1)[Ψ(τ)]n|∆τ
≤||Ψ||(b−a).
Therefore
b
R
a
g([±αS(τ) +M h(τ) +M EF x(τ)]1)[Ψ(τ)]n∆τ →g(±∞) b
R
a
[Ψ(τ)]n∆τ as
Theorem 4.8. Suppose that the kernel of (B+DΦ(b)) is one dimensional. If
i. [S(t)]1 is of one sign for all t ∈[a, b]T
ii. g :R→R is continuous iii. g(∞) and g(−∞) exist
iv. g(∞)g(−∞)<0
v.
b
R
a
hT(τ)Ψ(τ)∆τ = 0
vi.
b
R
a
[Ψ(τ)]n∆τ 6= 0
then there is at least one solution to equation (4.1)–(4.2).
Proof. For simplicity we will assume that g(∞)> g(−∞) and
b
R
a
[Ψ(τ)]n∆τ >0.
Let r= sup
z∈R
|g(z)|. Using Proposition 4.7 there is an α0 >0 such that for α > α0
b
Z
a
g([S(τ)α+M h(τ) +M EF(Sα+v)(τ)]1)[Ψ(τ)]n∆τ >0
and
b
Z
a
g([S(τ)(−α) +M h(τ) +M EF(Sα+v)(τ)]1)[Ψ(τ)]n∆τ <0
for v ∈ Im(I−P).
to (4.5)–(4.6). Let
B ={(v, α) :||v|| ≤ ||M h||+||M E||r, and |α| ≤δ where δ=α0+r(b−a)||Ψ||}.
Notice that
b
R
a
g([S(τ)(−α) +M h(τ) +M EF(Sα+v)(τ)]1)[Ψ(τ)]n∆τ
≤r(b−a)||Ψ||. Forα ∈[0, δ],
−δ ≤ −r(b−a)||Ψ|| ≤H1(α, v)≤α≤δ
and
−δ ≤ −α≤H1(−α, v)≤r(b−a)||Ψ|| ≤δ.
Now let (v, α)∈ B. Then
||H2(v, α)||=||M h+M EF(Sα+v)|| ≤ ||M h||+||M E||r.
Since H(B) ⊂ B by the Schauder Fixed Point Theorem there is at least one fixed point of H in B. If ( ˆα,v) is this fixed point, then ˆˆ v = M h+ M EFvˆ and
b
R
a
g([ ˆαS(τ) +M h(τ) +M EF( ˆαS+ ˆv)(τ)]1)[Ψ(τ)]n= 0. By Proposition 4.6
4.3
Periodic Boundary Conditions
In this section we establish the existence of solutions to periodic boundary value problems. We consider
(4.9) u∆∆(t) +βu∆(t) +γu(t) = q(t) +g(u(t)) t∈[a, b]T
subject to
(4.10) u(a)−u(a+T) = 0 and u∆(a)−u∆(a+T) = 0 where [a, a+T]T ⊂Tκ2
and β, γ ∈R whereγµ−β is regressive. We will assume that the solution space of
(4.11) u∆∆(t) +βu∆(t) +γu(t) = 0 t∈[a, a+T]T
subject to
(4.12) u(a)−u(a+T) = 0 and u∆(a)−u∆(a+T) = 0
is one dimensional. Let A=
0 1
−γ −β
. It is easily verified that the kernel of
the time scale exponential function [5]. If we impose the boundary conditions we find that the solution space of this scalar homogeneous boundary value problem is spanned by u(t) = 1 for t ∈ [a, a+T]T. Consequently the constant function [1,0]T
spans the kernel of L.
Now supposeAhas a repeated eigenvalue of zero. The solution to the correspond-ing homogeneous problem is u(t) =c1 +c2t. If we impose the boundary conditions we find that the solution space of this scalar homogeneous boundary value problem is spanned by u(t) = 1 for t∈[a, a+T]T. Consequently the constant function [1,0]T
spans the kernel of L in this case as well.
We can now write that the solutions to the corresponding homogeneous boundary value problem of (4.9)–(4.10) are real multiples of [1,0]T. Therefore [S(t)]
1 is of one sign for allt ∈[a, a+T]T.
Theorem 4.9. If
u∆∆(t) +βu∆(t) +γu(t) = q(t), t∈[a, a+T]T
subject to
u(a)−u(a+T) = 0 and u∆(a)−u∆(a+T) = 0
has a solution and g(∞) and g(−∞) exist where g(∞)g(−∞) < 0 then there is at least one solution to equation (4.1)–(4.2).
It is easy to verify that the most significant results in Etheridge and Rodr´ıguez [10] are a direct consequence of Theorem 4.9.
Corollary 4.10. Suppose the same conditions as in Theorem 4.9 are satisfied. If
i. q is periodic with period T
ii. T is a periodic time scale with period T, meaning if t∈T then t+T ∈T
then there exists at least one periodic solution to equation (4.9)–(4.10).
