CIMPA Research School on
Evolutionary Equations with Applications in the
Natural Sciences
(Muizenberg, South Africa, 2013)
Applying Functional Analytic Techniques to
Evolution Equations
Wilson Lamb
Department of Mathematics & Statistics
University of Strathclyde, Glasgow, Scotland, U.K.
July, 2013
Contents
I Preliminaries.
I Finite-Dimensional Dynamical Systems.
I Applications to Population Models.
I Infinite-Dimensional Dynamical Systems and Semigroups of Operators.
I Applications to Coagulation-Fragmentation Models.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Section 1: Preliminaries
Contents
I Mathematical Models and Dynamical Systems.
I Basic Concepts from Functional Analysis.
I Banach spaces and Operators.
I Calculus of Vector-Valued Functions.
I Contraction Mapping Principle.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Mathematical Models
I Mathematical models often involve equations that describe how the
phenomena under investigation evolve in time.
I Such evolution equations can arise in a number of different forms; e.g
differential equations.
I Typical procedure used in deriving and using mathematical models of
evolutionary processes involves the following steps.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Mathematical Models
• We make assumptions on the various factors that influence the evolution of
the time-dependent process that we are interested in.
• We obtain a ‘model’ by expressing these assumptions in terms of
mathematics.
• We use mathematical techniques to analyse our model. If the model takes
the form of an equation, then ideally we would like to obtain an explicit
formula for its solution (unfortunately, this can’t be done in the majority of
cases).
• Finally, we examine the outcome of our mathematical analysis and translate
this back into the real world situation to find out how closely the predictions
from our model agree with actual observations.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Mathematical Models
I A mathematical model is usually only an approximation to what is actually
happening in reality.
I Highly detailed models, incorporating many different factors, inevitably mean
very complicated mathematical equations which are difficult to analyse; crude
models, which are easy to analyse, are most likely to provide poor predictions
of actual behaviour.
I In practice, a compromise has to be reached; a small number of key
factors are identified and used to produce a model which is not excessively
complicated.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Well-Posed Problem
When faced with a specific mathematical problem that has emerged from
the modelling process, an important part of the mathematical analysis is to
establish that the problem has been correctly formulated.
The usual requirements for this to be the case are the following.
1. Existence of Solutions. We require at least one solution to exist.
2. Uniqueness of Solutions. There must be no more than one solution.
3. Continuous Dependence on the Problem Data. The solution should
depend continuously on any input data, such as initial or boundary
conditions.
Problems that meet these requirements are said to be well-posed.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Dynamical Systems and Semigroups
From a mathematical viewpoint, a dynamical system consists of two parts:
• a state vector which describes the state of the system at a given time,
• a function which maps the state at one instant of time to the state at a later
time.
Definition 1.1 Let X represent some state space and let J be a subset of R
(which we will assume contains 0). A function φ : J × X → X that has the
two properties
◦
◦
(i) φ(0, u) =u
◦
◦
(ii) φ(s, φ(t, u)) = φ(t+s, u) , for t, s, t+s ∈ J, (the semigroup property)
is called a dynamical system on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Dynamical Systems and Semigroups
I Throughout, we assume that X is a Banach space (i.e. a complete normed
vector space).
◦
I We can regard φ(t, u) as the state at time t of the system that initially was
◦
at state u.
I J is an interval in R, usually J = R+ = [0, ∞). The dynamical system is
then called a continuous-time (semi- or forward) dynamical system.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Dynamical Systems and Semigroups
In operator form, we can write
◦
◦
φ(t, u) = S(t) u,
where S(t) is an operator mapping X into X.
Note that S(0) = I (the identity operator on X) and the semigroup property
(in the case when J = R+) becomes
S(t)S(s) = S(t + s), ∀t, s ≥ 0.
The family of operators S = {S(t)}t≥0 is said to be a semigroup of operators
on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Typical Questions
◦
1. Given some initial condition u, can we determine the asymptotic (long–
◦
term) behaviour of φ(t, u) as t → ∞ ?
2. Can we identify particular initial values which give rise to the same
asymptotic behaviour?
◦
3. Can we say anything about the stability of the system? For example, if u
◦
◦
is “close to” v in X, what can be said about the distance between φ(t, u)
◦
and φ(t, v ) for future values of t?
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Typical Questions
In many situations, a dynamical system may also depend on a parameter (or
several parameters), i.e. the system takes the form φµ : J × X → X where
µ ∈ R represents the parameter. In such cases, the following questions would
also be of interest:
3. Can we determine what happens to the behaviour of the dynamical system
as the parameter varies?
4. Can we identify the values of the parameter at which changes in the
behaviour of the system occur (bifurcation values)?
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Banach Spaces
I Vector Space X (see §1.2.1,
I X is said to be finite-dimensional if ∃n ∈ N such that X contains a linearly
independent set of n vectors whereas any set of n + 1 or more vectors of X
is linearly dependent - in this case X is said to have dimension n.
I By definition, if X = {0} then dim X = 0.
I A norm on a vector space X is a mapping from X into R satisfying the
conditions
• kf k ≥ 0 for all f ∈ X and kf k = 0 ⇔ f = 0;
• kαf k = |α| kf k for all scalars α and f ∈ X;
• kf + gk ≤ kf k + kgk for all f, g ∈ X (the Triangle Inequality).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Banach Spaces
I (fn)∞
n=1 is convergent in X(= (X, k · k) if ∃f ∈ X such that
lim kfn − f k = 0.
n→∞
I (fn)∞
n=1 is Cauchy in X if for every > 0, ∃N ∈ N such that
kfm − fnk < for all m, n ≥ N.
I The normed vector space X is complete if every Cauchy sequence in X is
convergent.
I A complete normed vector space is called a Banach space.
I All finite-dimensional normed vector space are Banach spaces.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Banach Spaces
Example 1.3 Banach space (Rn, k · k).
kf k :=
q
f12 + f22 + · · · + fn2 ,
f = (f1, f2, . . . , fn) ∈ Rn,
Example 1.4
For fixed µ ≥ 0, we define a vector space of scalar-valued sequences (fi)∞
i=1
by
∞
X
µ
`1µ := {f = (fi)∞
:
i
|fi| < ∞}.
i=1
i=1
The norm on `1µ is given by
kf k1,µ =
∞
X
iµ|fi|.
i=1
(`1µ, k · k1,µ) is an infinite-dimensional Banach space.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Operators
I L : X → X is linear if
L(α1f1 + α2f2) = α1L(f1) + α2L(f2) for all f1, f2 ∈ X, and for all α1, α2.
I The set of all linear operators mapping X into X will be denoted by L(X).
I L(X) is a vector space (with (L1 + L2)(f ) := L1(f ) + L2(f ) and
(αL)(f ) := αL(f )).
I An operator T : X → X (T not necessarily linear) is continuous at f ∈ X
if and only if
fn → f in X ⇒ T (fn) → T (f ) in X.
I T is continuous on X if it is continuous at each f ∈ X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Bounded Operators
I T : X → X is said to be bounded on the normed vector space X if
kT (f )k ≤ M kf k for all f ∈ X,
where M is a positive constant that is independent of f .
I In the case of a linear operator L : X → X, continuity and boundedness
are equivalent as it can be proved that
the linear operator L : X → X is continuous on X ⇔ L is bounded on X.
I B(X) is the subspace of L(X) consisting of all bounded, linear operators
on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Normed Vector Space B(X)
I A norm can be defined on B(X) by
kLk := sup {kL(f )k : f ∈ X, kf k ≤ 1} .
I Equipped with this norm, B(X) is a normed vector space and is a Banach
space whenever X is a Banach space.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Calculus of Vector-Valued Functions
Let u : J → X where J ⊂ R is an interval.
I u is strongly continuous at c ∈ J if, for each ε > 0, a positive δ can be
found such that
ku(t) − u(c)k < ε whenever t ∈ J and |t − c| < δ.
I u is strongly differentiable at c ∈ J if there exists an element u0(c) ∈ X
such that
u(c + h) − u(c)
lim
= u0(c),
h→0
h
where the limit is with respect to the norm defined on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Calculus of Vector-Valued Functions
I Strong Riemann integrals of vector-valued functions are defined in an
analogous way to Riemann integrals of scalar valued functions and have
similar properties.
I In particular, suppose that u : [a, b] → X is strongly continuous on [a, b].
Then, for each t ∈ [a, b],
Z
t
u(s) ds exists ,
a
Z
t
u(s) dsk
dt
t
ku(s)k ds,
≤
a
a
d
Z
Z
t
u(s) ds = u(t).
a
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Contraction Mapping Principle
Theorem 1.1 (Banach Contraction Mapping Principle) Let (X, k · k) be a
Banach space and let T : X → X be an operator with the property that
kT f − T gk ≤ α kf − gk for all f, g ∈ X,
for some constant α < 1 (T is said to be a (strict) contraction). Then the
equation
Tf = f
has exactly one solution (called a fixed point of T) in X. Moreover, if we
denote this unique solution by f and use T iteratively to generate a sequence
of vectors (f1, T f1, T 2f1, T 3f1, . . .), where f1 is any given vector in X, then
T nf1 → f as n → ∞.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Contraction Mapping Principle
Proof. Let the sequence (fn)∞
n=1 be defined as in the statement of the
theorem. Then, for n ≥ 1,
kfn+1 − fnk = kT fn − T fn−1k ≤ αkfn − fn−1k ≤ · · · ≤ αn−1kf2 − f1k.
Hence, for m > n ≥ 1, we have
kfm − fnk = kfm − fm−1k + kfm−1 − fm−2k + · · · + kfn+1 − fnk
≤ (αm−2 + αm−3 + · · · + αn−1)kf2 − f1k
< αn−1(1 + α + α2 + · · · )kf2 − f1k =
αn−1
1−α
kf2 − f1k.
It follows that (fn)∞
n=1 is a Cauchy, and hence convergent, sequence in the
Banach space (X, k · k).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Contraction Mapping Principle
Let f ∈ X be the limit of this convergent sequence. Then, by continuity of the
operator T , we obtain
fn+1 = T fn ⇒ lim fn+1 = lim T fn = T
n→∞
n→∞
lim fn ⇒ f = T f,
n→∞
and so f is a fixed point of T .
To show that no other fixed point exists, suppose that both f and g are fixed
points, with f 6= g. Then
kf − gk = kT f − T gk ≤ αkf − gk.
Dividing each side by kf − gk (6= 0) leads to 1 ≤ α, which is a contradiction.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations