A fixed point theorem in locally convex spaces

Abstract

For a locally convex space
with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping
defined on some set
. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu,
Under some additional assumptions, one of which is the existence of a fixed point for the operator
, we prove that there exists a fixed point of
. For a class of elements satisfyingK
n
(p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.
We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.

Full text

Collect. Math. 61, 2 (2010), 223–239
c
2010 Universitat de Barcelona
A fixed point theorem in locally convex spaces
Vladimir Kozlov, Johan Thim, and Bengt Ove Turesson
Department of Mathematics, Link¨
oping University,
SE-581 83 Link¨
oping, Sweden
E-mail: vlkoz@mai.liu.se jothi@mai.liu.se betur@mai.liu.se
Received September 15, 2008. Revised March 4, 2009
Abstract
For a locally convex space Xwith the topology given by a family {p(·;α)}α∈Ω
of seminorms, we study the existence and uniqueness of fixed points for a
mapping K:DK→DKdefined on some set DK⊂X. We require that
there exists a linear and positive operator K, acting on functions defined on the
index set Ω, such that for every u, v ∈DK
p(K(u)−K(v) ; α)≤K(p(u−v;·))(α), α∈Ω.
Under some additional assumptions, one of which is the existence of a fixed
point for the operator K+p(K(0) ; ·), we prove that there exists a fixed point
of K. For a class of elements satisfying Kn(p(u;·))(α)→0as n→ ∞,
we show that fixed points are unique. This class includes, in particular, the class
for which we prove the existence of fixed points.
We consider several applications by proving existence and uniqueness of
solutions to first and second order nonlinear differential equations in Banach
spaces. We also consider pseudo-differential equations with nonlinear terms.
1. Introduction
In this article, we study existence and uniqueness of fixed points for a certain type
of mappings on locally convex spaces. Spaces of this type arise in many applica-
tions, where there is no natural Banach space to work in and the topology is given by
Keywords: Fixed point theorem; Locally convex spaces; Ordinary differential equations; Pseudo-
differential operators.
MSC2000: 47H10; 46N20; 47G30; 34A34.
223
Collectanea Mathematica (electronic version): http://www.collectanea.ub.edu

224 Kozlov, Thim, and Turesson
seminorms. Seminorms may also provide better understanding of local behaviour of
solutions to, e.g., integral and differential equations.
We let Xdenote a locally convex topological space, where the topology is given
by a family {p(·;α)}α∈Ωof seminorms that separates points. The index set Ω is not
assumed to have any specific structure. We want to solve the equation
K(u) = u,u∈DK, (1)
where K:DK→DKis a mapping defined on a subset DK⊂X. We assume
that 0 ∈DK. The novelty of our approach consists of the use of an auxiliary linear
and positive operator Kthat acts on functions defined on the index set Ω. More
specifically, we assume that K:DK→RΩ, where DK⊂RΩis a linear subspace.
By RΩwe denote the set of all real-valued functions on Ω, endowed with the topology
of pointwise convergence.
We suppose that the operator Kis subordinated to K; in particular, we require
that
p(K(u)−K(v) ; α)≤K(p(u−v;·))(α), α∈Ω.
Under some natural assumptions on the operator K(see (K1)–(K4)) which guarantee,
in particular, the existence of a minimal non-negative solution σin DKto the equation
σ(α) = Kσ(α) + p(K(0) ; α), α∈Ω, (2)
we prove that equation (1) has a solution in DK; see Theorem 2.4. In other words,
the existence of a fixed point to a “simpler” operator K+p(K(0) ; ·) implies the
existence of a fixed point of the operator K.
If the functions from DKsatisfy limn→∞ Kn(p(u;·))(α) = 0, we also prove that
the fixed point is unique; see Theorem 2.5.
In Section 2.6, we show how Banach’s fixed point principle for contractive map-
pings on Banach spaces may be deduced from our result. For other generalisations
of Banach’s contraction principle, we refer to Dugundji and Granas [1] and references
therein.
In Section 3, we then consider some applications, starting with two types of non-
linear differential equations in a Banach space. First, in Section 3.1, we treat a first
order equation, where the right-hand side satisfies a Lipschitz-Carath´eodory condition.
In Section 3.2, we then consider a second order equation of Sturm-Liouville type, where
the nonlinear term satisfies a similar condition. For both of these equations, we prove
existence and uniqueness results. In the last section, we consider a class of nonlinear
pseudo-differential equations on RN. For more examples of applications where the
above fixed point theorems are useful, we refer to Kozlov and Maz’ya [5, 6].
2. Main results
2.1 The operator K
We suppose that K:DK→RΩ, where DK⊂RΩand Kis linear, and require that K
is subject to the following conditions.

A fixed point theorem in locally convex spaces 225
(K1) Positivity of K.The operator Kis positive, i.e., if η∈DKis non-negative,
then Kη ≥0.
(K2) Fixed point inequality. The function k0(·) = p(K(0) ; ·)∈DK, and there
exists a non-negative function z∈DKsuch that
z(α)≥Kz(α) + k0(α), α∈Ω, (3)
and Kz ∈DK.
(K3) Monotone closedness of K.The operator Kis closed for non-negative,
increasing sequences: if {ηn}is a non-negative sequence in DKsuch that
ηn%η,η≤z, and Kηn%ζ, then η∈DKand Kη =ζ.
(K4) Invariance property. If η, ζ ∈DKsuch that 0 ≤η≤ζand Kζ ∈DK,
then Kη ∈DK.
The existence of a non-negative solution zto (3) allows us to prove the existence of a
non-negative solution to the equation
σ(α) = Kσ(α) + k0(α), α∈Ω, (4)
which is minimal in the sense that if η∈DKis another non-negative solution to (4),
then σ≤η.
Lemma 2.1
Suppose that (K1) to (K4) are satisfied. Then there exists a unique minimal
solution σ∈DKto (4) such that σ≤z. This solution is the limit of the iterations
σ0= 0 and σk+1 =Kσk+k0,k= 0,1,2, . . . ,(5)
which are well-defined and converge for all α∈Ω. Moreover, for n= 1,2, . . .,Knσ
belongs to DKand Knσ→0as n→ ∞.
Proof. To see that the iterations are well-defined, we proceed by induction. Obvi-
ously σ0belongs to DK,σ0≤z, and by (K2), σ1=k0∈DK. Assume that σk∈DK
and σk≤z. Then Kσkbelongs to DKby (K4) and hence, σk+1 belongs to DK
since DKis a linear space. Now, (K1) and (K2) imply that σk+1 ≤Kz +k0≤z.
Thus, the sequence {σk}is well-defined and σk≤zfor every k≥0. Furthermore, this
sequence is increasing. Indeed, σ1≥σ0. Assume that σk≥σk−1. Then, by (K1),
σk+1 −σk=Kσk−Kσk−1≥0.
This implies that σk(α) converges to a number σ(α) for every α∈Ω. Obviously σ≤z.
Moreover, we have Kσk=σk+1 −k0, and therefore Kσk%σ−k0. Hence, by (K3),
we obtain that σ∈DKand Kσ =σ−k0.
Let 0 ≤η∈DKbe another solution to (4). The argument above with zreplaced
by ηshows that σ≤η. This proves that σis minimal.
Let us finally show that Knσbelongs to DKand that Knσ→0 as n→ ∞. It is
clear that Kσ belongs to DKand that Kσ =σ−σ1. Assume that Knσ∈DKand
that Knσ=σ−σn. Then Knσ≤zand (K4) implies that Kn+1 σbelongs to DK. We
also obtain that
Kn+1σ=K(σ−σn) = σ−σn+1 . (6)
Thus, Knσis well-defined for all n∈Nand tends to zero as n→ ∞.

226 Kozlov, Thim, and Turesson
2.2 The operator K
Suppose that the operator Kmaps DKinto DK. We let σbe the minimal solution
to (4), and put
DK,σ =u∈DK:p(u;α)≤σ(α) for every α∈Ω.
We shall require the following properties to hold.
(K1) Subordination to K.If u, v belong to DK,σ , then p(u−v;·) belongs to DK,
and we have
p(K(u)−K(v) ; α)≤K(p(u−v;·))(α), α∈Ω. (7)
(K2) Closedness of DK,σ.If {vk}∞
k=0 is a sequence in DK,σ such that v0= 0 and
∞
X
k=0
p(vk+1 −vk;α)≤σ(α), α∈Ω, (8)
then the limit of vkexists and belongs to DK,σ .
As an example of the condition in (K2), consider the following.
Remark 2.2 Let Xbe a sequentially complete space and suppose that DKis sequen-
tially closed. If a sequence in DK,σ satisfies (8), then it is clearly a Cauchy sequence.
By completeness it converges to some element v∈Xfor which p(v;α)≤σ(α), α∈Ω.
Now, the fact that DKis sequentially closed immediately implies that v∈DK,σ.
Since 0 ∈DK,σ , (K1) implies that p(u;·)∈DKfor u∈DK. We also obtain
the following lemma concerning properties of elements in DK,σ.
Lemma 2.3
Suppose that (K1) holds. Then
(i) the operator Kmaps DK,σ into itself;
(ii) if u∈DK,σ, then Kn(p(u;·)) is well-defined for every non-negative integer n
and limn→∞ Kn(p(u;·))(α)=0for every α∈Ω.
Proof. Let u∈DK,σ . The assumption (K1) implies that p(u;·)∈DKand
p(K(u) ; α)≤p(K(u)−K(0) ; α) + k0(α)
≤K(p(u;·)) + k0(α)
≤Kσ(α) + k0(α)
=σ(α).
This proves (i). To prove (ii), we notice that K(p(u;·)) ∈DKby (K4). Assume
that Kn(p(u;·)) belongs to DK. Since p(u;·)≤σand (6) holds for every n∈N, it
follows that
Kn(p(u;·)) ≤Knσ=σ−σn≤σ≤z.
Then (K4) implies that Kn+1(p(u;·)) ∈DK. Moreover, by Lemma 2.1 we know
that σn→σ, which proves that Kn(p(u;·)) →0 by the previous inequality. 

A fixed point theorem in locally convex spaces 227
2.3 Existence of fixed points
We wish to find a fixed point of K. This will be accomplished by the following
iterations:
u0= 0 ∈DK,σ and uk+1 =K(uk), k= 0,1,2, . . . . (9)
By Lemma 2.3(i), this sequence is well-defined and every element of the sequence
belongs to DK,σ . We will now prove that the iterations converge to a solution to (1).
Theorem 2.4
Suppose that Ksatisfies (K1) to (K4) and that Ksatisfies (K1) and (K2).
Then there exists a fixed point of Kin DK,σ. This fixed point is the limit of the
iterations in (9).
Proof. We have u1=K(0), so
p(u1−u0;α) = k0(α) = σ1(α)−σ0(α).
We proceed by induction to show that
p(uk+1 −uk;α)≤σk+1(α)−σk(α), k= 0,1,2, . . . . (10)
Assume that p(uk+1 −uk;α)≤σk+1(α)−σk(α). Then, by (K1),
p(uk+2 −uk+1 ;α) = p(K(uk+1)−K(uk) ; α)
≤K(p(uk+1 −uk;·))(α)
≤Kσk+1(α)−Kσk(α).
Hence, (10) holds. Since σkconverges to σ, it follows that
∞
X
k=0
p(uk+1 −uk;·)≤σ−σ0=σ.
Thus (K2) implies that ukconverges to some element uin DK,σ. We also see that
p(u−uk;α)≤σ(α)−σk(α). (11)
By (11), (4) and the definition of σk, we now obtain that
p(K(u)−K(uk) ; α)≤K(p(u−uk;·))(α)
≤Kσ(α)−Kσk(α)
=σ(α)−σk+1(α).
Thus, K(uk)→K(u). Since we also know that K(uk) = uk+1 →u, it is clear that u
is a fixed point of K.

228 Kozlov, Thim, and Turesson
2.4 Uniqueness of fixed points
We now turn to prove a uniqueness result. Suppose that the operator Kmaps DK
into itself. We shall assume that the following conditions hold.
(I) If u∈DK, then Kn(p(u;·)) is defined and belongs to DKfor every non-
negative integer n, and limn→∞ Kn(p(u;·)) = 0.
(II) If η, ζ ∈DKsuch that 0 ≤η≤ζand Kζ ∈DK, then Kη ∈DK.
(III) If u, v belong to DK, then the function p(u−v;·) belongs to DK, and (7)
holds.
Theorem 2.5
Suppose that the operators Kand Ksatisfy (K1) and (I) to (III). Then there
exists at most one fixed point of K.
Proof. Let uand vbe two fixed points of K. Then (III) implies that
p(u−v;α) = p(K(u)−K(v) ; α)≤K(p(u−v;·))(α), α∈Ω. (12)
Since 0 ∈DK, it follows from (III) that p(u;·) and p(v;·) belong to DK. We
also have p(u−v;α)≤p(u;α) + p(v;α)∈DKand, by (I), K(p(u;·) + p(v;·))
also belongs to DK, so it follows from (II) that K(p(u−v;·)) ∈DK. Assume
that Kn(p(u−v;·)) belongs to DKfor some n≥1. Then
Kn(p(u−v;·)) ≤Kn(p(u;·)) + Kn(p(v;·)) ∈DK.
We also have K(Knp(u;·) + Knp(v;·)) ∈DK, so Kn+1(p(u−v;·)) belongs to DK
by (II). Thus, Kn(p(u−v;·)) is well-defined for every n∈N. Now, (12) implies that
p(u−v;α)≤Kn(p(u;·))(α) + Kn(p(v;·))(α), n∈N,α∈Ω.
The assumption (I) now implies that p(u−v;α) = 0 for every α∈Ω, which implies
that u=vsince the seminorms separates points. 
Let us restrict Kto DK,σ. Suppose that Kand Ksatisfy (K1) to (K4)
and (K1) to (K2), respectively. It is immediate that (K1) implies (III). Since
p(u;·)≤σfor every u∈DK, (K4) implies (II). Furthermore, (I) follows from
Lemma 2.3(ii). Theorem 2.5 now shows that the fixed point in Theorem 2.4 is unique
in the set DK,σ.
2.5 Error estimates
In the following theorem, σkare the iterations defined by (5) and σis the limit, which
exists by Lemma 2.1. Let {uk}be the iterations defined by (9) and let ube the limit.
Theorem 2.6
We assume that all assumptions in Theorem 2.4 are valid, that is, that Ksat-
isfy (K1)–(K4) and that Ksatisfy (K1) and (K2). Then we have the following a
priori estimate :
p(u−un;α)≤σ(α)−σn(α),n∈N,α∈Ω,(13)

A fixed point theorem in locally convex spaces 229
and the a posteriori estimate :
p(u−un+1 ;α)≤
∞
X
i=0
Ki+1(p(un−un+1 ;·))(α),n∈N,α∈Ω,(14)
where the series is finite for every α∈Ω.
Proof. Let m, n ∈N. From (10), it follows that
p(un+m−un;α)≤σn+m(α)−σn(α) (15)
If we let m→ ∞ in (15), (13) now follows.
To prove the a posteriori estimate, we observe that as in the proof of (ii) in
Lemma 2.3, we have Ki(p(un−um;·)) ∈DKfor every i∈N. This follows from (K4)
since p(un−um;·)≤σn−σmfor m≤n. Thus,
p(un+1 −un+m+1 ;α)≤
m
X
i=1
Ki(p(un+1 −un;·))(α)
=
m−1
X
i=0
KiK(p(un+1 −un;·))(α).
(16)
Using the same argument as in the proof of (6), we obtain that Kiσn=σn+i−σi, and
consequently
m−1
X
i=0
Ki(σn+2 −σn+1) =
m−1
X
i=0
(σn+i+2 −σn+i+1)
=σn+m+1 −σn+1.
Since K(p(un+1 −un;·)) ≤σn+2 −σn+1, this shows that if we let m→ ∞ in (16),
then the series in (14) is finite and the a posteriori estimate holds. 
2.6 Comparison with Banach’s fixed point theorem
We now demonstrate that our result is indeed a generalisation of Banach’s fixed point
theorem. Let Xbe a Banach space and K:DK→DKa Lipschitz mapping, with
constant γ, on a closed non-empty set DK⊂X. We have only one-seminorm, i.e.,
the norm |· | on X. Thus, the index set Ω consists of one point and RΩ=R. Let Kbe
the operator given by multiplication by γ. For a non-negative solution zto (3) to exist
in general, it is necessary that 0 ≤γ < 1, i.e., that Kis a contraction. The unique
solution to (4) is given by σ= (1 −γ)−1|K(0)| ∈ DK. Thus, if Kis a contraction,
then (K2) holds. Obviously, Kis linear and satisfies (K1), (K3), and (K4). It is also
obvious that (K1) and (K2) hold. Theorem 2.4 now shows that there exists a fixed
point xof K. Furthermore, Kn|y| → 0 for every yin X, so the fixed point is unique
in DKby Theorem 2.5. From the expressions in (13) and (14), we also obtain the
well-known prior and posterior estimates; see Zeidler [11, p. 19].

230 Kozlov, Thim, and Turesson
3. Applications
3.7 A first order differential equation
To show how to apply the abstract fixed point theorems, we give a new proof of a well-
known solvability result for first order differential equations in a separable Banach
space B, with the norm denoted by |·|. We consider the following equation:
x0(t) = A(x(t), t), t≥0, (17)
where
x(0) = a∈B.
We suppose that A:B×[0,∞)→Bsatisfies the following Lipschitz-Carath´eodory
condition:
(A) For every fixed x∈B,A(x, ·) is measurable1, and there exists a function ω
in L1
loc([0,∞)) such that for all x, y ∈Band every t≥0, we have
|A(x, t)−A(y, t)| ≤ ω(t)|x−y|. (18)
This condition implies, in particular, that the composition A(x(·),·) with a
measurable function x: [0,∞)→Bis again measurable. We also require that
|A(0,·)| ∈ L1
loc([0,∞)).
Integrating (17), we obtain
x(t) = a+Zt
0
A(x(τ), τ )dτ,t≥0. (19)
We let L1
loc([0,∞) ; B) denote the linear space of functions on [0,∞) into Bthat are
locally Bochner integrable (see Hille [3, p. 78]). It is clear that the assumptions above
imply that A(x(·),·)∈L1
loc([0,∞) ; B) if x∈L1
loc([0,∞) ; B), so equation (19) is
well-defined. Similarly, by L∞
loc([0,∞) ; B) we denote the linear space of measurable
functions xmapping [0,∞) into Bsuch that |x(·)|belongs to L∞
loc([0,∞)). If there ex-
ists a solution x∈L∞
loc([0,∞) ; B) to equation (19), xwill be continuous and solves (17)
in the sense of vector distributions; see Lions and Magenes [8, Section 1.3].
Theorem 3.1
There exists a unique solution xin L∞
loc([0,∞) ; B)to (19), and this solution
satisfies
|x(t)| ≤ |a|expZt
0
ω(τ)dτ,t≥0.
Proof. Let Xbe the vector space of all mappings from Ω = [0,∞) into B. We define
the topology on Xby the seminorms p(x;t) = |x(t)|,x∈X,t≥0. Obviously these
seminorms separate points, and it is easy to see that Xis a complete, locally convex
space with this topology.
1Since Bis separable, strong measurability is equivalent with weak measurability (cf. Hille [3],
Theorem 3.5.3).

A fixed point theorem in locally convex spaces 231
We first prove existence of a solution to (19). Let K:DK→Xbe defined by
the right-hand side in (19), with DKchosen as L∞
loc([0,∞) ; B). Define K:DK→RΩ
by
Kη(t) = Zt
0
ω(τ)η(τ)dτ,t≥0,
with DK=L∞
loc([0,∞)). Obviously, if x∈DK, then K(x) belongs to DK. The
condition (A) implies that (K1) holds. Clearly, Kis linear, positive, and closed for
non-negative increasing sequences (by the monotone convergence theorem), so (K1)
and (K3) hold. Moreover, the function σ∈DK, defined by
σ(t) = |a|expZt
0
ω(τ)dτ,t≥0, (20)
is the unique solution to σ=Kσ +|a|, so we may choose z=σ, which proves
that (K2) is valid. For η∈DK,Kη is continuous, so (K4) is satisfied. Suppose that the
sequence {xn}∞
n=0 satisfy the condition (8) in (K2). Clearly, this is a Cauchy sequence
in DK,σ. Then xnconverges to some measurable x∈Xsuch that |x| ≤ σ. From this
it follows that xbelongs to L∞
loc([0,∞) ; B), which proves that (K2) holds. Thus, all
requirements for Theorem 2.4 are satisfied, so there exists a fixed point x∈DKof K
such that |x| ≤ σ.
Next, we prove that this fixed point is unique. It is obvious that (II) and (III)
hold. To prove that (I) is valid as well, we will use the following identity:
Zt
0
ω(τn)Zτn
0
ω(tn−1)Zτn−1
0
· · · Zτ2
0
ω(τ1)dτ1dτ2· · · dτn=1
n!Zt
0
ω(τ)dτn
,
which holds for every t≥0 and n∈N. For continuous ω, this can be checked by
differentiation and for general ω, it follows by density of smooth functions. From this,
we obtain that, for η∈DK,
|Knη(t)| ≤ 1
n!kωkn
L1([0,t])kηkL∞([0,t]) (21)
for every t≥0 and n∈N. This shows that Kn(p(y;·)) →0 for every y∈DK, so the
fixed point is unique by Theorem 2.5. 
Remark 3.2 If the function ωin (A) instead belongs to L∞
loc([0,∞)), one can
choose DK=L1
loc([0,∞)) and DK=L1
loc([0,∞) ; B). With small changes to the
proof of Theorem 3.1, we obtain existence and uniqueness of a solution to (19) in the
space L1
loc([0,∞) ; B).
3.8 A second order differential equation
Let us consider a second order differential equation in the Banach space B:
−x00(t) + k2x(t) = A(x(t), t), t ∈R, (22)
where kis a positive constant and A:B×R→Bsatisfies the following Lipschitz-
Carath`eodory condition:

232 Kozlov, Thim, and Turesson
(A0) For every fixed x∈B,A(x, ·) is measurable, and there exists a function ω
in L∞(R) such that for all x, y ∈Band every t∈R, (18) holds.
It is easy to verify that the function g(t) = (2k)−1exp(−k|t|), t∈R, is a Green’s
function for the operator −∂2
t+k2. Using this, we can formally rewrite (22) as
x(t) = ZR
g(t−τ)A(x(τ), τ )dτ,t∈R. (23)
One can show that a solution in L1
loc(R;B) to (23) is continuous and satisfies (22) in
the sense of vector distributions.
In order to describe our results for (23), we introduce the auxiliary differential
equation
−w00(t) + k2w(t)−ω(t)w(t) = h(t), t∈R, (24)
where h(t) = |A(0, t)|for t∈R. We will require that
sup
t∈R
ω(t)< k2. (25)
Under this condition, the operator −∂2
t+k2−ωhas a Green’s function gω(t, τ ), re-
presented by the Neumann series
∞
X
k=0 ZRk
g(t−τ1)ω(τ1)g(τ1−τ2)· · · ω(τk)g(τk−τ)dτ1dτ2· · · dτk; (26)
see Kozlov and Maz’ya [4, Section 6]. This Green’s function is uniquely defined if we
require that gωis bounded. We also let w±be two positive solutions to the to (24)
corresponding homogeneous equation, such that
(i) w±(t)>0 for t∈R,w±(t)→0 as t→ ±∞,w±(t)→ ∞ as t→ ∓∞,
One can show that w±satisfy
(ii) |w∓(t)|+|∂tw∓(t)| ≤ Ce±kt for t≷0.
These solutions exist and are unique up to a positive constant factor; see the proof of
Theorem 6.4.1 in Kozlov and Maz’ya [4]. Using these, one can give another represen-
tation for gω:
gω(t, τ ) = (D w+(t)w−(τ), t≥τ,
D w−(t)w+(τ), t≤τ,(27)
where Dis some positive constant.
Theorem 3.3
Suppose that
ZR
gω(0, τ )|A(0, τ)|dτ < ∞.(28)
Then there exists a solution x∈L1
loc(R;B)to (23) that satisfies
|x(t)| ≤ ZR
gω(t, τ )|A(0, τ)|dτ ,t∈R,(29)

A fixed point theorem in locally convex spaces 233
and
|x(t)|=o(w±(t)) as t→ ∓∞.(30)
Moreover, a solution in L1
loc(R;B)to (23), that satisfies (30), is unique.
To verify (28) in specific cases, one can employ well-known asymptotic properties of
solutions to ordinary differential equations; see Eastham [2], Kozlov and Maz’ya [4],
and Wasov [10].
Proof. As in the proof of Theorem 3.1, we let Xbe the vector space of all mappings
from Ω = Rinto B. The topology on Xis given by the seminorms p(x;t) = |x(t)|,x∈
X,t∈R. With this topology, Xis locally convex and complete. We define the
operator K:DK→RΩby
Kη(t) = ZR
g(t−τ)ω(τ)η(τ)dτ,t∈R,
where the domain DKof Kis the set of measurable functions ηon Rsuch that
ZR
g(τ)ω(τ)|η(τ)|dτ < ∞. (31)
Furthermore, let Kbe the right-hand side in (23). It follows from (A0) that, for
example, if x∈L1
loc(Ω ; B) such that |x| ∈ DK, then K(x) is defined:
|K(x)(t)| ≤ ZR
g(t−τ)|A(x(τ), τ )−A(0, τ)|dτ +|K(0)(t)|
≤K|x|(t) + |K(0)(t)|.
(32)
The last term is finite since (28) holds.
We will next show that the functions w±both belong to DK. First of all,
e−kτ ω(τ)w−(τ) = −∂τ(e−kτ (∂τ+k)w−(τ))
and |(∂τ+k)w−(τ)| ≤ Cekτ for τ > 0. Thus,
ZM
0
g(τ)ω(τ)w−(τ)dτ =−ZM
0
∂τe−kτ (∂τ+k)w−(τ)dτ
is bounded with respect to M > 0. Moreover, w−(τ) is bounded for τ < 0, so
Z0
−∞
g(τ)ω(τ)w−(τ)dτ < ∞.
This shows that w−∈DK. In a similar manner, one can show that w+∈DK.
We now prove that (K2) is satisfied. Let h(t) = |A(0, t)|,t∈R. By Theorem 6.5.2
in Kozlov and Maz’ya [4], there exists a solution zto (24), which is given by
z(t) = ZR
gω(t, τ )h(t)dτ,t∈R, (33)

234 Kozlov, Thim, and Turesson
and this solution satisfies z(t) = o(w±(t)) as t→ ∓∞. It is also clear that zbelongs
to DKsince w±belong to DK. Moreover, the fact that gis a Green’s function for −∂2
t+
k2implies that zalso solves the equation
Kz(t) + h∗g(t) = z(t), t∈R,
and hence, (K2) is satisfied. Let η , ζ ∈DKsuch that 0 ≤η≤ζand Kζ belongs to DK.
Clearly Kη is measurable, so it follows that Kη ∈DKsince Kη ≤Kζ and Kζ ∈DK.
Therefore it is clear that (K4) holds. By monotone convergence, it follows that K
is closed for non-negative, increasing sequences. Hence, (K3) is valid. Thus, (K1)
to (K4) are satisfied, so Lemma 2.1 shows that a unique minimal σ∈DKexists.
Existence. We let DKbe defined as the set of those measurable functions xthat
maps RNinto Bfor which |x| ≤ σ. It follows from (18) that Kis defined on DK
since (A0) holds and σ∈DK; compare with (32). It is clear that K(x) is measurable
for every xin DK, and using the same argument as in the proof of (i) in Lemma 2.3,
we obtain that |K(x)| ≤ σif x∈DK. Thus, Kmaps DKinto itself. As above, we
also have |K(x)−K(y)| ≤ K|x−y|for all x, y ∈DK. Hence, (K1) holds.
We next show that (K2) is satisfied. Suppose that {xn}∞
n=0 satisfies condition (8)
in (K2). Then the sequence is a Cauchy sequence in DK,σ. Since Xis (sequentially)
complete, we have xn→xfor some measurable x∈X. Clearly |x| ≤ σ, so x∈DK,σ.
Thus, all requirements for Theorem 2.4 are satisfied and there exists a fixed point x
in DK,σ of K. Moreover, it is straightforward to verify, using (27) and the properties
of w±, that the estimate in (29) implies (30).
Uniqueness. Choose DKas the linear space of functions x∈L1
loc(R;B) such
that (30) holds. If x∈DK, then |x| ∈ DKsince w±∈DK, so K(x) is defined;
compare with (32). We next show that the operator Kmaps DKinto itself. To see
that this is true, let xbelong to DKand let ε > 0 be arbitrary. Choose M > 0 such
that |x(t)| ≤ εw∓(t) for t≷±M. Since (28) holds, it is sufficient to prove that
ZR
g(t−τ)ω(τ)|x(τ)|dτ =o(w±(t)) as t→ ∓∞. (34)
It is straightforward to show that
ZM
−M
g(t−τ)ω(τ)|x(τ)|dτ →0 as t→ ∓∞.
Since ω(τ)w±(τ)=(−∂2
t+k2)w±(τ), we also have
Z−M
−∞
g(t−τ)ω(τ)|x(τ)|dτ ≤εZ∞
−∞
g(t−τ)ω(τ)w+(τ)dτ
=εw+(t),
and similarly
Z∞
M
g(t−τ)ω(τ)|x(τ)|dτ ≤εw−(t).
This implies that (34) is valid. The relation in (34) also shows that (II) holds. The
fact that (III) holds follows from (A0) and the definition of DKand DK.

A fixed point theorem in locally convex spaces 235
We now turn to show that (I) holds. For an arbitrary but fixed function uin DK,
let η(t) = |u(t)|,t∈R. Then η∈DKand η(t) = o(w±(t)) as t→ ∓∞. We prove, for a
fixed t=t0, that Knη(t0)→0 as n→ ∞. Let ε > 0. Choose a positive number Mso
that |η(t)| ≤ εw∓(t) when t≷±M. Put ηM(t) = |η(t)|when |t| ≤ M,η−(t) = |η(t)|
when t < −M, and η+(t) = |η(t)|when t > M. All three functions are defined as zero
elsewhere.
For ηM, we multiply (26) by ηMand integrate. The monotone convergence theo-
rem implies that
∞
X
n=0
gn(t) = ZR
gω(t, τ )ηM(τ)dτ < ∞,t∈R, (35)
where
gn(t) = ZRn
g(t−τ1)ω(τ1)g(τ1−τ2)· · · ω(τn−1)g(τn−1−τn)ηM(τn)dτ1dτ2· · · dτn.
Since KnηM(t)≤k2gn(t) for every t, the convergence of the series in (35) implies
that |KnηM(t0)| ≤ εif nis large enough.
For η±, the fact that Kw±=w±shows that
Knη±(t0)≤εKnw±(t0) = εw±(t0).
Hence, we obtain that for all large n, we have
|Knη(t0)|<(1 + w−(t0) + w+(t0))ε.
This proves that Knη(t0)→0 as n→ ∞. Hence, we may apply Theorem 2.5, and find
that the fixed point is indeed unique. 
3.9 A pseudo-differential equation
Let 1 <p<∞and let Sbe a pseudo-differential operator on RNwith an invertible
symbol a(ξ) which is smooth outside the origin and positively homogeneous of order
zero. We consider the equation
Su(x) = Q(u)(x), x∈RN, (36)
where the mapping Q:Lp
loc(RN\ {0})→Lp
loc(RN\ {0}) is assumed to satisfy the
following Lipschitz type condition.
(Q) There exists some q∈L∞(0,∞) such that for all u, v ∈Lp
loc(RN\ {0}):
NpQ(u)−Q(v) ; r≤q(r)Npu−v;r,r > 0, (37)
where
Np(u;r) = 1
rNZr<|x|<2r
|u(x)|pdx1/p
,r > 0. (38)

236 Kozlov, Thim, and Turesson
Equations of this type occur, for example, when one solves boundary value problems
for partial differential equations with nonlinearities in the boundary condition. An
example of such an operator Qis Q(u)(x) = F(u, x), where Fsatisfies a Lipschitz-
Carath´eodory condition: the function F(u, ·) is measurable for every u∈Rand there
exists some lin L∞(RN) such that
|F(u, x)−F(v, x)| ≤ l(x)|u−v|,u, v ∈Rand x∈RN.
We also assume that F(0,·)∈Lp
loc(RN\ {0}). Then Qmaps Lp
loc(RN\ {0}) into itself
and satisfies (Q) with q(r) = ess supr<|x|<2rl(x).
Since the symbol a(ξ) is invertible, the operator Shas an inverse, which we denote
by T, with symbol 1/a(ξ). Formally applying Tto (36), we arrive at
u(x) = T Q(u)(x), x∈RN. (39)
Under natural assumptions, we will establish existence and uniqueness of solutions
to (39). The operator Tcan also be represented as a singular integral operator:
T v(x) = (V.P.) ZRN
K(x−y)v(y)dy,x∈RN, (40)
where the kernel Kis positively homogenous of order −N, infinitely differentiable
outside the origin, and satisfies a cancellation condition; see Stein [9, p. 26]. We
let µ(ρ) = ρNfor 0 ≤ρ < 1 and µ(ρ) = 1 for ρ≥1. The operator Tis defined for
functions v∈Lp
loc(RN\ {0}) such that
Z∞
0
µ(ρ)Np(v;ρ)dρ
ρ<∞. (41)
In fact, the following inequality holds:
Np(T u ;r)≤CTZ∞
0
µρ
rNp(u;ρ)dρ
ρ,r > 0, (42)
where the constant CTonly depends on N,p, and K. In Kozlov, Thim, and Tures-
son [7], this result is proved for the Riesz transform, but it holds with the obvious
modifications for all operators of the type given in (40).
In what follows, we require that
sup
r>0
q(r)<N
4CT
, (43)
and put ω(t) = NCTq(et) for t∈R. In order to formulate our results for (39), let us
introduce an auxiliary differential equation:
−(∂t+N)∂tw(t)−ω(t)w(t) = h(t), t∈R, (44)
where h(t) = NCTNp(Q(0) ; et), t∈R. A Green’s function for the differential op-
erator −(∂t+N)∂tis given by the function g(t) = N−1µ(e−t), t∈R. As before,
the Green’s function gω(t, τ ) for the differential operator −(∂t+N)∂t−ωmay be
represented as the Neumann series in (26) with gas above. To give another repre-
sentation for gω, we introduce two positive solutions v±to the to (44) corresponding
homogeneous equation such that

A fixed point theorem in locally convex spaces 237
(i) v±(t)>0 for t∈R,v±(t) = o(e−Nt/2) as t→ ±∞,eNt/2v±(t)→ ∞
as t→ ∓∞.
One can show that v±satisfy
(ii) |v∓(t)|+|∂tv∓(t)| ≤ Ce−Nt/2±Nt/2for t≷0.
Compare (i) and (ii) with w±in Section 3.2. We may then represent gωas
gω(t, τ ) = (D v+(t)eNτ v−(τ), t≥τ,
D v−(t)eN τ v+(τ), t≤τ,(45)
where Dis some positive constant. Observe that eN tv±(t) are solutions to the to (44)
corresponding homogeneous equation for the formal adjoint operator, i.e., the equation
−w00(t) + N w0(t)−ω(t)w(t) = 0.
Theorem 3.4
Suppose that
Z∞
0
gω(0,log ρ)Np(Q(0) ; ρ)dρ
ρ<∞.
Then there exists a solution u∈Lp
loc(RN\ {0})to (39). This solution satisfies
Np(u;r)≤Z∞
0
gω(log r, log ρ)Np(Q(0) ; ρ)dρ
ρ,r > 0,
and
Np(u;r) = o(v+(log r)) as r→0+,
o(v−(log r)) as r→ ∞.(46)
A solution in Lp
loc(RN\ {0})to (39) that satisfies (46) is unique.
Proof. Let X=Lp
loc(RN\ {0}) and put Ω = (0,∞). We let {Np(·;r)}r>0be the
seminorms on this space. We start by defining K:DK→RΩby
Kη(r) = CTZ∞
0
µρ
rω(log ρ)η(ρ)dρ
ρ,r > 0,
where CTis the constant in (42). The domain DKof Kis the set of functions η
in L1
loc(0,∞) such that
Z∞
0
µ(ρ)ω(log ρ)η(ρ)dρ
ρ<∞. (47)
Using monotone convergence, it follows that Kis closed for non-negative increasing
sequences. The functions v±(log ρ), ρ > 0, both belong to DK; this follows by the
same kind of argument that was used in the proof of Theorem 3.3, after first making
the substitution ρ=et. Next, we define Kas the right-hand side in (39).
To see that (K2) is satisfied, let h(t) = N CTNp(Q(0) ; et), t∈R. By Theo-
rem 6.5.2 in Kozlov and Maz’ya [4], there exists a solution wto (44), given by (33)
with gωas in (45), and this solution satisfies
|w(t)|=o(v±(t)) as t→ ∓∞.

238 Kozlov, Thim, and Turesson
The function r7→ w(log r) belongs to DKsince we know that v±◦log ∈DK. Further-
more, g(t) is a Green’s function for −(∂t+N)∂t, so after a change of variables, we see
that zsolves the equation
Kz(r)+(h∗g)(log(r)) = z(r), r > 0,
and hence, (K2) is satisfied. It is also clear that (K4) is valid. Thus, (K1) to (K4)
are all satisfied, so we may apply Lemma 2.1 to obtain a unique minimal σ∈DKthat
solves (4).
Existence. We let DKbe the linear space of those x∈Lp
loc(RN\ {0}) such
that Np(u;r)≤σ(r), r > 0. Inequality (42) implies that K(u)∈Lp
loc(RN\ {0})
for u∈DK. From (4) it now follows that Kmaps DK,σ into DK,σ. The condition
in (Q) also shows that (K1) holds.
Let {un}be a Cauchy sequence in DK,σ. Since Lp
loc(RN\ {0}) is (sequentially)
complete, we have un→ufor some u∈Lp
loc(RN\ {0}). Furthermore, from the facts
that σ∈DKand Np(u;·)≤σwe obtain that Np(u;·)∈DKby dominated conver-
gence. Hence, u∈DK,σ. This proves that (K2) is satisfied. Thus, all requirements
for Theorem 2.4 are satisfied and there exists a fixed point u∈DK,σ of Ksuch that
the inequality Np(u;·)≤σis holds.
Uniqueness. As in the proof of uniqueness in Theorem 3.3, one can show that
with DKas those functions u∈Lp
loc(RN\ {0}) such that (46) holds, K(u) is defined
and satisfies (46). It is now straightforward to verify that (II) and (III) are valid. After
the substitution t= log rin the proof of uniqueness in Theorem 3.3, and using v±in the
place of w±, we obtain that Knη(r)→0 as n→ ∞ for all r > 0. Thus, Theorem 2.5
is applicable, which proves that the solution is unique. 
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