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CUBO A Mathematical Journal
Vol.13, No¯ 02, (37–57). June 2011

Remarks on the Generation of Semigroups of Nonlinear
Operators on p-Fréchet Spaces, 0 < p < 1

Sorin G. Gal

University of Oradea,

Department of Mathematics,

Str. Universitatii 1, 410087 Oradea, Romania

email: galso@uoradea.ro

ABSTRACT

In this paper we study the convergence properties of the Crandall-Liggett sequence

Jn
t/n

(A)(x) =
(

I − t
n

A
)−n

(x), n ∈ N, for A a nonlinear operator on some important

non-locally convex F-spaces (called p-Fréchet spaces with 0 < p < 1) and the generation

of the corresponding strongly continuous one-parameter nonlinear semigroups.

RESUMEN

En este trabajo se estudian las propiedades de convergencia de la secuencia de Crandall-

Liggett Jn
t/n

(A)(x) =
(

I − t
n

A
)−n

(x), n ∈ N para A un operador lineal en algunos

importantes F-espacios no-localmente convexos (llamado p-Fréchet espacios con 0 <

p < 1) y la generación de los correspondientes semigrupos fuertemente continuos no

lineales con un parámetro.

Keywords and phrases:: p-Fréchet space, 0 < p < 1, Cauchy problem, affine semigroup, non-

linear semigroup, Crandall-Liggett type theorem.

Mathematics Subject Classification: 47H06, 47H20.



38 Sorin G. Gal CUBO
13, 2 (2011)

1. Introduction

It is well known that an F-space (X, +, ·, || · ||) is a linear space (over the field K = R or

K = C) such that ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ X, ||x|| = 0 if and only if x = 0, ||λx|| ≤ ||x||, for

all scalars λ with |λ| ≤ 1, x ∈ X, and with respect to the metric d(x, y) = ||x − y||, X is a complete

metric space (see e.g. [4, p. 52] or [7]).

In addition, if there exists 0 < p < 1 with ||λx|| = |λ|p||x||, for all λ ∈ K, x ∈ X, then || · || will be

called a p-norm and X will be called p-Fréchet space. (This is only a slight abuse of terminology.

Note that in e.g. [1] these spaces are called p-Banach spaces).

It is known that the F-spaces are not necessarily locally convex spaces. Three classical examples

of p-Fréchet spaces, non-locally convex, are the Hardy space Hp with 0 < p < 1 that consists in

the class of all analytic functions f : D → C, D = {z ∈ C; |z| < 1} with the property

||f|| =
1

2π
sup{

∫ 2π

0

|f(reit)|pdt; r ∈ [0, 1)} < +∞,

the sequence space

lp = {x = (xn)n; ||x|| =

∞∑

n=1

|xn|
p < ∞}

for 0 < p < 1, and the Lp[0, 1] space, 0 < p < 1, given by

Lp = Lp[0, 1] = {f : [0, 1] → R; ||f|| =
∫ 1

0

|f(t)|pdt < ∞.}

Some important characteristics of the F-spaces are given by the following remarks.

Remarks. 1) Three of the basic results in Functional Analysis hold in F-spaces too : the

Principle of Uniform Boundedness (see e.g. [4, p. 52]), the Open Mapping Theorem and the Closed

Graph Theorem (see e.g. [7, p. 9-10]).

2) The Hahn-Banach Theorem fails in non-locally convex F-spaces. More exactly, if in an F-

space the Hahn-Banach theorem holds, then that space is a necessarily locally convex space (see

e.g. [6, Chapter 4]).

The beginning of a theory of semigroups of linear operators on p-Fréchet spaces, 0 < p < 1,

was developed in the very recent paper [5]. One of the main result in [5] is the Chernoff-type

formula etA(x) = ĺımn→ ∞
(

I − t
n

A
)−n

(x), for A a bounded linear operator on a p-Fréchet space

with 0 < p < 1.

The aim of the present paper is to look for similar results, that is for convergence properties

of the sequence Jn
t/n

(A)(x) =
(

I − t
n

A
)−n

(x), n ∈ N, in the case when A is a nonlinear operator

on a p-Fréchet space with 0 < p < 1. A very careful examination of the proofs in [3] shows us that

because of the property ||λx|| = |λ|p||x|| with 0 < p < 1, the estimate for ||Jn
t/n

(A)(x) − Jm
t/m

(A)(x)||

does not converges to zero as m, n → ∞ and in fact the sequence Jn
t/n

(A)(x), n ∈ N, is not, in

general, a convergent one.



CUBO
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Remarks on the Generation of Semigroups . . . 39

However, by using techniques in Functional Analysis, we will be able to prove that the sequence

(Jn
t/n

(A)(x))n∈N contains some convergent subsequences in the spaces l
p and Hp with 0 < p < 1,

while this kind of result seems to fail in the space Lp[0, 1], 0 < p < 1. Moreover, in the simplest

nonlinear case when A is an affine operator, we prove that the sequence (Jn
t/n

(A)(x))n∈N is still

convergent and some results in the case of Banach spaces in [6] will be extended to p-Fréchet spaces

(0 < p < 1) too.

The plan of the paper goes as follows. In Section 2 we study the case when A is an affine

operator on an arbitrary p-Fréchet space, 0 < p < 1, Section 3 deals with the case when A is a

nonlinear Lipschitz operator on lp, 0 < p < 1, while the Sections 4 and 5 deal with the similar

problem in the spaces Hp and Lp[0, 1], respectively, with 0 < p < 1.

2. Affine Semigroups

As we will see, the affine case is closely connected to the linear case.

First we need a result in operator theory on p-Fréchet spaces (well-known in the case of

classical Banach spaces).

Lemma 2.1 Let A, B : X → X be bounded linear operators on the p-Fréchet space (X, || · ||),
0 < p < 1. If A is bijection and |||A−1B||| < 1 then A + B is bounded linear bijection on X.

Proof. Since A is a bijection, as a consequence of the Open Mapping Theorem it follows that

A−1 is a bounded linear operator (see e.g. [1, Theorem 14, p. 20 and Corollary 2, p. 23]).

Next we reason as in the case of Banach spaces. Let y ∈ X be arbitrary fixed and define

Ty(x) = A
−1(y) − (A−1B)(x). Then the equation (A + B)(x) = y is equivalent to the equation

Ty(x) = x. But ||Ty(x1) − Ty(x2)|| ≤ |||A
−1B||| · ||x1 − x2||, which shows that Ty is a contraction in

the complete metric space X (with respect to the metric d(x1, x2) = ||x1 − x2||). Therefore it has a

unique fixed point x, which shows that A + B is bijective and the lemma is proved.

The first result on affine semigroups is the following.

Theorem 2.2 Let (X, || · ||) be a p-Fréchet space, 0 < p < 1, A(x) = B(x) + x0, where x0 ∈ X

is fixed and B : X → X is a bounded, linear and strictly dissipative operator, i.e |||(I − λB)−1||| < 1,
for all λ > 0 sufficiently small. Then B is invertible and if we define

Jλ(A)(x) = (I − λA)
−1(x),

(here I defines the identity operator) then

T (t)(x) = ĺım
n→ +∞

Jnt/n(A)(x) = e
tB(x) + B−1[etB(x0)] − B

−1(x0),

is a strongly continuous semigroup of nonlinear (affine) operators on X, (where according to [4],

etB(x) = ĺımn→ +∞ J
n
t/n

(B)(x) is a strongly continuous semigroup of linear operators on X).



40 Sorin G. Gal CUBO
13, 2 (2011)

Proof. By easy calculation we can write

Jλ(A)(x) = Jλ(B)(x + λx0) = (I − λB)
−1(x + λx0),

and in general

Jnλ (A)(x) = (I − λB)
−n(x) + λ

[

n∑

k=1

(I − λB)−k(x0)

]

.

But it is easy to show that for any operator G we have the identity

(I − G)(I + G + G2 + ... + Gn−1) = I − Gn.

Replacing G by Jλ(B), by Lemma 2.1 it follows that I − Jλ(B) is invertible and we immediately

obtain
[

n∑

k=1

(I − λB)−k(x0)

]

= λJλ(B)[I − Jλ(B)]
−1[I − Jnλ (B)]

−n(x0).

But

Jλ(B)[I − Jλ(B)]
−1 = (I − λB)−1[I − Jλ(B)]

−1 =

{[I − (I − λB)−1][I − λB]}−1 =

{−λB}−1 = −
1

λ
B−1,

which implies that
[

n∑

k=1

(I − λB)−k(x0)

]

= −B−1[(I − λB)−n](x0).

Taking λ = t
n

, passing to limit with n → +∞ and taking into account the important Remark
after the Theorem 2.11 in [4] which says that

etB(x) = ĺım
n→ +∞

(I −
t

n
B)−n,

we arrive at

ĺım
n→ +∞

Jnt/n(A)(x) = e
tB + B−1[etB(x0)] − B

−1(x0).

Also, simple calculations show that if we denote T (t)(x) = etB(x) + B−1[etB(x0)] − B
−1(x0),

then T (0) = I, {T (t), t ≥ 0} has the semigroup property, T (·)(x) is continuous as function of t,

A(x) = ĺımhց 0
T (h)(x)−x

h
, for all x ∈ X and T ′(t)(x) = B[T (t)(x)] + x0, which proves the theorem.

Remarks. 1) According to Theorem 2.2, T (t)(u0) is the unique solution of the Cauchy problem

u′(t) = B[u(t)] + x0, u(0) = u0.

(The uniqueness of the solution follows from Lemma 2.12 in [5] concerning the uniqueness of the

solution for the inhomogeneous Cauchy problem in p-Fréchet spaces, 0 < p < 1.)



CUBO
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Remarks on the Generation of Semigroups . . . 41

2) Let us give a simple example satisfying Theorem 2.2. Let (X, || · ||) be a p-Fréchet space,

0 < p < 1, and define A : X → X by A(x) = B(x) + x0, where B(x) = −x for all x ∈ X and x0 ∈ X
is fixed. B obviously is strictly dissipative and A obviously is nonlinear, strictly dissipative, with

||(I − λA)−1||Lip =
1

1 + λ
< 1,

for all λ > 0.

We see that B−1 = B, etB(x) = xe−t and T (t)(x) = (x − x0)e
−t + x0 and in this case T (t)(u0)

is the unique solution to the nonlinear Cauchy problem

du

dt
= −u(t) + x0, u(0) = u0.

3) From the proof of Theorem 2.2, it easily follows the following.

Corollary 2.3 In the case when (X, ||·||) is a Banach space (i.e. a p-Fréchet space with p = 1),

the statement of Theorem 2.2 still remains true.

In what follows, let us consider some concepts introduced in [6] for Banach spaces. They

remain unchanged for the case of p-Fréchet spaces too.

Definition 2.4 By an affine semigroup (S(t) : t ≥ 0) on a p-Fréchet space X, 0 < p < 1, we

mean a family of continuous affine transformations on X with the properties :

(i) S(0)=I, S(t+s)=S(t)[S(s)], for all t, s ≥ 0 ;

(ii) For each x ∈ X, t → S(t)(x) is a continuous function from [0, +∞) into X.
(iii) Any family (S(t) : t ≥ 0) of affine transformations on X can be written in the form

S(t)(x) = T (t)(x) + z(t), for all t ≥ 0, x ∈ X, where T (t)(x) = S(t)(x) − S(t)(0) is its linear part

and z(t) = S(t)(0) is its translation part (z : [0, +∞) → X).
(iv) Let us denote by X = X × R. It is a p-Fréchet space, endowed with the p-norm ||(x, r)|| =

máx{||x||, |r|p}. If (S(t) : t ≥ 0) is a family of affine transformations on X of the form S(t)(x) =

T (t)(x)+z(t), for all t ≥ 0, x ∈ X, where T (t)(x) is its linear part and z(t) is its translation part, the

augmented family associated with (S(t) : t ≥ 0), is a family (T (t); t ≥ 0) of linear transformations

on X, defined by

T (t)[x, r] = [T (t)(x) + rz(t), r].

Having introduced these concepts, Propositions 1.1 and 1.2 proved in [6] for Banach spaces,

hold (with the same proofs) for p-Fréchet spaces too, summarized as follows.

Theorem 2.5 (i) Let (S(t) : t ≥ 0) be a family of affine transformations on the p-Fréchet

space X, 0 < p < 1, with its linear part (T (t) : t ≥ 0) and its translation part z(t); t ≥ 0). Then

(S(t) : t ≥ 0) is an affine semigroup on X if and only if (T (t) : t ≥ 0) is a linear semigroup on X

and z(·) is a continuous map from [0, +∞) into X satisfying

z(t + s) = T (t)[z(s)] + z(t), s, t ≥ 0.



42 Sorin G. Gal CUBO
13, 2 (2011)

(ii) Let (S(t) : t ≥ 0) be a family of affine transformations on the p-Fréchet space X, 0 < p < 1,

and let (T (t) : t ≥ 0) be the augmented family on X, associated with S(·). Then (S(t) : t ≥ 0) is an

affine semigroup on X, if and only if (T : t ≥ 0) is a linear semigroup on X.

Remark. While Proposition 2.1 in [6] remains valid in the case of p-Fréchet spaces too,

0 < p < 1, the other results in [6, Section 2] (i.e. Corollary 2.2, Proposition 2.3, Proposition 2.4

and Corollary 2.5) seem to be not valid. The reason is that they use the Fundamental Theorem of

Calculus in Banach spaces, which, as it was pointed out in [5], does not hold in p-Fréchet spaces,

0 < p < 1.

It would be of interest to see what other results for affine semigroups on Banach spaces in [6],

would remain valid for p-Fréchet spaces too, 0 < p < 1.

3. Nonlinear Semigroups on lp, 0 < p < 1

Before to starting the study in the concrete lp-case, 0 < p < 1, let us briefly recall the problem

and make a useful remark, valid in any p-Fréchet space, 0 < p < 1.

For (X, || · ||X) a p-Fréchet space, 0 < p ≤ 1 (the case p = 1 means that X is a Banach space),

let A : X → X be a nonlinear operator and let us consider the abstract Cauchy problem

d

dt
u(t) = A[u(t)], t ≥ 0,

u(0) = x,

where the solution is u : R+ → X and x ∈ X is fixed. The nonlinear operator A is considered a
Lipschitz mapping, that is

||A(x) − A(y)||X ≤ |||A|||Lip||x − y||X, for every x, y ∈ X,

where |||A|||Lip = sup{||A(x)−A(y)||X/||x−y||X; x, y ∈ X, x 6= y} < +∞. If we replace this differential
equation by the difference equation

1

ε
[uε(t) − uε(t − ε)] = A[uε(t)], t ≥ 0,

with initial condition uε(s) = x, −ε ≤ s ≤ 0, then we easily get by recurrence that uε(t) =
(

I − t
n

A
)−n

(x), for ε = t
n

.

Remark. Without loss of generality, we may suppose A(0) = 0. Indeed, if we suppose that

A(0) 6= 0, then denoting B(u) = A(u) − A(0) we get B(0) = 0 and if v(t) is solution of the abstract

Cauchy problem
d

dt
v(t) = B[v(t)], v(0) = u0,

then u(t) = v(t)+tA(0) is a solution of the above (in A) mentioned problem. Moreover, if for a fixed

ω ∈ R, the operator A − ωI is dissipative, then B − ωI also is dissipative. Indeed, from B − ωI =



CUBO
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Remarks on the Generation of Semigroups . . . 43

(A − ωI) − A(0), since A − ωI is injective and surjective, it easily follows that B − ωI is injection

and surjection and , in addition, from the relationship (B − ωI)−1(y) = (A − ωI)−1(y + A(0)), we

get ||(B − ωI)−1||Lip = ||(A − ωI)
−1||Lip ≤ 1.

In what follows, we denote the p-norm in lp by || · ||p. The first main result of this section is

the following.

Theorem 3.1 Let A : lp → lp, 0 < p < 1, be nonlinear and Lipschitz, such that A(0) = 0
and there exists ω ∈ R with A − ωI dissipative. Then the sequence in lp defined by Jn

t/n
(x) =

(I − t
n

A)−n(x), n ∈ N, t ≥ 0, x ∈ lp, contains a subsequence J
nk
t/nk

(x), k ∈ N (the same subsequence

for all x ∈ lp and all t ∈ R+ = the set of all rational numbers ≥ 0), convergent to an element of

lp in the weak topology of lp.

Proof. By A(0) = 0 we get (I − t
n

A)−n(0) = 0, for all n ∈ N. The dissipative property implies

||[I − t(A − ωI)]−1||Lip ≤ 1, for all t ≥ 0, which is equivalent to ||(I −
t

1+tω
A)−1||Lip ≤ |1 + tω|

p, for

all t ≥ 0 with 1 + tω 6= 0. For λ = t
1+tω

we get ||(I − λA)−1||Lip ≤ (1 − λω)
−p, in particular for all

λ > 0 with λω < 1. (Note that for n sufficiently great, depending on t and ω, we have t
n

ω < 1.)

Therefore,

||(I −
t

n
A)−1||Lip ≤ (1 −

t

n
ω)−p

and by mathematical induction

||(I −
t

n
A)−n||Lip ≤ (1 −

t

n
ω)−np.

But it is known that the sequence (1+ s
n

)n converges (for n → +∞) to es, and for any s ∈ R it
is monotonically increasing, for all n ≥ [|s|]+1 (see e.g. [10, p. 263]), which implies that (1− t

n
ω)−np

converges to etωp, monotonically decreasing, for all n ≥ [|tw|] + 1. Therefore, the greatest value of

(1 − t
n

ω)−np is for n = [|tω|] + 1, which means that there exists M = M(t, p, ω) > 0 (depending

only on t, p and ω) such that

||(I −
t

n
A)−n||Lip ≤ M,

for all n ∈ N.

We obtain

||Jnt/n(x) − J
n
t/n(y)||p ≤ (1 −

t

n
ω)−np||x − y||p ≤ M||x − y||p, (3.1)

for all x, y ∈ lp. Taking y = 0 we have Jn
t/n

(0) = 0 and denoting Jn
t/n

(x) = (gn,r(t)(x))r ∈ l
p, we

obtain ||Jn
t/n

(x)||p ≤ M||x||p, i.e.

+∞∑

r=1

|gn,r(t)(x)|
p ≤ M||x||p < +∞, (3.2)

for all n ∈ N.



44 Sorin G. Gal CUBO
13, 2 (2011)

Now, since lp, 0 < p < 1, has a Schauder basis (see e.g. [7, p. 20]), it follows that it is separable,

denote by Y a countable dense subset of lp. Also, denote by R+, the set of all positive nonnegative

rational numbers and define Gn : N×R+ ×Y → K (where K = R or C), by Gn(r, t, y) = gn,r(t)(y).
Since E := N ×R+ × Y is countable and by (2) the sequence (Gn)n is pointwise bounded on E,

by the Cantor’s diagonal process (see e.g. [11, p. 156-157]), there exists a subsequence Gnk , k ∈ N,

pointwise convergent on E. Denote gr(t)(y) = ĺımk→ +∞ gnk,r(t)(y), for all (r, t, y) ∈ E.

We will show that in fact there exists the limit ( in R ), ĺımk→ ∞ gnk,r(t)(x), for all r ∈ N,

t ∈ R+ and x ∈ l
p. For this purpose, we will show that (gnk,r(t)(x))k is a Cauchy sequence in R

(i.e. it is convergent).

For this purpose, let x ∈ lp and y ∈ Y. We have

|gnk,r(t)(x) − gns,r(t)(x)| ≤ |gnk,r(t)(x) − gnk,r(t)(y)|+

|gnk,r(t)(y) − gns,r(t)(y)| + |gns,r(t)(y) − gns,r(t)(x)|.

Taking into account (1) too, we immediately obtain

|gnk,r(t)(x) − gns,r(t)(x)| ≤ 2M
1/p · ||x − y||1/pp + |gnk,r(t)(y) − gns,r(t)(y)|.

Now, since Y is dense in lp, for x ∈ lp and ε > 0, let y ∈ Y such that 2M1/p||x − y||
1/p
p <

ε
2
,

which implies

|gnk,r(t)(x) − gns,r(t)(x)| <
ε

2
+ |gnk,r(t)(y) − gns,r(t)(y)|.

But for this y, the sequence (gnk,r(t)(y))k is convergent, i.e. it is a Cauchy sequence, which

implies that there exists l0 such that for all k, s ≥ l0 we have

|gnk,r(t)(y) − gns,r(t)(y)| <
ε

2
.

This leads to

|gnk,r(t)(x) − gns,r(t)(x)| < ε,

for all k, s ≥ l0, i.e. (gnk,r(t)(x))k is a Cauchy sequence in R. Therefore, we can write

gr(t)(x) = ĺım
k→ ∞

gnk,r(t)(x),

for all t ∈ R+ and x ∈ l
p.

By (2) it follows
m∑

r=1

|gnk,r(t)(x)|
p ≤ M||x||p < +∞,

for all k, m ∈ N. Passing here to limit with k → ∞, we get
m∑

r=1

|gr(t)(x)|
p ≤ M||x||p < +∞,



CUBO
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Remarks on the Generation of Semigroups . . . 45

for all m ∈ N, which obviously implies

∞∑

r=1

|gr(t)(x)|
p ≤ M||x||p < +∞,

i.e. g(t)(x) := (gr(t)(x))r belongs to l
p.

Now, we will show that for any x∗ ∈ (l1)∗, i.e. of the form (see e.g. [8, pp. 36-37]) x∗(z) =∑∞
j=1

ujzj, for all z ∈ l
1, where (uj)j ∈ m, with m denoting the space of all bounded sequences, we

have x∗(J
nk
t/nk

(x)) → x∗(g(t)(x)), when k → ∞, for any fixed t ∈ R+, x ∈ lp. Note that x∗(g(t)(x))
has sense for g(t)(x) ∈ lp, because lp ⊂ l1.

It is obvious that each functional of the form x∗i (x) = xi, for all x = (xi)i ∈ l
1, is linear

and continuous on l1, since |x∗i (x)| = |xi| ≤
∑∞

j=1
|xj| = ||x||l1 and for k → ∞, x∗i [J

nk
t/nk

(x)] =

gnk,i(t)(x) → gi(t)(x) = x∗i (g(t)(x)), for all i ∈ N.
Then, obviously that for any y∗ ∈ span{x∗1, ..., x

∗
i , ..., } =: Y

∗ we also have y∗[J
nk
t/nk

(x)] →
y∗[g(t)(x)], for k → ∞.

We show that Y∗ is dense in (l1)∗ in the weak topology on (l1)∗. Indeed, let x∗ ∈ (l1)∗ be

arbitrary, x∗(u) =
∑∞

i=1
αiui, for all u = (uj)j ∈ l

1, where α = (αj)j ∈ m. Since z
∗
n(u) =∑n

j=1
αjuj =

∑n
j=1

αjx
∗
j (u), it follows z

∗
n ∈ Y

∗ and we get

|x∗(u) − z∗n(u)| ≤

+∞∑

j=n+1

|αjuj| ≤ ||α||m

+∞∑

i=n+1

|ui| ≤ M0

+∞∑

i=n+1

|ui| → 0,

for n → ∞.
This implies that z∗n → x∗ in the weak topology (i.e. the density of Y∗ in (l1)∗ in the weak

topology) and that for any ε > 0 and any u1, u2 ∈ l
1, x∗ ∈ (l1)∗, there exists y∗ ∈ Y∗, such that

|x∗(uj) − y
∗(uj)| < ε, j = 1, 2.

For u1 = J
nk
t/nk

(x) and u2 = g(t)(x), we get

|x∗[J
nk
t/nk

(x)] − x∗[g(t)(x)]| ≤ |x∗[J
nk
t/nk

(x)] − y∗[J
nk
t/nk

(x)]|+

|y∗[J
nk
t/nk

(x)] − y∗[g(t)(x)]| + |y∗[g(t)(x)] − x∗[g(t)(x)]| <

2ε + |y∗[J
nk
t/nk

(x)] − y∗[g(t)(x)]| < 3ε,

for all k > k0, with k0 depending on ε, t and x.

This shows that for any x∗ ∈ (l1)∗, if k → ∞ then we have x∗[Jnk
t/nk

(x)] → x∗[g(t)(x)], for any
fixed t ∈ R+ and x ∈ l

p.

Finally, since according to [7], p. 27, l1 is the so-called Banach envelope of lp and (l1)∗ = (lp)∗

(with the same dual norms too), the theorem is proved.

Remarks. 1) We may repeat the reasonings in the proof of Theorem 3.1 for the sequence

(Jn
t/n

(x), n ∈ N, n 6= nk), where nk is the subsequence in Theorem 3.1, so that by mathematically



46 Sorin G. Gal CUBO
13, 2 (2011)

induction we easily obtain that the sequence (Jn
t/n

(x))n∈N has at most a countable set of limit

points in the weak topology of lp, denote that set by T ∗(t)(x), where t ∈ R+, x ∈ l
p. For any fixed

t ∈ R+, an element a ∈ T
∗(t) is in fact a mapping a : lp → lp.

2) From the proof of Theorem 3.1, we easily can derive that in addition, the functions gn,r(t) :

lp → R are Lipschitz functions, i.e. |gn,r(t)(x) − gn,r(t)(y)| ≤ M1/p||x − y||1/pp , which implies
that the family (gn,r(t))n,r∈N is equicontinuous. Also, for any x ∈ l

p, t ∈ R+, the sequence

(gn,r(t)(x))n,r∈N is bounded . Unfortunately we cannot apply the classical Arzela-Ascoli theorem

in lp, because lp is not locally compact.

However, we may impose some additional properties to the nonlinear operator A, which could

imply better convergence results in Theorem 3.1, as follows.

Consider on lp the so called lexicographic order, i.e. for x = (xj)j, y = (yj)j ∈ l
p, we write

x ≤ y if and only if xj ≤ yj, for all j ∈ N and x < y if and only if x ≤ y and there is a j with

xj < yj.

The following simple result holds.

Lemma 3.2 Suppose that A : lp → lp is a dissipative nonlinear operator, A(0)=0, A is convex
and non-increasing with respect to the above order, i.e.

A[αx + (1 − α)y] ≤ αA(x) + (1 − α)A(y),

for all x, y ∈ lp, α ∈ [0, 1] and x < y implies A(x) ≥ A(y). We have :

(i) I − λA is concave and non-decreasing, for any λ > 0 ;

(ii) B := (I − λA)−1 is convex and non-decreasing, for any λ > 0 ;

(iii) Bn is convex and non-decreasing, for any λ > 0.

The proof is an easy exercise and it is left to the reader.

Remark. Lemma 3.2 says that if A is convex and non-increasing, then so is Jn
t/n

(x), which

obviously implies that the functions gn,r(t) : l
p → R in the proof of Theorem 3.1 are convex and

non-decreasing.

Corollary 3.3 Denote by T (t)(x) ∈ lp the weak limit in lp of the sequence (J
nk
t/nk

(x))k, for

all t ∈ R+ and x ∈ l
p, where (nk)k is the subsequence in Theorem 3.1. We have

(i) T (0) = I ;

(ii) ||T (t)(x) − T (t)(y)||l1 ≤ e
tω||x − y||

1/p
p , for all t ∈ R+, x, y ∈ l

p ⊂ l1 ;

(iii) For any t, s ∈ R+ and a ∈ T
∗(t + s), there exist b ∈ T ∗(t) and c ∈ T ∗(s) such that

a(x) = b[c(x)], for all x ∈ lp.

Proof. (i) It is obvious by the definition of J
nk
t/nk

(x) ;

(ii) First, passing to limit with k → +∞ in the following inequality in the proof of Theorem



CUBO
13, 2 (2011)

Remarks on the Generation of Semigroups . . . 47

3.1

||J
nk
t/nk

(x) − J
nk
t/nk

(y)||p ≤ (1 −
t

nk
ω)−nkp||x − y||p,

we easily get

limk→ ∞ ||J
nk
t/nk

(x) − J
nk
t/nk

(y)||p ≤ limk→ ∞ (1 −
t

nk
ω)−nkp||x − y||p = e

tωp
||x − y||p.

Let x∗ ∈ (lp)∗ be with |||x∗|||(lp)∗ ≤ 1. According to [7, p. 27], it can be extended to a

x∗ ∈ (l1)∗, preserving its norm, i.e. |||x∗|||(l1)∗ ≤ 1. We have

|x∗[T (t)(x)] − x∗[T (t)(y)]| = |x∗[T (t)(x) − T (t)(y)]| ≤

|x∗[T (t)(x)] − x∗[J
nk
t/nk

(x)]| + |x∗[J
nk
t/nk

(x)] − x∗[J
nk
t/nk

(y)]|+

|x∗[J
nk
t/nk

(y)] − x∗[T (t)(y)]| :=

ak + |x
∗[J

nk
t/nk

(x)] − x∗[J
nk
t/nk

(y)]| + bk,

where limk→ ∞ ak = limk→ ∞ bk = 0 by the definitions of T (t)(x) and T (t)(y).

On the other hand,

|x∗[J
nk
t/nk

(x)] − x∗[J
nk
t/nk

(y)]| ≤ |||x∗|||(l1)∗ ||J
nk
t/nk

(x) − J
nk
t/nk

(y)||l1 ≤

||J
nk
t/nk

(x) − J
nk
t/nk

(y)||l1 ≤ ||J
nk
t/nk

(x) − J
nk
t/nk

(y)||1/pp .

Passing in the above two inequalities to limit with k → ∞, we get

|x∗[T (t)(x)] − x∗[T (t)(y)]| ≤ etω||x − y||1/pp ,

for all x∗ ∈ (l1)∗, with |||x∗|||(l1)∗ ≤ 1. Passing here to supremum with such x
∗, by a classical result

in functional analysis for normed spaces, it follows

sup|||x∗|||
(l1)∗

≤1|x
∗[T (t)(x)] − x∗[T (t)(y)]| = ||T (t)(x) − T (t)(y)||l1 ≤ e

tω
||x − y||1/pp .

(iii) Let q ∈ N, t ∈ R+, x ∈ l
p be fixed. For any x∗ ∈ (lp)∗, we have

limk→ +∞ x
∗[J

nk
qt/nk

(x)] = x∗([T (qt)](x)),

which immediately implies

limk→ +∞ x
∗(J

qnk
qt/qnk

(x)) = limk→ +∞ x
∗(J

qnk
t/nk

(x)) = ĺım
k→ +∞

x∗([T (t)]q(x)).

Applying the same reasonings as in the proof of Theorem 3.1, there exists a subsequence of (qnk)k,

let us denote it by (qk)k, such that

ĺım
k→ +∞

x∗[J
qk
qt/qk

(x)] = x∗([T (t)]q(x)),



48 Sorin G. Gal CUBO
13, 2 (2011)

which shows that for all t ∈ R+, if a ∈ T
∗(qt), then there exists b ∈ [T ∗(t)]q with a = b.

Then, for l, k, r, s ∈ N, we easily get that for any a ∈ T ∗( l
k

+ r
s
) = T ∗

(

ls+rk
ks

)

, there exists

d ∈ [T ∗( 1
ks

)]ls+kr with a = d. On the other hand, denoting ks = t, we have J
n(ls+kr)

t/n
(x) = (I −

t
n

A)−n(ls+kr)(x) = (I − t
n

A)−nls[(I − t
n

A)−nkr(x)], so for the above d, there exists a subsequence

(nj)j with d = ĺımj→ +∞ x
∗(J

nj(ls+kr)

t/nj
(x)) and there exist b ∈ [T ∗(t)]ls and c ∈ [T ∗(t)]kr such that

d(x) = b[c(x)]. The corollary is proved.

The next result shows that for some particular nonlinear operators, the whole sequence

(Jn
t/n

(x))n is convergent in the l
p space.

Theorem 3.4 Let A : lp → lp, 0 < p < 1, be a nonlinear operator of the form A(x) =
(fk(xk))k∈N, for all x = (xk)k∈lp , where fk : R → R are non-increasing continuous functions,
fk(0) = 0 and there exists M > 0 such that |fk(α) − fk(β)| ≤ M|α − β|, for all k ∈ N, α, β ∈ R.

Then, for any t ≥ 0 and x ∈ lp, the sequence Jn
t/n

(x) = (I− t
n

A)−n(x), n ∈ N is strongly convergent

to a limit in lp.

Proof. First by definition it easily follows that A is a Lipschitz operator with respect to the

|| · ||p-norm in l
p. Then, we can write Jn

t/n
(x) = (gn,k(t)(xk))k, where gn,k(t) : R → R are given

by gn,k(t)(u) = (I −
t
n

fk)
−n(u). By the hypothesis, it follows that each sequence (gn,k(t)(u))k is

convergent in the Banach space R, denote gk(t)(u) = ĺımn→ +∞ gn,k(t)(u).

We know that lp has the basis {e1, e2, ..., en, ..., }, where ei = (δin)n∈N. Due to the particular

form of Jn
t/n

(x) = (gn,k(t)(xk))k, we have gn,k(t)(0) = 0 for all k ∈ N and it is obvious that if

x ∈ span{e1, ..., ei, ...} =: Y, then J
n
t/n

(x) becomes a sequence with only a finite number of non-zero

elements. This means that for such x, Jn
t/n

(x) is convergent in lp. Also, obviously Y is dense in lp.

Let x ∈ lp and ε > 0 be arbitrary. There exists y ∈ Y such that ||x − y||p < ε. We get

||Jnt/n(x) − J
m
t/m(x)||p ≤ ||J

n
t/n(x) − J

n
t/n(y)||p + ||J

n
t/n(y) − J

m
t/m(y)||p+

||Jmt/m(y) − J
m
t/m(x)||p ≤

2M||x − y||p + ||J
n
t/n(y) − J

m
t/m(y)||p < 2Mε + ||J

n
t/n(y) − J

m
t/m(y)||p,

where ||Jn
t/n

||Lip ≤ M = 1 (see the proof of Theorem 3.1, where we take ω = 0).

Since (Jn
t/n

(y))n is convergent in l
p, it is a Cauchy sequence and therefore given δ > 0, there is

a n0 such that ||J
n
t/n

(y) − Jm
t/m

(y)||p < δ, for all m, n > n0. Together with the above inequality this

implies that (Jn
t/n

(x))n is a Cauchy sequence in the complete metric space l
p, i.e. it is convergent

in lp. The theorem is proved.

As an application of Theorem 3.1, we obtain the following

Corollary 3.5 Let A : lp → lp, 0 < p < 1, be nonlinear, Lipschitz, such that A(0) = 0, there
exists ω ∈ R with A − ωI dissipative and A is weakly continuous (that is for any x∗ ∈ (lp)∗, if

limn→ ∞ x
∗(an) = x

∗(a), then limn→ ∞ x
∗[A(an)] = x

∗[A(a)] ).



CUBO
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Remarks on the Generation of Semigroups . . . 49

For x ∈ lp and t ∈ R+, let us consider as in the statement and proof of Theorem 3.1, the

sequence in lp, uk(x)(t) = J
nk
t/nk

(x) = (gk,r(x)(t))r∈N, convergent (as k → ∞) in the weak topology
of lp, to u(x)(t) = (gr(x)(t))r∈N.

Let us suppose that for all r ∈ N, x ∈ lp, the real functions gr(x)(t) are left differentiable with

respect to t ∈ R+, that is there exists (finite)

[gr(x)]
′
−(t) = limh→ 0,h∈R+

gr(x)(t) − gr(x)(t − h)

h
, t ∈ R+,

and that for all k, r ∈ N, x ∈ lp, the real functions gk,r(x)(t) are differentiable (in the classical

sense) with respect to t ∈ [0, σ), satisfying in addition the relation

limtkր t[gk,r(x)]
′(tk) = [gr(x)]

′
−(t),

for all t ∈ [0, σ) ∩ R+ and all tk ∈ [0, σ) with tk ր t. Here, for s < 0 we take by conven-
tion gr(x)(s) = gr(x)(0), gk,r(x)(s) = gk,r(x)(0), which gives sense to [gr(x)]

′
−(0) = 0 and

[gk,r(x)]
′(s) = 0, s ≤ 0.

Then, v(t) = u(x)(t) is a solution of the Cauchy problem

v′−(t) = A[v(t)], t ∈ [0, σ) ∩ R+,

v(0) = x,

where v′−(t) is defined componentwise as above and v(s)=v(0), for s < 0.

Proof. Let x∗ ∈ (lp)∗ be arbitrary. According to [7, p. 27], it can be extended to a x∗ ∈ (l1)∗,

preserving its norm. By the considerations from the beginning of this section, it follows that

uk(x)(t) satisfies the difference equation

uk(x)(t) − uk(x)(t − t/nk)
t

nk

= A[uk(x)(t)], t ≥ 0.

This obviously implies

x∗

[

uk(x)(t) − uk(x)(t − t/nk)
t

nk

− A(uk(x)(t))

]

= 0, t ≥ 0.

But by Theorem 3.1 we have limk→ ∞ x
∗[uk(x)(t)] = x

∗[u(x)(t)], for all t ∈ R+, x ∈ l
p. Taking

into account the weak continuity of A, first we obtain limk→ ∞ x
∗(A[uk(x)(t)]) = x

∗(A[u(x)(t)]).

Next we will show that

limk→ ∞ x
∗

[

uk(x)(t) − uk(x)(t − t/nk)
t

nk

]

= x∗([u(x)]′−(t)) (3),

for all t ∈ R+, x ∈ l
p.



50 Sorin G. Gal CUBO
13, 2 (2011)

For this purpose, we reason as in the proof of Theorem 3.1, that is first we prove (3) for any

x∗r ∈ (l
p)∗, r ∈ N of the form x∗r(x) = xr, for all x = (x1, ..., xr, ...) ∈ l

p. This one reduces to

limk→ ∞
gk,r(x)(t) − gk,r(x)(t − t/nk)

t
nk

= [gr(x)]
′
−(t),

for all t ∈ R+, x ∈ l
p, r ∈ N.

By the mean value theorem, there exists ξt,k ∈ (t−t/nk, t) such that
gk,r(x)(t)−gk,r(x)(t−t/nk)

t
nk

=

[gk,r(x)]
′(ξt,k), which by the hypothesis immediately implies that at the limit with k → ∞ we

obtain (3).

Also, it is clear that (3) holds for any y∗ ∈ span{x∗1, ..., x
∗
r, ...} = Y

∗. Reasoning now exactly

as at the end of proof in Theorem 3.1 (since Y∗ is dense in (l1)∗ in the weak topology on (l1)∗),

we easily get that (3) is satisfied for all x∗ ∈ (l1)∗.

In conclusion, we get

x∗[(u(x))′−(t) − A(u(x)(t))] = 0,

for all t ∈ R+ and all x
∗ ∈ (l1)∗. Passing here to supremum with |||x∗|||(l1)∗ ≤ 1 and taking into

account a classical result in functional analysis (since l1 is a normed space), we obtain

||[u(x)]′−(t) − A(u(x)(t))||l1 = 0, t ∈ R+, x ∈ l
p,

which implies [u(x)]′−(t) = A(u(x)(t)), for all t ∈ R+, x ∈ l
p. Also, obviously u(x)(0) = x, which

proves the corollary.

A consequence of Theorem 3.4 is the following

Corollary 3.6 For x = (x1, ..., xk, ...) ∈ l
p and 0 < p < 1, let us consider as in the statement

and proof of Theorem 3.4, the operator A, the sequence un(t) := J
n
t/n

(x) = (gn,k(t)(xk))k∈N ∈ l
p,

where gn,k(t) : R → R are given by gn,k(t)(u) = (I − tn fk)
−n(u) and u(t) = (gk(t)(xk))k∈N ∈ l

p

with limn→ ∞ ||un(t) − u(t)||p = 0, for all t ≥ 0.

If, in addition, un(t), u(t) are differentiable with respect to t ∈ [0, σ] such that

limn→ ∞

∞∑

k=1

||g′n,k(xk) − g
′
k(xk)||

p = 0,

where ||g′n,k(xk) − g
′
k(xk)|| := supt∈[0,σ]|g

′
n,k(t)(xk) − g

′
k(t)(xk)|, then v(t) = u(t) represents the

unique solution of the nonlinear Cauchy problem

d

dt
v(t) = A[v(t)], t ∈ [0, σ],

v(0) = x.



CUBO
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Remarks on the Generation of Semigroups . . . 51

Proof. By the considerations from the beginning of this section, it follows that un(t) = J
n
t/n

(x)

satisfies the difference equation

un(t) − un(t − t/n)
t
n

= A[un(t)], t ≥ 0.

Passing here to limit (with n → ∞) in the || · ||p-norm in lp, since A is Lipschitz in lp (see the
proof of Theorem 3.4 ), it follows that limn→ ∞ A(un(t)) = A(u(t)), for all t ∈ [0, σ].

For the left-hand side, we have

∥

∥

∥

∥

u′(t) −
un(t) − un(t − t/n)

t
n

∥

∥

∥

∥

p

≤

∥

∥

∥

∥

u′(t) −
u(t) − u(t − t/n)

t
n

∥

∥

∥

∥

p

+

∥

∥

∥

∥

u(t) − u(t − t/n)
t
n

−
un(t) − un(t − t/n)

t
n

∥

∥

∥

∥

p

,

where limn→ ∞ ||u
′(t) −

u(t)−u(t−t/n)
t
n

||p = 0 by the definition of derivative, while by the mean

value theorem we obtain
∥

∥

∥

∥

u(t) − u(t − t/n)
t
n

−
un(t) − un(t − t/n)

t
n

∥

∥

∥

∥

p

=

∞∑

k=1

∣

∣

∣

∣

[gk(t)(xk) − gn,k(t)(xk)] − [gk(t − t/n)(xk) − gn,k(t − t/n)(xk)]

t/n

∣

∣

∣

∣

p

=

∞∑

k=1

|g′k(ξt,k,n)(xk) − g
′
n,k(ξt,k,n)(xk)|

p
≤

∞∑

k=1

||g′k(xk) − g
′
n,k(xk)||

p,

which by the hypothesis implies

limn→ ∞

∥

∥

∥

∥

u(t) − u(t − t/n)
t
n

−
un(t) − un(t − t/n)

t
n

∥

∥

∥

∥

p

= 0

and proves the corollary.

Example. A simple example satisfying the conditions (and the conclusions) in Corollary 3.6

is given as follows. Define the non-linear strictly decreasing continuous function f : R → R, by
f(x) = −x if x < 0, f(x) = −2x if x ≥ 0 and A : lp → lp by A(x) = (f(x1), ..., f(xk), ...), for all
x = (x1, ..., xk, ...) ∈ l

p.

It is easy to check that |f(α) − f(β)| ≤ 2|α − β|, for all α, β ∈ R, which implies that A is

Lipschitz nonlinear operator. Also, it is easy to check that for all λ > 0, the operator I − λA is

invertible, with x = (x1, ..., xk, ...), (I − λA)
−1(x) = (g(x1), ..., g(xk), ...), g(xk) =

xk
1+λ

if xk < 0,

g(xk) =
xk

1+2λ
if xk ≥ 0, and

||(I − λA)−1||Lip ≤

(

1

1 + λ

)p

≤ 1,



52 Sorin G. Gal CUBO
13, 2 (2011)

which shows that A is dissipative.

Simple calculation shows that un(t) = (gn(t)(x1), ..., gn(t)(xk), ...), where gn(t)(xk) =
xk

(1+(t/n))n

if xk < 0, gn(xk) =
xk

(1+2(t/n))n
if xk ≥ 0, u(t) = (g(t)(x1), ..., g(t)(xk), ...), where g(t)(xk) =

xke
−t if xk < 0, g(t)(xk) = xke

−2t if xk ≥ 0.

It is easy to prove that all the conditions in Corollary 3.6 are satisfied with σ = 1, which shows

that u(t) defined as above is the unique solution of the nonlinear Cauchy problem

d

dt
v(t) = A[v(t)], t ∈ [0, 1],

v(0) = x.

4. Nonlinear Semigroups on Hp, 0 < p < 1

In this section we consider the Hp space, 0 < p < 1, where we denote its p-norm by || · ||p.

The main result is the following.

Theorem 4.1 Let A : (Hp, || · ||p) → (Hp, || · ||p), 0 < p < 1, be nonlinear and Lipschitz, such
that A(0) = 0 and there exists ω ∈ R with A − ωI dissipative. Then the sequence in Hp defined

by Jn
t/n

(x) = (I − t
n

A)−n(x), n ∈ N, t ≥ 0, x ∈ Hp, contains a subsequence J
nk
t/nk

(x), k ∈ N (the

same subsequence for all x ∈ Hp and all t ∈ R+ = the set of all rational numbers ≥ 0), uniformly

convergent on compacts in D.

Proof. By A(0) = 0 we get (I − t
n

A)−n(0) = 0, for all n ∈ N. The dissipative property implies

||I − t(A − ωI)−1||Lip ≤ 1, for all t ≥ 0, which is equivalent to ||(I −
t

1+tω
A)−1||Lip ≤ |1 + tω|

p, for

all t ≥ 0 with 1 + tω 6= 0. For λ = t
1+tω

we get ||(I − λA)−1||Lip ≤ (1 − λω)
−p, in particular for all

λ > 0 with λω < 1. (Note that for n sufficiently great, depending on t and ω, we have t
n

ω < 1.)

Therefore,

||(I −
t

n
A)−1||Lip ≤ (1 −

t

n
ω)−p

and by mathematical induction

||(I −
t

n
A)−n||Lip ≤ (1 −

t

n
ω)−np.

But it is known that the sequence (1+ s
n

)n converges (for n → +∞) to es, and for any s ∈ R is
monotonically increasing, for all n ≥ [|s|] + 1 (see e.g. [10, p. 263]), which implies that (1 − t

n
ω)−np

converges to etωp, monotonically decreasing, for all n ≥ [|tw|] + 1. Therefore, the greatest value of

(1 − t
n

ω)−np is for n = [|tω|] + 1, which means that there exists M = M(t, p, ω) > 0 (depending

only on t, p and ω) such that

||(I −
t

n
A)−n||Lip ≤ M,

for all n ∈ N.



CUBO
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Remarks on the Generation of Semigroups . . . 53

We obtain

||Jnt/n(x) − J
n
t/n(y)||p ≤ (1 −

t

n
ω)−np||x − y||p ≤ M||x − y||p, (4)

for all x, y ∈ Hp. Taking y = 0 we have Jn
t/n

(0) = 0 and we obtain

||Jnt/n(x)||p ≤ M(t, p, ω)||x||p < +∞, (5)

for all n ∈ N.

Since Hp, 0 < p < 1, has a Schauder basis (see e.g. [9]), it follows that it is separable,

denote by Y a countable dense subset of Hp. Also, denote by R+, the set of all nonnegative

rational numbers and define E = R+ × Y. Obviously E is a countable set, let us denote it by

E = {e1, ..., ej, ..., } with the distinct elements two by twos, ej = (rj, yj) and for each e = (t, y) ∈ E,

denote Sn(e) = J
n
t/n

(y) ∈ Hp. Obviously, Sn(e) are analytic functions in D, for all n ∈ N and all

e ∈ E.

According to e.g. [7, p. 35, (3.4)], the point evaluations ϕz(x) = x(z), z ∈ D, are linear and

bounded functionals on Hp, 0 < p < 1 and the following inequality holds

|x(reiθ)| ≤ 21/p||x||p(1 − r)
−1/p, for all x ∈ Hp and z = reiθ.

Together with (5), this implies that for all z = reiθ, |z| ≤ r0 < 1, i.e. 0 < r ≤ r0, e = (t, y) ∈ E, we

obtain

|Sn(e)(z)| = |ϕz[Sn(e)]| ≤ 2
1/p

||Sn(e)||p
1

(1 − r)1/p
≤

21/p
1

(1 − r0)
1/p

M(t, p, ω)||y||p.

In other words, for any fixed e = (t, y) ∈ E, the sequence of analytic functions (Sn(e))n∈N, is

uniformly bounded on each compact subset of D, which by the classical Montel’s theorem implies

that it contains a subsequence uniformly convergent on compact subsets of D.

For e1 ∈ E, there exists a subsequence of (Sn(e1))n∈N, denoted by (S1,n(e1))n∈N, which is

uniformly convergent on compact subsets of D.

For e2 ∈ E, reasoning analogously, the sequence (S1,n(e2))n∈N contains in turn, a subsequence

denoted by (S2,n(e2))n∈N, which is uniformly convergent on compact subsets of D.

In general, for em ∈ E, there exists a subsequence of the previous one, (Sm,n(em))n∈N, uni-

formly convergent on compact subsets of D.

Continuing this process gives rise to the infinite array of analytic functions in D,

S1,1, S1,2, S1,3, ...,

S2,1, S2,2, S2,3, ...,

S1,1, S1,2, S3,3, ...,



54 Sorin G. Gal CUBO
13, 2 (2011)

.........................................

S1,m, S2,m, S3,m, ...,

and so on, such that the first row means that (S1,n(e1))n∈N uniformly converges on compact subsets

of D, the second row means that (S2,n(ej))n∈N is uniformly convergent on compact subsets of D

for j = 1, 2, the third row means that (S3,n(ej))n∈N is uniformly convergent on compact subsets of

D, for j = 1, 2, 3, and so on.

As a consequence, we can consider the diagonal sequence (Sn,n)n∈N, which has the prop-

erty that (Sn,n(ej))n∈N is uniformly convergent on compact subsets of D, for all j ∈ N, that is

(Sn,n(e))n∈N is uniformly convergent on compact subsets of D, for all e ∈ E.

Now, let us denote A = R+ × H
p. We will show that in fact (Sn,n(e))n∈N is uniformly

convergent on compact subsets of D, for all e ∈ A. Indeed, let e = (r, x) ∈ A and since Y is dense

in Hp, let yk ∈ Y, k ∈ N, satisfying ||x − yk||p → 0, when k → ∞. Denoting ak = (r, yk) ∈ E, by
(4) we have

||Sn,n(e) − Sn,n(ak)||p ≤ M(r, p, ω)||x − yk||p,

for all k ∈ N. It is enough to show that (Sn,n(e)(z))n∈N is a Cauchy sequence with respect to the

uniform norm (denoted by || · ||) in each compact disk, Dr in D. Indeed, this is immediate by the

inequalities

||Sn,n(e) − Sm,m(e)|| ≤ ||Sn,n(e) − Sn,n(ak)|| + ||Sn,n(ak) − Sm,m(ak)|| + ||Sm,m(ak) − Sm,m(e)|| ≤

2M(r, p, ω)||x − yk||p + ||Sn,n(ak) − Sm,m(ak)||

and by the above properties.

The theorem is proved.

Remarks. 1) By relation (3.4) in [7, p. 35], it is evident that if limn→ ∞ ||xn − x||p = 0, then

(xn)n is uniformly convergent on compact subsets of D. In general, the converse is not valid. As a

consequence, if we denote by x the uniform limit of (xn)n on compact subsets of D, then x is an

analytic function in D, but in general it does not belong to Hp, 0 < p < 1.

2) We can repeat the reasonings in the proof of Theorem 4.1 for the sequence (Jn
t/n

(x), n ∈

N, n 6= nk), where nk is the subsequence in Theorem 4.1, so that by mathematically induction

we easily obtain that the sequence (Jn
t/n

(x))n∈N has at most a countable set of limit points in

the locally convex topology of uniform convergence on compact subsets in D. If we denote that

set by T ∗(t)(x), where t ∈ R+, x ∈ H
p, then for any fixed t ∈ R+, an element a ∈ T

∗(t) is in

fact a mapping a : Hp → Hol(D), where Hol(D) denotes the spaces of all holomorphic (analytic)
functions in D.

Corollary 4.2. Let A : (Hp, || · ||p) → (Hp, || · ||p), 0 < p < 1, be nonlinear and Lipschitz, such
that A(0) = 0, there exists ω ∈ R with A−ωI dissipative and A : (Hp, T ) → (Hp, T ) is continuous,
where T represents the locally convex topology of uniform convergence on compact subsets in D.



CUBO
13, 2 (2011)

Remarks on the Generation of Semigroups . . . 55

For x ∈ Hp and t ∈ R+, let us consider as in the statement of Theorem 4.1, the sequence in

Hp, uk(x)(t) = J
nk
t/nk

(x), k ∈ N, uniformly convergent on compacts in D (as k → ∞) to u(x)(t).
Let us suppose that for x ∈ Hp and all z ∈ D, the complex valued function [u(x)(t)](z) is left

derivable with respect to the real variable t ∈ R+, that is there exists (finite)

∂[u(x)(t)(z)]−

∂t
= limh→ 0,h∈R+

[u(x)(t)](z) − [u(x)(t − h)](z)

h
, t ∈ R+,

and also suppose that

limk→ ∞
[uk(x)(t)](z) − [uk(x)(t − t/nk)](z)

t
nk

=
∂[u(x)(t)(z)]−

∂t
,

for all t ∈ R+ ∩ [0, σ), x ∈ H
p, z ∈ D.

Here, for s < 0 we take by convention [u(x)(s)](z) = [u(x)(0)](z), [uk(x)(s)](z) = [uk(x)(0)](z),

for all z ∈ D, which gives sense to
∂[u(x)(0)(z)]−

∂t
, for all z ∈ D.

Then, v(t) = u(x)(t) is a solution (analytic in D but not necessarily in Hp) of the Cauchy

problem
∂[v(t)(z)]−

∂t
= A[v(t)](z), t ∈ [0, σ) ∩ R+, z ∈ D

v(0)(z) = x(z), z ∈ D,

where
∂[v(t)(z)]−

∂t
is defined as above and v(s)=v(0), for s < 0.

Proof. By the considerations from the beginning of the Section 3, it follows that uk(x)(t)

satisfies the difference equation

uk(x)(t) − uk(x)(t − t/nk)
t

nk

= A[uk(x)(t)], t ≥ 0.

But by Theorem 4.1 and by the continuity assumption on A, we have limk→ ∞ A[uk(x)(t)](z) =

A[u(x)(t)](z), for all z ∈ D.

Therefore, passing to limit with k → ∞ in the above difference equation, by the hypothesis
we immediately obtain

∂[u(x)(t)(z)]−

∂t
= A[u(x)(t)](z), t ∈ [0, σ) ∩ R+, z ∈ D,

z ∈ D.

Also, obviously u(x)(0) = x, which proves the corollary.

5. Nonlinear Semigroups on Lp[0, 1], 0 < p < 1

In this section we consider the Lp[0, 1] space, 0 < p < 1, where we denote its p-norm by || · ||p.

The main result is the following.



56 Sorin G. Gal CUBO
13, 2 (2011)

Theorem 5.1. Let A : (Lp[0, 1], || · ||p) → (Lp[0, 1], || · ||p), 0 < p < 1, be nonlinear and
Lipschitz, such that A(0) = 0 and there exists ω ∈ R with A − ωI dissipative. Then, for any fixed

t ≥ 0, x ∈ Lp[0, 1], the sequence in Lp[0, 1] defined by Jn
t/n

(x) = (I − t
n

A)−n(x), n ∈ N, contains a

subsequence ak(t, x) := J
nk
t/nk

(x), k ∈ N, such that

limk→ ∞
1

k1/p

k∑

i=1

ai(t, x)(s) = 0,

a.e. s ∈ [0, 1].

Proof. Reasoning exactly as in the proof of Theorem 4.1, relations (4)-(5), we get that

||Jnt/n(x)||p ≤ M(t, p, ω)||x||p,

where || · ||p is the p-norm in L
p[0, 1]. In other words, for any fixed t ≥ 0 and x ∈ Lp[0, 1], the

sequence (Jn
t/n

(x))n is bounded in the p-norm of L
p[0, 1], 0 < p < 1.

According to [2], this implies that for any t ≥ 0 and x ∈ Lp[0, 1], there exists a subsequence

ai(t, x) := J
ni
t/ni

(x), i ∈ N, such that

limn→ ∞
1

n1/p

n∑

i=1

ai(t, x)(s) = 0,

a.e. s ∈ [0, 1].

Remark. Unfortunately, a sequence (ai(t, x))i∈N, satisfying the relation proved by Theorem

5.1, can satisfy (in the sense that does not produce a contradiction) limi→ ∞ ai(t, x, ω)(s) = ∞,
a.e. s ∈ [0, 1], which is the worst possible divergence result. If to this fact we add that the dual

space of Lp[0, 1], 0 < p < 1, is {0}, then it seems that in this space, in general we cannot derive

any result on the convergence of some subsequences of (Jn
t/n

(x))n.

Acknowledgement. The author thanks the referee for the very useful remarks.

Received: September 2009. Revised: December 2009.

Referencias

[1] A. Bayoumi, Foundations of Complex Analysis in Non-Locally Convex Spaces, North-Holland

Mathematics Studies, Elsevier, vol. 193, Amsterdam, 2003.

[2] S.D. Chatterji, A general strong law, Inventiones Math., 9(1970), 235-245.

[3] M.G. Crandall and T.M. Liggett, Generation of semigroups of non-linear transforma-

tions on general Banach spaces, Amer. J. Math., 93(1971), 265-298.

[4] N. Dunford and J.T. Schwartz, Linear Operators, Part I, Interscience, New York, 1964.



CUBO
13, 2 (2011)

Remarks on the Generation of Semigroups . . . 57

[5] S.G. Gal and J.A. Goldstein, Semigroups of linear operators on p-Fréchet spaces, 0 < p <

1, Acta Math. Hungar., 114(1-2)(2007), 13-36.

[6] J.A. Goldstein, S. Oharu and A. Vogt, Affine semigroups on Banach spaces, Hiroshima

Math. J., 18(1988), No. 2, 433-450.

[7] N.J. Kalton, N.T. Peck and J.W. Roberts, An F-Space Sampler, London Mathematical

Society Lecture Notes Series, vol. 89, Cambridge University Press, 1984.

[8] I. Muntean, Course and Problems in Functional Analysis, Vol. II (Romanian), Faculty of

Mathematics, ”Babes-BolyaiÜniversity Press, Cluj, 1988.

[9] P. Oswald, On Schauder bases in Hardy spaces, Proc. Roy. Soc. Edinburg, Sect. A,

93(1982/83), no. 3-4, 259-263.

[10] T. Popoviciu, Course of Mathematical Analysis, Part 2, Functions. Limits. (Romanian),

”Babes-BolyaiÜniversity, Faculty of Mathematics-Mechanics, Cluj, 1972.

[11] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill Inc., 1976.


	Introduction
	Affine Semigroups
	Nonlinear Semigroups on lp, 0<p<1
	Nonlinear Semigroups on Hp, 0<p<1
	Nonlinear Semigroups on Lp[0, 1], 0<p<1