THE ABSTRACT RENEWAL EQUATION

Štefan Schwabik

Praha

Abstract. The abstract Perron-Stieltjes integral defined in the Kurzweil-Henstock sense is used for introducing Stieltjes convolutions. The corresponding facts on integration are given in [6], [7] and [8].

The approach is used for obtaining the basic existence result for the abstract renewal equation which was studied e. g. by Diekmann, Gyllenberg and Thieme in

[1] and [2].

For a given Banach space X let L(X) be the Banach space of all bounded linear operators A : X → X with the uniform operator topology.

For B : L(X) × X → X given by B(A, x) = Ax ∈ X for A ∈ L(X) and x ∈ X, we obtain the bilinear triple (L(X), X, X) because we have

∥B(A, x)∥_{X} ≤ ∥A∥_{L(X)}∥x∥_{X}

for the bilinear form B. Similarly, if we define the bilinear form B^{∗} : L(X)×L(X) → L(X) by B^{∗}(A, C) = AC ∈ L(X) for A, C ∈ L(X) where AC is the composition of the linear operators A and C we get the bilinear triple (L(X), L(X), L(X)) because

∥B^{∗}(A, C)∥_{L(X)} ≤ ∥AC∥_{L(X)} ≤ ∥A∥_{L(X)}∥C∥_{L(X)}.

Assume that the interval [0, b] ⊂ ℝ is bounded.

Given A : [0, b] → L(X), the function A is of bounded variation on [0, b] if

var_{[0,b]}(A) = sup{∑_{j=1}^{k} ∥A(α_{j}) − A(α_{j−1})∥_{L(X)}} < ∞,

where the supremum is taken over all finite partitions

D : 0 = α_{0} < α_{1} < ⋯ < α_{k−1} < α_{k} = b

of the interval [0, b]. The set of all functions A : [0, b] → L(X) with var_{[0,b]}(A) < ∞ will be denoted by BV ([0, b]; L(X)).

For A : [0, b] → L(X) and a partition D of the interval [0, b] define

V_{0}^{b}(A, D) = sup{∥ ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}∥_{X}},

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 1 where the supremum is taken over all possible choices of x_{j} ∈ X, j = 1, …, k with ∥x_{j}∥_{X} ≤ 1.

The paper is in final form and no version of it will be submitted for publication elsewhere.

This work was supported by the grant 201/97/0218 of the Grant Agency of the Czech Republic.

Let us set

s var_{[0,b]}(A) = sup V_{0}^{b}(A, D)

where the supremum is taken over all finite partitions D of the interval [0, b].

An operator valued function A : [0, b] → L(X) with s var_{[0,b]}(A) < ∞ is called a function of bounded semi-variation on [0, b] (cf. [4]).

We denote by BSV ([0, b]; L(X)) the set of all functions A : [0, b] → L(X) with

s var_{[0,b]}(A) < ∞.

Assume that η ≥ 0 is given and define

var_{[0,b]}^{(η)}(A) = sup{∑_{j=1}^{k} e^{−ηα_{j−1}}∥A(α_{j}) − A(α_{j−1})∥_{L(X)}}

where the supremum is taken over all finite partitions D of the interval [0, b]. Similarly define

V_{0}^{b}(η, A, D) = sup{∥ ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X}},

where the supremum is taken over all possible choices of x_{j} ∈ X, j = 1, …, k with ∥x_{j}∥_{X} ≤ 1 and set

s var_{[0,b]}^{(η)}(A) = sup V_{0}^{b}(η, A, D)

where the supremum is taken over all finite partitions D of the interval [0, b]. Since for every j = 1, …, k we have

e^{−ηb} ≤ e^{−ηα_{j−1}} ≤ 1

we get

(1)  e^{−ηb} var_{[0,b]}(A) ≤ var_{[0,b]}^{(η)}(A) ≤ var_{[0,b]}(A)

and also

e^{−ηb}V_{0}^{b}(A, D) ≤ V_{0}^{b}(η, A, D) ≤ V_{0}^{b}(A, D).

The last inequalities lead immediately to

(2)  e^{−ηb}s var_{[0,b]}(A) ≤ s var_{[0,b]}^{(η)}(A) ≤ s var_{[0,b]}(A).

Let us mention that

var_{[0,b]}^{(0)}(A) = var_{[0,b]}(A) and s var_{[0,b]}^{(0)}(A) = s var_{[0,b]}(A).

It is well known that BV ([0, b]; L(X)) with the norm

∥A∥_{BV} = ∥A(0)∥_{L(X)} + var_{[0,b]}(A)

is a Banach space and in [8] it was shown that with the norm

∥A∥_{SV} = ∥A(0)∥_{L(X)} + s var_{[0,b]}(A)

the space BSV ([0, b]; L(X)) is also a Banach space.

Taking into account the inequalities (1) and (2) we get the following statement. EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 2 1. Proposition. For every η ≥ 0 the space BV ([0, b]; L(X)) with the norm

∥A∥_{BV,η} = ∥A(0)∥_{L(X)} + var_{[0,b]}^{(η)}(A)

is a Banach space and the space BSV ([0, b]; L(X)) with the norm

∥A∥_{SV,η} = ∥A(0)∥_{L(X)} + s var_{[0,b]}^{(η)}(A)

is also a Banach space.

The norms ∥A∥_{BV,η} and ∥A∥_{BV} are equivalent on BV ([0, b]; L(X)) and the norms ∥A∥_{SV,η} and ∥A∥_{SV} are equivalent on BSV ([0, b]; L(X)).

Given x : [0, b] → X, the function x is called regulated on [0, b] if it has one–sided limits at every point of [0, b], i.e. if for every s ∈ [0, b) there is a value x(s+) ∈ X such that

lim_{t→s+} ∥x(t) − x(s+)∥_{X} = 0

and if for every s ∈ (0, b] there is a value x(s−) ∈ X such that

lim_{t→s−} ∥x(t) − x(s−)∥_{X} = 0.

The set of all regulated functions x : [0, b] → X will be denoted by G([0, b]; X). The space G([0, b]; X) endowed with the norm

∥x∥_{G([0,b];X)} = sup_{t∈[0,b]} ∥x(t)∥_{X}, x ∈ G([0, b]; X)

is known to be a Banach space (see [4, Theorem 3.6]).

It is clear that the space C([0, b]; X) of continuous functions x : [0, b] → X is a closed subspace of G([0, b]; X), i.e.

C([0, b]; X) ⊂ G([0, b]; X).

We are using the concept of abstract Perron-Stieltjes integral based on the Kurzweil-Henstock definition presented via integral sums (for more detail see e.g. [5], [6], [7]).

A finite system of points

{α_{0}, τ_{1}, α_{1}, τ_{2}, …, α_{k−1}, τ_{k}, α_{k}}

such that

0 = α_{0} < α_{1} < ⋯ < α_{k−1} < α_{k} = b

and

τ_{j} ∈ [α_{j−1}, α_{j}] for j = 1, …, k

is called a P –partition of the interval [0, b].

Any positive function δ : [0, b] → (0, ∞) is called a gauge on [0, b] .

For a given gauge δ on [0, b] a P –partition {α_{0}, τ_{1}, α_{1}, τ_{2}, …, α_{k−1}, τ_{k}, α_{k}} of [0, b] is called δ–fine if

[α_{j−1}, α_{j}] ⊂ (τ_{j} − δ(τ_{j}), τ_{j} + δ(τ_{j})) for j = 1, …, k.

Definition. Assume that functions A, C : [0, b] → L(X) and x : [0, b] → X are given.

We say that the Stieltjes integral ∫_{0}^{b} d[A(s)]x(s) exists if there is an element J ∈ X such that for every ε > 0 there is a gauge δ on [0, b] such that for

S(dA, x, D) = ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x(τ_{j})

we have

∥S(dA, x, D) − J∥_{X} < ε

provided D is a δ–fine P –partition of [0, b]. We denote J = ∫_{0}^{b} d[A(s)]x(s).

Analogously we say that the Stieltjes integral ∫_{0}^{b} d[A(s)]C(s) exists if there is an element J ∈ L(X) such that for every ε > 0 there is a gauge δ on [0, b] such that for

S(dA, C, D) = ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]C(τ_{j})

we have

∥S(dA, C, D) − J∥_{L(X)} < ε

provided D is a δ–fine P –partition of [0, b].

Similarly we can define the Stieltjes integral ∫_{0}^{b} A(s)d[C(s)] using Stieltjes integral sums of the form

S(A, dC, D) = ∑_{j=1}^{k} A(τ_{j})[C(α_{j}) − C(α_{j−1})].

Assume that U, V : [0, ∞) → L(X) and x : [0, ∞) → X are given and define the convolutions

(U ∗ x)(t) = ∫_{0}^{t} d[U(s)]x(t − s)

and

(U ∗ V)(t) = ∫_{0}^{t} d[U(s)]V (t − s)

for t ∈ [0, ∞).

Let us denote by BSV_{loc}([0, ∞), L(X)) the set of all U : [0, ∞) → L(X) for which U ∈ BSV ([0, b], L(X)) for every b > 0.

In [8] it was shown that if U, V ∈ G([0, ∞), L(X)) ∩ (B)BV_{loc}([0, ∞), L(X)) and x ∈ G([0, ∞), X) then the convolutions (U ∗ x)(t) and (U ∗ V)(t) are well defined for every t ∈ [0, ∞) when the abstract Perron-Stieltjes integral is used.

It was also shown in [8] that

(3)  ∥(U ∗ V)(t)∥_{L(X)} ≤ ∥U∥_{SV} .∥V ∥_{SV}

holds for every t ≥ 0.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 4 2. Lemma. Assume that

U ∈ G([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X)), f ∈ G([0, ∞), X)

and that η ≥ 0 is given.

Then the integral ∫_{0}^{b} d[U(s)]e^{−ηs}f(s) ∈ X exists for every b > 0 and

(4)  ∥ ∫_{0}^{b} d[U(s)]e^{−ηs}f(s)∥_{X} ≤ s var_{[0,b]}^{(η)}(U) · sup_{s∈[0,b]} ∥f(s)∥_{X}

holds.

Proof. The existence of the integral ∫_{0}^{b} d[U(s)]e^{−ηs}f(s) is clear because the function e^{−ηs}f(s) is regulated on [0, ∞) (c.f. [6, Proposition 15]).

Assume that b > 0 is fixed. By the existence of the integral, for any ε > 0 there is a gauge δ on [0, b] such that for every δ- fine P - partition

D = {0 = α_{0}, τ_{1}, α_{1}, τ_{2}, …, α_{k−1}, τ_{k}, α_{k} = b}

of [0, b] the inequality

∥ ∫_{0}^{b} d[U(s)]e^{−ηs}f(s) − ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ητ_{j}}f(τ_{j})∥_{X} < ε

holds. Hence

(5) ∥ ∫_{0}^{b} d[U(s)]e^{−ηs}f(s)∥_{X} < ε + ∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ητ_{j}}f(τ_{j})∥_{X}.

Let us choose a fixed δ- fine P - partition D of [0, b] for which α_{j−1} < τ_{j} for every j = 1, …, k. Then

∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ητ_{j}}f(τ_{j})∥_{X} =

= ∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ηα_{j−1}}e^{−η(τ_{j}−α_{j−1})}f(τ_{j})∥_{X} =

= ∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ηα_{j−1}} \frac{e^{−η(τ_{j}−α_{j−1})}f(τ_{j})}{∥f(τ_{j}∥_{X}}∥f(τ_{j})∥_{X}∥_{X}

and we have

∥\frac{e^{−η(τ_{j}−α_{j−1})}f(τ_{j})}{∥f(τ_{j})∥_{X}}∥_{X} ≤ 1

for j = 1, …, k.

Hence

∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ηα_{j−1}} \frac{e^{−η(τ_{j}−α_{j−1})}f(τ_{j})}{∥f(τ_{j})∥_{X}}∥f(τ_{j}∥_{X}∥_{X} ≤

≤ sup_{j=1,...,k} ∥f(τ_{j}∥_{X} · ∥ ∑_{j=1}^{k}[U(α_{j}) − U(α_{j−1})]e^{−ηα_{j−1}} \frac{e^{−η(τ_{j}−α_{j−1})}f(τ_{j})}{∥f(τ_{j})∥_{X}}∥_{X} ≤

≤ sup_{s∈[0,b]} ∥f(s)∥_{X} · s var_{[0,b]}^{(η)}(U)

and this together with (5) gives the result.

3. Proposition. Assume that U, V ∈ G([0, ∞), L(X))∩BSV_{loc}([0, ∞), L(X)) and that U(0) = V (0) = 0.

Then the convolution

(U ∗ V)(t) = ∫_{0}^{t} d[U(s)]V (t − s) ∈ L(X)

is well defined for every t ∈ [0, ∞), and for every b > 0, η ≥ 0 the inequality

(6)  s var_{[0,b]}^{(η)}(U ∗ V) ≤ s var_{[0,b]}^{(η)}(U).s var_{[0,b]}^{(η)}(V)

holds.

Proof.

Define

Ṽ (σ) = V (σ) for σ ≥ 0

and

Ṽ (σ) = 0 for σ < 0.

Assume that b ≥ 0 and let 0 = α_{0} < α_{1} < ⋯ < α_{k} = b be an arbitrary partition of [0, b].

Using the definition of Ṽ we have for every α ∈ [0, b] the equality

∫_{0}^{α} d[U(s)]V (α − s) = ∫_{0}^{b} d[U(s)]Ṽ (α − s)

and therefore we obtain for any choice of x_{j} ∈ X, ∥x_{j}∥_{X} ≤ 1, j = 1, …, k the equalities

∥ ∑_{j=1}^{k}[(U ∗ V)(α_{j}) − (U ∗ V)(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} =

= ∥ ∑_{j=1}^{k}[∫_{0}^{α_{j}} d[U(s)]V (α_{j} − s) − ∫_{0}^{α_{j−1}} d[U(s)]V (α_{j−1} − s)]x_{j}e^{−ηα_{j−1}}∥_{X} =

= ∥ ∑_{j=1}^{k} ∫_{0}^{b} d[U(s)][Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−ηα_{j−1}}∥_{X} =

(7)  = ∥ ∫_{0}^{b} d[U(s)]e^{−ηs} ∑_{j=1}^{k}[Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−η(α_{j−1}−s)}∥_{X}.

The function

s ↦ ∑_{j=1}^{k}[Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−η(α_{j−1}−s)} ∈ X

is evidently regulated on [0, b] because V ∈ G([0, b], L(X)) and therefore by Lemma 2 we obtain

∥ ∫_{0}^{b} d[U(s)]e^{−ηs} ∑_{j=1}^{k}[Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−η(α_{j−1}−s)}∥_{X} ≤

≤ s var_{[0,b]}^{(η)}(U) · sup_{s∈[0,b]} ∥ ∑_{j=1}^{k}[Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−η(α_{j−1}−s)}∥_{X}.

On the other hand, for every s ∈ [0, b] we have

∥ ∑_{j=1}^{k}[Ṽ (α_{j} − s) − Ṽ (α_{j−1} − s)]x_{j}e^{−η(α_{j−1}−s)}∥_{X} ≤ s var_{[0,b]}^{(η)}(V)

and this gives

∥ ∑_{j=1}^{k}[(U ∗ V)(α_{j}) − (U ∗ V)(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} ≤

≤ s var_{[0,b]}^{(η)}(U) · s var_{[0,b]}^{(η)}(V)

and by the definition also

s var_{[0,b]}^{(η)}(U ∗ V) ≤ s var_{[0,b]}^{(η)}(U) · s var_{[0,b]}^{(η)}(V).

This inequality yields by (2) also that

s var_{[0,b]}(U ∗ V) < ∞,

i.e. that

(8)  U ∗ V ∈ BSV_{loc}([0, ∞), L(X))

because b ≥ 0 can be taken arbitrarily.

Analogously it can be proved that the following statement holds.

4. Proposition. Assume that U, V ∈ BV_{loc}([0, ∞), L(X)) and that U(0) = V (0) = 0.

Then the convolution

(U ∗ V)(t) = ∫_{0}^{t} d[U(s)]V (t − s) ∈ L(X)

is well defined for every t ∈ [0, ∞) and for every b > 0, η ≥ 0 the inequality

(9)  var_{[0,b]}^{(η)}(U ∗ V) ≤ var_{[0,b]}^{(η)}(U) · var_{[0,b]}^{(η)}(V)

holds.

In [8] the following result has been proved.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 7 5. Proposition. For every b > 0 the set of all U : [0, b] → L(X) with U ∈ C([0, b], L(X)) ∩ BSV ([0, b], L(X)) and U(0) = 0 is a Banach algebra with the Stieltjes convolution U ∗ V as multiplication and s var_{[0,b]}(U) as the norm.

See [8 ,Theorem 15].

6. Remark. Unfortunately a statement of the form:

For every b > 0 the set of all U : [0, b] → L(X) with U ∈ BV ([0, b], L(X)) and U(0) = 0 is a Banach algebra with the Stieltjes convolution

(U ∗ V)(t) = ∫_{0}^{t} d[U(s)]V (t − s)

as multiplication and var_{[0,b]}(U) as the norm.

does not hold because in this case the multiplication given by the convolution is not associative.

It was also shown [8 ,Proposition 12 and 13] that the following two statements hold.

7. Proposition. If U, V ∈ BV_{loc}([0, ∞), L(X)) and U(0) = V (0) = 0 then U ∗V ∈ BV_{loc}([0, ∞), L(X)).

8. Proposition. If U, V ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X)) and U(0) = V (0) = 0 then U ∗ V ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X)).

9. Lemma. Assume that A ∈ BSV ([0, b], L(X)) for some b > 0. Then for every η ≥ 0 and c ∈ (0, b] we have

(10)  s var_{[0,b]}^{(η)}(A) ≤ s var_{[0,c]}^{(η)}(A) + e^{−ηc}s var_{[c,b]}^{(η)}(A).

Proof. Assume that D is a partition of [0, b] given by the points

0 = α_{0} < α_{1} < ⋯ < α_{k} = b

and that x_{j} ∈ X with ∥x_{j}∥_{X} ≤ 1 for j = 1, …, k. Then there is an index l = 1, …, k such that c ∈ (α_{l−1}, α_{l}] and

∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}} =

= ∑_{j=1}^{l−1}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}} + [A(α_{l}) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}}+

+ ∑_{j=l+1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}.

Taking into account that

[A(α_{l}) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}} =

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 8

= [A(α_{l}) − A(c)]x_{l}e^{−ηα_{l−1}} + [A(c) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}}

we obtain

∥ ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} =

= ∥ ∑_{j=1}^{l−1}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}} + [A(c) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}}+

+[A(α_{l}) − A(c)]x_{l}e^{−ηα_{l−1}} + ∑_{j=l+1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} ≤

≤ ∥ ∑_{j=1}^{l−1}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}} + [A(c) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}}∥_{X}+

+∥[A(α_{l}) − A(c)]x_{l}e^{−ηα_{l−1}} + ∑_{j=l+1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X}.

For the first term on the right hand side of this inequality we have evidently

∥ ∑_{j=1}^{l−1}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}} + [A(c) − A(α_{l−1})]x_{l}e^{−ηα_{l−1}}∥_{X} ≤

≤ s var_{[0,c]}^{(η)}(A)

and for the second

∥[A(α_{l}) − A(c)]x_{l}e^{−ηα_{l−1}} + ∑_{j=l+1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} =

= ∥[A(α_{l}) − A(c)]x_{l}e^{−ηα_{l−1}} + e^{−ηc} ∑_{j=l+1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−η(α_{j−1}−c)}∥_{X} ≤

≤ e^{−ηc}V_{c}^{b}(η, A, D_{+}) ≤ e^{−ηc} var_{[c,b]}^{(η)}(A)

(D_{+} is the partition of [c, b] given by the points c ≤ α_{l} < ⋯ < α_{k} = b). Hence

∥ ∑_{j=1}^{k}[A(α_{j}) − A(α_{j−1})]x_{j}e^{−ηα_{j−1}}∥_{X} ≤

≤ s var_{[0,c]}^{(η)}(A) + e^{−ηc} var_{[c,b]}^{(η)}(A)

and the lemma is proved.

Similarly it can be shown that the following statement is valid. EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 9 10. Lemma. Assume that A ∈ BV ([0, b], L(X)) for some b > 0. Then for every η ≥ 0 and c ∈ (0, b] we have

(11)  var_{[0,b]}^{(η)}(A) ≤ var_{[0,c]}^{(η)}(A) + e^{−ηc} var_{[c,b]}^{(η)}(A).

11. Proposition. If A ∈ C([0, b], L(X)) ∩ BSV ([0, b], L(X)), A(0) = 0 and if there is a c ∈ (0, b] such that

(12)  s var_{[0,c]}(A) < 1,

then there exists a unique R ∈ C([0, b], L(X)) ∩ BSV ([0, b], L(X)) with R(0) = 0 such that

(13)  R(t) − ∫_{0}^{t} d[A(s)]R(t − s) = A(t), t ∈ [0, b]

and

(14)  R(t) − ∫_{0}^{t} d[R(s)]A(t − s) = A(t), t ∈ [0, b].

Proof. By Lemma 9, (2) and (12) we have

s var_{[0,b]}^{(η)}(A) ≤ s var_{[0,c]}^{(η)}(A) + e^{−ηc}s var_{[c,b]}^{(η)}(A) ≤
≤ s var_{[0,c]}(A) + e^{−ηc}s var_{[c,b]}(A)

and this yields that taking η > 0 sufficiently large we get

(15)  s var_{[0,b]}^{(η)}(A) < 1.

Let us now define A_{0}(t) = A(t) and A_{n+1}(t) = (A ∗ A_{n})(t), t ∈ [0, b] and put

(16)  R(t) = ∑_{n=0}^{∞} A_{n}(t).

By (6) from Proposition 3 we get the inequalities

s var_{[0,b]}^{(η)}(A_{n}) ≤ (s var_{[0,b]}^{(η)}(A))^{n}, n ∈ ℕ.

Since (15) holds, this implies the convergence of the series (16) in BSV ([0, b], L(X)) and by Proposition 8 also the continuity of its sum R(t), i. e. R ∈ C([0, b], L(X)) ∩ BSV ([0, b], L(X)) and clearly also R(0) = 0.

By the definitions we have

((∑_{n=0}^{N} A_{n}) ∗ A)(t) = (A ∗ (∑_{n=0}^{N} A_{n}))(t) = ∑_{n=1}^{N+1} A_{n}(t) = ∑_{n=0}^{N+1} A_{n}(t) − A(t)

for every N ∈ ℕ and passing to the limit for N → ∞ we obtain (13) and (14). Concerning the uniqueness let us assume that

Q ∈ C([0, b], L(X)) ∩ BSV ([0, b], L(X))

also satisfies (13) and (14). Then

Q − A ∗ Q = A and R − R ∗ A = A.

Using the associativity of convolution products we get

R = A + R ∗ A = A + R ∗ (Q − A ∗ Q) = A + R ∗ Q − R ∗ A ∗ Q =
= A + (R − R ∗ A) ∗ Q = A + A ∗ Q = Q

and the unicity is proved.

12. Corollary. Assume that A : [0, ∞) → L(X), A(0) = 0. If

A ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X))

and if there is a c ∈ (0, b] such that

s var_{[0,c]}(A) < 1

then there exists a unique R : [0, ∞) → L(X),

R ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X))

with R(0) = 0 such that for every b > 0 (13) and (14) hold.

R ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X)) given in Corollary 12 is called the resolvent of A ∈ C([0, ∞), L(X)) ∩ BSV_{loc}([0, ∞), L(X)).

13. Theorem. Assume that A : [0, ∞) → L(X), A(0) = 0, A ∈ C([0, ∞), L(X))∩ BSV_{loc}([0, ∞), L(X)) and that there is a c ∈ (0, b] such that

s var_{[0,c]}(A) < 1.

Then for every F ∈ G([0, ∞), L(X)) and f ∈ G([0, ∞), X) there exist unique solutions X : [0, ∞) → L(X) and x : [0, ∞) → X for the abstract renewal equations

(17)  X(t) = F (t) + ∫_{0}^{t} d[A(s)]X(t − s)

and

(18)  x(t) = f(t) + ∫_{0}^{t} d[A(s)]x(t − s),

respectively, and the relations

(19)  X(t) = F (t) + ∫_{0}^{t} d[R(s)]F (t − s),

(20)  x(t) = f(t) + ∫_{0}^{t} d[R(s)]f(t − s)

hold for t > 0 where R is the resolvent of A.

Proof. The expression on the right hand side of (19) is well defined and it reads X(t) = F (t) + (R ∗ F)(t).

Hence using (13) we obtain

A∗X(t) = A∗F (t)+(A∗(R∗F))(t) = ((A+A∗R)∗F)(t) = (R∗F)(t) = X(t)−F (t)

and this yields that by (19) a solution of (17) is given.

The analogous result for (18) can be shown similarly.

For renewal equations see also the excellent book [3].

The author expresses his thanks to the referee for pointing out that the statement given in Remark 6 is not valid.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 26, p. 11

References

[1] O. Diekmann, M. Gyllenberg, H. R. Thieme, Perturbing semigroups by solving Stieltjes renewal equations, Differential Integral Equations 6 (1993), 155–181.

[2] O. Diekmann, M. Gyllenberg, H. R. Thieme, Perturbing evolutionary systems by step responses on cumulative outputs, Differential Integral Equations 8 (1995), 1205–1244.

[3] G. Grippenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.

[4] Ch. S. H”nig, Volterra – Stieltjes Integral Equations, North – Holland Publ. Comp., Amsterdam, 1975.

[5] J. Kurzweil, Nichtabsolut konvergente Integrale, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980.

[6] Š. Schwabik, Abstract Perron – Stieltjes integral, Mathematica Bohemica 121 (1996), 425-447.

[7] Š. Schwabik, A note on integration by parts for abstract Perron - Stieltjes integrals, Mathematica Bohemica (to appear).

[8] Š. Schwabik, Operator-valued functions of bounded semivariation and convolutions, Mathematica Bohemica (to appear).

Žitná 25, 115 67 Praha 1, Czech Republic

   E-mail address: schwabik@math.cas.cz