EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5560
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Approximation Theorems for Exponentially Bounded
K-Convoluted C-Cosine Functions

Youssef Bajjou1,∗, Abdelkhalek El Amrani1, Aziz Blali3

1 Department of Mathematics, Dhar El Mahraz Faculty of Sciences, Sidi Mohamed Ben
Abdellah University, Atlas Fez, Morocco
2 Department of Mathematics, Higher Normal School, Sidi Mohamed Ben Abdellah
University, B.P. 5206 Bensouda-Fez, Morocco

Abstract. Let C : E → E be a bounded linear operator on a complex Banach space E and
K : [0,+∞[→ C a locally integrable function. The aim of this paper, based on the theory of K-
convoluted C-cosine functions, is to study the approximation theorem for K-convoluted C-cosine
functions by showing the relation between the convergence of the sequence of C-resolvent and the
exponentially bounded sequence of K-convoluted C-cosine functions.

2020 Mathematics Subject Classifications: 46A32, 47D09, 47A58, 60J35

Key Words and Phrases: K-convoluted C-cosine functions, C-resolvent, Approximation

1. Introduction

Throughout this paper E denote a non-trivial complex Banach space, L(E) denotes
the Banach algebra of bounded linear operators from E into E, C is an injective element
of L(E). For a linear operator A acting on E, D(A), N(A), R(A) and ρC(A), denotes its
domain (equipped with the graph norm), kernel, range and the C-resolvent set ofA, defined
by ρC(A) := {λ ∈ C | R(C) ⊆ R(λI−A) and λI−A is injective in B(E)} and if λ ∈ ρC(A)
then we denoted by RC(λ,A) the C-resolvent defined by RC(λ,A) = (λI − A)−1C. If
t ∈ R, ⌊t⌋ = sup{n ∈ Z, n ≤ t} denotes the integer part of t. K is a complex-valued locally
integrable function in [0,+∞[ (ie K ∈ L1

loc([0,+∞[)), not identical to zero such that:

• (P): K is Laplace transformable, that is to say there exists β ∈ R so that L(K)(λ) =∫ +∞
0 e−λtK(t)dt < +∞ for all λ ∈ C with Re(λ) > β. Put abs(K) := inf{Re(λ) :
L(K)(λ) < +∞}.

• (Q): For all λ > abs(K), L(K)(λ) ̸= 0.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5560

Email addresses: youssefbajjou2017@gmail.com (Y. Bajjou),
abdelkhalek.elamrani@usmba.ac.ma (A. El Amrani), aziz.blali@usmba.ac.ma (A. Blali)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 2 of 15

• (R): 0 ∈ supp(K) (According to Titchmarsh’s theorem [[2]], for every φ ∈ C([0,+∞[,
the assumption
for all t ∈ [0,+∞[,

∫ t
0 K(t− s)φ(s)ds = 0 implies φ ≡ 0).

For example the following function is a kernel:

K(t) :=
1

2
√
2πt3

e
−1
4t if t > 0 and K(0) = 0

see [1]. We can define on [0,+∞[, the absolutely continuous function by

for all t ≥ 0, Θ(t) :=

∫ t

0
K(s)ds,

then
for all t ≥ 0, Θ′(t) = K(t) a.e t ∈ [0,+∞[.

We let
l∞(E) = {(xk)k∈N : xk ∈ E and sup

k∈N
| xk |< +∞}

the Banach space equipped with the norm

∥ (xk)k∈N ∥= sup
k∈N

| xk |

for all sequence x = (xk)k∈N ∈ l∞(E) and c(E), the closed subspace of l∞(E), defined by

c(E) = {(xk)k∈N : xk ∈ E and lim
k→∞

xk exists}.

See [1] for more details.
In this work we will use the theory of integration in the sense of Bochner.

2. K-convoluted C-cosine function

A strongly continous operator family (C(t))t≥0 such that:

• For all t ≥ 0 C(t)A ⊆ AC(t),

• For all t ≥ 0 C(t)C ⊆ CC(t),

• For all x ∈ E and t ≥ 0∫ t
0 (t− s)C(s)xds ∈ D(A) and A

∫ t
0 C(s)xds = C(t)x−Θ(t)Cx,

• There exist M ≥ 1, there exist ω ≥ 0 : for all t ≥ 0, ∥ C(t) ∥≤ Meωt,



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 3 of 15

is called an exponentially bounded K−convoluted C−cosine function with subgenerator
A. We can prove that CA ⊂ AC see [9]. For example, if K(t) = tα−1

Γ(α) for some α ≥
0 a K−convoluted C−cosine function on E is called an α−times integrated C−cosine
function on E see [10] and [13] for more details. We say that (C(t))t≥0 is non-degenerate
if additionnally C(t)x = 0 for all t ≥ 0 implies that x = 0, since C is injective then each
K−convoluted C-cosine function is no degenerate (see [11], [3], [6], [12], [8], [7] and [4]).
If (C(t))t≥0 is K−convoluted C-cosine function then the following formulae holds:

2C(t)C(s)x = {
∫ t+s

0
−
∫ t

0
−
∫ s

0
}K(t+ s− r)C(r)xdr +

∫ t

|t−s|
K(s− t+ r)C(r)Cx (1)

+

∫ s

|t−s|
K(t− s+ r)C(r)Cxdr +

∫ |t−s|

0
K(| t− s | +r)C(r)Cxdr

for all t, s ≥ 0 and x ∈ E, see chapter 2 theorem 2.1.13 of [5].
For a K−convoluted C−cosine function (C(t))t≥0, we define its integral generator  :
D(Â) ⊂ E → E by

D(Â) = {x ∈ E : (∃yx ∈ E) : C(t)x−Θ(t)Cx =

∫ t

0
(t− s)C(s)yxds for all t ≥ 0}

and Âx = yx for all x ∈ D(Â). Â is a closed operator which is an extension of any
subgenerator of (C(t))t≥0, C

−1AC = Â and (C(t))t≥0 is uniquely determined by one of
its subgenerators see [9]. In the rest of this part, let M > 0, ω ≥ max(0, abs(K)) and let’s
put ω1 = max(ω, abs(K)). Suppose that (A,D(A)) is closed linear operator and (C(t))t≥0

is strongly continuous operator family and for all t ≥ 0 ∥ C(t) ∥≤ Meωt then we have the
following useful properties:

(i) • (i) Assume thatA is a subgenerator of an exponentially bounded,K−convoluted
C−cosine function (C(t))t≥0 then

{λ2 : ℜ(λ) > ω1 L(K)(λ) ̸= 0} ⊂ ρC(A), (2)

and

λ(λ2 −A)Cx =
1

L(K)(λ)

∫ +∞

0
e−λtC(t)xdt, x ∈ E,ℜ(λ) > ω1, L(K)(λ) ̸= 0.

(3)
For more details see [9] and [5].

• (ii) Suppose that the family (C(t))t≥0 satisfies the two conditions (2)-(3), then
(C(t))t≥0 is an exponentially bounded, K−convoluted C−cosine function with
subgenerator A. For more details see [9] and [5].

• (iii) Assume that (2)-(3) hold only for real values of λ’s, then (C(t))t≥0 is still
an exponentially bounded, K−convoluted C−cosine function with subgenerator
A. For more details see [[9]].



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 4 of 15

(ii) Put for all x ∈ E and λ > ω, Rλ2x := 1
λL(K)(λ)

∫ +∞
0 e−λtC(t)xdt. Then for all

λ, µ > ω and all x ∈ E, (λ2 − µ2)Rλ2Rµ2x = Rµ2Cx − Rλ2Cx if the formula (1)
holds for all x ∈ E and s ≥ 0. For more details see [13].

Remark 1. (i) If for all t ≥ 0, CC(t) = C(t)C then for all λ > ω, CRλ2 = Rλ2C.
Indeed for all x ∈ E and all λ > ω we have:

Rλ2Cx =
1

λL(K)(λ)

∫ +∞

0
e−λtC(t)Cxdt

=
1

λL(K)(λ)

∫ +∞

0
Ce−λtC(t)xdt

= C
1

λL(K)(λ)

∫ +∞

0
e−λtC(t)xdt

= CRλ2x.

(ii) We assumed that for all λ, µ > ω and all x ∈ E,

(λ2 − µ2)Rλ2Rµ2 = Rµ2Cx−Rλ2Cx,

then for all λ, µ > ω Rλ2Rµ2 = Rµ2Rλ2 . Indeed for all x ∈ E and all λ, µ > ω we
have

0 = (R2
λCx−R2

µCx) + (R2
µCx−R2

λCx)

= (µ2 − λ2)Rλ2Rµ2x+ (λ2 − µ2)Rµ2Rλ2

= (λ2 − µ2)(Rµ2Rλ2x−Rλ2Rµ2x)

(iii) We assumed that CC(.) = C(.)C and for all λ, µ > ω and all x ∈ E,

(λ2 − µ2)Rλ2Rµ2 = Rµ2Cx−Rλ2Cx,

then

• N(Rλ2) is independent of λ > ω. Indeed for all λ > ω, all x ∈ E such that Rλ2x =
0 and for all µ > ω we have

0 = CRλ2x

= Rλ2Cx

= (Rλ2Cx−Rµ2Cx) +Rµ2Cx

= (µ2 − λ2)Rλ2Rµ2x+Rµ2Cx

= (µ2 − λ2)Rµ2Rλ2x+Rµ2Cx

= 0 +Rµ2Cx

= Rµ2Cx

= CRµ2x,

hence Rµ2x = 0 since C is injective.



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 5 of 15

• If R(Rλ2) ⊂ R(C) then R(Rλ2) is independent of λ > ω. Indeed for all λ >
ω, all y ∈ R(Rλ2) such that y = Rλ2x and for all µ > ω we have

CRµ2(x+ (µ2 − λ2)C−1y) = CRµ2x+ (µ2 − λ2)CRµ2C−1y

= R2
µCx+ (µ2 − λ2)R2

µCC−1y

= Rµ2Cx+ (µ2 − λ2)Rµ2y

= Rµ2Cx+ (µ2 − λ2)Rµ2Rλ2x

= Rµ2Cx+ (Rλ2Cx−Rµ2Cx)

= Rλ2Cx

= CRλ2x

= Cy,

hence y = Rµ2(x+ (µ2 − λ2)C−1y) ∈ R(Rµ2) since C is injective.

• If R(Rλ2) ⊂ R(C) and there exists µ > ω such that N(Rµ2) = {0} then there
is a linear operator (A,D(A)) such that RC(λ,A) = Rλ2. Indeed

for all λ, µ > ω, for all y ∈ R(Rλ2), there is a unique xλ2 , xµ2) ∈ E2 : y = Rλ2xλ2 = Rµ2xµ2 .

On the other hand, if we put

W = Rλ2Rµ2((µ2y − Cxµ2)− (λ2y − Cxλ2)),

then we have :

W = Rλ2Rµ2((µ2 − λ2)y − C(xµ2 − xλ2))

= (µ2 − λ2)Rλ2Rµ2y − CRλ2Rµ2(xµ2 − xλ2)

= (Rλ2Cy −Rµ2Cy)− (CRλ2y − CRµ2y)

= (Rλ2Cy −Rµ2Cy)− (Rλ2Cy −Rµ2Cy)

= 0

It is (µ2y−Cxµ2) = (λ2y−Cxλ2) since Rλ2Rµ2 is injective, so we cane defined
the operator (A,D(A)) by D(A) = R(Rµ2) and for all y ∈ R(Rλ2), Ay =

λ2y − CR−1
λ2 y, since Rλ2 ∈ B(E), moreover for all y ∈ R(Rλ2) we have

CR−1
λ2 (y) = λ2y −Ay

= (λ2I −A)y

as result Rλ2 = (λ2I −A)−1C = RC(λ
2, A).

(iv) If (C(t))t≥0 is K-convoluted C−cosine function with subgenerator (A,D(A)) such
that for all t ≥ 0, ∥ C(t) ∥≤ Meωt, then for all λ > ω, Rλ2 = RC(λ

2, A).



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 6 of 15

3. Main results

Theorem 1. Let (C(t))t≥0 be a K−convoluted C−cosine functions with subgenerator A.
For each n ∈ N, let (Cn(t))t≥0 a K−convoluted C−cosine function with subgenerator
(An,D(An)) and suppose that there exist ω ≥ 0 and M > 0 such that for all t ≥ 0
and all x ∈ E, ∥ C(t)x ∥≤ Meωt and for all n ∈ N ∥ Cn(t)x ∥≤ Meωt. If we put
ω1 = max(ω + 1, abs(K)), then the following statements are equivalent:

(i) There exist λ0 > ω1) : for all x ∈ E lim
n→+∞

RC(λ
2
0, An)x = RC(λ

2
0, A)x and

(Cn(.)x)n∈N is equicontinuous.

(ii) There exist λ0 > ω1 : for all y ∈ R(C)) lim
n→+∞

(λ2I − An)
−1y = (λ2I − A)−1y and

for all x ∈ E, (Cn(.)x)n∈N is equicontinuous.

(iii) For all λ > ω1, for all y ∈ R(C)), lim
n→+∞

(λ2I −An)
−1y = (λ2I −A)−1y and for all

x ∈ E, (Cn(.)x)n∈N is equicontinuous.

(iv) For all λ > ω1, for all, x ∈ E, lim
n→+∞

RC(λ
2, An)x = RC(λ

2, A)x and (Cn(.)x)n∈N

is equicontinuous.

(v) For all t ≥ 0, for allx ∈ E, lim
n→+∞

Cn(t)x = C(t)x, the convergence is uniform on

any compact of [0,+∞[.

Proof. 1 ⇒ 2 |
Like RC(λ

2
0, An) = (λ2

0I − An)
−1C and RC(λ

2
0, A) = (λ2I − A)−1C, then the proof is

obvious.
2 ⇒ 3 |
Let’s pose

U = {λ > ω1 : L(K)(λ) ̸= 0} (=]ω1,+∞[)

and
V = {λ ∈ U : for all y ∈ R(C), lim

n→+∞
(λ2I −An)

−1y = (λ2I −A)−1y}.

According to the statements of (2), V is a nonempty set.
Let be λ ∈ V fixed and n ∈ N, then for µ in the open set Oλ, where

Oλ := {µ ∈ U :∥ (µ2 − λ2)(λ2I −An)
−1 ∥< 1

4
},

we have ∥ (µ2 − λ2)(λ2I −An)
−1 ∥< 1

4 < 1 so I − (µ2 − λ2)(λ2I −An)
−1 is invertible, the

series
∑
k≥0

(
(µ2 − λ2)(λ2I −An)

−1
)k

is uniformly convergent and

(I − (µ2 − λ2)(λ2I −An)
−1)−1 =

+∞∑
k=0

{(µ2 − λ2)(λ2I −An)
−1}k.



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 7 of 15

But

µ2I −An = (µ2 − λ2)I + (λ2I −An)

=
(
(µ2 − λ2)(λ2I −An)

−1 + I
)
(λ2I −An)

=
(
(I − (µ2 − λ2)(λ2I −An)

−1
)
(λ2I −An),

so

(µ2I −An)
−1 = (λ2I −An)

−1
(
(I − (µ2 − λ2))(λ2I −An)

−1
)−1

= (λ2I −An)
−1

+∞∑
k=0

(
(µ2 − λ2)(λ2I −An)

−1
)k

=
+∞∑
k=0

(µ2 − λ2)k
(
(λ2I −An)

−1
)k+1

,

and for all y ∈ R(C)

lim
n→+∞

(µ2I −An)
−1y = lim

n→+∞

+∞∑
k=0

(µ2 − λ2)k
(
(λ2I −An)

−1
)k+1

y

=
+∞∑
k=0

lim
n→+∞

(µ2 − λ2)k
(
(λ2I −An)

−1
)k+1

y

=

+∞∑
k=0

(µ2 − λ2)k
(
(λ2I −A)−1

)k+1
y

= (λ2I −A)−1
+∞∑
k=0

(µ2 − λ2)
(
(λ2I −A)−1

)k
y

= (µ2I −A)−1y

(because lim
n→+∞

∥ (µ2 − λ2)(λ2I −An)
−1 ∥< 1).

So we can conclude that for all λ ∈ V there exists an open set Oλ such that Oλ ⊂ V ;
therefore V is an open set.
Let be (λk)k∈N a sequence in V such that lim

k→+∞
λk = λ and λ ∈ U, let’s show that λ ∈ V.

Like for n ∈ N, the open set O′
λ := {µ ∈ U :∥ (µ2−λ2)(µ2I−An)

−1 ∥< 1
4} contains λ ther-

fore there exists λk0 ∈ U such that λk0 ∈ O′
λ; but (λ

2I−An)
−1 =

+∞∑
k=0

(
(λ2 − λ2

k0)
k(λ2

k0I −An)
−1

)k+1

whose series converges uniformly on O′
λ and as the function β 7−→ (β2I − An)

−1 is con-
tinuous from ]ω1,+∞[ to B(E) then

lim
n→+∞

(λ2I −An)
−1 = lim

n→+∞

+∞∑
k=0

(
(λ2 − λ2

k0)
k(λ2

k0I −An)
−1

)k+1



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 8 of 15

=
+∞∑
k=0

lim
n→+∞

(λ2 − λ2
k0)

k
(
(λ2

k0I −An)
−1

)k+1

=
+∞∑
k=0

(λ2 − λ2
k0)

k
(
(λ2

k0I −A)−1
)k+1

= (λ2I −A)−1,

the last equality is due to the following inequality

lim
n→+∞

∥ (λ2 − λ2
k0)(λ

2
k0I −An)

−1 ∥≤ 1

4
< 1,

so λ ∈ V. Therefore V is relatively closed from U.
Finally the set V is both an open and a closed of the connected set U, whence V = U.
3 ⇒ 4 |
Obvious.
4 ⇒ 5 |
Suppose that the conditions of statement 4 are satisfied.
Let x ∈ E be fixed. We define, for each n ∈ N, the following functions:
fn : R+ → E, t 7→ Cn(t)x.
f : R+ → l∞(E), t 7→ (fn(t))n∈N.
Fn :]ω1,+∞[→ E, λ 7→ λL(K)(λ)RC(λ

2, An)x.
F :]ω1,+∞[→ l∞(E), λ 7→ (Fn(λ)n∈N).
gn : R+ → E, t 7→

∫ t
0 (t− s)fn(s)ds.

g : R+ → E, t 7→ (gn(t))n∈N.

(i) • a) f is well defined.
Let t be a positive real.
We have

for all n ∈ N, ∥ fn(t) ∥≤ M ∥ x ∥ eω1t,

so for all t ≥ 0
∥ (fn(t))n∈N ∥∞≤ M ∥ x ∥ eω1t < +∞,

therfore the function f is well defined, and since the sequence (fn)n∈N is equicon-
tinuous, the function f is continuous.

• b) F has value in c(E).
Let λ in ]ω1,+∞[.
We now have Theorem 1, for all n ∈ N,

Fn(λ) = λL(K)(λ)RC(λ
2, An)x

=

∫ +∞

0
e−λtC(t)xdt

=

∫ +∞

0
e−λtfn(t)dt,



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 9 of 15

so For all n ∈ N, ∥ Fn(λ) ∥≤ M∥x∥
λ−ω , therefore

∥ F (λ) ∥∞ = ∥ (Fn(λ))n∈N ∥∞

≤ M ∥ x ∥
λ− ω1

< +∞,

and by hypothesis

lim
n→+∞

Fn(λ) = lim
n→+∞

λL(K)(λ)RC(λ
2, An)x = λL(K)(λ)RC(λ

2, A)x

which give F (λ) ∈ c(E).

• c) F ∈ C∞(]ω1,+∞[, l∞(E)) and for all k ∈ N for all λ > ω1 F (k)(λ) ∈ c(E).
Let t and h be a positive reals. For all n ∈ N we have

∥ gn(t) ∥ ≤
∫ t

0
(t− s) ∥ fn(s) ∥ ds

≤ M ∥ x ∥
ω

teωt

≤ M ∥ x ∥
ω

e(ω+1)t

≤ M ∥ x ∥
ω

eω1t.

So ∥ g(t) ∥∞=∥ (gn(t))n∈N ∥∞≤ M∥x∥
ω eω1t < +∞

which gives that g(t) ∈ l∞(E). And for all n ∈ N:

∥ gn(t+ h)− gn(t) ∥ = ∥
∫ t+h

t
(t− s)fn(s)ds+ h

∫ t+h

0
fn(s)ds ∥

≤ h{
∫ t+h

t
∥ fn(s) ∥ ds+

∫ t+h

0
∥ fn(s) ∥ ds}

≤ 2hM ∥ x ∥
ω1

eω1(t+h).

So ∥ g(t+ h)− g(t) ∥≤ 2hM∥x∥
ω eω(t+h), from where g is continuous at t. So the

function g is well defined and continuous.
On the other hand, the function Id : R+ → R+, t 7→ t is continuous and for all
λ > 0 we have

L(Id)(λ) =

∫ +∞

0
e−λssds =

∫ +∞

0
se−λsds = L(Id)(λ) =

1

λ2
,

then the proposition 1.6.4 from [1] give L(Id ∗ f)(λ) exists for all λ > ω1 and

L(Id ∗ f)(λ) = L(Id)(λ)L(f)(λ)



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 10 of 15

=
1

λ2
L(f)(λ)

=
1

λ2

∫ +∞

0
e−λtf(t)dt

=
1

λ2

∫ +∞

0
(e−λtfn(t))n∈Ndt

=
1

λ2
(

∫ +∞

0
e−λtfn(t)dt)n∈N

=
1

λ2
(Fn(λ)n∈N

=
1

λ2
F (λ),

but

L(Id ∗ f)(λ) =

∫ +∞

0
e−λt(Id ∗ f)(t)dt

=

∫ +∞

0
e−λt

∫ t

0
(t− s)f(s)dsdt

=

∫ +∞

0
e−λtg(t)dt

= L(g)(λ)

from which follows the equality F (λ) = λ2L(g)(λ), according to Theorem 1.5.1
of [1], L(g) (so F ) is infinitely differentiable on ]ω1,+∞[ and since c(E) is closed
of l∞(E) then

For all k ∈ N, for all λ ∈]ω1,+∞[, F (k)(λ) ∈ c(E).

• d) lim
n→+∞

Cn(t)xdt = C(t)x.

We have

For all t > 0, there exist kt ∈ N : for all k ≥ kt, (−1)k
1

k!
(
k

t
)k+1F (k)(

k

t
) ∈ c(E),

it is that for all t > 0 there is kt ∈ N such that

((−1)k
1

k!
(
k

t
)k+1F (k)(

k

t
))k≥kt

is a sequence of elements of c(E). f is continuous on R+, so each t > 0 is a
Lebesgue point of f, the Post-Widder theorem (see theorem 1.7.7 of [1]) give
for t > 0

f(t) = lim
k→+∞

(−1)k
1

k!
(
k

t
)k+1f̂ (k)(

k

t
) = lim

k→+∞
(−1)k

1

k!
(
k

t
)k+1F (k)(

k

t
). But c(E)

is closed then f(t) = (fn(t))n∈N ∈ c(E) therefore lim
n→+∞

fn(t) exist and this



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 11 of 15

for all t > 0 but fn(0) = Cn(0)) = 0, then if we noted by h the function

h : R+ → E, t 7→ h(t) =

{
lim

k→+∞
fn(t), t > 0;

0, t=0.
Then (fn)n∈N is a sequence of

equicontinuous functions which converges pointwise to h, then h is continuous
in R+. The convergence dominate theorem give that

lim
n→+∞

∫ +∞

0
e−λtCn(t)xdt =

∫ +∞

0
e−λth(t)dt,

but

lim
n→+∞

∫ +∞

0
e−λtCn(t)xdt = lim

n→+∞
λL(K)(λ)RC(λ

2, An)x

= λL(K)(λ)RC(λ
2, A)x,

then

λL(K)(λ)RC(λ
2, A)x =

∫ +∞

0
e−λth(t)dt,

but {λ2 : λ > ω1 and L(K)(λ) ̸= 0} ⊂ ρC(A), then by the properties 1.(b)
and 1.(c), we can deduce that h is K-convoluted C-cosine function generated
by A, and like (C(t))t≥0 is uniquely determined by one of its subgenerators,
then h(.) = C(.)x, it is lim

n→+∞
Cn(t)x = C(t)x, and this for all t ≥ 0.

(ii) Let H be a compact of [0,+∞[ and x ∈ E.
Like H ⊂ [0, sup (H)] then it suffices to prove that the convergence is uniform on
the compact [0, sup(H)]. For that let ε > 0, (Cn(.)x)n∈N is equicontinuous at all
t ∈ [0,+∞[ so (Cn(.)x)n∈N is equicontinuous in [0, sup(H)] which is compact, then
(Cn(.)x)n∈N is uniformly equicontinuous in [0, sup(H)] which implies the existence
of η > 0 such that

(∀s, t ≥ 0) | t− s |< η =⇒ (∀n ∈ N) ∥ Cn(t)x− Cn(s)x ∥< ε

3
. (4)

For n0 = ⌊ sup(H)
η ⌋+1 ∈ N∗, we have sup(H)

n0
< η (is therefore for all n ≥ n0,

sup(H)
n ≤

sup(H)
n0

< η). For all i ∈ {0, ..., n0}, ti = i
n0

sup(H) ∈ [0, sup(H)]. So for each
t ∈ [0, sup(H)] there is i ∈ {0, ..., n0 − 1} such that ti ≤ t ≤ ti+1. For all i ∈
{1, ..., n0}, (Cn(ti)x)n∈N is a Cauchy sequence since it is convergent, therefore there
exist mi ∈ N∗ such that for all n,m ≥ mi ∥ Cn(ti)x− Cm(ti)x ∥≤ ε

3 .
If we posed N0 = max

0≤i≤n0

{mi}, then for all m,n ≥ N0 and all t ∈ [0, sup(H)],

there exist i0 ∈ {0, ..., n0 − 1} such that ti0 ≤ t ≤ ti0+1, and then if An,m =∥
Cn(t)x− Cm(t)x ∥ we have

An,m = ∥ Cn(t)x− Cn(ti0)x+ Cn(ti0)x− Cm(ti0)x+ Cm(ti0)x− Cm(t)x ∥



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 12 of 15

≤ ∥ Cn(t)x− Cn(ti0)x ∥ + ∥ Cn(ti0)x− Cm(ti0)x ∥ +

∥ Cm(ti0)x− Cm(t)x ∥

≤ ε

3
+ ∥ Cn(ti0)x− Cm(ti0)x ∥ + ∥ Cm(ti0)x− Cm(t)x ∥

(because | t− ti0 |< η)

≤ ε

3
+

ε

3
+ ∥ Cm(ti0)x− Cm(t)x ∥ (because m ≥ N0 ≥ mi)

≤ ε

3
+

ε

3
+

ε

3
(because | t− ti0 |< η)

≤ ε.

So the uniform Cauchy criterion implies that (Cn(.)x)n∈N converge uniformly on H
towards C(.)x).
5 ⇒ 1 |
Let x ∈ E; like (Cn(.)x)n∈N converge uniformly of any compact in [0,+∞[ then
(Cn(.)x)n∈N is equicontinuous in [0,+∞[. Let λ > ω1, then for all n ∈ N, we have
λL(K)(λ)RC(λ

2, An)x =
∫ +∞
0 e−λtCn(t)dt. The convergence dominate theorem give

that

lim
n→+∞

λL(K)(λ)(λ)RC(λ
2, An)x = lim

n→+∞

∫ +∞

0
e−λtCn(t)xdt

=

∫ +∞

0
e−λtC(t)xdt

= λL(K)(λ)RC(λ
2, A)x.

Corollary 1. Let (C(t))t≥0 be an α−times integrated cosine function (for some α ≥ 0)
with generator A, and for each n ∈ N, let (Cn(t))t≥0 an α−times integrated cosine function
with generator (An,D(An)) such that:

there exist ω ≥ 0, there exist M > 0 : for all t ≥ 0, for all x ∈ E, ∥ C(t)x) ∥≤ Meωt

and for all n ∈ N) ∥ Cn(t)x) ∥≤ Meωt.

Then if we put ω1 = max(ω + 1, abs(K)), the following statements are equivalent:

(i) There exist λ0 > ω1 such that for all x ∈ E,

lim
n→+∞

R(λ2
0, An)x = R(λ2

0, A)x

and (Cn(.)x)n∈N is equicontinuous.

(ii) There exist λ0 > ω1 such that for all y ∈ R(C),

lim
n→+∞

(λ2I −An)
−1y = (λ2I −A)−1y

and for all x ∈ E, (Cn(.)x)n∈N is equicontinuous.



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 13 of 15

(iii) For all t ≥ 0 and x ∈ E, lim
n→+∞

Cn(t)x = C(t)x, the convergence is uniform on any

compact of [0,+∞[.

Proof. Let α ≥ 0. Then if K(t) = tα−1

Γ(α) and C = I, then the α−times integrated cosine
function is a K-convoluted C-cosine function on E, thus Theorem 1 gives the results.

Corollary 2. Let (C(t))t≥0 be a K−convoluted C−cosine function with subgenerator A
and for each n ∈ N let (Cn(t))t≥0 a K−convoluted C−cosine function with subgenerators
(An,D(An)) such that there exist

ω ≥ 0, there exist M > 0 : for all t, h ≥ 0, ∥ C(t+ h)− C(t) ∥≤ Mheω(t+h)

and for all n ∈ N, ∥ Cn(t+ h)− Cn(t) ∥≤ Mheω(t+h).

Then if we put ω1 = max(ω + 1, abs(K)), the following statements are equivalent:

(i) There exist λ0 > ω1 such that L(K)(λ0) ̸= 0 and for all x ∈ E,

lim
n→+∞

RC(λ
2
0, An)x = RC(λ

2
0, A)x.

(ii) There exist λ0 > ω1 such that L(K)(λ0) ̸= 0 and for all y ∈ R(C),

lim
n→+∞

(λ2I −An)
−1y = (λ2I −A)−1y.

(iii) For all λ > ω1 such that L(K)(λ) ̸= 0), for all y ∈ R(C),

lim
n→+∞

(λ2I −An)
−1y = (λ2I −A)−1y.

(iv) For all λ > ω1 such that K(λ) ̸= 0, for all x ∈ E,

lim
n→+∞

RC(λ
2, An)x = RC(λ

2, A)x.

(v) For all t ≥ 0 and all x ∈ E, lim
n→+∞

Cn(t)x = C(t)x, the convergence is uniform on

any compact of [0,+∞[.

Proof. The condition

There exist (ω,M) ∈ R+×R+
∗ , for all n ∈ N, for all t, h ≥ 0 ∥ Cn(t+h)−Cn(t) ∥≤ Mheω(t+h)

implie that for t = 0 and h ≥ 0,

∥ Cn(h)) ∥≤ Mheωh ≤ Me(ω+1)h ≤ Meω1h

and for all x ∈ E, (Cn(.)x)n∈N is equicontinuous, then Theorem 1 gives the result.



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 14 of 15

Theorem 2. Suppose that (C(t))t≥0 is a strongly continuous operators family such that
for all t ≥ 0, ∥ C(t) ∥≤ Meωt and CC(.) = C(.)C. For all x ∈ E and λ > ω, put
Rλ2x := 1

λL(K)(λ)

∫ +∞
0 e−λtC(t)xdt.

For each n ∈ N, note (An,D(An)) the subgenerators of some K−convoluted C−cosine
function (Cn(t))t≥0, such that

• i) There exist ω ≥ 0 there exist M > 0 : for all n ∈ N ∥ Cn(t)) ∥≤ Meωt.

• ii) For all x ∈ E) (Cn(.)x)n∈N is equicontinuous.

• iii) There exist λ > ω such that lim
n→+∞

RC(λ
2, An)x = Rλ2x, R(Rλ2) ⊂ R(C) and

N(Rλ2) = {0}.

Then there is a linear operator A which is subgenerator of a K−convoluted C− cosine
function (C(t))t≥0 such that for all t ⩾ 0 and all x ∈ E,

lim
n→+∞

Cn(t)x = C(t)x,

the convergence is uniform on any compact of [0,+∞[.

Proof. As lim
n→+∞

R(λ2, An)x = Rλ2x then by Theorem 1 and Remark 1, we have for

all λ, µ > ω and all n ∈ N,

(λ2 − µ2)R(λ2, An)R(µ2, An) = R(λ2, An)Cx−R(λ2, An)Cx,

then passing to the limit as n tends to +∞, we get for all λ, µ > ω

(λ2 − µ2)Rλ2Rµ2 = Rµ2Cx−Rλ2Cx.

The remark 1 implies that there is a linear operator A (D(A) = R(Rλ2)), such that
Rλ2x = (λ2 −A)−1Cx = RC(λ

2, A). By definition we know that

λL(K)(λ)RC(λ
2, An)x =

∫ +∞

0
e−λtCn(t)xdt,

but

lim
n→+∞

λL(K)(λ)RC(λ
2, An)x = λL(K)(λ)Rλ2x

= λL(K)(λ)RC(λ
2, A)x,

by the proof of Theorem 1, we obtain that lim
n→+∞

Cn(t)x = C(t)x, hence λL(K)(λ)RC(λ
2, A)x =∫ +∞

0 e−λtC(t)xdt, then A is subgenerator of K−convoluted C−cosine function (C(t))t≥0,
such that lim

n→+∞
Cn(t)x = C(t)x for all x ∈ E, and the convergence is uniform on any

compact of [0,+∞[.



Y. Bajjou, A. El Amrani, A. Blali / Eur. J. Pure Appl. Math, 18 (1) (2025), 5560 15 of 15

4. Conclusion

Among the things that math people like is finding necessary an sufficient conditions
so that the limit of a sequence of mathematical objects having specific properties has
same properties. Thi is exactly what we did in this article, it is to find necessary and
sufficient conditions so that the limit of equicontinuous sequence K-convoluted C-cosine
functions (respectively the C-resolvent operators) is also equicontinuous K-convoluted
C-cosine functions ( respectively C-resolvent operators) and treated the equivalence be-
tween the convergence of equicontinuous sequence K-convoluted C-cosine functions and
the convergence of the associated C−resolvent sequence.

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