EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 538-547
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Results of Semigroup of Linear Operators Generating a
General Class of Semilinear Initial Value Problems

O. Y. Saka-Balogun1, F. H. Oyelami1, A. Y. Akinyele2,∗, J. B. Omosowon2

1 Department of Mathematical and Physical Sciences, Afe Babalola University, Ado-Ekiti,
Nigeria
2 Department of Mathematics, University of Ilorin, Ilorin, Nigeria

Abstract. This paper present results of ω-order preserving partial contraction mapping generating
a general class of semilinear initial value problems. We consider the use of fractional powers
of unbounded linear operators for its application by starting with some results concerning such
fractional powers. We assume A to be the infinitesimal generator of an analytic semigroup in a
Banach space X, 0 ∈ ρ(A) and defined the fractional powers of A for 0 < α ≤ 1. We also show that
Aα is a closed linear operator whose domain D(Aα) ⊃ D(A) is dense in X. Finally we established
that the operator is bounded, continuous and Holder continuous.

2020 Mathematics Subject Classifications: 06F15, 06F05, 20M05

Key Words and Phrases: ω-OCPn, Strongly Elliptic, C0-semigroup, Analytic Semigroup

1. Front Matter

Assume Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω and let

A(x,D) =
∑

|α|≤2m

aα(x)D
α (1)

be a strongly elliptic differential operator in Ω. For 1 < p < ∞ we associate with A(x,D)
and operator Ap in Lp(Ω) by

D(Ap) = W 2m,p(Ω) ∩Wm,p
0 (Ω) (2)

and
Apu = A(x,D)u (3)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4659

Email addresses: balogunld@yahoo.com (O. Y. Saka-Balogun),
adefolajufunmilayo@gmail.com (F. H. Oyelami), jbo0011@mix.wvu.edu (J. B. Omosowon),
olaakinyele04@gmail.com (A. Y. Akinyele∗)

https://www.ejpam.com 538 © 2023 EJPAM All rights reserved.



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 539

for u ∈ D(Ap) and A ∈ ω − OCPn. Suppose Ap is the infinitesimal generator of an
analytic semigroup on Lp(Ω). By adding to A(x,D), and hence to Ap, a positive multiple
of identity, we obtain an infinitesimal generator −(Ap + KI) of an analytic semigroup,
which is invertible. In the sequel we will tactically assume that this has been done and
thus assume directly that Ap itself is invertible. Let A be a strongly elliptic operator of
order 2m on a bounded domain Ω with smooth boundary ∂Ω in Rn and let 1 < p < ∞.
There exists a constant C such that

∥u∥2m,p ≤ C(∥Au∥0,p + ∥u∥0,p) (4)

for every u ∈ D(Ap) and Ap ∈ ω −OCPn.
Since we assume now that Ap is invertible in Lp(Ω) it follows readily that C∥u∥0,p ≤

∥Apu∥0,p for some constant C > 0 and therefore we have

∥u∥2m,p ≤ C∥Apu∥0,p for u ∈ D(Ap). (5)

Suppose X is a Banach space, Xn ⊆ X is a finite set, ω −OCPn the ω-order preserv-
ing partial contraction mapping, Mm be a matrix, L(X) be a bounded linear operator on
X, Pn a partial transformation semigroup, ρ(A) a resolvent set, σ(A) a spectrum of A.
This paper consist of results of ω-order preserving partial contraction mapping generat-
ing a general class of semilinear initial value problems. Akinyele et al. [1], characterized
ω-order reversing partial contraction mapping as a compact semigroup of linear operator
and also in [2], Akinyele et al., obtained differentiable and analytic results on ω-order
preserving partial contraction mapping in semigroup of linear operator. Balakrishnan [3],
presented an operator calculus for infinitesimal generators of semigroup. Banach [4], es-
tablished and introduced the concept of Banach spaces. Brezis and Gallouet [5], generated
nonlinear Schrödinger evolution equation. Chill and Tomilov [6], presented some resolvent
approach to stability operator semigroup. Davies [7], obtained linear operators and their
spectra. Engel and Nagel [8], introduced one-parameter semigroup for linear evolution
equations. Omosowon et al. [13], generated some analytic results of semigroup of linear
operator with dynamic boundary conditions, and also in [11], Omosowon et al., introduced
dual Properties of ω-order Reversing Partial Contraction Mapping in Semigroup of Lin-
ear Operator. Omosowon et al. [10], established a regular weak*-continuous semigroup
of linear operators, and also in [9], Omosowon et al., obtained a quasilinear equations
of evolution on semigroup of linear operator.reversing partial contraction mapping gen-
erating a differential operator. Omosowon et al. [12], deduced results of semigroup of
linear equation generating a wave equation. Pazy [14], presented asymptotic behavior of
the solution of an abstract evolution and some applications and also in [15], obtained a
class of semi-linear equations of evolution. Rauf and Akinyele [16], introduced ω-order
preserving partial contraction mapping and obtained its properties, also in [17], Rauf et
al., established some results of stability and spectra properties on semigroup of linear op-
erator. Vrabie [18], proved some results of C0-semigroup and its applications. Yosida [19],
deduced some results on differentiability and representation of one-parameter semigroup
of linear operators.



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 540

2. Preliminaries

Definition 2.2 (C0-Semigroup) [18]
A C0-Semigroup is a strongly continuous one parameter semigroup of bounded linear op-
erator on Banach space.

Definition 2.3 (ω-OCPn) [16]
A transformation α ∈ Pn is called ω-order preserving partial contraction mapping if
∀x, y ∈ Domα : x ≤ y =⇒ αx ≤ αy and at least one of its transformation
must satisfy αy = y such that T (t + s) = T (t)T (s) whenever t, s > 0 and otherwise for
T (0) = I.
Definition 2.4 (Evolution Equation) [14]
An evolution equation is an equation that can be interpreted as the differential law of the
development (evolution) in time of a system. The class of evolution equations includes,
first of all, ordinary differential equations and systems of the form

u = f(t, u), u = f(t, u, u),

etc., in the case where u(t) can be regarded naturally as the solution of the Cauchy
problem; these equations describe the evolution of systems with finitely many degrees of
freedom.
Definition 2.5 ( Mild Solution) [15]
A continuous solution u of the integral equation.

u(t) = T (t− t0)u0 +

∫ t

t0

T (t− s)f(s, u(s))ds

will be called a mild solution of the initial value problem{
du(t)
dt +Au(t) = f(t, u(t)), t > t0

u(t0) = u0

if the solution is a Lipschitz continuous function.

Definition 2.6(Analytic Semigroup) [18]
We say that a C0-semigroup {T (t); t ≥ 0} is analytic if there exists 0 < θ ≤ π, and a
mapping S : C̄θ → L(X) such that:
(i) T (t) = S(t) for each t ≥ 0;
(ii) S(z1 + z2) = S(z1)S(z2) for z1, z2 ∈ C̄θ;
(iii) limz1∈C̄θ,z1→0S(z1)x = x for x ∈ X; and
(iv) the mapping z1 → S(z1) is analytic from C̄θ to L(X). In addition, for each 0 < δ < θ,
the mapping z1 → S(z1) is bounded from Cδ to L(X), then the C0-Semigroup {T (t); t ≥ 0}
is called analytic and uniformly bounded.
Definition 2.7(Strongly Elliptic) [15]
The operator A(x,D) is strongly elliptic if there exists a constant C > 0 such that

Re(−1)mA1(x, ξ) ≥ C|ξ|2m



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 541

for all x ∈ Ω and ξ ∈ Rn.

Example 1

For every 2× 2 matrix in [Mm(Rn)].
Suppose

A =

(
2 0
∆ 2

)
and let T (t) = etA, then we have

etA =

(
e2t I
e∆t e2t

)
.

Example 2
For every 3× 3 matrix in [Mm(C)], we have
for each λ > 0 such that λ ∈ ρ(A) where ρ(A) is a resolvent set on X.
Suppose we have

A =

 2 2 I
2 2 2
∆ 2 2


and let T (t) = etAλ , then we have

etAλ =

e2tλ e2tλ I
e2tλ e2tλ e2tλ

e∆tλ e2tλ e2tλ

 .

Example 3
Let X = Cub(N∪{0}) be the space of all bounded and uniformly continuous function from
N∪{0} to R, endowed with the sup-norm ∥ · ∥∞ and let {T (t); t ∈ R+} ⊆ L(X) be defined
by

[T (t)f ](s) = f(t+ s)

For each f ∈ X and each t, s ∈ R+, one may easily verify that {T (t); t ∈ R+} satisfies
Examples 1 and 2 above.

Lemma 2.1
Let Ω be a bounded domain in Rn with boundary ∂Ω of class Cm and let u ∈ Wm,r(Ω) ∩
Lq(Ω) where 1 ≤ r, q ≤ ∞. For any integer j, 0 ≤ j < m and any j

m ≤ ϑ ≤ 1 we have

∥Dju∥0,p ≤ C∥u∥ϑm,r∥u∥1−ϑ
0,q (6)

provided that
1

p
=

j

n
+ ϑ

(
1

r
− m

n

)
+ (1− ϑ)

1

q
(7)



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 542

and m− j − n
r is not a nonnegative integer, the (6) holds with ϑ = j

m .
Theorem 2.2 [Sobolev]
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω (e.g. ∂Ω is of class Cm),
then

W k,p ⊂ Lnp/(n−kp)(Ω) for kp < n (8)

and
W k,p(Ω) ⊂ Cm(Ω) for 0 ≤ m < k − n

p
. (9)

Moreover, there exists a constant C1 and C2 such that for any u ∈ Wm,p(Ω)

∥u∥0,n,p/(n−kp) ≤ C1∥u∥k,p for kp < n (10)

and
sup(|Dα

αu(x)| : |α| ≤ m,x ∈ Ω) ≤ C2∥u∥k,p for 0 ≤ m < k − n

p
. (11)

3. Main Results

This section present results of semigroup of linear operator by using ω-OCPn to gen-
erates a general class of semilinear initial value problems:

Theorem 3.1
Suppose Ap : D(Ap) ⊆ X → X is the infinitesimal generator of an analytic semigroup
{T (t); t ≥ 0}. Assume 1 < p < ∞ and let Ap be the operator defined in Lemma 2.1. For
any multi-index β, |β| = j < 2m and any j/2m < α ≤ 1 we have

∥DβA−α
p u∥0,p ≤ C∥u∥0,p (12)

for u ∈ D(Ap) and Ap ∈ ω −OCPn.
Proof:
Set B = Dβ. Since |β| < 2m, it is clear that D(B) ⊃ D(Ap) for all A,B ∈ ω − OCPn.
From Lemma 2.1, we have

∥Dβu∥0,p ≤ C∥u∥j/2m2m,p ∥u∥
1−j/2m
0,p . (13)

Polarization of (13) together with estimate (5) yields

∥Dβu∥0,p ≤ C(ρ−1+j/2m∥Apu∥0,p + ρj/2m∥u∥0,p) (14)

for p > 0, u ∈ D(Ap) and Ap ∈ ω−OCPn. Suppose B is a closed linear operator satisfiying
D(B) ⊃ D(A). If for some γ, 0 < γ < 1, and every ρ ≥ ρ0 > 0 we have

∥Bx∥ ≤ C(ργ∥x∥+ ργ−1∥Ax∥) (15)



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 543

for x ∈ D(A) and A ∈ ω −OCPn, then

D(B) ⊃ D(Aα) for every γ ≤ 1. (16)

It follows now that D(B) ⊃ D(Aα
p ) for j/2m < α≤ 1, that, is BA−α

p is bounded for these
values of α and this achieved the proof.

Theorem 3.2
Assume Ap : D(Ap) ⊆ Lp(Ω) → Lp(Ω) is the infinitesimal generator of a semigroup
{T (t); t ≥}. Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω such that
A ∈ ω −OCPn. If 0 ≤ α ≤ 1, then

Xα ⊂ W k,q(Ω) for k − n

q
< 2mα− n

p
, q ≥ p (17)

Xα ⊂ Cp(Ω) for 0 ≤ v < 2mα− n

p
, (18)

and the embeddings are continuous.
Proof:
From Theorem 3.1 it follows readily that Xα ⊂ W j,p(Ω) provided that j < 2mα and the
imbedding is continuous. Since Ω is a bounded domain in Rn with smooth boundary ∂Ω,
assume ∂Ω is of class C ′ and let 1 ≤ r, p < ∞. If j,m are integers such that 0 ≤ r, j < m
and

1

p
>

1

r
+

j

n
− m

n
(19)

then Wm,r(Ω) ⊃ W j,p(Ω) and the imbedding is compact. It follows that W j,p(Ω) is con-
tinuously imbedded in W k,q(Ω) provided that k − n/q < j − n/p and (17) follows. From
Theorem 2.2 (Sobolev), it follows that W j,p(Ω) is continuously imbedded in Cv(Ω) for
0 ≤ v < j − n/p and (18) follows. Hence, the proof is completed.

Theorem 3.3
Let A(x,D) be a strongly elliptic operator given by

A(x,D) = −
3∑

k,l=1

∂

∂xk
ak,l(x)

∂

∂xl
.

Let Ω be a bounded domain in R3 with smooth boundary ∂Ω such that A ∈ ω −
OCPn where ak,l(x) = al,k(x) are real valued and continuously differentiable in Ω. Let
f(t, x, u, p), p ∈ R3, be locally Lipschitz continuous function of all its arguments and
assume further that there is a continuous function ρ(t, r) : R × R → R+ and a real
constant 1 ≤ r < 3 such that

|f(t, x, u, p)| ≤ ρ(t, |u|)(1 + |p|γ) (20)

|f(t, x, u, p)− f(t, x, u, q)| ≤ ρ(t, |u|)(1 + |p|γ−1 + |q|γ−1)|p− q| (21)



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 544

|f(t, x, u, p)− f(t, x, v, p)| ≤ ρ(t, |u|+ |v|)(1 + |p|γ)|u− v|. (22)

Then for every u0 ∈ H2(Ω) ∩H1
0 (Ω), the initial value problem

∂u
∂t = A(x,D)u+ f(t, x, u, grad u) in Ω
u(t, x) = 0 on ∂Ω
u(0, x) = u0(x) in Ω

(23)

has a unique local strong solution in L2(Ω).
Proof:
We recall that with the strongly elliptic operator A(x,D), we associate an operator A in
L2(Ω) by

D(A) = H2(Ω) ∩H1
0 (Ω)

and
Au = A(x,D)u for u ∈ D(A) and A ∈ ω −OCPn.

Let 1 < p < ∞, the operator A is the infinitesimal generator of an analytic semigroup of
contractions on Lp(Ω), then it follows that A is the infinitesimal generator of an analytic
semigroup on L2(Ω). From the strong ellipticity together with Poincare’s inequality it
follows readily that A is also invertible. From Theorem 3.2 it follows that if α > 3/4, then
Xα ⊂ L∞(Ω) and if also 1/q > (5− 4α)/6, then Xα ⊂ W 1,q(Ω). Thus for max(3/4, (5γ −
3)/4γ) < a < 1, we have

Xα ⊂ W 1,2γ(Ω) ∩ L∞(Ω). (24)

In order to show that the initial value problem has a unique local solution, we have to
show that the mapping

F (t, u)(x) = f(t, x, x(x),∇u(x)), x ∈ Ω (25)

is well defined on R+ ×Xα and satisfies a local Holder condition. From (20) and (24), we
have for every u ∈ Xα

∥F (t, u)∥0,2 ≤ 2p(t, ∥u∥0,∞)(M1/2 + ∥u∥γ1,2γ)

where M is the measure of Ω. Therefore F is well defined on R+ ×Xα. To show that F
satisfies a local Hölder condition we note that

∥F (t, u)− F (t, v)∥20,2 ≤ 2

∫
0

|f(t, x, u,∇u)− f(t, x, u,∇v)|2dx

+ 2

∫
Ω

|f(t, x, u,∇v)− f(t, x, v,∇v)|2dx (26)

and estimate each of the two terms on the right of (26) separately. From (21) and (23)
we have∫

Ω

|f(t, x, u,∇u)− f(t, x, u,∇v)|2dx



A. Y. Akinyele et al. / Eur. J. Pure Appl. Math, 16 (1) (2023), 538-547 545

≤ C · ρ(t, ∥u∥0,∞)2
∫
Ω

(1 + |∇u|2γ−2 + |∇v|2γ−2)|∇(u− v)|2dx

≤ C · ρ(t, ∥u∥0,∞)2(M1 + ∥∇u∥2γ−2
0,2γ + ∥∇v∥2γ−2

0,2γ )∥∇(u− v)∥20,2γdx
≤ L(∥u∥α, ∥v∥α)∥u− v∥21,2γ ≤ L(∥u∥α, ∥v∥α)∥u− v∥2α (27)

where ∥ ∥α denotes the norm in Xα and L is a constant depending on ∥u∥α and ∥v∥α.
To obtain the second inequality we used Hölder’s inequality. The last inequality (27) is
a consequence of the continuous imbedding of Xα in W 1,2γ(Ω). Similarly for the second
term we have (22) and (24), then we have∫

Ω

|f(t, x, u,∇v)− f(t, x, v,∇v)|2dx

≤ Cρ(t, ∥u∥0,∞ + ∥v∥0,∞)2
∫
Ω

(1 + |∇u|2γ)|u− v|2dx

≤ Cρ(t, ∥u∥0,∞ + ∥v∥0,∞)2∥u− v∥20,∞(1 + ∥v∥2γ1,2γ)
≤ L(∥u∥α, ∥v∥α)∥u− v∥2α (28)

and therefore,
∥F (t, u)− F (t, v)∥0,2 ≤ L(∥u∥α, ∥v∥α)∥u− v∥ (29)

and the existence of the strong local solution (23) is a direct consequence that the initial
value problem (23) has a unique local solution u. Hence the proof is completed.

Theorem 3.4
Assume A : D(A) ⊆ H2(Ω) → H2(Ω) is the infinitesimal generator of a C0-semigroup
{T (t)t≥0}. Let

f(u) =
3∑

i=1

u
∂u

∂xi
. (30)

If γ > 3
4 , u ∈ D(A) and A ∈ ω −OCPn, then f(u) is well defined and

∥f(u)∥ ≤ C∥Aγu∥∥A
1
2u∥. (31)

If u, v ∈ D(A) and A ∈ ω −OCPn, then

∥f(u)− f(v)∥ ≤ C(∥Aγu∥∥A
1
2u−A

1
2 v∥+ ∥A

1
2 v∥∥Aγu−Aγv∥). (32)

Proof:
Since D(A) ⊂ H2(Ω), then it follows from Sobolev’s theorem which states that if Ω is a
bounded domain in Rn with a smooth boundary ∂Ω of class Cm, then

W k,p(Ω) ⊂ Lnp/(n−kp)(Ω) for kp < n (33)



REFERENCES 546

and
W k,p(Ω) ⊂ Cm(Ω) for 0 ≤ m < K − n

p
. (34)

Moreover, there exist constants C1 and C2 such that for any u ∈ Wm,p(Ω),

∥u∥0,np/(n−kp) ≤ C1∥u∥k,p for kp < n (35)

and
sup{|Dαu(x)| : |α| ≤ m, x ∈ Ω ≤ C2∥u∥k,p} (36)

for 0 ≤ m < K − n
p and it follows that u ∈ L∞(Ω) and therefore f(u) ∈ L2(Ω) is thus

well-defined. Moreover, from Theorem 3.3 we have

∥f(u)∥ ≤ ∥u∥0,∞∥∇u∥ ≤ C∥Aγu∥∥∇u∥ = C∥Aγu∥∥A
1
2u∥.

Also

∥f(u)− f(v)∥ ≤ ∥u∥0,∞∥∇(u− v)∥+ ∥u− v∥0,∞∥∇u∥

≤ C(∥Aγu∥∥A
1
2u−A

1
2
vv∥+ ∥A

1
2 v∥∥Aγu−Aγv∥). (37)

Hence the proof in completed.

4. Conclusion

In this paper, it has been established that ω-order preserving partial contraction map-
ping generates some results of a general class of semilinear initial value problems.

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