EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 1297-1314
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On a System of Linear Singular Partial Differential
Equations with Weight Functions

Euler Yoland B. Guerrero
1 Department of Mathematics and Statistics, College of Science and Mathematics,
MSU-Iligan Institute of Technology, Iligan City, Philippines

Abstract. Let X be a Banach space, Ω an open bounded subset of X, and Y a complex Banach
space. We consider a Volevič system of singular linear partial differential equations of the form

t
∂ui
∂t

=

N∑
j=1

aij(t, x)uj(t, x) +
∑

(j,k)∈N (i)

bjk(t, x)((µ0(t)D)kuj(t, x) · x(k)k )(j,k) + gi(t, x), (1)

1 ≤ i ≤ N , in the unknown function u = (u1, u2, ..., uN ) ∈ Y N of t ≥ 0 and x ∈ Ω, where
aij , bjk ∈ C, xk = (x, ..., x) (x is k times) D denotes the Frechet differentiation with respect to x,
and

N (i) = {(j, k) : j and k are integers, 1 ≤ j ≤ N, 0 < k ≤ n(i, j)}, (2)

n(i, j) = n(i)− n(j) + 1, where n(i), i = 1, 2, ..., N , are nonnegative integers. The map µ0 belongs
to C0([0, T ],C). We express growth estimates in terms of weight functions and we establish an
existence and uniqueness theorem for our system in the class of ultradifferentiable maps with
respect to the space variable x.

2010 Mathematics Subject Classifications: 35A01, 35A02, 35A10

Key Words and Phrases: System of partial differential equations, ultradifferentiable, weight
functions

1. Introduction

The study of partial differential equations have been a very fruitful endeavor both in
pure and applied mathematics. Its practical use cannot be underestimated as many recent
scientific and engineering works such as in [8] uses partial differential equations to model
real-world problems.

Gerard and Tahara [2], and Baouendi and Goulaouic [1] were some of the authors who
worked on nonlinear or linear differential equations with singularity. Lope [5], extended

DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3498

Email addresses: euleryoland.guerrero@g.msuiit.edu.ph (EY Guerrero)

http://www.ejpam.com 1297 c© 2019 EJPAM All rights reserved.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1298

the work of Baouendi and Galaouic using the concept of weight functions. These weight
functions are used to describe growth estimates on the coefficients of the partial Taylor
expansion of a function.

In [3], Koike considered a Volevič system of singular nonlinear partial differential equa-
tions with general singularity. He established the existence and uniqueness theorem of
the solution in the ultradifferentiable class using the Banach fixed point theorem and
Nirenberg-Nishida [6, 7] iteration method. This method was also used in [4].

In this paper, we will establish an existence and uniqueness theorem on (1) in the
ultradifferentiable class with growth estimates in terms of weight functions.

2. Preliminaries

We first give the definition of a weight function as defined by Tahara [9]. We then give
the definitions and basic results about ultradifferentiable maps as proved by Koike [3].

Definition 1. Let T > 0. we say that µ(t) is a weight function on [0, T ] if it is continuous,
nonnegative, increasing function on (0, T ] such that∫ T

0

µ(t)

t
dt < +∞.

Let V and W be Banach spaces, and U be an open subset of V . We denote by
C0(V,W ) the set of all continuous mappings from V to W and L(V,W ) the Banach space
of all bounded (continuous) linear mappings from V to W . Moreover, we let Lp(U,W ) to
be the space of all p-linear continuous mappings of Up into W .

Definition 2. Let Mj, j = 0, 1, ..., be a sequence of positive numbers with

M0 = M1 = 1.

A map v ∈ C∞(Ω, Y ) is said to belong to the ultradifferentiable class {Mp}(Ω, Y ) (or
{Mp} for short) if

‖Djv(x)‖ ≤ C1+jMj

x ∈ Ω, j = 0, 1, 2, ..., and constant C.

As was done in Koike’s paper, in our problem, we impose on the sequence {Mp} the
following conditions:

(C1) If
n∑
i=1

ki = n, ki ≥ 0, n = 1, 2, ..., then
n∏
i=1

Nki+1 ≤ Nn+1, where Np =
Mp

p!
.

(C2) There is a constant K such that Mj+1 ≤ K(j + 1)Mj , j = 0, 1, 2, ....



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1299

For s > 0, we write

‖u‖s = ‖u‖s(U) = sup
x∈U

∞∑
j=0

‖Dju(x)‖sj

Mj
,

‖u‖′s = ‖u‖′s(U) = sup
x∈U

∞∑
j=1

‖Dju(x)‖sj

Mj
,

and

Bs(U, V ) = {u ∈ C∞(U, V ) : ‖u‖s(U) <∞},

where V is a subset of a Banach space.

Remark 1. It is not difficult to show that u ∈ {Mp}(U, V ) if and only if u ∈ Bs(U, V )
for some s > 0.

Let X , Y, and Z be Banach spaces, U an open subset of X , and V an open subset of
Y. The next theorem states the multiplication-closedness of the {Mp} class.

Theorem 1. Let G ∈ C∞(U,Lm(Y,Z)), ui ∈ C∞(U,Y), i = 1, 2, ...,m(m = 1, 2, ...).
Then

‖Gu1, ..., um‖s/H(U) ≤ Cm1 ‖G‖s(U)

m∏
i=1

‖ui‖s(U),

where (Gu1, ..., um)(x) = G(x)u1(x), ...., um(x) and C1 = max{ 1
N2
, 1}.

Theorem 2. Let f ∈ C∞(V,Z) and u ∈ C∞(U, V ). If ‖u‖s′(U) ≤ R for some s > 0 and
R > 0, then

‖f ◦ u‖s/H(U) ≤ ‖f‖R(V ).

Corollary 1. Let f ∈ C∞(V,Z), and u, v ∈ C∞(U, V ). Then

‖f ◦ u− f ◦ v‖s/H2(U) ≤ C1‖Df‖R(V )‖u− v‖s/H(U)

if ‖u‖′s(U) ≤ R and ‖v‖′s(U) ≤ R.

Theorem 3. Assume (C2). Then there exists Kn > 0 such that

‖Dnu‖r ≤ Kn(s− r)−n‖u‖s (3)

0 < r < s ≤ s1, where Kn is independent of u, r and s.

Remark 2. The preceding theorem implies that if u ∈ {Mp}, then Du ∈ {Mp}.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1300

We will now give our assumptions for (1). Let Y be a complex Banach space and
Lk(X,Y ) the Banach space of all bounded multi-k-linear maps from Xk to Y , while
L0(X,Y ) denotes Y . Let Ω be an open subset of X and Ui a neighborhood of the origin
in the Banach space {(ξjk)(j,k)∈N (i) : ξjk ∈ Lk(X,Y )}, where N (i) is the set defined in
(2). Let

fi(u,w)(t, x) =
N∑
j=1

aij(t, x)uj(t, x) +
∑

(j,k)∈N (i)

bjk(t, x)((µ0(t)D)kuj(t, x) · x(k)k )(j,k).

We work on (1) under the following assumptions:

(A1) µ0 belong to C0([0, T ],C) for a T > 0 and fi ∈ C0([0, T ], Bs1(Ω× Ui, Y )), for some
s1 > 0.

(A2) fi(0, 0)(0, x) = 0, for all x ∈ Ω, 1 ≤ i ≤ N

(A3) The spectrum of the N ×N matrix A(x) = (Aij(x)) ∈ L(Y N ), where

Aij = −Dujfi(u,w)(0, x)|(u,w)=(0,0)

is contained in the half plane{z ∈ C : Rez > b0} for a positive number b0.

(A4) For some κ ∈ (0, 1), ∫ T

0

(µ(t))κ

t
dt <∞,

where µ(t) = sup0≤τ≤t |µ0(τ)|.

Condition (A1) states that fi is continuous in t and ultradifferentiable in the other
variables.

The next results are proved in [3] assuming (C1), (C2), and (A1)-(A4).
Let κ be the number as in (A2) and

c = max
1≤i≤N

{n(i)}+ 1 +
κ

1− κ
d = max

1≤i,j≤N
{n(i, j)}. (4)

Then c ≥ d + 1 and d ≥ 1. The function ω in the following lemma plays an important
role.

Lemma 1. There exists a function ω ∈ C0([0, T ],R) ∩ C1((0, T ],R) such that ω(0) = 0,

ω(t)cω′(t) ≥
µ(t)

κ

t
(5)

and

ω(t)c ≥ µ(t)κ (6)

for t ∈ (0, T ].



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1301

Now put

ρ(i, j) = max{n(j, i), 1}

ν(τ, t) = ln

(
t

τ

)
and

E(τ, t)(x) = (Eij(τ, t)(x)) = exp

[
ln
τ

t
A(x)

]
∈ L(Y N )

for (τ, t) ∈ ∆, where A(x) = (Aij(x)) is the matrix operaton as in (A3) and

∆ = {(τ, t) : 0 = τ < t ≤ T or 0 < τ ≤ t ≤ T}.

Lemma 2. There exists a positive number b such that for every x0 ∈ Ω there are pos-
itive numbers s0(s0 < s1), C0 and an open neighborhood U ⊂ Ω of x0 such that E ∈
C0(∆, Bs0(U,L(Y N ))),

‖E(τ, t)‖s0(U) ≤ C0

(
τ

t

)b
(7)

and

‖Eij(τ, t)‖s0(U) ≤ C0eρ(i,j)(τ, t), (8)

where

eρ(i,j)(τ, t) =

(
τ

t

)b
ν(τ, t)ρ(i,j)−1

(ρ(i, j)− 1)!
.

Note that 0 does not belong to the spectrum of A(x), thus the map A : x → A(x)−1

is well-defined and ultradifferentiable with respect to x, that is, we can assume that A ∈
Bs0(Ω, L(Y N )), for some s0 > 0.

Lemma 3. Let s ∈ (0, s0], δ ∈ (0, T ] and v ∈ C0([0, δ), Bs(U, Y
N )). Then u(0) = Av(0)

and

u(t) =

∫ t

0

1

τ
E(τ, t)v(τ)dτ

for t ∈ (0, δ), if and only if u ∈ C0([0, δ), Bs(U, Y
N )) and

t
∂u

∂t
(t) +Au(t) = v(t)

for t ∈ (0, δ).



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1302

We write, for t > 0,

H[h](t) =

∫ t

0

τ b−1

tb
h(τ)dτ.

Note that H[h](0) = h(0)/b. We may assume b ≤ 1 without loss of generality. Note that
H[1](t) = 1/b.

Lemma 4. Let δ ∈ (0, T ], a > 0, β ≥ 0and γ ≥ 1, and let m = 0 or m = 1. If α ≥ κm,
ω(t) < a and h(t) ≤ µ(t)αω(t)β(1− ω(t)/a))−γ for t ∈ [0, δ), then

H[h](t) ≤ Cγamµ(t)α−κmω(t)β+cm

(
1− ω(t)

a

)−Max{1,γ−m}

for t ∈ [0, δ), where Cγ = max

{
1

γ − 1
,
1

b

}
if γ > 1 and Cγ =

1

b
if γ = 1.

Lemma 5. Let h ∈ C0([0, δ),R), δ ∈ (0, T ]. Then it holds that∫ t

0

1

τ
ep(τ, t)h(τ)dτ = Hp[h](t)

for t ∈ (0, δ) and p = 1, 2, . . . .

3. Existence and Uniqueness Theorem

We first state our main theorem and then prove the existence and uniqueness parts in
two sections.

Theorem 4 (Main Theorem). Let C1, C2, andA1 − A4 hold and α ∈ (0, 1]. For every
x0 ∈ Ω, there exists a positive number R small enough and a neighborhood U ⊂ Ω such
that if the map gi : t→ (x 7→ gi(t, x)) belongs to C0([0, T ], Bs1(Ω, Y ) for some s1 > 0 with

‖gi(t, x)‖s1(Ω) ≤ CRµ(t)α, (t, x) ∈ [0, T ]× Ω, 1 ≤ i ≤ N

for some constant C > 0, then (1) has a unique solution u = (u1, ..., uN ) in [0, T0)×U for
a positive number T0 ≤ T and a neighborhood U ⊂ Ω of x0, satisfying

uj ∈ C0([0, T0), Bs(U, Y )) ∩ C1((0, T0), Bs(U, Y )), 1 ≤ j ≤ N

and

‖uj(t, x)‖s(U) ≤ Rµ(t)α and ‖((µ0D)kuj(t, x))(j,k)∈N (i)‖s(U) ≤ Rµ(t)α,

for all t ∈ [0, δ) and some s > 0.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1303

3.1. Existence

Let α ∈ [0, 1], µ(t) be a weight function, and x0 ∈ Ω. Let U be the set obtained by
Lemma 2. For brevity, we abbreviate ‖ · ‖s(U) to ‖ · ‖s, and (t, x) to (t) if t is the only
variable needed in our analysis. We let wjk(t, x) = ((µ0D)kuj(t, x))(j,k) then

fi(u,w)(t, x) =
N∑
j=1

aij(t)uj(t, x) +
∑

(j,k)∈N (i)

bjkw(j,k)(t, x),

and write
Fi(u,w)(t, x) = fi(t, x, uj(t, x), wjk(t, x))

for u = (uj)1≤j≤N and w = (wjk)(j,k)∈N (i), where the values of uj and wjk belong to Y

and Lk(X,Y ), respectively. Further, we set F = (Fi)1≤i≤N and

Ψ(u,w)(t) =

∫ t

0

E(τ, t)

τ
(F (u,w)(τ) +Au(τ) + g(τ))dτ.

We need to show that for a fixed w, the operator Ψ(·, w) is a contraction mapping
from a function space to itself. Let u = (u1, ..., uN ) and WT be the set

WT = {u ∈ C0([0, T ), (Bs(U, Y ))N ) : ‖u(t)‖s ≤ Cµ(t)α for some C > 0}.

For a u ∈WT we define the norm ‖u‖W as

‖u(t)‖W = max
1≤j≤N

‖uj(t)‖s.

Then (WT , ‖ · ‖W ) is a Banach space. For R > 0, we set

WT,R = {u ∈WT : ‖u‖W ≤ Rµ(t)α}.

This is a closed subset of WT and so it is a complete metric space. WT,R will be the form
of our function space. We note that if u ∈WT,R, then ‖u(t)‖s ≤ Rµ(t)α.

Similarly, we define W ′T,R by just replacing (Bs(U, Y ))N in our definition of WT by

Bs(U,L
k(X,Y )).

Let C2 = sup0≤t≤T ‖Dwfi(t)‖s

(
Ω × Y N ×

∏
(j,k)∈N (i),k>0

Lk(X,Y )

)
. This is finite by

(A1) and Remark 2.10. Further, we let C ′ = N2C2
1C0C2, where C1 and C0 are the

constants in Theorem 2.6 and Lemma 2.12, respectively.
Set r0 = min{bd/C ′, 1} and b is the positive constant obtained in Lemma 2.

Proposition 1. There exists T0 ∈ (0, T ] and R < s1 such that if

‖gi(t)‖s(Ω) ≤ br2(1− r)
C0

Rµ(t)α, t ∈ [0, T ], ‖x(k)k ‖ ≤ 1



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1304

and fixed w ∈W ′T0,R with

‖wjk(t)‖s ≤ r0rRµ(t)α, t ∈ [0, T0), (9)

then the following are true:

(a) Ψ[·, w] is a mapping from WT0,R to itself.

(b) Ψ[·, w] is a contraction map.

Proof. By Remark 2 and (A1), Dufi ∈ {Mp}. Hence, fi is continuous with respect
to t, u, and w. Thus, we can find T0 ∈ [0, T ] and R < s1 such that if u, v ∈ WT0,R and
w,w ∈W ′T0,R, then

N2C2
1C0‖Dufi(τ, P,Q)−Dufi(0, 0, 0)‖s ≤ rbd (10)

where P = θu+ (1− θ)v, Q = θw + (1− θ)w. Now, since Fi(0, 0, 0) = 0

Fi(u,w)(t) = Fi(u, v)(t)− Fi(0, 0)(t)

=
N∑
j=1

∫ 1

0
Dufi(t, θu, θw)uj(t)dθ +

∑
(j,k)∈N (i)

∫ 1

0
Dwfi(t, θu, θw)wjk(t) · x

(k)
k dθ.

Using the definition of A we may rewrite Aijuj(t) as

Aijuj(t) = −
∫ 1

0
Dufi(0, 0, 0) · uj(t)dθ.

Hence,

Fi(u,w)(t) +
N∑
j=1

Aijuj(t) + gi(t) =
N∑
j=1

∫ 1

0
[Dufi(t, θu, θw)−Dufi(0, 0, 0)]uj(t)dθ

+
∑

(j,k)∈N (i)

∫ 1

0
Dwfi(t, θu, θw)wjk(t) · x

(k)
k dθ + gi(t).

Thus,∥∥∥∥∥Fi(u,w)(t) +
N∑
j=1

Aijuj(t) + gi(t)

∥∥∥∥∥
s

≤
N∑
j=1

C1‖Dufi(t, θu, θw)−Dufi(0, 0, 0)]‖s‖uj(t)‖s

+
∑

(j,k)∈N (i)

C1‖Dwfi(t, θu, θw)‖s‖wjk(t) · x
(k)
k ‖s

+‖gi(t)‖s.

Using Lemma 2, we have

‖Ψi(u,w)(t)‖s ≤
∫ t

0

∥∥∥∥∥E(τ, t)

(
Fi(u,w)(τ) +

N∑
j=1

Aijuj(τ) + gi(τ)

)∥∥∥∥∥
s

dτ

τ
.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1305

≤
∫ t

0

{
C0
τ b−1

tb

(
N∑
j=1

C1‖Dufi(t, θu, θw)−Dufi(0, 0, 0)]‖s‖uj(τ)‖s

+
∑

(j,k)∈N (i)

C1‖Dwfi(t, θu, θw)‖s‖wjk(τ) · x(k)k ‖s + ‖gi(τ)‖s

)}
dτ

=

(
N∑
j=1

C0C1‖Dufi(t, θu, θw)−Dufi(0, 0, 0)]‖s‖uj(t)‖s

+
∑

(j,k)∈N (i)

C0C1‖Dwfi(t, θu, θw)‖s‖wjk(t)‖s‖x
(k)
k ‖s + C0‖gi(t)‖s

)
1

b
.

Note also that bd < b ≤ 1 and d > 1. Thus, by our assumptions, (10) and our defined
constant C ′,

‖Ψi(u,w)(t)‖s ≤

(
NC0C1‖Dufi(t, θu, θw)−Dufi(0, 0, 0)]‖s max

1≤j≤N
‖uj(t)‖s

+NC0C1‖Dwfi(t, θu, θw)‖s max
(j,k)∈N(i)

‖wjk(t)‖s + C0‖gi(t)‖s

)
1

b

≤

(
rb max

1≤j≤N
‖uj(t)‖s + C ′ max

(j,k)∈N(i)
‖wjk(t)‖s + C0

br2(1− r)
C0

Rµ(t)α

)
1

b

≤ r max
1≤j≤N

‖uj(t)‖s +
C ′

b
max

(j,k)∈N(i)
‖wjk(t)‖s + r2(1− r)Rµ(t)α

≤ sup
0≤τ≤t

{
r max
1≤j≤N

‖uj(t)‖s +
C ′

b
max

(j,k)∈N(i)
‖wjk(t)‖s

}
+ r2(1− r)Rµ(t)α.

Thus, using the definition of r0 and with r ≤ 1
3 , we have

‖Ψi(u,w)(t)‖ ≤ rRµ(t)α +
C ′

b
rr0Rµ(t)α + r2(1− r)Rµ(t)α

≤ 1

3
Rµ(t)α +

C ′

b
· b

d

C ′
1

3
Rµ(t)α +

1

3
Rµ(t)α

≤

(
1

3
+
C ′

b
· b

d

3C ′
+

1

3

)
Rµ(t)α

= Rµ(t)α

proving (a). Furthermore, note that

Fj(u,w)(t) − Fj(v, w)(t)

=

N∑
k=1

∫ 1

0

[
Dufj(t, P,Q) · (uk − vk)(t) +Dwfj(t, P,Q)(wkη − wkη)

]
dθ.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1306

Hence, similar to the previous approach,

Fj(u,w)(t) − Fj(v, w)(t) +

N∑
k=1

Ajk(uk(t)− vk(t))

=
N∑
k=1

∫ 1

0

[
Dufj(t, P,Q)−Dufj(0, 0, 0)](uk − vk)(τ)]dθ

+
∑
(k,η)

Dwfj(t, P,Q)(wkη − wkη)(τ)dθ

Thus, by Lemma 2 and (10) we have∥∥∥∥∥
N∑
j=1

Eij

[
Fj(u,w)(t)− Fj(v, w)(t) +

N∑
k=1

Ajk(uk(t)− vk(t))

]∥∥∥∥∥
s

≤
N∑
j=1

C0eρ(i,j)(τ, t)

[
N∑
k=1

C1‖Dufj(t, P,Q)−Dufj(0, 0, 0)‖s‖(uk − vk)(t)‖s

+
∑
(k,η)

C1‖Dwfj(t, P,Q)‖s‖(wkη − wkη)(t)‖s
]

≤ N max
1≤j≤N

eρ(i,j)(τ, t)

[
NC1C0‖Dufj(t, P,Q)−Dufj(0, 0, 0)‖s

× max
1≤k≤N

‖(uk − vk)(t)‖s +NC1C0‖Dwfj(t, P,Q)‖s

× max
(k,η)∈N (i)

‖(wkη − wkη)(t)‖s

]

≤ rbd max
1≤j,k≤N

eρ(i,j)(τ, t)‖(uk − vk)(t)‖s + C ′ max
1≤j≤N

(k,η)∈N (i)

[
eρ(i,j)(τ, t)

×‖(wkη − wkη)(t)‖s

]

By Lemma 5,

‖Ψi(u,w)(t)−Ψi(v, w)(t)‖ ≤ rbd max
1≤j,k≤N

Hρ(i,j)[‖uk − vk‖s](t)

+C ′ max
1≤j≤N

(k,η)∈N (i)

Hρ(i,j)[‖wlη − wlη‖s](t). (11)

Hence, when w = w, we have

‖Ψi(u,w)(t)−Ψi(v, w)‖ ≤ rbd max
1≤j≤N

Hρ(i,j)[‖uk − vk‖s](t),



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1307

proving (b).

It follows from the Banach fixed point theorem that there exists a unique u ∈ WT0,R

such that u = Ψ(u,w). Denote this u by S[w]. We have by (11),

‖Si[w](t)− Si[w(t)]‖s‖ = ‖Ψi(S[w], w)(t)−Ψi(S[w], w)(t)‖
≤ rbd max

1≤j≤N
Hρ(i,j)[‖Sk[w]− Sk[w]‖s](t)

+C ′ max
(k,η)∈N (i)

Hρ(i,j)[‖wkη − wkη‖s](t).

Using (11) n-times, we get

‖Si[w](t)− Si[w(t)]‖s‖ ≤ rn+1bd max
1≤j≤N

sup
0≤τ≤t

Hρ(i,j)[‖Sk[w]− Sk[w]‖s](τ)

+
n∑
p=0

rpC ′ max
(k,η)∈N (i)

sup
0≤τ≤t

Hρ(i,j)[‖wkη − wkη‖s](τ).

As n→∞, we have the following Proposition:

Proposition 2. For w,w ∈WT0,R satisfying (9), we have

‖Si[w](t)− Si[w(t)]‖s‖ ≤ C max
(k,η)∈N (i)

sup
0≤τ≤t

Hρ(i,j)[‖wkη − wkη‖s](τ), (12)

where C = C ′/(1− r).

From (11), when w = 0 and u ∈WT0,R, we have

‖Si[0](t)‖s = ‖Ψi(S[0], 0)(t)‖s
≤ r‖Si[0](t)‖s + r2(1− r)Rµ(t)α.

Hence, since r ∈ (0, 1), we have

(1− r)‖Si[0](t)‖s ≤ r2(1− r)Rµ(t)α

‖Si[0]‖s ≤ r2Rµ(t)α. (13)

To solve the equation u = S[((µ0D)ηuk)(k,η)∈M] we use the method of Nirenberg-
Nishida. We define un = (un,1, un,2..., un,N ), n = 0, 1, ..., recursively by

u0 = 0, un+1 = S[((µ0D)ηun,k)(k,η)∈N (i)] (n = 0, 1, ...).

we write vn = un+1 − un. Let a0 ∈ (0, 1) be a small number to be determined later and

an = a0

n∏
j=1

(1 + j−2)−1.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1308

Then, {an}n≥0 is a decreasing sequence of positive numbers tending to a positive limit
a∞. Observe that

a∞ = a0

∞∏
j=1

(1 + j−2)−1 = a0

( ∞∏
j=1

(1 + j−2)

)−1
.

Since
∑∞

j=1 j
−2 is convergent, a∞ is convergent.

Corresponding to each an, we have the t-interval

In(s) = {t ≥ 0 : ω(t) < an(s0 − s)} (0 < s < s0),

and

σn,s(t) =

(
1− ω(t)

an(s0 − s)

)−1
.

Note that for all n, σn,s(t) ≥ 1 and In+1(s) ⊂ In(s). Let a0s0 ≤ w(T0). Then

I0(s) ⊂ [0, T0). Put s(t) = (s0 + s − ω(t)
an

)/2. Then, for 0 < s < s(t) < s0, we have the
following remark.

Remark 3. If t ∈ In(s), then

(1) t ∈ In(s(t))

(2) σn,s(t) ≤ 2σn,s(t)

(3) (s(t)− s)−η = 2η(s0 − s)−ησn,s(t)η

(4) 1 ≤ σn,s(t) ≤ (n+ 1)2 + 1.

(5) (s0 − s)−η ≤
a0ω(t)cη

ω(t)ηµ(t)κη

We now prove the following proposition. Proving it means proving the convergence of
our solution u(t, x) = lim

n→∞
un(t, x), for x ∈ U and

t ∈ I∞(s) = {t ≥ 0 : ω(t) < a0(s0 − s)} (0 < s < s0).

Proposition 3. Let vn,i = un+1,i − un,i. For n ≥ 0 the following hold:

(a) un+1,i := Si[(µ0D)ηun,k](k,η)∈N (i) exists on In(s)× Ui.

(b) For t ∈ In(s),

‖vn,i(t)‖s ≤ Rrn+2µ(t)(1−κ)nω(t)nσn,s(t)
dnµ(t)α.



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1309

(c) For t ∈ In(s),

‖(µ0(t)D)ηvn,i(t)‖s ≤ Rrn+22dn+ηKη(s0 − s)−ηµ(t)(1−κ)n+ηω(t)nσn,s(t)
dn+ηµ(t)α.

implying that for t ∈ In+1,

‖(µ0(t)D)ηvn,i(t)‖s ≤ Rrn+22dn+ηKηa0µ(t)(1−κ)n+(1−κ)ηωn+(c−1)ησn,s(t)
dn+ηµ(t)α

and thus,
‖(µ0(t)D)ηun+1,i‖s ≤ Rµ(t)α.

Proof. Since u0,i = 0, Proposition 1 assures us that u1,i = Si[0] exists for t ∈ I0(s). By
(13),

‖v0,i(t)‖s = ‖u1,i(t)− u0,i(t)‖s
= ‖u1,i(t)‖s
= ‖Si[0](t)‖s
≤ Rr2µ(t)α.

By (3) and Remark 3 (3) we have

‖(µ0D)ηv0,i(t)‖s ≤ µ(t)ηKη(s(t)− s)−η‖v0,i(t)‖s(t)
≤ µ(t)ηKη2

η(s0 − s)−ησ0,s(t)ηRr2µ(t)α

= Rr22ηKη(s0 − s)−ηµ(t)ηση0,sµ(t)α.

Hence, by Remark 3 (5), for t ∈ I1(s), we have

‖(µ0D)ηv0,i(t)‖s = ‖(µ0D)ηu1,i(t)‖
≤ Rr22ηKη(s0 − s)−ηµ(t)ηση0,sµ(t)α

≤ Rr22ηKη
aη0ω(t)cη

ω(t)ηµ(t)κη
µ(t)ηση0,sµ(t)α

≤ Rr22ηKηa0µ(t)(1−κ)ηω(t)(c−1)ηση0,sµ(t)α

≤ Rµ(t)α,

provided a0 is small enough.
Suppose (a)-(c) hold for n = 0, 1, ..., p with n ≤ l. Proposition 1 and (c) imply that

up+2,i = S[(µ0D)ηup+1,k] exists for t ∈ Ip+1(s), showing (a) for n = p + 1. Now, for
t ∈ Ip+1(s) and Proposition 2,

‖vp+1,i(t)‖s = ‖Si[((µ0D)ηup+1,k)](t)− Si[((µ0D)ηup,k)](t)‖s
≤ C max

(k,η)∈N (i)
sup

0≤τ≤t
Hρ(i.j)[‖((µ0D)ηvp,k)‖s](τ).



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1310

Using Lemma 4 ρ(i, j)-times, we have by Proposition 2 and (c) that

‖vp+1,i(t)‖s ≤ max
(k,η)∈N (i)

(i,j)

min
m

hC(γ)(s0 − s)−η(ap+1(s0 − s))mµ(t)(1−κ)p+η−κm

×ω(t)p+cmσp+1,s(t)
max{1,dp+η−m}µ(t)α,

where
(i,j)

min
m

= min
0≤m≤min{ρ(i,j),α+η

κ
}

m an integer, and C(γ) depends only on γ. Thus, since w(t) < a0(s0 − s) and

ap+1(s0 − s) < 1,

we have

‖vp+1,i(t)‖s ≤ max
(k,η)∈N (i)

(i,j)

min
m

hC(γ)a0µ(t)(1−κ)p+η−κm

×ω(t)p+cm−ησp+1,s(t)
max{1,dp+η−m}µ(t)α.

If m = 1, then

‖vp+1,i(t)‖s ≤ Rrp+3µ(t)(1−κ)p+1−κω(t)p+1σp+1,s(t)
d(p+1)µ(t)α,

where h = Rr2, C(γ)a0 ≤ rp+1, since σp+1,s ≥ 1 and dp ≤ d(p+ 1). Thus,

‖vp+1,i(t)‖s ≤ Rrq+3µ(t)(1−κ)(p+1)ω(t)p+1σp+1,s(t)
d(p+1)µ(t)α.

Hence, by (3) we have

‖(µ(t)D)ηvp+1,i‖s ≤ µ(t)ηKη(s(t)− s)−η‖vp+1,i‖s(t)
≤ Rrp+3Kη2

d(p+1)+ηa0µ(t)(1−κ)(p+1)+(1−κ)η

×ω(t)(p+1)+(c−1)ησp+1,s(t)
d(p+1)+ηµ(t)α.

Then, by (3) and Remark 3.1.3 , we have for t ∈ Ip(s) (n < p ≤ l)

‖(µ0D)ηup,i(t)‖s =

∥∥∥∥∥
p−1∑
n=0

(µ0D)ηvn,i

∥∥∥∥∥
s

≤
p−1∑
n=0

Rrn+2Kn2dn+ηa0µ(t)(1−k)n+(1−k)ηω(t)n+(c−1)ησn,s(t)
dn+ηµ(t)α.

Thus, by Remark 3.1.3(4),

‖(µ0D)ηup,i(t)‖s ≤ Rr2a02
dl+d((l + 1)2 + 1)dl+dKηµ(t)α

p−1∑
n=0

rn



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1311

≤ Rµ(t)α,

provided r and a0 are small enough.
Now let l be an arbitrary integer satisfying l ≥ cd+c

1−κ . We prove by induction on (p, q)
(p ≥ 0, 0 ≤ q ≤ c) that the estimation

‖vl+pc+q,i(t)‖s ≤ Rrl+pc+q+2 (q,i)
max
G,φ

min
0≤L≤G

µ(t)α(φ,L)ω(t)β(φ,L)σl+pc+q,s(t)
γ(φ,L)µ(t)α (14)

holds for t ∈ Il+pc+q(s), where

α(φ,L) = cd+ φ− κL, β(φ,L) = cd+ c− φ+ L, γ(φ,L) = ld+ φ− L,

and
(q,i)
max
G,φ

= max
q≤G≤qd,

q≤φ≤n(i)+G

,

with G,L denoting integers and φ a real number. When (p, q)=(0,0),

‖vl,i(t)‖s ≤ Rrl+2µ(t)cdω(t)cd+cσl,s(t)
ldµ(t)α,

where G = 0 = L and φ = 0. Thus,

‖vl,i(t)‖s ≤ Rr1+2µ(t)(1−κ)lω(t)lσl,s(t)
ldµ(t)α,

since l(1 − κ) ≥ cd + c ≥ cd and l ≥ cd+c
1−k ≥ cd + c. Hence, (b) shows that (14) holds.

Assume that (14) holds for some (p, q) with q < c. If

a0 max{C(γ(φ,L)) : 0 ≤ φ ≤ cd+ c, 0 ≤ L ≤ cd} ≤ r,

then, applying Lemma 4 ρ(i, j)-times, we have

‖vl+pc+q+1,i(t)‖s ≤ Rrl+pc+q+3 max
(k,η)∈M(j),

1≤j≤N

(q,k)
max
G,φ

(i,j)

min
m

min
0≤L≤G

µ(t)α(φ+η,L+m)

×ω(t)β(φ+η,L+m)σ1+pc+q,s(t)
γ(φ+η,L+m)µ(t)α,

and
(q,i)
max
G,φ

= max
q+1≤G+ρ(i,j)≤qd+d,

q+1≤φ+η≤n(i)+(G+ρ(i,j))

,

which implies (14) for (p, q + 1), since the conditions (k, η) ∈ M(j) and m ≤ ρ(i, j) yield
that q + 1 ≤ φ+ η,

φ+ η ≤ (n(k) +G) + n(j, k)

= n(k) +G+ n(j)− n(k) + 1

= G+ n(j) + 1



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1312

= n(i) +G+ n(j)− n(i) + 1

= n(i) +G+ n(j, i)

≤ n(i) + (G+ ρ(i, j)),

L+m ≤ G+ ρ(i, j)

and

q + 1 ≤ G+ ρ(i, j)

≤ qd+ d.

Now, assume that (14) holds for (p, c) (i.e., c = q) with some p. Then

0 ≤ c− n(i) ≤ φ− n(i) ≤ G,

so we can put
L = φ− n(i)

(−[−z]) is the smallest integer which is not less than z. We have then

α(φ,L) ≥ cd+ φ− κ(φ+ 1− n(i))

= cd+ n(i) + (1− κ)

(
φ− n(i)− κ

1− κ

)

≥ α(n(i), 0) + (1− κ)

(
c− n(i)− κ

1− κ

)
≥ α(n(i), 0),

β(φ,L) ≥ cd+ c− φ+ (φ− n(i))

= β(n(i), 0)

and

γ(φ,L) ≤ γ(n(i), 0).

Therefore, (14) holds for (p+1, 0). This completes the proof of (14). Note that α(φ,L) ≥ 0,
β(φ,L) ≥ 0 and γ(φ,L) is bounded (indeed γ(φ,L) ≤ ld+ cd+ c). Hence, for n ≥ l,

‖(µ0D)ηun+1,i(t)‖s =

∥∥∥∥∥
n∑
x=0

(µ0D)ηvx,i(t)

∥∥∥∥∥
s

≤
n∑
x=0

µ(t)η‖Dηvx,i‖s



E. Y. Guerrero / Eur. J. Pure Appl. Math, 12 (3) (2019), 1297-1314 1313

≤
n∑
x=0

µ(t)ηKη(s(t)− s)−η‖vx,i(t)‖s(t)µ(t)α

≤
n∑
x=0

Kη2
ηa0µ(t)(1−κ)ηω(t)(c−1)ησx,s(t)

η

×Rrx+2 (q,i)
max
G,φ

min
0≤L≤G

µ(t)α(φ,L)ω(t)β(φ,L)σx,s(t)(t)
γ(φ,L)µ(t)α

≤
n∑
x=0

Kη2
ηa0µ(t)(1−κ)ηω(t)(c−1)ησx,s(t)

η

×Rrx+2 (q,i)
max
G,φ

min
0≤L≤G

µ(t)α(φ,L)ω(t)β(φ,L)2γ(φ,L)σx,s(t)
γ(φ,L)µ(t)α.

Thus,

‖(µ0D)ηun+1
i (t)‖s ≤ Rr2a0Kη2

d+ld+φ((x+ 1)2 + 1)d+ld+φ
n∑
x=0

rx

≤ Rµ(t)α.

Therefore, we have shown the well-definedness of un+1(t) for n ≥ l and un(t) converges to
u(t) ∈ Bs uniformly in I(s) = {t ≥ 0 : ω(t) < limn→∞ an(s0 − s)}. This u ∈ C0(I(s), Bs)
is the solution of (1).

3.2. Uniqueness of the Solution

The next proposition implies the uniqueness of our solution, but the proof is similar
to that of Proposition 3 and so we omit it here.

Proposition 4. Suppose un = (un,i)1≤n≤N and vn = (vn,i)1≤n≤N are two solutions of (1)
in

C0(I∞, (Bs(U, Y ))N )

with estimate
{‖un,i‖s, ‖vn,i‖s} ≤ Rµ(t)α

for all t ∈ I∞(s). Then, for t ∈ I∞(s), n = 0, 1, 2, . . . , we have

‖(un,i − vn,i)(t)‖s ≤ 2Rrn+2µ(t)(1−κ)nω(t)nσn,s(t)
dnµ(t)α

and

‖(µ0(t)D)η(un,i − vn,i)(t)‖s ≤ Rrn+22dn+η+1Kηa0µ(t)(1−κ)n+(1−κ)η

×ω(t)n(c−1)ησn,s(t)
dn+ηµ(t)α.



REFERENCES 1314

Acknowledgements

The author is supported by the Commission on Higher Education (CHED) of the
Philippines.

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