Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 31, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LOWER AND UPPER SOLUTIONS FOR DELAY EVOLUTION

EQUATIONS WITH NONLOCAL AND IMPULSIVE

CONDITIONS

XUPING ZHANG

Abstract. In this article, we apply the method of lower and upper solutions

for studying delay evolution equations with nonlocal and impulsive conditions

in infinite dimensional Banach spaces. Under wide monotone conditions and
noncompactness measure condition of nonlinear term, we obtain the existence

of extremal solutions and a unique solution between lower and upper solutions.
A concrete application to partial differential equations is considered.

1. Introduction and main results

Many complex process in nature and technology are described by functional
differential equations which are dominant nowadays because the functional compo-
nents in equations allow one to consider after-effect or prehistory influence. Delay
evolution equation is one of the important type of functional differential equations,
in which the response of system depends not only on the current state of sys-
tem, but also on the past history of system. For more details on this topic, see
[1, 13, 18, 21, 22, 23, 25, 26, 27, 28] and the references therein.

The study of abstract nonlocal Cauchy problem was initiated by Byszewski in
[5]. It is demonstrated that the nonlocal problems have better effects in applica-
tions than the traditional Cauchy problems, differential equations with nonlocal
conditions were studied by many authors and some basic results on nonlocal prob-
lems have been obtained, see [6, 8, 9, 15, 17, 24, 29, 30] and the references therein
for more comments and citations. Particularly, there has been a significant devel-
opment in the theory of impulsive evolution equations with nonlocal conditions in
Banach spaces. In 2009, Liang, Liu and Xiao [24] combined impulsive conditions
and nonlocal conditions, and investigated the nonlocal impulsive evolution equation
in Banach spaces. Later, Balachandran, Kiruthika and Trujillo [4], Chang, Kavitha
and Mallike Arjunan [7], Chen and Li [9], Debbouche and Baleanu [11], Ji, Li and
Wang [16], Fan and Li [17] studied the impulsive evolution equation with nonlocal
conditions.

2020 Mathematics Subject Classification. 35R12, 35K90, 47D06.
Key words and phrases. Evolution equations; delay; nonlocal and impulsive conditions;

lower and upper solutions; measure of noncompactness.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted June 2, 2021. Published April 18, 2022.

1



2 X. ZHANG EJDE-2022/31

We mention that in 2012, Chuong and Ke [10] studied the delay evolution inclu-
sions involving nonlocal and impulsive conditions

u′(t) +Au(t) ∈ F (t, u(t), ut), t ∈ [0, a], t 6= tk,

u(t+k ) = u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m,

u(s) + g(u)(s) = ϕ(s), s ∈ [−r, 0],

(1.1)

where X is a Banach space, F : [0, a] × X × C([−r, 0], X) → P (X) is a multi-
valued map, P (X) stands for the collection of all nonempty subsets of X, A is a
closed linear operator on X. By using the fixed point theory for multi-valued maps
and the theory of differential inclusions, the authors obtain the existence of mild
solutions for nonlocal problem (1.1). Furthermore, by applying corresponding mea-
sure of noncompactness estimates, they also proved the continuity of the solution
mapping, which demonstrates that the solution set depends continuously on initial
data. But so far we have not seen relevant papers that study delay evolution equa-
tions involving nonlocal and impulsive conditions by applying the iterative method,
perturbation technique and the method of lower and upper solutions.

Let X be a Banach space with norm ‖ · ‖, and a and h positive constants. We
denote by PC([−h, a], X) the space of piecewise continuous functions u : [−h, a]→
X such that u(t) is continuous at t 6= tk, left continuous at t = tk, and u(t+k )
exists for k = 1, 2, . . . ,m. Evidently, PC([−h, a], X) is a Banach space with norm
‖u‖PC = supt∈[−h,a] ‖u(t)‖. Let ut(τ) = u(t + τ) for τ ∈ [−h, 0]. In this space B
is considered as a Banach space of piecewise continuous functions v : [−h, 0] → X
with the norm ‖v‖B = sup−h≤s≤0 ‖v(s)‖.

In this article, we use the method of lower and upper solutions to discuss the
existence of solutions to the delay evolution equations with nonlocal and impulsive
conditions in an order space X,

u′(t) +Au(t) = f(t, u(t), ut), t ∈ [0, a], t 6= tk,

u(t+k )− u(t−k ) = Ik(u(tk)), k = 1, 2, . . . ,m,

u(s) = g(u)(s) + φ(s), s ∈ [−h, 0],

(1.2)

where A : D(A) ⊂ X → X is a closed linear operator and −A generates a positive
strongly continuous semigroup (positive C0-semigroup, in short) T (t) (t ≥ 0) on X;
f : [0, a]×X×B → X is a Carathéodory continuous; 0 < t1 < t2 < · · · < tm < a are
pre-fixed numbers, Ik ∈ C(X,X) is an impulsive function, k = 1, 2, . . . ,m, u(t+k )

and u(t−k ) represent the right and the left limits of u(t) at t = tk, respectively;
φ ∈ C([−h, 0], X) is a priori given history, while the function g : PC([−h, a], X)→
C([−h, 0], X) implicitly defines a complementary history, chosen by the system
itself; ut denotes the function in B defined as ut(τ) = u(t + τ) for τ ∈ [−h, 0] and
ut(·) represent the time history of the state from the time t − h up to the present
time t.

We denote J0 = [−h, 0], J1 = [0, t1], Jk = (tk−1, tk], k = 2, 3, . . . ,m+ 1, tm+1 =
a, I ′ = [−h, a]\{t1, t2, . . . , tm} and I ′′ = [−h, a]\{0, t1, t2, . . . , tm}, and use X1 to
denote the Banach space D(A) with the graph norm ‖ · ‖1 = ‖ · ‖ + ‖A · ‖. An
abstract function u ∈ PC([−h, a], X)∩C1(I ′′, X)∩C(I ′, X1) is called a solution of
the delay evolution equations with nonlocal and impulsive conditions (1.2) if u(t)
satisfies all the equalities in (1.2).



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 3

Let X be an ordered Banach space with partial order ≤, whose positive cone
P = {u ∈ X | u ≥ θ} is normal with normal constant N . Evidently, PC([−h, a], X)
and B are also order Banach spaces with partial order “ ≤ ” reduced by the positive
function cones KPC = {u ∈ PC([−h, a], X) : u(t) ≥ θ, t ∈ [−h, a]} and KB =
{u ∈ B | u(s) ≥ θ, s ∈ [−h, 0]} (θ is the zero element of X) respectively. For
v, w ∈ PC([−h, a], X) with v ≤ w, we use [v, w] to denote the order interval
{u ∈ PC([−h, a], X) | v ≤ u ≤ w}, and [v(t), w(t)] to denote the order interval
{u ∈ X : v(t) ≤ u(t) ≤ w(t), t ∈ [−h, a]}. If a function u ∈ PC([−h, a], X) ∩
C1(I ′′, X) ∩ C(I ′, X1) satisfies

u′(t) +Au(t) ≤ f(t, u(t), ut), t ∈ [0, a], t 6= tk,

u(t+k )− u(t−k ) ≤ Ik(u(tk)), k = 1, 2, . . . ,m,

u(s) ≤ g(u)(s) + φ(s), s ∈ [−h, 0],

(1.3)

we call it a lower solution of problem (1.2); if all inequalities of (1.3) are reversed,
we call it an upper solution of problem (1.2).

The method of lower and upper solutions is an important method for seeking
solutions of differential equations in abstract spaces. Early on, Du and Lakshmikan-
tham [14] built the method of lower and upper solutions for addressing the initial
ordinary differential equations in Banach space. Latter, Guo and Liu [19] built
an upper and lower solution method for the initial value problem (IVP) impulsive
differential equations in an ordered Banach space X,

u′(t) = f(t, u(t), Gu(t)), t ∈ [0, a], t 6= tk,

u(t+k )− u(t−k ) = Ik(u(tk)), k = 1, 2, . . . ,m,

u(0) = x0,

(1.4)

where f ∈ C([0, a]×X ×X,X), a > 0, 0 < t1 < t2 < · · · < tm < a, Ik ∈ C(X,X),
k = 1, 2, . . . ,m,

Gu(t) =

∫ t

0

K(t, s)u(s)ds, K ∈ C(∆,R+), ∆ = {(t, s) | 0 ≤ s ≤ t ≤ a}.

They proved that if IVP (1.4) has a lower solution v(0) and an upper solution w(0)

with v(0) ≤ w(0), and the nonlinear term f and impulsive function Ik satisfy the
monotonicity condition

f(t, x̄, ȳ)− f(t, x, y) ≥ −C(x̄− x)− C∗(ȳ − y), Ik(x̄) ≥ Ik(x),

v(0)(t) ≤ x ≤ x̄ ≤ w(0)(t), Gv(0)(t) ≤ y ≤ ȳ ≤ Gw(0)(t), ∀t ∈ [0, a],
(1.5)

with positive constant C and C∗, and noncompactness measure conditions

α(f(t, U, V )) ≤ L1α(U) + L2α(V ), (1.6)

α(Ik(D)) ≤Mkα(D), (1.7)

where U, V,D ⊂ X are arbitrarily sets, α(·) denotes the Kurataowski measure of
noncompactness in X, L1, L2 and Mk are positive constants and satisfy

2a(M + L1 + aK0L2) +

m∑
k=1

Mk < 1, (1.8)

where K0 = max(t,s)∈∆K(t, s), then IVP (1.4) has a minimal and maximal solu-

tions, which can be obtained by a monotone iterative procedure staring from v(0)



4 X. ZHANG EJDE-2022/31

and w(0) respectively. Recently, Chen and Li [9] extended the results of [19] to the
nonlocal impulsive problem evolution equations without delay in X,

u′(t) +Au(t) = f(t, u(t), Gu(t)), t ∈ J, t 6= tk,

u(t+k )− u(t−k ) = Ik(u(tk)), k = 1, 2, . . . ,m,

u(0) = g(u) + x0,

(1.9)

where g constitutes a nonlocal condition, x0 ∈ X.
The purpose of this article is to improve and extend the above-mentioned results

to the delay evolution equation involving nonlocal and impulsive conditions (1.2).
We will delete the noncompactness measure condition (1.7) for impulsive function
Ik and the strong restriction condition (1.8) for the constants. Our main results
are as follows.

Theorem 1.1. Let X be an ordered Banach space and its positive cone P be normal.
Assume that problem (1.2) has a lower solution v(0) ∈ PC([−h, a], X)∩C1(I ′′, X)∩
C(I ′, X1) and an upper solution w(0) ∈ PC([−h, a], X)∩C1(I ′′, X)∩C(I ′, X1) with
v(0) ≤ w(0). Suppose also that the following conditions are satisfied:

(H1) There exists a constant C > 0 such that

f(t, x̄, ȳ)− f(t, x, y) ≥ −C(x̄− x),

for ∀ t ∈ [0, a], x, x̄ ∈ X and y, ȳ ∈ B with v(0)(t) ≤ x ≤ x̄ ≤ w(0)(t) and
(v(0))t ≤ y ≤ ȳ ≤ (w(0))t;

(H2) Ik is increasing on order interval [v(0)(t), w(0)(t)] for t ∈ [0, a], k = 1, 2, . . . ,m;
(H3) The nonlocal function g(u) is continuous and compact and is increasing on

order interval [v(0), w(0)];
(H4) There exists a constant Lf > 0, such that for every t ∈ [0, a],

α
({
f(t, u(n)(t), (u(n))t)

})
≤ Lf

[
α
({
u(n)(t)

})
+ sup
−h≤τ≤0

α
({
u(n)(t+ τ)

})]
,

where {u(n)} ⊂ [v(0), w(0)] is countable and increasing or decreasing mono-
tonic set and {(u(n))t} ⊂ B.

Then problem (1.2) has minimal and maximal mild solutions between v(0) and w(0),
which can be obtained by a monotone iterative procedure starting from v(0) and w(0)

respectively.

Theorem 1.1 improves the main results in [9] and [19]. In Theorem 1.1, if X
is a weakly sequentially complete Banach space, then the condition (H4) holds
automatically. Hence, we can easily obtain the following result from Theorem 1.1.

Theorem 1.2. Let X be an ordered and weakly sequentially complete Banach space
and its positive cone P be normal. Assume that the problem (1.2) has a lower
solution v(0) ∈ PC([−h, a], X)∩C1(I ′′, X)∩C(I ′, X1) and an upper solution w(0) ∈
PC([−h, a], X)∩C1(I ′′, X)∩C(I ′, X1) with v(0) ≤ w(0), and conditions (H1)–(H3)
are satisfied, then problem (1.2) has minimal and maximal mild solution between
v(0) and w(0), which can be obtained by a monotone iterative procedure starting
from v(0) and w(0) respectively.

If we replace assumption (H4) by the assumption



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 5

(H5) There exist positive constants Lf , Mf and Lg <
1

NM such that for any

u, v ∈ [v(0), w(0)] and s ∈ [−h, 0],

f(t, u(t), ut)− f(t, v(t), vt)

≤Mf (u(t)− v(t)) + Lf (u(t+ τ)− v(t+ τ)), ∀t ∈ [0, a],

g(u)(s)− g(v)(s) ≤ Lg(u(s)− v(s)), ∀ s ∈ [−h, 0],

then we have the following uniqueness result.

Theorem 1.3. Let X be an order Banach space and its positive cone P be normal.
If problem (1.2) has a lower solution v(0) ∈ PC([−h, a], X)∩C1(I ′′, X)∩C(I ′, X1)
and an upper solution w(0) ∈ PC([−h, a], X) ∩ C1(I ′′, X) ∩ C(I ′, X1) with v(0) ≤
w(0), such that conditions (H1)–(H3), (H5) hold, then problem (1.2) has a unique
solution between v(0) and w(0), which can be obtained by a monotone iterative pro-
cedure starting from v(0) or w(0).

The proofs of Theorem 1.1 and 1.3 will be shown in the next section.

2. Proof of main results

Throughout this paper, let A : D(A) ⊂ X → X be a closed linear operator and
let −A generate a positive C0-semigroup T (t) (t ≥ 0) on ordered Banach space X.
Then there exist constants M1 ≥ 1 and δ ∈ R such that

‖T (t)‖ ≤M1e
δt, t ≥ 0. (2.1)

Denote L(X) by the Banach space of all bounded linear operators from X to X
equipped with its natural topology. From (2.1) we know that

M := sup
t∈[0,a]

‖T (t)‖L(X) ≥ 1 (2.2)

is a finite number. One can easily to see that for any constant C ≥ 0, −(A + CI)
also generates a positive C0-semigroup S(t) = e−CtT (t) (t ≥ 0) in X, and

sup
t∈[0,a]

‖S(t)‖L(X) = sup
t∈[0,a]

‖e−CtT (t)‖L(X) = M ≥ 1. (2.3)

Definition 2.1. A function u ∈ PC([−h, a], X) is said to be a mild solution of
(1.2) if it satisfies the equation

u(t) =


g(u)(t) + φ(t), t ∈ [−h, 0],

T (t)[g(u)(0) + φ(0)] +
∫ t

0
T (t− s)f(s, u(s), us)ds

+
∑

0<tk<t
T (t− tk)Ik(u(tk)), t ∈ [0, a].

(2.4)

We use α(·) denote the Kuratowski measure of noncompactness on the bounded
set of X. For the details about the definition and properties of the measure of
noncompactness, we refer to the monographs by Ayerbe, Domı́nguez and Lópezwo
[2], Banas̀ and Goebel [3], Deimling [12]. The following lemma will be used in proof
of Theorem 1.1.

Lemma 2.2 ([20]). Let D = {un}∞n=1 ⊂ PC([0, a], X) be a bounded and countable
set. Then α(D(t)) is Lebesgue integrable on [0, a], and

α
({∫ a

0

un(t)dt | n ∈ N
})
≤ 2

∫ a

0

α(D(t))dt.



6 X. ZHANG EJDE-2022/31

Proof of Theorem 1.1. It is easy to see that problem (1.2) is equivalent to the fol-
lowing delay evolution equations involving nonlocal and impulsive conditions

u′(t) +Au(t) + Cu(t) = f(t, u(t), ut) + Cu(t), t ∈ [0, a], t 6= tk,

u(t+k )− u(t−k ) = Ik(u(tk)), k = 1, 2, . . . ,m,

u(s) = g(u)(s) + φ(s), s ∈ [−h, 0],

(2.5)

for any constant C > 0. Therefore, we consider F : [v(0), w(0)] → PC([−h, a], X)
defined by

(Fu)(t) =


g(u)(t) + φ(t), if t ∈ [−h, 0],

S(t)[g(u)(0) + φ(0)] +
∫ t

0
S(t− s)[f(s, u(s), us) + Cu(s)]ds

+
∑

0<tk<t
S(t− tk)Ik(u(tk)), if t ∈ [0, a],

(2.6)

where S(t) = e−CtT (t) (t ≥ 0) is the positive C0-semigroup generated by−(A+CI).
By Definition 2.1, the mild solution of problem (1.2) is equivalent to the fixed point
of F defined by (2.6).

First, we prove that the operator F : [v(0), w(0)] → PC([−h, a], X) defined by
(2.6) is continuous. For this purpose, let {u(n)}∞n=1 ⊂ [v(0), w(0)] be a sequence such
that limn→∞ u(n) = u in [v(0), w(0)]. Then for any t ∈ [0, a], limn→∞(u(n))t = ut.
If t ∈ [−h, 0], by (2.6) and the continuity of the nonlocal function g, we have that

‖(Fu(n))(t)− (Fu)(t)‖ = ‖g(u(n))(t)− g(u)(t)‖ → 0 textas n→∞, (2.7)

and if t ∈ [0, a], by the Carathéodory continuity of the nonlinear function f , and
the continuity of the impulsive function Ik for k = 1, 2, . . . ,m, we obtain

lim
n→∞

‖f(s, u(n)(s), (u(n))s) + Cu(n)(s)− f(s, u(s), us)− Cu(s)‖ = 0 (2.8)

a.e. s ∈ [0, t]; and

lim
n→∞

‖Ik(u(n)(tk))− Ik(u(tk))‖ = 0 for k = 1, 2, . . . ,m. (2.9)

Applying assumption (H1), we know that for any u ∈ [v(0), w(0)] and s ∈ [0, t],
t ∈ [0, a],

f(s, v(0)(s), (v(0))s) + Cv(0)(s) ≤ f(s, u(s), us) + Cu(s)

≤ f(s, w(0)(s), (w(0))s) + Cw(0)(s).

The above inequality combined with the normality of the positive cone P , we know
that there exists a constant C1 > 0, such that

‖f(s, u(s), us) + Cu(s)‖ ≤ C1, s ∈ [0, t], t ∈ [0, a]. (2.10)

By (2.3), (2.6), (2.8)-(2.10) and the Lebesgue dominated convergence theorem, we
know that for any t ∈ [0, a],

‖(Fu(n))(t)− (Fu)(t)‖

≤M‖g(u(n))(0)− g(u)(0)‖

+M

∫ t

0

‖f(s, u(n)(s), (u(n))s) + Cu(n)(s)− f(s, u(s), us)− Cu(s)‖ds

+M
∑

0<tk<t

‖Ik(u(n)(tk))− Ik(u(tk))‖ → 0 as n→∞.

(2.11)



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 7

Hence, from (2.6), (2.8) and (2.11) we obtain

‖Qu(n) −Qu‖PC → 0 as n→∞,

which means that F : [v(0), w(0)]→ PC([−h, a], X) is a continuous operator.
Secondly, we prove that F maps [v(0), w(0)] to [v(0), w(0)] is a monotonic increas-

ing operator. By assumptions (H1)–(H3), F is increasing in [v(0), w(0)], and maps
any bounded set in [v(0), w(0)] into a bounded set. Next, we show that v(0) ≤ Fv(0)

and Fw(0) ≤ w(0). Letting

h(t) = (v(0))′(t) +Av(0)(t) + Cv(0)(t), t ∈ [0, a], t 6= tk, k = 1, 2, . . . ,m.

By Definition 2.1, we obtain that h ∈ PC([0, a], X) and h(t) ≤ f(t, v(0)(t), (v(0))t)+
Cv(0)(t) for t ∈ [0, a]. Therefore, by Definitions 2.1, the definition of lower solution
and the positivity of the C0-semigroup S(t) (t ≥ 0), we obtain that for t ∈ [0, a],

v(0)(t) = S(t)v(0)(0) +

∫ t

0

S(t− s)h(s)ds+
∑

0<tk<t

S(t− tk)[v(0)(t+k )− v(0)(t−k )]

≤ S(t)[g(v(0))(0) + ϕ(0)] +

∫ t

0

S(t− s)[f(s, v(0)(s), (v(0))s) +Mv(0)(s)]ds

+
∑

0<tk<t

S(t− tk)Ik(v(0)(tk))

= (Fv(0))(t), t ∈ [0, a],

and we know that for t ∈ [−h, 0],

v(0)(t) ≤ g(v(0))(t) + φ(t) = (Fv(0))(t).

The two inequalities above imply that v(0) ≤ Fv(0). Similarly, it can be shown that
Fw(0) ≤ w(0). Therefore, F : [v(0), w(0)] → [v(0), w(0)] is a monotonic increasing
operator.

Now, we define two sequences {v(n)} and {w(n)} in the ordered interval [v(0), w(0)]
by the following iterative scheme:

v(n) = Fv(n−1), w(n) = Fw(n−1), n = 1, 2, . . . . (2.12)

From the monotonicity of F , it follows that

v(0) ≤ v(1) ≤ v(2) ≤ · · · ≤ v(n) ≤ · · · ≤ w(n) ≤ · · · ≤ w(2) ≤ w(1) ≤ w(0). (2.13)

In what follows, we prove that {v(n)} and {w(n)} are convergent on [−h, a]. For
convenience, let U = {v(n) | n ∈ N} and U∗ = {v(n−1) | n ∈ N}. Then U = F(U∗).
From U = U∗ ∪ {v(0)} it follows that α(U(t)) = α(U∗(t)) for t ∈ [−h, a]. Let

ϕ(t) := α(U(t)) = α(U∗(t)), t ∈ [−h, a].

Going from J0 to Jm+1 interval by interval we show that ϕ(t) ≡ 0 in [−h, a].
For t ∈ J0, by (2.6) and assumption (H3), we have that

ϕ(t) = α(U(t)) = α(F(U∗)(t)) = α
{
g(v(n−1))(t) + φ(t)

}
= 0.



8 X. ZHANG EJDE-2022/31

For t ∈ J1, by (2.6), Lemma 2.2, the assumptions (H3) and (H4), we have

ϕ(t) = α(U(t)) = α(F(U∗)(t))

= α
({
S(t)[g(v(n−1))(0) + φ(0)]

+

∫ t

0

S(t− s)[f(s, v(n−1)(s), (v(n−1))s) + Cv(n−1)(s)]
})

≤ 2M

∫ t

0

α
({
f(s, v(n−1)(s), (v(n−1))s) + Cv(n−1)(s)

})
ds

≤ 2M

∫ t

0

[
Lfα(U∗(s)) + Lf sup

−h≤τ≤0
α(U∗(s+ τ)) + Cα(U∗(s))

]
ds

= 2M

∫ t

0

[
Lfϕ(s) + Lf sup

−h≤τ≤0
ϕ(s+ τ) + Cϕ(s)

]
ds.

(2.14)

Let ρ1(t) = sup{ϕ(s) : −h ≤ s ≤ t}, t ∈ J1. Then we have for any t ∈ J1,

ϕ(t) ≤ ρ1(t), sup
−h≤τ≤0

ϕ(t+ τ) ≤ ρ1(t). (2.15)

Combining with (2.14) and (2.15), we obtain that for any t ∈ J1,

ρ1(t) ≤ 2M(2Lf + C)

∫ t

0

ρ1(s)ds.

Hence, by the Gronwall’s inequality on gets that ρ1(t) ≡ 0 on J1. This means that
ϕ(t) ≡ 0 on J1. In particular, α(U(t1)) = α(U∗(t1)) = ϕ(t1) = 0, this implies that
U(t1) and U∗(t1) are precompact on X. Thus I1(U∗(t1)) is precompact on X and
α(I1(U∗(t1))) = 0.

Now, for t ∈ J2, by (2.6) and the above argument, we have

ϕ(t) = α(U(t)) = α(F(U∗)(t))

= α
({
S(t)[g(v(n−1))(0) + φ(0)]

})
+ α

({∫ t

0

S(t− s)[f(s, v(n−1)(s), (v(n−1))s) + Cv(n−1)(s)]ds
})

+ α({I1(v(n−1)(t1))})

≤ 2M

∫ t

0

[
Lfϕ(s) + Lf sup

−h≤τ≤0
ϕ(s+ τ) + Cϕ(s)

]
ds+ α(I1(U∗(t1)))

= 2M

∫ t

t1

[
Lfϕ(s) + Lf sup

−h≤τ≤0
ϕ(s+ τ) + Cϕ(s)

]
ds.

(2.16)

Let ρ2(t) = sup{ϕ(s) : −h ≤ s ≤ t}, t ∈ J2. Then for any t ∈ J2, we have

ϕ(t) ≤ ρ2(t), sup
−h≤τ≤0

ϕ(t+ τ) ≤ ρ2(t). (2.17)

Combining with (2.16) and (2.17), we obtain that for any t ∈ J2,

ρ2(t) ≤ 2M(2Lf + C)

∫ t

t1

ρ2(s)ds.

Again by the Gronwall’s inequality one gets that ρ2(t) ≡ 0 on J2. Thus ϕ(t) ≡ 0
on J2, from which we obtain that α(U∗(t2)) = 0 and α(I2(U∗(t2))) = 0.



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 9

Continuing such a process interval by interval up to Jm+1, we can prove that
ϕ(t) ≡ 0 on every Jk (k = 0, 1, . . . ,m+ 1). Thus, {vn(t)} is precompact on X and
{v(n)(t)} has a convergent subsequence for every t ∈ [−h, a]. Combining this with
the monotonicity (2.13), we easily prove that {v(n)(t)} itself is convergent for any
t ∈ [−h, a]. Using a similar argument with {v(n)(t)}, we can prove that {w(n)(t)}
is also convergent for any t ∈ [−h, a]. Set

lim
n→∞

v(n)(t) = u(t), lim
n→∞

w(n)(t) = u(t), t ∈ [−h, a].

Evidently, {v(n)} ⊂ PC([−h, a], X), so u is bounded and integrable on [−h, a].
By (2.6), we have that

v(n)(t) = F(v(n−1))(t)

=


g(v(n−1))(t) + φ(t), if t ∈ [−h, 0],

S(t)[g(v(n−1))(0) + φ(0)] +
∫ t

0
S(t− s)[f(s, v(n−1)(s), (v(n−1))s)

+Cv(n−1)(s)]ds+
∑

0<tk<t
S(t− tk)Ik(v(n−1)(tk)), if t ∈ [0, a].

Letting n → ∞ in the above equalities, by the Lebesgue dominated convergent
theorem, we have

u(t) = (Fu)(t)

=


g(u)(t) + φ(t), if t ∈ [−h, 0],

S(t)[g(u)(0) + φ(0)] +
∫ t

0
S(t− s)[f(s, u(s), (u)s) + Cu(s)]ds

+
∑

0<tk<t
S(t− tk)Ik(u(tk)), if t ∈ [0, a].

This means that u ∈ PC([−h, a], X) and u = Fu. Similarly, we can prove that
u ∈ PC([−h, a], X) and u = Fu. Letting n → ∞ in (2.13), we see that v(0) ≤
u ≤ u ≤ w(0). By the monotonicity of F , it is easy to see that u and u are the
minimal and maximal fixed points of F in [v(0), w(0)]. Therefore, u and u are the
minimal and maximal mild solutions of the problem (1.2) in [v(0), w(0)], and they
can be obtained by a monotone iterative procedure (2.12) starting from v(0) and
w(0), respectively. This completes the proof. �

Proof of Theorem 1.3. We firstly proof that (H1) and (H5) imply (H4). For this
purpose, let {u(n)} ⊂ [v(0), w(0)] be a increasing sequence. For m,n ∈ N with
m > n, by (H1) and (H5), we obtain that for every t ∈ [0, a] and τ ∈ [−h, 0],

θ ≤ f(t, u(m)(t), (u(m))t)− f(t, u(n)(t), (u(n))t) + C[u(m)(t)− u(n)(t)]

≤ (C +Mf )[u(m)(t)− u(n)(t)] + Lf [u(m)(t+ τ)− u(n)(t+ τ)].

By this and the normality of cone P , we have that for any t ∈ [0, a] and some
τ ∈ [−h, 0],

‖f(t, u(m)(t), (u(m))t)− f(t, u(n)(t), (u(n))t)‖

≤ N(C +Mf )‖u(m)(t)− u(n)(t)‖

+NLf‖u(m)(t+ τ)− u(n)(t+ τ)‖+ C‖u(m)(t)− u(n)(t)‖

≤ [N(C +Mf ) + C]‖u(m)(t)− u(n)(t)‖

+NLf sup
−h≤τ≤0

‖u(m)(t+ τ)− u(n)(t+ τ)‖.



10 X. ZHANG EJDE-2022/31

From this inequality and the definition of the measure of noncompactness, it follows
that for every t ∈ [0, a],

µ
({
f(t, u(n)(t), (u(n))t)

})
≤ Lf

[
µ
({
u(n)(t)

})
+ sup
−h≤τ≤0

µ
({
u(n)(t+ τ)

})]
,

where Lf = max
{
N(C+Mf ) +C,NLf

}
. If {u(n)} ⊂ [v(0), w(0)] is decreasing, the

above inequality is also valid. Hence (H4) holds.
Therefore, by Theorem 1.3, problem (1.2) has minimal mild solution u and max-

imal mild solution u in [v(0), w(0)]. Next, going from J0 to Jm+1 interval by interval
we show that u(t) ≡ u(t) on every Jk (k = 0, 1, . . . ,m+ 1).

For t ∈ J0, by (2.6) and assumption (H5), we obtain

θ ≤ u(t)− u(t) = Fu(t)−Fu(t) = g(u)(t)− g(u)(t) ≤ Lg(u(t)− u(t)).

Combining this and the normality of cone P , we have

‖u(t)− u(t)‖ ≤ NLg‖u(t)− u(t)‖, t ∈ J0.

Hence, for any t ∈ J0,

(1−NLg)‖u(t)− u(t)‖ ≤ 0.

From the above inequality and the assumption 1−NLg > 0, one gets that u(t) ≡
u(t) on J0.

For t ∈ J1, by (2.6) and assumption (H5), we have

θ ≤ u(t)− u(t) = Fu(t)−Fu(t)

= S(t)[g(u)(0)− g(u)(0)]

+

∫ t

0

S(t− s)[f(s, u(s), (u)s)− f(s, u(s), (u)s) + C(u(s)− u(s))]ds

≤ LgS(t)(u(0)− u(0))

+

∫ t

0

S(t− s)[(C +Mf )(u(s)− u(s)) + Lf (u(s+ τ)− u(s+ τ))]ds.

Since u(0) = u(0), using the above inequality and normality of cone P , we can
prove that

‖u(t)−u(t)‖ ≤ NM
∫ t

0

[(C+Mf )‖u(s)−u(s)‖+Lf‖u(s+τ)−u(s+τ)‖]ds. (2.18)

We define a non-negative function ρ′1(t) = sup{‖u(s)− u(s)‖ : −h ≤ s ≤ t} on J1.
Then, for any t ∈ J1,

‖u(t)− u(t)‖ ≤ ρ′1(t), sup
−h≤τ≤0

‖u(t+ τ)− u(t+ τ)‖ ≤ ρ′1(t). (2.19)

From (2.18) and (2.19)) it follows that

ρ′1(t) ≤ NM(C +Mf + Lf )

∫ t

0

ρ′1(s)ds, t ∈ J1.

By this fact and Gronwall’s inequality, we obtain that u(t) ≡ u(t) on J1.



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 11

For t ∈ J2, since I1(u(t1)) = I1(u(t1)), using completely similar argument as
above for t ∈ J1, we can prove that

‖u(t)− u(t)‖

≤ NM
∫ t

0

[(C +Mf )‖u(s)− u(s)‖+ Lf‖u(s+ τ)− u(s+ τ)‖]ds

≤ NM
∫ t

t1

[(C +Mf )‖u(s)− u(s)‖+ Lf‖u(s+ τ)− u(s+ τ)‖]ds.

(2.20)

Defining a non-negative function ρ′2(t) = sup{‖u(s) − u(s)‖ : −h ≤ s ≤ t} on J2.
We know that for t ∈ J2,

‖u(t)− u(t)‖ ≤ ρ′2(t), sup
−h≤τ≤0

‖u(t+ τ)− u(t+ τ)‖ ≤ ρ′2(t). (2.21)

Combining (2.20) and (2.21), we have

ρ′2(t) ≤ NM(C +Mf + Lf )

∫ t

t1

ρ′2(s)ds, t ∈ J2.

Again by the Gronwall’s inequality, we obtain that ρ′2(t) ≡ 0 on J2. Hence, u(t) ≡
u(t) on J2.

Continuing such a process interval by interval up to Jm+1, we see that u(t) ≡ u(t)
over the whole of [−h, a]. Therefore, ũ := u = u is the unique mild solution
of problem (1.2) in [v(0), w(0)], which can be obtained by the monotone iterative
procedure (2.13) starting form v(0) or w(0). This completes the proof. �

3. Example

In this section, we give an example to illustrate the applicability of our abstract
results obtained in Section 2. We consider the delay parabolic partial differential
equation involving nonlocal and impulsive conditions of the form

∂

∂t
w(x, t)− ι ∂

2

∂x2
w(x, t) = L

( |w(x, t)|
1 + |w(x, t)|

)
+

∫ 0

−h
G(s)w(x, t+ s)ds,

x ∈ [c, d], t ∈ [0, a], t 6= tk,

w(x, t+k ) = w(x, t−k ) +

√
|w(x, tk)|

1 + |w(x, tk)|
, x ∈ [c, d], k = 1, 2, . . . ,m,

]w(c, t) = w(d, t) = 0, t ∈ [0, a],

w(x, s) =

∫ a

0

ρ(s, t) lg(1 + |w(x, t)|)dt+ φ(x, s), x ∈ [c, d], s ∈ [−h, 0],

(3.1)

where ι > 0 is the coefficient of heat conduction, a, h, L > 0 are all constants,
0 < t1 < t2 < · · · < tm < a, G ∈ L([−h, 0],R+), ρ(s, t) is a continuous function
from [−h, 0]× [0, a] to R+, φ ∈ C([c, d]× [−h, 0],R+).

Let X = L2([c, d],R) with the norm ‖ · ‖2 and let P = {v ∈ L2([c, d],R) : v(x) ≥
0 a.e. x ∈ [c, d]}. Then X is a Banach space, P is a normal cone of X with normal
constant N = 1. Define an operator A : D(A) ⊂ X → X by

Aw = −ι ∂
2

∂x2
w, w ∈ D(A).

The domain D(A) is defined by

D(A) = H2(c, d) ∩H1
0 (c, d).



12 X. ZHANG EJDE-2022/31

It is well know that A has discrete spectrum with eigenvalues λn = ιn2π2(d −
c)2 (n ∈ N) and the corresponding normalized eigenvectors are en(x) =

√
2/z sinnπx(d−

c), z =
√
d− c+ (sin 2nπc− sin 2nπd)/(2nπ), the set {en : n ∈ N} is an orthonor-

mal basis of X and

Aw =

∞∑
n=1

(w, en)en, w ∈ D(A).

Furthermore, −A generates a positive C0-semigroup T (t) (t ≥ 0) in X, which is
given by

T (t)w =

∞∑
n=1

e
− ιn2π2t

(d−c)2 (w, en)en, w ∈ X, t > 0.

and ‖T (t)‖ ≤ e−
ιπ2t

(d−c)2 , for any t ≥ 0. Let

u(t) = w(·, t), t ∈ [−h, a],

f(t, u(t), ut) = L
( |w(·, t)|

1 + |w(·, t)|

)
+

∫ 0

−h
G(s)w(·, t+ s)ds, t ∈ [0, a],

Ik(u(tk)) =

√
|w(x, tk)|

1 + |w(x, tk)|
, k = 1, 2, . . . ,m,

g(u)(s) =

∫ a

0

ρ(s, t) lg(1 + |w(·, t)|)dt, φ(s) = φ(·, s), s ∈ [−h, 0].

Then the delay parabolic partial differential equation involving nonlocal and im-
pulsive conditions (3.1) can be transformed into the abstract form of problem (1.2).

Theorem 3.1. Assume that there exists a function v = v(x, t) ∈ PC([c, d] ×
[−h, a],R) ∩ C1([0, 1]× I ′′,R) such that

∂

∂t
v(x, t)− ι ∂

2

∂x2
v(x, t) ≥ L

( |v(x, t)|
1 + |v(x, t)|

)
+

∫ 0

−h
G(s)w(v, t+ s)ds,

t ∈ [0, a], t 6= tk,

v(x, t+k ) ≥ v(x, t−k ) +

√
|w(x, tk)|

1 + |w(x, tk)|
, k = 1, 2, . . . ,m,

v(0, t) = v(1, t) = 0, t ∈ [0, a],

v(x, s) ≥
∫ a

0

ρ(s, t) lg(1 + |v(x, t)|)dt+ φ(x, s), s ∈ [−h, 0].

Then the delay parabolic partial differential equation involving nonlocal and impul-
sive conditions (3.1) exist a minimal mild solution and a maximal mild solution
between 0 and v(x, t), which can be obtained by a monotone iterative procedure
starting from 0 and v(x, t), respectively.

Proof. From the assumption and the definition of nonlinear term f , impulsive func-
tion Ik for k = 1, 2, . . . ,m and nonlocal function g, we can verify that that v(0) = 0
and w(0) = v(x, t) are the lower and the upper solutions of the delay parabolic
partial differential equation involving nonlocal and impulsive conditions (3.1) re-
spectively. Furthermore, the conditions (H1), (H2) and (H3) are satisfied with
C = 1

2 and the nonlocal function g : PC([−h, a], X)→ B is completely continuous.



EJDE-2022/31 NONLOCAL EVOLUTION EQUATIONS WITH DELAY 13

On the other hand, from the definition of nonlinear term f , we know that

θ ≤ f(t, v2, ϕ2)−f(t, v1, ϕ1) ≤ L
( |v2|

1 + |v2|
− |v1|

1 + |v1|

)
+

∫ 0

−h
G(s)(ϕ2(s)−ϕ1(s))ds,

for all t ∈ [0, a], v1, v2 ∈ X and ϕ1, ϕ2 ∈ B with θ ≤ ϕ1 ≤ ϕ2. Combining this and
the normality of cone P , we have

‖f(t, v2, ϕ2)− f(t, v1, ϕ1)‖2

≤ L(‖v2 − v1‖2) +

∫ 0

−h
G(s)‖ϕ2(s)− ϕ1(s)‖2ds

≤ L(‖v2 − v1‖2) +

∫ 0

−h
G(s)ds sup

−h≤s≤0
‖ϕ2(s)− ϕ1(s)‖2.

Then for every t ∈ [0, a],

µ
({
f(t, u(n)(t), (u(n))t)

})
≤ max

{
L,

∫ 0

−h
G(s)ds

}[
µ
({
u(n)(t)

})
+ sup
−h≤s≤0

µ
({
u(n)(t+ s)

})]
,

where {u(n)} ⊂ [v(0), w(0)] is countable and increasing or decreasing monotonic

set. This means that the condition (H4) holds with Lf = max
{
L,
∫ 0

−hG(s)ds
}

.
Therefore, our conclusion follows from Theorem 1.1. This completes the proof. �

Acknowledgments. The authors would like to express sincere thanks to the
anonymous referee for his/her carefully reading the manuscript and valuable com-
ments and suggestions. This research was supported by the NSF of Gansu Province
(No. 20JR5RA522), NNSF of China (No. 12061063), Science Research Project for
Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-
LKQN2019-13 and Project of NWNU-LKQN2019-3.

References

[1] S. Antontsev, J. Ferreira, E. Piskin, H. Yuksekkaya, M. Shahrouzi; Blow up and asymptotic
behavior of solutions for a p(x)-Laplacian equation with delay term and variable exponents,

Electron. J. Differential Equations, 2021 no. 84 (2021), 1–20.
[2] J. M. Ayerbe, T. Domı́nguez, G. López; Measures of Noncompactness in Metric Fixed Point

Theory, Adv. Appl., vol. 99, Birkhäuser Verlag, Basilea, 1997.

[3] J. Banas̀, K. Goebel; Measures of Noncompactness in Banach Spaces, In Lecture Notes in
Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980.

[4] K. Balachandran, S. Kiruthika, J. J. Trujillo; On fractional impulsive equations of Sobolev
type with nonlocal condition in Banach spaces, Comput. Math. Appl., 62 (2011), 1157–1165.

[5] L. Byszewski; Theorems about the existence and uniqueness of solutions of a semilinear
evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494–505.

[6] L. Byszewski; Application of preperties of the right hand sides of evolution equations to an
investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413–426.

[7] Y. K. Chang, V. Kavitha, M. Mallike Arjunan; Existence results for impulsive neutral dif-
ferential and integrodifferential equations with nonlocal conditions via fractional operators,
Nonlinear Anal.: Hybrid Syst., 4 (2010), 32–43.

[8] P. Chen, Y. Li; Monotone iterative technique for a class of semilinear evolution equations
with nonlocal conditions, Results Math., 63 (2013), 731–744.

[9] P. Chen, Y. Li; Monotone iterative method for abstract impulsive integro-differential equa-

tions with nonlocal conditions in Banach spaces, Appl. Math., 59 (2014), 99–120.
[10] N. M. Chuong, T. D. Ke; Generalized Cauchy problems involving nonlocal and impulsive

conditions, J. Evol. Equ., 12 (2012), 367–392.



14 X. ZHANG EJDE-2022/31

[11] A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasi-

linear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442–1450.

[12] K. Deimling; Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
[13] Q. X. Dong, G. Li; Measure of noncompactness and semilinear nonlocal functional differential

equations in Banach spaces, Acta Math. Sin. (Engl. Ser.), 31 (2015) 140–150.

[14] S.W. Du, V. Lakshmikantham, Monotone iterative technique for differential equations in
Banach spaces, J. Math. Anal. Appl., 87 (1982), 454–459.

[15] K. Ezzinbi, X. Fu, K. Hilal; Existence and regularity in the α-norm for some neutral partial

differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613–1622.
[16] S. Ji, G. Li, M. Wang; Controllability of impulsive differential systems with nonlocal condi-

tions, Appl. Math. Comput., 217 (2011), 6981–6989.

[17] Z. Fan, G. Li; Existence results for semilinear differential equations with nonlocal and im-
pulsive conditions, J. Functional Anal., 258 (2010), 1709–1727.

[18] X. Fu; Approximate controllability of semilinear non-autonomous evolution systems with
state-dependent delay, Evol. Equ. Control Theory, 6 (2017): 517–534.

[19] D. Guo, X. Liu; Extremal solutions of nonlinear impulsive integro-differential equations in

Banach spaces, J. Math. Anal. Appl., 177 (1993), 538–552.
[20] H.P. Heinz; On the behaviour of measure of noncompactness with respect to differentiation

and integration of rector-valued functions, Nonlinear Anal. 7, (1983), 1351–1371.

[21] Y. Li; Existence and asymptotic stability of periodic solution for evolution equations with
delays, J. Functional Anal., 261 (2011), 1309–1324.

[22] K. Li, J. Jia; Existence and uniqueness of mild solutions for abstract delay fractional differ-

ential equations, Comput. Math. Appl., 62 (2011), 1398–1404.
[23] J. Liang, T. J. Xiao; Solvability of the Cauchy problem for infinite delay equations, Nonlinear

Anal., 58 (2004), 271–297.

[24] J. Liang, J. Liu, T. J. Xiao; Nonlocal impulsive problems for integrodifferential equations,
Math. Comput. Modelling, 49 (2009), 798–804.

[25] C. C. Travis, G. F. Webb; Existence and stability for partial functional differential equations,
Trans. Amer. Math. Soc. 200 (1974) 395–418.

[26] I. I. Vrabie; Existence in the large for nonlinear delay evolution inclusions with nonlocal

initial conditions, J. Functional Anal., 262 (2012), 1363-1391.
[27] I. I. Vrabie; Delay evolution equations with mixed nonlocal plus local initial conditions, Com-

mun. Contemp. Math., 17(2015), 1350035, 22 pp.

[28] J. Wang, Y. Zhou; Existence of mild solutions for fractional delay evolutions systems, Appl.
Math. Comput., 218 (2011), 357–367.

[29] R. N. Wang, T. J. Xiao, J. Liang; A note on the fractional Cauchy problems with nonlocal

initial conditions, Appl. Math. Lett., 24 (2011), 1435–1442.
[30] X. Zhang, Y. Li, Fractional retarded evolution equations with measure of noncompactness

subjected to mixed nonlocal plus local initial conditions, Int. J. Nonlinear Sci. Numer. Simul.,

19 (2018), 69–81.

Xuping Zhang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Email address: lanyu9986@126.com


	1. Introduction and main results
	2. Proof of main results
	3. Example
	Acknowledgments

	References