Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 44, pp. 1–15.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.44

EXISTENCE OF POSITIVE S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS

OF TIME-SPACE FRACTIONAL NONLOCAL REACTION-DIFFUSION

EQUATIONS

XUPING ZHANG, KAIBO DING, PENGYU CHEN

Abstract. This article studies the asymptotically periodic problem of time-space fractional
reaction-diffusion equations with nonlocal initial conditions on infinite intervals. Without the

assumption of upper and lower S-asymptotically ω-periodic solutions, the existence results of

positive S-asymptotically ω-periodic solutions for a class of abstract time-space fractional evolu-
tion equations with nonlocal initial conditions under growth and order conditions are obtained

by using the theory of operator semigroups and the method of monotone iteration. Finally, the

abstract results were applied to time-space fractional reaction-diffusion equations with nonlocal
initial conditions and some new results were obtained.

1. Introduction

In this article, we study the positive S-asymptotically ω-periodic solutions for the following
time-space fractional reaction-diffusion equation with nonlocal initial conditions

cDα
t u(t, x) + (−∆)βu(t, x) = F (t, u(t, x)), (t, x) ∈ [0,+∞)× Ω,

u(t, x) = 0, (t, x) ∈ [0,+∞)× ∂Ω,

u(0, x) = u0(x) +

m∑
k=1

aku(Tk, x), x ∈ Ω,

(1.1)

where cDα
t is the Caputo fractional derivative of order 0 < α < 1, (−∆)β is a fractional Laplacian

with 0 < β < 1 , Ω is a bounded open domain in Rn, 0 < T1 < T2 < · · · < Tm < +∞, ak ̸= 0 are
real numbers, k = 1, 2, . . . ,m, F : [0,+∞)× R → R is a continuous function.

It is well known that many realistic models are not strictly periodic. Therefore, since the concept
of S-asymptotically ω-periodic function was introduced in [20], asymptotically periodic problems
have been rapidly developed due to their broad physical background and realistic mathematical
models. In particular, the hereditary and memorability of fractional derivatives provides an ideal
tool for describing many phenomena and processes. It is worth noting that there are many relevant
results on the existence and uniqueness of S-asymptotically ω-periodic solutions to fractional
differential equations, one can refer to [3, 5, 6, 7, 22, 26] and references therein.

In addition, in many specific system applications, sometimes only positive solutions are sig-
nificant. In recent years, there are many results on the existence of positive solutions for frac-
tional differential equations, one can see [14, 17, 21, 25, 33]. However, the existence of positive
S-asymptotically ω-periodic solutions on infinite intervals are few. Shu [29] investigated the exis-
tence of the positive S-asymptotically ω-periodic solutions to a class of semilinear neutral Caputo
fractional differential equations with infinite delay. Li et al. [23] discussed the asymptotically
periodic problem for the abstract fractional evolution equation under order conditions and growth
conditions as well as obtained some new results on the existence of the positive S-asymptotically

2020 Mathematics Subject Classification. 35R11, 35B09, 34G20, 47J35.
Key words and phrases. Time-space fractional reaction-diffusion equation; nonlocal initial conditions;

positive S-asymptotically ω-periodic mild solutions; monotone iterative technique.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 13, 2024. Published April 24, 2025.

1



2 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

ω-periodic mild solutions. Gou [18] investigated the existence of minimal positive S-asymptotically
ω-periodic mild solution for structural damped elastic systems with delay and nonlocal conditions
on infinite interval. Besides, Gou [19] studied the existence of minimal positive S-asymptotically
ω-periodic mild solution for abstract evolution equation with delay on infinite interval. Further-
more, compared to the classical conditions, the nonlocal initial conditions are more practical when
describe some physical phenomena. It is worth noting that there are many relevant results on
nonlocal problems. For more details of nonlocal conditions, one can see [9, 10, 11, 12, 34] and
references therein.

Inspired by the above work, we are concerned about the positive S-asymptotically ω-periodic
solutions of nonlocal problem (1.1). The organization of this paper can be described as follows.
In the Section 2, we collect some necessary definitions and preliminary facts. In Section 3, we
present our abstract results. In the last section, applying our abstract results to nonlocal problem
(1.1), we prove the existence of positive S-asymptotically ω-periodic solutions.

2. Preliminaries

Unless stated otherwise, we will assume that (E, ∥ · ∥) is an ordered Banach space with partial
order “≤”, whose positive cone P = {u ∈ E : u ≥ θ} is normal with normal constant N , θ is the
zero element of E. Combining property of exponential functions, define a Banach space

Ce([0,∞), E) = {u ∈ C([0,∞), E) : lim
t→∞

e−t∥u(t)∥ = 0}

with the norm ∥ · ∥e = supt∈R+ e−t∥u(t)∥. We define a positive cone Pe ⊂ Ce(E) by

Pe = {u ∈ Ce(E) : u(t) ≥ θ, t ∈ [0,∞)}.

Then, Pe is normal and Ce(E) is an ordered Banach space, whose partial order ”≤” is induced
by the cone Pe. Now, we present an important result that will play an important role in the
subsequent proof.

Lemma 2.1 ([8]). The set Ξ ⊂ Ce([0,∞), E) is relatively compact if and only if the following
conditions hold:

(a) for each a > 0, the set Ξ is equicontinuous on [0, a];
(b) for any t ∈ [0,∞), Ξ(t) = {u(t) : u ∈ Ξ} is relatively compact in E;
(c) limt→∞ e−t∥u(t)∥ = 0 uniformly for u ∈ Ξ.

Next, let A : D(A) ⊂ E → E and −A generates an exponentially stable analytic semigroup
T (t)(t ≥ 0) in E. As we all know, for a general C0-semigroup, there exist constants M ≥ 1 and
ν ∈ R such that

∥T (t)∥ ≤Meνt, t ≥ 0.

In particular, let growth exponent

ν0 := inf{ν ∈ R : ∃M ≥ 1 such that ∥T (t)∥ ≤Meνt, t ≥ 0} < 0,

the semigroup T (t)(t ≥ 0) is said to be exponentially stable. It is well known [30] that if the
semigroup T (t) is continuous in the uniform operator topology for t > 0 in E, then ν0 can
obtained by the spectrum σ(A) of the operator A,

ν0 = − inf{Reλ | λ ∈ σ(A)}. (2.1)

By Blakrishnan’s definition [4, 35], the fractional power Aβ is well defined as

Aβu :=
sin(βπ)

π

∫ ∞

0

λβ−1(λ I +A)−1Audλ, 0 < β < 1, u ∈ D(A). (2.2)

From [35] one know that −Aβ is a closed densely defined operator, which generates a bounded
analytic semigroup Tβ(t)(t ≥ 0), which can be expressed as

Tβ(t) =

∫ ∞

0

gβ,t(s)T (s)ds, t > 0,



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 3

where gβ,t(·) is defined by the inverse Laplace integral

gβ,t(s) =
1

2πi

∫ σ+i∞

σ−i∞
ezs−tzβ

dz, σ > 0,

and the brach of zβ is so taken that Re(zβ) > 0 for Re(z) > 0. Moreover, one can see gβ,t(s) ≥ 0
for s > 0 and

∫∞
0
gβ,t(s)ds = 1 in [35]. It is valuable to note that there is an important lemma

about Tβ(t).

Lemma 2.2 ([24]). If the semigroup T (t)(t ≥ 0) generated by −A is exponentially stable and
compact, then the semigroup Tβ(t)(t ≥ 0) generated by −Aβ is exponentially stable and compact.

In the following, consider a probability density function hα(τ) defined by

hα(τ) =
1

πα

∞∑
n=1

(−τ)n−1Γ(nα+ 1)

n!
sin(nπα), τ ∈ (0,∞).

Obviously,

hα(τ) ≥ 0,

∫ ∞

0

hα(τ)dτ = 1,

∫ ∞

0

τhα(τ)dτ =
1

Γ(1 + α)
, τ ∈ (0,∞). (2.3)

Based on thesestatements, for t ≥ 0, we define the two operators:

Jα,β(t) =

∫ ∞

0

hα(τ)Tβ(t
ατ)dτ, Kα,β(t) = α

∫ ∞

0

τhα(τ)Tβ(t
ατ)dτ.

Similar to the proof in [1, 13, 32, 36], one has the following results.

Lemma 2.3. The operators Jα,β(t)(t ≥ 0) and Kα,β(t)(t ≥ 0) have the following properties.

(1) The operators Jα,β(t) and Kα,β(t) are strongly conntinuous operators, this indicates that
for any x ∈ E and 0 ≤ t1 ≤ t2, ∥Jα,β(t2)x−Jα,β(t1)x∥ → 0 and ∥Kα,β(t2)x−Kα,β(t1)x∥ →
0 as t2 − t1 → 0.

(2) Jα,β(t) and Kα,β(t) are linear bounded operators for any fixed t ∈ R+,

∥Jα,β(t)x∥ ≤M∥x∥ , ∥Kα,β(t)x∥ ≤ M

Γ(α)
∥x∥.

(3) Jα,β(t) and Kα,β(t) are uniformly continuous for every t > 0.
(4) If semigroup Tβ(t)(t ≥ 0) is compact, then Jα,β(t) and Kα,β(t) are compact operators for

every t > 0.
(5) If semigroup Tβ(t)(t ≥ 0) is positive, then Jα,β(t) and Kα,β(t) are positive operators.
(6) If semigroup Tβ(t)(t ≥ 0) is exponentially stable with the growth exponent −|ν0|β, then

∥Jα,β(t)∥ ≤MEα(−|ν0|βtα), ∥Kα,β(t)x∥ ≤MEα,α(−|ν0|βtα) (2.4)

for every t ≥ 0, where Eα(·) and Eα,α(·) are the Mittag-Leffler functions.

Lemma 2.4 ([31]). Eα(−µ) =
∫∞
0
τhα(τ)e

−µτdτ , Eα,α(−µ) = α
∫∞
0
τhα(τ)e

−µτdτ .

Now, we provide a definition of S-asymptotically ω-periodic function. Let Cb([0,∞), E) denote
the Banach space of all bounded and continuous functions from [0,∞) to E equipped with the
norm ∥u∥C = supt∈R+ ∥u(t)∥.

Definition 2.5 ([20]). A function f ∈ Cb([0,∞), E)is said to be S-asymptotically ω-periodic if
there exists ω > 0 such that limt→∞ ∥f(t + ω) − f(t)∥ = 0. In this case we say that ω is an
asymptotic period of f .

Let SAPω(E) represent the subspace of Cb([0,∞), E) consisting of all the E-value S-asymptotically
ω-periodic functions endowed with the uniform convergence norm denoted by ∥u∥C . Then SAPω(E)
is a Banach space.

Lemma 2.6. [27] Let Π be a convex, bounded and closed subset of a Banach space E. If Θ : Π → Π
is a condensing map, then Θ has a fixed poind in Π.



4 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

3. Abstract results

In this section, we discuss the positive S-asymptotically ω-periodic mild solutions for the fol-
lowing abstract time-space fractional evolution equations with nonlocal conditions

cDα
t u(t) +Aβu(t) = G(t, u(t)), t ∈ [0,+∞),

u(0) = u0 +

m∑
k=1

aku(Tk),
(3.1)

where cDα
t is the Caputo fractional derivative of the order 0 < α < 1, A : D(A) ⊂ E → E is a

closed linear operator and −A generates an exponentially stable analytic semigroup T (t)(t ≥ 0)
in E, Aβ is the fractional power operator of A for 0 < β < 1, 0 < T1 < T2 < · · · < Tm < +∞ and
ak are real numbers, G : [0,∞)× E → E is a continuous function.

Definition 3.1. A function u : [0,∞) → E is said to be a mild solution of the nonlocal problem
(3.1) if u ∈ C([0,∞), E) and satisfies

u(t) = Jα,β(t)Λu0 +

m∑
k=1

akJα,β(t)Λ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds

+

∫ t

0

(t− s)α−1Kα,β(t− s)G(s, u(s))ds.

(3.2)

Moreover, if u(t) ≥ θ for all t ≥ 0, then it is said to be a positive mild solution of nonlocal problem
(3.1).

To prove the main result, we also need the following assumption:

(H0)
∑m

k=1 |ak| <
1
M .

It follows from Lemma 2.3 (2) that ∥
∑m

k=1 akJα,β(Tk)∥ ≤ M
∑m

k=1 |ak| < 1. By the operator
spectral theorem, (H0) give a sufficient condition to guarantee the operator Λ on E given by

Λ =
(
I −

m∑
k=1

akJα,β(Tk)
)−1

exists and be bounded, where I is the identity operator. Indeed, by Neumann formula, Λ can be
expressed by

Λ =

∞∑
n=0

( m∑
k=1

akJα,β(Tk)
)n

.

Therefore,

∥Λ∥ ≤
∞∑

n=0

∥
m∑

k=1

akJα,β(Tk)∥n =
1

1− ∥
∑m

k=1 akJα,β(Tk)∥
≤ 1

1−M
∑m

k=1 |ak|
. (3.3)

Theorem 3.2. Let E be an ordered Banach space, whose positive cone P is normal, A : D(A) ⊂
E → E be a closed linear operator and −A generate an exponentially stable, positive, and compact
analytic semigroup T (t)(t ≥ 0) in E, whose growth exponent ν0 < 0, the nonlinear function
G : R+ × E → E be a continuous function. If the conditions (H0) and the following 3 conditions
hold:

(H1) for t ≥ 0 and x ∈ E, there exist positive constants A0 ≥ 0 and
A1 ∈ (0, (1−M

∑m
k=1 |ak|)|ν0|β/M) such that

∥G(t, etx)∥ ≤ A1∥x∥+A0,

(H2) G is nondecreasing with respect to the second variable, i.e., for x2 ≥ x1 ≥ θ,

G(t, x2) ≥ G(t, x1) ≥ θ, t ≥ 0,

(H3) there exists ω > 0, for every t ∈ [0,∞), x ∈ E,

lim
t→∞

∥G(t+ ω, x)−G(t, x)∥ = 0,



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 5

then there exist a minimal positive S-asymptotically ω-periodic mild solution ũ of nonlocal problem
(3.1).

Proof. Consider the operator Θ on Ce(E) defined by

(Θu)(t) = Jα,β(t)Λu0 +

∫ t

0

(t− s)α−1Kα,β(t− s)G(s, u(s))ds

+

m∑
k=1

akJα,β(t)Λ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds.

(3.4)

By (3.3), (3.4) and (H1), one can conclude that

e−t∥(Θu)(t)∥

≤ e−t∥Jα,β(t)Λu0∥+ e−t

∫ t

0

(t− s)α−1∥Kα,β(t− s)∥∥G(s, u(s))∥ds

+ e−t
m∑

k=1

|ak|∥Jα,β(t)∥∥Λ∥
∫ Tk

0

(Tk − s)α−1∥Kα,β(Tk − s)∥∥G(s, u(s))∥ds

≤ e−tM∥u0∥
1−M

∑m
k=1 |ak|

+
e−tM

∑m
k=1 |ak|

1−M
∑m

k=1 |ak|

× αM

∫ Tk

0

∫ ∞

0

τhα(τ)(Tk − s)α−1e−|ν0|β(Tk−s)ατ (A1∥u∥e +A0)dτds

+ e−tαM

∫ t

0

∫ ∞

0

τhα(τ)(t− s)α−1e−|ν0|β(t−s)ατ (A1∥u∥e +A0)dτds

≤ e−tM∥u0∥
1−M

∑m
k=1 |ak|

+
e−tM

∑m
k=1 |ak|

1−M
∑m

k=1 |ak|
M(A1∥u∥e +A0)

∫ ∞

0

hα(τ)dτ

∫ ∞

0

e−|ν0|βsds

+ e−tM(A1∥u∥e +A0)

∫ ∞

0

hα(τ)dτ

∫ ∞

0

e−|ν0|βsds

≤ e−tM∥u0∥
1−M

∑m
k=1 |ak|

+
e−tM

∑m
k=1 |ak|

1−M
∑m

k=1 |ak|
M(A1∥u∥e +A0)

|ν0|β

+
e−tM(A1∥u∥e +A0)

|ν0|β

≤ e−tM∥u0∥
1−M

∑m
k=1 |ak|

+ e−t
( M

∑m
k=1 |ak|

1−M
∑m

k=1 |ak|
+ 1

)M(A1∥u∥e +A0)

|ν0|β

≤ e−tM∥u0∥
1−M

∑m
k=1 |ak|

+
e−tM(A1∥u∥e +A0)(
1−M

∑m
k=1 |ak|

)
|ν0|β

. (3.5)

Thus, we can conclude that

∥(Θu)(t)∥e ≤
M∥u0∥

1−M
∑m

k=1 |ak|
+

M(A1∥u∥e +A0)

(1−M
∑m

k=1 |ak|)|ν0|β
:= φ+ ψ∥u∥e, (3.6)

where

φ =
|ν0|βM∥u0∥+MA0

(1−M
∑m

k=1 |ak|)|ν0|β
, ψ =

MA1

(1−M
∑m

k=1 |ak|)|ν0|β

are positive with ψ < 1. Hence, limt→∞ e−t∥(Θu)(t)∥ = 0, which implies that Θ : Ce(E) → Ce(E)
is well defined.

Next we prove that Θ is continuous on Ce(E). Let {un} ⊂ Ce(E) such that un → u as n→ ∞
in Ce(E). From the continuity of G, it can be obtained that

sup
s∈[0,∞)

∥G
(
s, un(s)

)
−G

(
s, u(s)

)
∥ → 0 as n→ ∞.

Then by the Lebesgue dominated convergence theorem,

∥(Θun)(t)− (Θu)(t)∥



6 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

≤M

m∑
k=1

|ak|∥Λ∥
∫ Tk

0

(Tk − s)α−1∥Kα,β(Tk − s)∥∥G(s, un(s))−G(s, u(s))∥ds

+

∫ t

0

(t− s)α−1∥Kα,β(Tk − s)∥∥G(s, un(s))−G(s, u(s))∥ds

≤
M

∑m
i=1 |ak|

1−M
∑m

k=1 |ak|
M

∫ ∞

0

hα(τ)dτ

∫ ∞

0

e−|ν0|βs∥G(s, un(s))−G(s, u(s))∥ds

+M

∫ ∞

0

hα(τ)dτ

∫ ∞

0

e−|ν0|βs∥G(s, un(s))−G(s, u(s))∥ds

≤ M(
1−M

∑m
k=1 |ak|

)
|ν0|β

sup
s∈[0,∞)

∥G(s, un(s))−G(s, u(s))∥

→ 0 as n→ ∞.

Hence,

∥(Θun)(t)− (Θu)(t)∥e = sup
t∈[0,∞)

e−t∥(Θun)(t)− (Θu)(t)∥ → 0 (n→ ∞),

which implies that Θ : Ce(E) → Ce(E) is a continuous operator. Therefore, one can deduced that
the fixed points of Θ are mild solutions to nonlocal problem (3.1).

Based on this fact, we first prove that Θ(SAPω(E)) ⊂ SAPω(E). For any ϵ > 0 and u ∈
SAPω(E), there exists a constant t1ϵ > 0, for t ≥ t1ϵ , have ∥u(t+ω)− u(t)∥ ≤ ϵ. On the one hand,
by continuity of G, for t > t1ϵ ,

∥G(t, u(t+ ω))−G(t, u(t))∥ ≤ |ν0|β

M
ϵ. (3.7)

On the other hand, by (H3), there exists a constant t2ϵ such that for t > t2ϵ ,

∥G(t+ ω, u(t+ ω))−G(t, u(t+ ω))∥ ≤ |ν0|β

M
ϵ. (3.8)

According to (2.4), we let

M0 =M max
{
sup
t≥0

Eα(−|ν0|βtα)(1 + t)α, sup
t≥0

Eα,α(−|ν0|βtα)(1 + t)2α
}
,

then

∥Jα,β(t)∥ ≤ M0

(1 + t)α
, ∥Kα,β(t)∥ ≤ M0

(1 + t)2α
, t ≥ 0. (3.9)

Hence, for t > max{t1ϵ , t2ϵ}, it follows from (3.4) that

(Θu)(t+ ω)− (Θu)(t) =

4∑
i=1

Bi(t),

where

B1(t) =
(
Jα,β(t+ ω)− Jα,β(t)

)
(
Λu0 +

m∑
k=1

akΛ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds
)
,

B2(t) =

∫ ω

0

(t+ ω − s)α−1Kα,β(t+ ω − s)G(s, u(s))ds,

B3(t) =

∫ t

0

(t− s)α−1Kα,β(t− s)
(
G(s, u(s+ ω))−G(s, u(s))

)
ds,

B4(t) =

∫ t

0

(t− s)α−1Kα,β(t− s)
(
G(s+ ω, u(s+ ω))−G(s, u(s+ ω))

)
ds.



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 7

This implies that

∥(Θu)(t+ ω)− (Θu)(t)∥ ≤
4∑

i=1

∥Bi(t)∥.

Let us start with estimations of ∥B1(t)∥ and ∥B2(t)∥. By (3.9), one can see that

∥B1(t)∥

= ∥
(
Jα,β(t+ ω)− Jα,β(t)

)
∥∥Λu0 +

m∑
k=1

|ak|Λ
∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds∥

≤ 2M0

(1 + t)α

(
∥Λu0∥+ ∥Λ∥

m∑
k=1

|ak|
∫ Tk

0

(Tk − s)α−1∥Kα,β(Tk − s)∥∥G(s, u(s))∥ds
)

and

∥B2(t)∥ = ∥
∫ ω

0

(t+ ω − s)α−1Kα,β(t+ ω − s)G(s, u(s))ds∥

≤
∫ ω

0

(t+ ω − s)α−1∥Kα,β(t+ ω − s)∥∥G(s, u(s))∥ds

≤
∫ ω

0

(t+ ω − s)α−1 (A1∥u(s)∥+A0)M0

(1 + t+ ω − s)2α
ds

≤ (A1∥u∥C +A0)
M0((t+ ω)α − tα)

α(1 + t)2α

≤ (A1∥u∥C +A0)
M0ω

α

α(1 + t)2α
.

By (H1), (3.7) and (3.9), one can obtain that

∥B3(t)∥ = ∥
∫ tϵ

0

(t− s)α−1Kα,β(t− s)
(
G(s, u(s+ ω))−G(s, u(s))

)
ds∥

+ ∥
∫ t

tϵ

(t− s)α−1Kα,β(t− s)
(
G(s, u(s+ ω))−G(s, u(s))

)
ds∥

≤
∫ tϵ

0

(t− s)α−1∥Kα,β(t− s)∥∥G(s, u(s+ ω))−G(s, u(s))∥ds

+

∫ t

tϵ

(t− s)α−1∥Kα,β(t− s)∥∥G(s, u(s+ ω))−G(s, u(s))∥ds

≤ 2M0

∫ tϵ

0

(t− s)α−1

(1 + t− s)2α
(A1∥u(s)∥+A0)ds

+

∫ t

0

(t− s)α−1∥Kα,β(t− s)∥ds |ν0|
β

M
ϵ

≤ 2M0(A1∥u∥C +A0)
(t− tϵ)

−α − t−α

α

+Mα

∫ t

0

((t− s)α−1

∫ ∞

0

τhα(τ)e
−|ν0|β(t−s)ατdτ)ds

|ν0|β

M
ϵ

≤ 2M0(A1∥u∥C +A0)
(t− tϵ)

−α − t−α

α
+ ϵ,

which implies that ∥B3(t)∥ tend to 0 as t→ ∞. Similarly, By (H1), (3.8) and (3.9), we can get that
∥B4(t)∥ tend to 0 as t→ ∞. Summing up, it follows from above results for ∥Bi(t)∥(i = 1, 2, 3, 4)
that

Θu ∈ SAPω(E),

which justifies the following inclusion, that is

Θ(SAPω(E)) ⊂ SAPω(E).



8 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

In what follows, we prove the existence of positive solutions by a monotone iterative technique.
For any u, v ∈ Pe with u ≤ v, by (H2), (3.4), u0 ≥ θ, the positivity of Jα,β(t) and Kα,β(t), one can
find that for all t ∈ [0,∞),

θ ≤ (Θu)(t) ≤ (Θv)(t).

Thus Θ is a monotonically increasing operator.
Let v0 = θ ∈ Pe ∩ SAPω(E) and define a sequence {vn} by

vn = Θvn−1, n = 1, 2, . . . . (3.10)

It follows from the monotonicity of Θ, (3.6) and (3.10) that {vn} ⊂ Pe ∩ SAPω(E) and

v0 ≤ v1 ≤ · · · ≤ vn ≤ . . . , (3.11)

∥vn∥e ≤ φ+ ψ∥vn−1∥e. (3.12)

Since ∥v0∥e ≡ 0, by (3.12), one can find that

∥vn∥e ≤ φ+ φψ + φψ2 + · · ·+ φψn−1 = φ
1− ψn

1− ψ
≤ φ

1− ψ
, (3.13)

which implies that the sequence {vn} is uniformly bounded. At this level, we verify that the
sequence {vn} is uniformly convergent.

Next, suppose that 0 < a < +∞ is an arbitrary constant, we need to verify {vn} ⊂ Pe∩SAPω(E)
is locally equicontinuous in [0, a]. For any u ∈ {vn} and 0 ≤ t1 ≤ t2 ≤ a, a direct computation
allows us to obtain

∥(Θu)(t2)− (Θu)(t1)∥ ≤
5∑

i=1

Di,

where

D1 = ∥Jα,β(t2)Λu0 − Jα,β(t1)Λu0∥,

D2 = ∥Jα,β(t2)− Jα,β(t1)∥
m∑

k=1

|ak|∥Λ∥
∫ Tk

0

(Tk − s)α−1∥Kα,β(Tk − s)∥∥G(s, u(s))∥ds,

D3 =

∫ t1

0

((t2 − s)α−1 − (t1 − s)α−1)∥Kα,β(t2 − s)∥∥G(s, u(s))∥ds,

D4 =

∫ t1

0

(t1 − s)α−1∥Kα,β(t2 − s)− Kα,β(t1 − s)∥∥G(s, u(s))∥ds,

D5 =

∫ t2

t1

(t2 − s)α−1∥Kα,β(t2 − s)∥∥G(s, u(s))∥ds.

We just need to examine that Di tend to 0 independently of u ∈ {vn} as t2 − t1 → 0 for i =
1, 2, 3, 4, 5. Thus, by Lemma 2.3, we obtain

D1 = ∥Jα,β(t2)Λu0 − Jα,β(t1)Λu0∥
≤ ∥Jα,β(t2)− Jα,β(t1)∥∥Λ∥∥u0∥

→ 0 as t2 − t1 → 0.

Similarly,

D2 ≤ ∥Jα,β(t2)− Jα,β(t1)∥
m∑

k=1

|ak|∥Λ∥
∫ Tk

0

(Tk − s)α−1∥Kα,β(Tk − s)∥∥G(s, u(s))∥ds

→ 0 ast2 − t1 → 0.

For D3, it follows from (H1) and (3.13) that

D3 =

∫ t1

0

((t2 − s)α−1 − (t1 − s)α−1)∥Kα,β(t2 − s)∥∥G(s, u(s))∥ds

≤ M

Γ(α+ 1)

(
A1

φ

1− ψ
+A0

)
(tα1 − tα2 + (t2 − t1)

α)



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 9

≤ M

Γ(α+ 1)

(
A1

φ

1− ψ
+A0

)
(t2 − t1)

α

→ 0 a st2 − t1 → 0.

For t1 = 0 and t2 > 0, it is conspicuous that D4 = 0. Now, for t1 > 0 and ϵ > 0 small enough, by
(H1), (3.13) and Lemma 2.3(2), we obtain

D4 ≤
∫ t1−ε

0

(t1 − s)α−1∥(Kα,β(t2 − s)− Kα,β(t1 − s))∥∥G(s, u(s))∥ds

+

∫ t1

t1−ε

(t1 − s)α−1∥(Kα,β(t2 − s)− Kα,β(t1 − s))∥∥G(s, u(s))∥ds

≤ (A1
φ

1− ψ
+A0) sup

s∈[0,t1−ε]

∥(Kα,β(t2 − s)− Kα,β(t1 − s))∥
∫ t1−ε

0

(t1 − s)α−1ds

+
2M

Γ(α)
(A1

φ

1− ψ
+A0)

∫ t1

t1−ε

(t1 − s)α−1ds

≤ (A1
φ

1− ψ
+A0)

(
sup

s∈[0,t1−ε]

∥(Kα,β(t2 − s)− Kα,β(t1 − s))∥ t
α
1 − εα

α
+

2M

Γ(α+ 1)
εα

)
→ 0 as t2 − t1 → 0, ϵ→ 0.

For D5, we observe that

D5 ≤
∫ t2

t1

(t2 − s)α−1∥Kα,β(t2 − s)∥∥G(s, u(s))∥ds

≤ M

Γ(α+ 1)
(A1

φ

1− ψ
+A0)(t2 − t1)

α

→ 0 as t2 − t1 → 0.

Combining all the above arguments, one can deduced that

∥(Θu)(t2)− (Θu)(t1)∥ → 0 as t2 − t1 → 0,

which means that the operator Θ is locally equicontinuous in [0, a] for arbitrary constant 0 < a <
+∞.

Subsequently, we need to prove {vn(t)} is relatively compact on E for t ∈ [0,∞). Let V = {vn}
and V0 = V ∪ {v0}. Obviously, V(t) = (ΘV0)(t) for t ∈ [0,∞). It is easy to prove that {vn(0)}
is relatively compact on E. We only consider the case t > 0, for all ∀ϵ ∈ (0, t) and δ > 0, define
Θϵ,δvn by

(Θϵ,δvn)(t) = Jα,β(t)Λvn−1(0) + α

m∑
k=1

akΛJα,β(t)

×
∫ Tk

0

∫ ∞

0

(Tk − s)α−1τhα(τ)Tβ((Tk − s)ατ)G(s, vn−1(s))dτds

+ α

∫ t−ϵ

0

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, vn−1(s))dτds

= Jα,β(t)Λvn−1(0) + α

m∑
k=1

akΛJα,β(t)

×
∫ Tk

0

∫ ∞

0

(Tk − s)α−1τhα(τ)Tβ((Tk − s)ατ)G(s, vn−1(s))dτds

+ αTβ(ϵ
αδ)

∫ t−ϵ

0

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ − ϵαδ)G(s, vn−1(s))dτds.



10 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

The compactness of Jα,β(t) and Tβ(ϵ
αδ) implies that the set (Θϵ,δV0)(t) is relatively compact in

E. Moreover, for ∀vn ∈ V0 and t ∈ (0,∞), one can obtain that

∥(Θvn)(t)− (Θϵ,δvn)(t)∥ = ∥α
∫ t

0

∫ δ

0

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, vn−1(s))dτds∥

+ ∥α
∫ t

t−ϵ

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, vn−1(s))dτds∥

≤
(
A1

φ

1− ψ
+A0

)
α

∫ t

0

∫ δ

0

(t− s)α−1τhα(τ)∥Tβ((t− s)ατ)∥dτds

+
(
A1

φ

1− ψ
+A0

)
α

∫ t

t−ϵ

∫ ∞

δ

(t− s)α−1τhα(τ)∥Tβ((t− s)ατ)∥dτds

≤M(A1
φ

1− ψ
+A0)

·
(∫ t

0

(t− s)α−1ds

∫ δ

0

τhα(τ)dτ +

∫ t

t−ϵ

(t− s)α−1ds

∫ ∞

δ

τhα(τ)dτ
)

→ 0 as ϵ→ 0, δ → 0.

We conclude that there is a relatively compact set (Θϵ,δV0)(t) arbitrarily close to the set (ΘV0)(t)
on E for t ∈ (0,∞). Consequently, we can obtain that {vn(t)} is relatively compact on E for
t ∈ [0,∞).

Further, for any u ∈ {vn}, by (3.5) and (3.13), we can easily get that

lim
t→∞

e−t∥(Θu)(t)∥ = 0.

Hence, it follows from Lemma 2.1 that {vn} is relatively compact in Pe ∩ SAPω(E). Hence, there
exist convergent subsequence in {vn}. As a result, one can obtain that {vn} itself is uniformly
convergent through the monotonicity of sequence and the normality of cone, which means that
there exist ũ ∈ Pe ∩ SAPω(E) such that limn→∞ vn = ũ. Moreover, taking the limit in (3.10),
we can obtain ũ = Θũ. Therefore, ũ ∈ Pe ∩ SAPω(E) is fixed point of Θ, which is a positive
S-asymptotically ω-periodic mild solution of nonlocal problem (3.1).

We need to verify that ũ is the minimal positive S-asymptotically ω-periodic mild solution. Let
û ∈ Pe ∩ SAPω(E) be a positive S-asymptotically ω-periodic mild solution of nonlocal problem
(3.1), which means that û(t) = Θû(t) for every t ∈ [0,∞). Obviously, û(t) ≥ v0 = 0. Taking into
account the monotonicity of Θ, one can deduced that

û(t) = (Θû)(t) ≥ (Θv0)(t) = v1(t), (3.14)

it follows that û > v1. Repeat this process, one can see û > vn, n = 1, 2, . . . . It’s worth noting
that one can obtain û > ũ through taking the limit in (3.14) as n → ∞, which means that ũ is
the minimal positive S-asymptotically ω-periodic mild solution of nonlocal problem (3.1). This
completes the proof of Theorem 3.2. □

Now, we assume that the cone P is a regeneration cone on E and T (t)(t ≥ 0) generated by −A
is a positive semigroup, it follows that λ0I+A has positive bounded inverse operator (λ0I+A)

−1 if
λ0 > − inf{Reλ | λ ∈ σ(A)} is sufficiently large through the characteristic of positive semigroups.
Since σ(A) ̸= ∅, the spectral radius

r((λ0I +A)−1) =
1

dist(−λ0, σ(A))
> 0.

Based on the famous Krein-Rutman theorem(see [15, 16]), A has the first eigenvalue λ1 > 0,
associated a positive eigenfunction e1, and

λ1 = inf{Reλ | λ ∈ σ(A)}.

Therefore, it follows from (2.1) that ν0 = −λ1. By Theorem (3.2), we have the following results.



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 11

Corollary 3.3. Let E be an ordered Banach space, whose positive cone P is a regeneration cone,
let A : D(A) ⊂ E → E be a closed linear operator and −A generate an exponentially stable,
positive, and compact semigroup T (t)(t ≥ 0) in E, u0 ≥ θ. Assume that G : [0,∞)× E → E is a
continuous function, and let conditions (H0), (H2), (H3), and

(H4) for t ≥ 0 and x ∈ E, there exist positive constants A0 ≥ 0 and

A1 ∈ (0, (1−M
∑m

k=1 |ak|)λ
β
1/M) such that

∥G(t, etx)∥ ≤ A1∥x∥+A0,

hold, then there exist a minimal positive S-asymptotically ω-periodic mild solution ũ of nonlocal
problem (3.1).

Theorem 3.4. Let E be an ordered Banach space, whose positive cone P is normal, A : D(A) ⊂
E → E be a closed linear operator and −A generate an exponentially stable, positive and compact
analytic semigroup T (t)(t ≥ 0) in E, whose growth exponent ν0 < 0, the nonlinear function
G : R+ × E → E be a continuous mapping. If the conditions (H0),(H3),(H4)and

(H5) for each u ∈ Ce(E) with u(t) ≥ ςe1, there is a constant ς > 0 such that

G(t, u(t)) ≥ G(t, ςe1) ≥ λβ1 ςe1,

hold and u(0) ≥ ςe1, then the nonlocal problem (3.1) has at least one positive S-asymptotically
ω-periodic mild solution.

Proof. Let Θ be defined by (3.4), it follows from the proof of Theorem (3.2) that

Θ(SAPω(E)) ⊂ SAPω(E).

We denote

BR0
:= {u ∈ Ce(E) | ∥u∥e ≤ R0, u(t) ≥ ςe1, t ≥ 0} (3.15)

which is a nonempty bounded convex closed set for

R0 ≥ M(λβ1∥u0∥+A0)(
1−M

∑m
k=1 |ak|

)
λβ1 −MA1

.

Hence, for any u ∈ BR0
and t ≥ 0, exploiting (H4), according to e−t ≤ 1, one can obtain

∥(Θu)(t)∥e = sup
t∈R+

e−t∥(Θu)(t)∥

≤ ∥(Θu)(t)∥

≤ M∥u0∥
1−M

∑m
k=1 |ak|

+
M(A1∥u∥e +A0)

(1−M
∑m

k=1 |ak|)λ
β
1

≤ R0.

Let w0 = ςe1. Then w0(t) = ςe1 for any t ≥ 0, and

η(t) :=c Dα
t w0(t) +Aβw0(t) = λβ1 ςe1 ≤ G(t, ςe1), t ≥ 0.

By the positivity of semigroup Tβ(t)(t ≥ 0), condition (H5) and (3.4), for any u ∈ BR0
and t ≥ 0,

one can see that

ςe1 = w0(t)

= Jα,β(t)Λw0(0) +

m∑
k=1

akJα,β(t)Λ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)η(s)ds

+

∫ t

0

(t− s)α−1Kα,β(t− s)η(s)ds

≤ Jα,β(t)Λςe1 +

m∑
k=1

akJα,β(t)Λ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, ςe1)ds

+

∫ t

0

(t− s)α−1Kα,β(t− s)G(s, ςe1)ds



12 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

≤ Jα,β(t)Λu0 +

m∑
k=1

akJα,β(t)Λ

∫ Tk

0

(Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds

+

∫ t

0

(t− s)α−1Kα,β(t− s)G(s, u(s))ds

= (Θu)(t).

Thus, Θ(BR0
) ⊂ BR0

and (Θu)(t) ≥ ςe1 for any u ∈ BR0
and t ≥ 0.

Next, we prove that Θ : BR0
→ BR0

is a completely continuous operator. From assumptions
(H3) and (H4), there is a constant W such that for all u ∈ BR0

,

sup
t∈[0,∞)

∥G(t, u(t))∥ ≤ W. (3.16)

It should be noted that the set Θ(BR0
) is locally equicontinuous on E by using the method similar

to Theorem (3.2) and for any u ∈ BR0
,

lim
t→∞

e−t∥(Θu)(t)∥ = 0.

So we only need to show that for any t ∈ [0,∞), {(Θu)(t) | u ∈ BR0
} is relatively compact in E.

Obviously, {(Θu)(0) : u ∈ BR0} is relatively compact in E. We only consider the case t > 0, for
all δ > 0 and ϵ ∈ (0, t), define (Θϵ,δu) by

(Θϵ,δu)(t) = Jα,β(t)Λu0 + α

m∑
k=1

akΛJα,β(t)

×
∫ Tk

0

∫ ∞

0

(Tk − s)α−1τhα(τ)Tβ((Tk − s)ατ)G(s, u(s))dτds

+ α

∫ t−ϵ

0

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, u(s))dτds

= Jα,β(t)Λu0 + α

m∑
k=1

akΛJα,β(t)

×
∫ Tk

0

∫ ∞

0

(Tk − s)α−1τhα(τ)Tβ((Tk − s)ατ)G(s, u(s))dτds

+ αTβ(ϵ
αδ)

∫ t−ϵ

0

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ − ϵαδ)G(s, u(s))dτds.

From the compactness of Jα,β(t) and Tβ(ϵ
αδ), one gets that {(Θϵ,δu)(t) | u ∈ BR0} is relatively

compact in E. Thus, for every u ∈ BR0 , it follows from (3.16) that

∥(Θu)(t)− (Θϵ,δu)(t)∥

= ∥α
∫ t

0

∫ δ

0

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, u(s))dτds∥

+ ∥α
∫ t

t−ϵ

∫ ∞

δ

(t− s)α−1τhα(τ)Tβ((t− s)ατ)G(s, u(s))dτds∥

≤ W
∫ t

0

∫ δ

0

(t− s)α−1τhα(τ)∥Tβ((t− s)ατ)∥dτds

+W
∫ t

t−ϵ

∫ ∞

δ

(t− s)α−1τhα(τ)∥Tβ((t− s)ατ)∥dτds

≤MW
∫ t

0

(t− s)α−1ds

∫ δ

0

τhα(τ)dτ +MW
∫ t

t−ϵ

(t− s)α−1ds

∫ ∞

δ

τhα(τ)dτ

→ 0 as ϵ→ 0, δ → 0,

which implies that there is a relatively compact set {(Θϵ,δu)(t) | u ∈ BR0
} arbitrarily close to the

set {(Θu)(t) | u ∈ BR0
} in E for t ∈ (0,∞). Therefore, the set {(Θu)(t) | u ∈ BR0

} is relatively



EJDE-2025/44 TIME-SPACE FRACTIONAL REACTION-DIFFUSION EQUATIONS 13

compact on E for t ∈ [0,∞). Moreover, it follows from Lemma 2.1 that Θ(BR0
) is relatively

compact in Ce(E).

Based on above results, one can find that Θ : BR0
∩ SAPω(E) → BR0

∩ SAPω(E) is a com-

pletely continuous operator, which implies that Θ is a condensing mapping from BR0
∩ SAPω(E)

into BR0 ∩ SAPω(E). Therefore, Lemma 2.6 implies that Θ has a fixed point ũ ∈ BR0 ∩ SAPω(E).
We need to verify that ũ ∈ SAPω(E). Let {un} ⊂ BR0 ∩ SAPω(E) converge to ũ, it follows

from the continuity of Θ and (3.15) that {Θun} converges to Θũ = ũ uniformly in [0,∞) and
ũ ≥ ςe1, which implies that ũ ∈ SAPω(E) is a positive S-asymptotically ω-periodic mild solution
of nonlocal problem (3.1). This completes the proof of Theorem 3.4. □

4. Application to nonlocal problem (1.1)

Let E = L2(Ω) with the L2-norm ∥ · ∥2 and partial order ≤, P = {u ∈ L2(Ω) | u(x) ≥
0, a.e.x ∈ Ω} is a normal cone in L2(Ω), then P is a regular cone of E. We define the operator
A : D(A) ⊂ E → E as follows:

D(A) =W 2,2(Ω) ∩W 1,2
0 (Ω), Au = −∆u. (4.1)

Let u(t, x) = u(t)(x) and

F (t, u(t, x)) = G(t, u(t))(x), u0 +

m∑
k=1

aku(Tk, x) = u0 +

m∑
k=1

aku(Tk)(x). (4.2)

Then the nonlocal problem (1.1) can be rewritten as an abstract evolution equation with nonlocal
conditions (3.1) in L2(Ω). According to (2.2), the fractional Laplacian is well defined. Besides,
if λn(n = 1, 2 . . . ) are the eigenvalues of −∆ with homogeneous Dirichlet boundary conditions
cnsidered in L2(Ω) and en as its corresponding eigenfuction, it follows that

(−∆)βen = λβnen, x ∈ Ω, e
∣∣
∂Ω

= 0,

which λn = n2π2 and corresponding eigenfunctions en(x) =
√
2 sin(nπx), n = 1, 2 . . . .

Hence, based on Corollary 3.3 and Theorem 3.4, we can establish the following results.

Theorem 4.1. Let nonlinear function F : [0,∞) × P → P be a continuous mapping. If the
followinf 4 conditions hold:

(K0)
∑m

k=1 |ak| < 1,
(K1) there are nonnegative constants A1 ∈ (0, (1−

∑m
k=1 |ak|)π2β), A0 ≥ 0 and a nondecreasing

function et ∈ C(R+, [1,∞)) with limt→∞ et = +∞ such that

∥F (t, etξ)∥2 ≤ A1∥ξ∥2 +A0, t ≥ 0, ξ ∈ E,

(K2) for any ξ1, ξ2 ∈ E with ξ2 ≥ ξ1 ≥ θ,

F (t, ξ2) ≥ F (t, ξ1) ≥ θ, t ≥ 0,

(K3) there exist ω > 0 such that

lim
t→∞

∥F (t+ ω, ξ)− F (t, ξ)∥2 = 0, ξ ∈ E, t ≥ 0,

then nonlocal problem (1.1) exist a minimal positive S-asymptotically ω-periodic solution.

Proof. From [2] one can see −A generates a uniformly bounded analytic semigroup T (t)(t ≥ 0)
in E , and T (t)(t ≥ 0) is contractive in E means that ∥T (t)∥ ≤ 1 for t ≥ 0. In addition, from
[28] the operator A has compact resolvent in L2(Ω) implies that the semigroup T (t)(t ≥ 0) is
compact. Besides, λI + A has a positive bounded inverse operator (λI + A)−1 for λ > 0 implies
that T (t)(t ≥ 0) is a positive semigroup. Therefore, based on the argument in preliminaries and
the properties of the semigroup T (t)(t ≥ 0) generated by −A, one can deduce that the analytic
semigroup Tβ(t)(t ≥ 0) generated by −Aβ is compact, positive and exponentially stable on E as
well as ∥Tβ(t)∥ ≤ 1 for all t ≥ 0. Let M = 1 and ν0 = −λ1 = −π2, by conditions (K0) and
(K1), we can deduced that conditions (H0) and (H4) hold. From the conditions (K2) and (K3),
we can deduced that conditions (H2) and (H3) hold. Thus, by Corrollary 3.3 one can deduced
that nonlocal problem (1.1) exist a minimal positive S-asymptotically ω-periodic solution. □



14 X. ZHANG, K. DING, P. CHEN EJDE-2025/44

Based on the proof of this theorem, it is not difficult to obtain the following result.

Theorem 4.2. Let nonlinear function F : [0,∞) × P → P be a continuous mapping. If the

conditions (K0)–(K4) hold for any ξ ∈ E with ξ ≥ ς
√
2 sin(πx), there is a constant ς > 0 such that

F (t, ξ) ≥ F (t, ς
√
2 sin(πx)) ≥ π2βς

√
2 sin(πx),

hold, and u0(x) ≥ ς
√
2 sin(πx), then nonlocal problem (1.1) has at least one positive S-asymptotically

ω-periodic solution.

Acknowledgments. This work was supported by the the National Natural Science Founda-
tion of China (No. 12061063), by the Outstanding Youth Science Fund of Gansu Province (No.
24JRRA122), by the Young Doctor Fund Project of Gansu Provincial Department of Education
(No. 2023QB-111), by the Funds for Innovative Fundamental Research Group Project of Gansu
Province (No. 23JRRA684), and by the Natural Science Foundation of Gansu Province (No.
24JRRA780) and Project 2024KGLX01017.

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Xuping Zhang (corresponding author)

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Email address: lanyu9986@126.com

Kaibo Ding

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Email address: dingkb583x@163.com

Pengyu Chen
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China.

Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou 730070,

China
Email address: chpengyu123@163.com


	1. Introduction
	2. Preliminaries
	3. Abstract results
	4. Application to nonlocal problem (1.1)
	Acknowledgments

	References