Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 43, pp. 1–15.

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

POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC

SOLUTIONS FOR SYSTEMS OF NONLINEAR DELAY

INTEGRAL EQUATIONS

ABDELLATIF SADRATI, ABDERRAHIM ZERTITI

Abstract. In this article we show the existence of positive Stepanov-like al-
most automorphic solutions for systems of nonlinear delay integral equations.

To do this, we apply the well-known Guo-Krasnosel’skii fixed point theorem

for cone expansion and compression.

1. Introduction

Almost automorphic functions, as a generalization of the classical periodic and
almost periodic functions, was introduced by Bochner in the earlier sixties [2, 3, 4]
to avoid some assumptions of uniform convergence that arise when using almost
periodic functions. From that time, the theory of almost automorphic functions has
been generalized and developed extensively because of its applications in mathe-
matical biology, physics, control theory, and other fields. For more details on these
functions we refer the reader to [5, 6, 7, 14, 20] and the references therein.

The study of almost automorphic solutions of various types of integral equations
and systems of integral equations is new and is an attractive area of research. A
comprehensive theory of almost automorphy and the applications can be found in
[12, 21, 22, 24] and references therein.

However, the concept of Stepanov-like almost automorphic functions, which was
introduced by N’Guérékata and Pankov [23], is more general than that of almost
automorphic functions. Such a notion was then utilized to study the existence
of weak Stepanov-like almost automorphic solutions to some parabolic evolution
equations. Since then, these functions have generated lot of developements and
applications.

In this work we consider a system of nonlinear delay integral equations where
the delays are specified functions. Models of this form play a fundamental role in
many biological systems and thus occur in many applications such as the evolution
in time of populations, the spread of infectious disease, etc. Our gool in this paper
is to study the existence of positive Stepanov-like almost automorphic solutions to

2010 Mathematics Subject Classification. 45G15, 47H10, 47H30.
Key words and phrases. Nonlinear delay integral system; fixed point in cones;

Stepanov-like almost automorphic solution.
c©2021 Texas State University.

Submitted October 4, 2019. Published May 25, 2021.

1



2 A. SADRATI, A. ZERTITI EJDE-2021/43

the type

x(s) =

∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ, (1.1)

where x = (x1, . . . , xn) : R→ Rn+, τ = (τ1, . . . , τn) : R→ Rn+ and f = (f1, . . . , fn) :
R × R+ × Rn+ → Rn+ are appropriate functions specified later. Hence system (1.1)
means that for each i ∈ {1, 2, . . . , n},

xi(s) =

∫ τi(s)

0

fi(s, σ, x1(s− σ − l), . . . , xn(s− σ − l)) dσ.

As an application, problem (1.1) models the evolution in time, of n species
x1, . . . , xn (n ≥ 2) with interaction. In the case n = 2, this equation generalizes
the one studied in [27], if one does the change of variable s− σ = u and sets l = 0,

x(s) =

∫ s

s−τ1(s)

f̃(σ, x(σ), y(σ)) dσ,

y(s) =

∫ s

s−τ2(s)

g̃(σ, x(σ), y(σ)) dσ.

(1.2)

In deed, this system also generalizes the system proposed by Cooke and Kaplan
[11] when τ1(t) = τ1, τ2(t) = τ2, and l = 0. There have been many papers concern-
ing the existence of positive periodic, positive almost periodic, positive weighted
pseudo almost automorphic solutions, etc, for various system. We refer the reader
to [8, 9, 25, 26, 27, 28]. However, the existence of Stepanov-like almost automorphic
solution to (1.1) is an untreated topic and this is the main motivation of the present
work. This article is organized as follows. In section 2, we recall some basic facts
about the notions of almost automorphy and Stepanov-like almost automorphy. In
section 3, we prove our results for the existence of positive Stepanov-like almost
automorphic solutions.

2. Preliminaries

This section includes notation, definitions, lemmas and preliminary facts which
will be used later.

Throughout this paper, p ∈ [1,+∞). We denote by R the set of real numbers,
by R+ the set of nonnegative real numbers and for x = (x1, . . . , xn) ∈ Rn, ‖x‖ =∑n
i=1 |xi|. Let measE be the Lebesgue measure for a subset E ⊂ R. Lploc(R,Rn)

denotes the space of all equivalence classes of measurable functions f : R → Rn
such that the restriction of f to every bounded subinterval of R is in Lp(R,Rn).

Lp,1loc(R×R+,Rn) denotes the space of all equivalence classes of measurable functions
f : R×R+ → Rn, (s, σ) 7→ f(s, σ), such that the restriction of f to every bounded
subset of R× R+ is in Lp,1(R× R+,Rn) = Lp(R, L1(R+,Rn)).

Definition 2.1 ([1]). A continuous function f : R→ Rn is called almost automor-
phic if for every sequence of real numbers (s′n)n there exists a subsequence (sn)n
such that

lim
m→+∞

lim
n→+∞

f(t+ sn − sm) = f(t), ∀t ∈ R.

This limit means that

f∗(t) = lim
n→+∞

f(t+ sn)



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 3

is well defined for each t ∈ R and

f(t) = lim
n→+∞

f∗(t− sn), ∀t ∈ R.

The collection of all such functions will be denoted by AA(R,Rn).

Note that some fundamental properties of almost periodic functions are not
satisfied by the almost automorphic functions, as example the property of uniform
continuity. A classical example of almost automorphic function which is not almost
periodic, as it is not uniformly continuous, is the function

f(t) = sin big(
1

2 + cos t+ cos
√

2t

)
, t ∈ R.

Remark 2.2. The function f∗ obtained in Definition 2.1 is measurable but not
necessarily continuous. Moreover, if f∗ is continuous, then f is uniformly continuous
[22, Theorem 2.6 ]. If the convergence in Definition 2.1 is uniform in t ∈ R, then f
is almost periodic [15].

Lemma 2.3 ([21]). Assume that f, g ∈ AA(R,Rn) and λ is a scalar. Then the
following statements hold:

(i) f + g, λf , fτ (t) = f(t+ τ), f̃(t) = f(−t) are almost automorphic.
(ii) The range Rf = {f(t) : t ∈ R} is precompact in Rn, and so f is bounded

in norm.
(iii) If {fn} is a sequence of almost automorphic functions and fn → f uniformly

on R, then f is almost automorphic.
(iv) AA(R,Rn) is a Banach space with the supremum norm

‖f‖∞ = sup
t∈R
‖f(t)‖ .

Definition 2.4 ([1]). A continuous function f : R× R+ × Rn+ → Rn is said to be
almost automorphic if f(s, σ, u) is almost automorphic in s ∈ R uniformly for all
(σ, u) ∈ K, where K is any bounded subset of R+ ×Rn+. The collection of all such
functions will be denoted by AA(R× R+ × Rn+,Rn).

Among others things, almost automorphic functions satisfy the following prop-
erty:

Let K = K1 ×K2 where, K1 ⊂ R+, K2 ⊂ Rn+ are compact subsets, and Ω ⊂ R.
We denote by CK(Ω × R+ × Rn+,Rn) the set of all functions f : Ω × R+ × Rn+ →
Rn such that f(s, ·, ·) is uniformly continuous on K uniformly for s ∈ Ω. If we

consider x ∈ AA(R,Rn), K1 is a compact subset of R+, K2 = {x(s) : s ∈ R} ⊂ Rn+
and f ∈ AA(R × R+ × Rn+,Rn) ∩ CK(Ω × R+ × Rn+,Rn), then the function s 7→
f(s, σ, x(s−σ− l) belong to AA(R,Rn), for a fixed constant l ∈ R+ and all σ ∈ K1.

Definition 2.5 ([13]). For a function f : R→ Rn, with t ∈ R, and s ∈ [0, 1], The
Bochner transform is defined as

f b(t, s) := f(t+ s).

Remark 2.6 ([13]). Note that a function ϕ(t, s), is the Bochner transform of a
certain function f(t), ϕ(t, s) = f b(t, s), if, and only if ϕ(t + τ, s − τ) = ϕ(s, t) for
all t ∈ R, s ∈ [0, 1] and τ ∈ [s− 1, s].



4 A. SADRATI, A. ZERTITI EJDE-2021/43

Definition 2.7 ([13]). For a function f : R×R×Rn → Rn, with t ∈ R, s ∈ [0, 1],
(σ, u) ∈ R× Rn, The Bochner transform is defined as

f b(t, s, σ, u) := f(t+ s, σ, u).

Definition 2.8 ([23]). Let p ∈ [1,+∞).

(i) The space BSp(R,Rn) of all Stepanov bounded functions, with the expo-
nent p, consists of all measurable functions f on R with values in Rn such
that f b ∈ L∞(R, Lp([0, 1],Rn)). This is a Banach space with the norm

‖f‖Sp = ‖f b‖L∞(R,Lp) = sup
t∈R

(∫ t+1

t

‖f(s)‖pds
)1/p

.

(ii) The space BSp(R×R+×Rn+,Rn) of all Stepanov bounded functions, with
the exponent p, consists of all measurable functions f : R×R+×Rn+ → Rn
such that

f b(·, ·, σ, u) ∈ L∞(R, Lp([0, 1],Rn)), t 7→ f b(t, ., σ, u) ∈ Lp([0, 1],Rn),

for each t ∈ R and each (σ, u) ∈ R+ × Rn+.

One can see that the Bochner transform f b is a continuous function on R with
values in Lp([0, 1],Rn), and thus f b ∈ BC(R, Lp([0, 1],Rn)). In fact, for p ≥ 1 we
have that (BC(R,Rn), ‖ · ‖BC) is continuously embeded in (BSp(R,Rn), ‖ · ‖Sp).

Definition 2.9 ([23]). The space ASp(R,Rn) of Stepanov-like almost automorphic
functions (or Sp-almost automorphic) consists of all f ∈ BSp(R,Rn) such that
f b ∈ AA(R, Lp([0, 1],Rn)).

In other words, a function f ∈ Lploc(R,Rn) is said to be Sp-almost automorphic

if its Bochner transform f b : R→ Lp([0, 1],Rn) is almost automorphic in the sense
that for every sequence of real numbers (s′n)n, there exist a subsequence (sn)n and
a function f∗ ∈ Lploc(R,Rn) such that(∫ t+1

t

‖f(s+ sn)− f∗(s)‖pds
)1/p

→ 0,(∫ t+1

t

‖f∗(s− sn)− f(s)‖pds
)1/p

→ 0

(2.1)

as n→ +∞ pointwise on R.

Lemma 2.10 ([23]). (i) (ASp(R,Rn), ‖.‖Sp) is a Banach space.
(ii) AA(R,Rn) is continuously embeded in ASp(R,Rn).

Remark 2.11. (1) The operator J : ASp(R,Rn) → ASp(R,Rn) such that
(Jx)(s) := x(−s) is well defined and linear. Moreover it is an isometry and
J2 = I.

(2) the operator Ta defined by (Tax)(s) := x(s + a) for a fixed a ∈ R leaves
ASp(R,Rn) invariant.

Definition 2.12 ([23]). A function f : R × R+ × Rn+ → Rn, (s, σ, u) → f(s, σ, u)
with f(., σ, u) ∈ Lploc(R,Rn) for each (σ, u) ∈ R+ × Rn+ is said to be Stepanov-like
almost automorphic in s ∈ R uniformly for (σ, u) ∈ R+ × Rn+, if s → f(s, σ, u) is
Stepanov-like almost automorphic for each (σ, u) ∈ R+ × Rn+. That is, for every



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 5

sequence of real numbers (s′n)n, there exist a subsequence (sn)n and a function
f∗ : R× R+ × Rn+ → Rn with f∗(·, σ, u) ∈ Lploc(R,Rn) such that(∫ t+1

t

‖f(s+ sn, σ, u)− f∗(s, σ, u)‖pds
)1/p

→ 0,(∫ t+1

t

‖f∗(s− sn, σ, u)− f(s, σ, u)‖pds
)1/p

→ 0

(2.2)

as n→ +∞ for all t ∈ R and (σ, u) ∈ R+×Rn+. We denote by ASp(R×R+×Rn+,Rn)
the set of all such functions.

Definition 2.13. Let E be a real Banach space. A closed convex set P in E is
called a convex cone if the following conditions are satisfied:

(1) if x ∈ P, then λx ∈ P for any λ ∈ R+;
(2) if x ∈ P and −x ∈ P, then x = 0.

A cone P induces a partial ordering ≤ in E by x ≤ y if and only if y − x ∈ P.
A cone P is called normal if there exists a constant N > 0 such that 0 ≤ x ≤ y

implies ‖x‖ ≤ N‖y‖, where ‖.‖ is the norm on E. We denote by
◦
P the interior set

of P. A cone P is called a solid cone if
◦
P 6= ∅.

We shall prove the existence of a positive solution of (1.1) by using the well-
known Guo-Krasnosel’skii fixed point theorem of cone expansion and compression.

Theorem 2.14 ([16]). Let E be a Banach space and P ⊂ E be a cone. Suppose Ω1

and Ω2 are two bounded open sets in Banach space E such that θ ∈ Ω1, Ω1 ⊂ Ω2

and suppose that the operator T : P ∩ (Ω2 \ Ω1)→ P is completely continuous such
that

(1) ‖Tx‖ ≤ ‖x‖,∀x ∈ P ∩ ∂Ω1 and ‖Tx‖ ≥ ‖x‖,∀x ∈ P ∩ ∂Ω2 or
(2) ‖Tx‖ ≥ ‖x‖,∀x ∈ P ∩ ∂Ω1 and ‖Tx‖ ≤ ‖x‖,∀x ∈ P ∩ ∂Ω2.

Then T has a fixed point in P ∩ (Ω2 \ Ω1).

3. Existence of positive Stepanov-Like almost automorphic solutions

In this section, we study the existence of positive Stepanov-like almost automor-
phic solution to the system (1.1) . For that, we need firstly to prove a composition
theorem.

Consider the set of all bounded functions BASp(R,Rn) ⊂ ASp(R,Rn), that
is, for each x ∈ BASp(R,Rn) we have ‖x‖∞ = sups∈R ‖x(s)‖ < ∞. It is clear
that (BASp(R,Rn), ‖.‖Sp) is a Banach space. Let ASp,1(R×R+ ×Rn+,Rn) be the
subset of ASp(R × R+ × Rn+,Rn) consists of all functions f such that f(·, ·, u) ∈
Lp,1loc(R× R+,Rn) for all u ∈ Rn+.

In the rest of this paper, we assume that the following holds

(H1) For each compact subset K ⊂ Rn+, there exist constants LK ,MK > 0 such
that

(i) for all u, v ∈ K, all σ1, σ2 ∈ R+ and all s ∈ R, it holds

‖f(s, σ1, u)− f(s, σ2, v)‖ ≤ LK(|σ1 − σ2|+ ‖u− v‖) .

(ii) for all (s, σ) ∈ R× R+ and all u ∈ K, it holds

‖f(s, σ, x)‖ ≤MK‖u‖,



6 A. SADRATI, A. ZERTITI EJDE-2021/43

Theorem 3.1. Assume that τ, x ∈ BASp(R,Rn) and f ∈ ASp,1(R×R+×Rn+,Rn)
such that (H1) holds. Then, the function Tx : R→ Rn defined by

Tx(s) =

∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ

is in BASp(R,Rn).

As in [15], to prove the above theorem we introduce some lemmas.

Lemma 3.2. Assume that f ∈ ASp,1(R × R+ × Rn+,Rn), K is a compact subset
of Rn+, τ ∈ R+ and (H1)(i) holds. Then, for each t ∈ R and each sequence of real
numbers (sn), there exist a subsequence (sm) and a set E ⊂ [0, 1] with measE = 0
such that limm→+∞ f(t + s + sm, σ, u) exists for each σ ∈ [0, τ ], u ∈ K and s ∈
[0, 1] \ E.

The proof of the above lemma is similar to that of [15, Lemma 2.1], we omit it
here.

Lemma 3.3. Assume that f ∈ ASp,1(R × R+ × Rn+,Rn), K is a compact subset
of Rn+, τ ∈ R+ and (H1) holds. Then, for each sequence of real numbers (s′n),
there exist a subsequence (sn), a function f∗ : R×R+×Rn+ → Rn with f∗(·, ·, u) ∈
Lp,1loc(R× R+,Rn) and a set E ⊂ R with measE = 0 such that for all σ ∈ [0, τ ], all
u, v ∈ K and s ∈ R \ E we have

‖f∗(s, σ, u)− f∗(s, σ, v)‖ ≤ LK‖u− v‖,
‖f∗(s, σ, u)‖ ≤MK‖u‖.

Moreover (2.2) holds.

The proof of the above lemma is similar to that of [15, Lemma 2.2], we omit it
here.

Lemma 3.4. Assume that f ∈ ASp,1(R × R+ × Rn+,Rn), K1, K2 are compact
subsets of Rn+ and (H1) holds. Then, for every sequence of real numbers (s′n) there
exist a subsequence (sn) and a function f∗ : R × R+ × Rn+ → Rn with f∗(., ., x) ∈
Lp,1loc(R× R+,Rn) such that

lim
n→∞

[ ∫ t+1

t

( sup
(w,u)∈K

‖
∫ w

0

(f(s+ sn, σ, u)− f∗(s, σ, u)) dσ‖)pds
]1/p

= 0,

lim
n→∞

[ ∫ t+1

t

( sup
(w,u)∈K

‖
∫ w

0

(f∗(s− sn, σ, u)− f(s, σ, u)) dσ‖)pds
]1/p

= 0,

for each t ∈ R, where K = K1 ×K2.

Proof. Let f∗ be as in Lemma 3.3. Then for every sequence of real numbers (s′n)
there exist a subsequence (sn) such that

lim
n→∞

(∫ t+1

t

‖f(s+ sn, σ, u)− f∗(s, σ, u)‖pds
)1/p

= 0,

lim
n→∞

(∫ t+1

t

‖f∗(s− sn, σ, u)− f(s, σ, u)‖pds
)1/p

= 0,



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 7

for each t ∈ R, σ ∈ R+ and u ∈ Rn+. In addition, there exists a set E ⊂ R with
measE = 0 such that for all σ ∈ [0, ‖K1‖], where ‖K1‖ = supw∈K1

‖w‖, for all
u, v ∈ K2 and s ∈ R \ E, we have

‖f∗(s, σ, u)− f∗(s, σ, v)‖ ≤ LK2
‖u− v‖. (3.1)

Fix t ∈ R. For any ε > 0, there exist (τ1, x1), . . . , (τm, xm) ∈ K1 ×K2 = K such
that K ⊂ ∪mi=1B((τi, xi),

ε
‖K‖ ), where ‖K‖ = sup(w,u)∈K{‖w‖+ ‖u‖}.

For the above ε, there exists a n0 ∈ N such that

(∫ t+1

t

‖f(s+ sn, σ, xi)− f∗(s, σ, xi)‖pds
)1/p

<
ε

m
, (3.2)

for n > n0 and i ∈ {1, 2, . . . ,m}.
For (w, u) ∈ K there exists i0 ∈ {1, 2, . . . ,m} such that (w, u) ∈ B((τi0 , xi0), ε

‖K‖ );

that is, ‖w−τi0‖ < ε
‖K‖ and ‖u−xi0‖ < ε

‖K‖ . From (H1) and (3.1), for each n > n0

and s ∈ [0, 1] with t+ s /∈ E, we have

‖
∫ w

0

[f(t+ s+ sn, σ, u)− f∗(t+ s, σ, u)] dσ‖

≤ ‖
∫ w

0

f(t+ s+ sn, σ, u) dσ −
∫ τi0

0

f(t+ s+ sn, σ, xi0) dσ‖

+ ‖
∫ τi0

0

[f(t+ s+ sn, σ, xi0)− f∗(t+ s, σ, xi0)] dσ‖

+ ‖
∫ τi0

0

f∗(t+ s, σ, xi0) dσ −
∫ w

0

f∗(t+ s, σ, u) dσ‖

≤ ‖
∫ τi0

0

[f(t+ s+ sn, σ, u)− f(t+ s+ sn, σ, xi0 ] dσ‖

+ ‖
∫ w

τi0

f(t+ s+ sn, σ, u) dσ‖

+

∫ ‖τi0‖
0

‖f(t+ s+ sn, σ, xi0)− f∗(t+ s, σ, xi0)‖ dσ

+ ‖
∫ w

0

[f∗(t+ s, σ, xi0)− f∗(t+ s, σ, u)] dσ‖+ ‖
∫ τi0

w

f∗(t+ s, σ, xi0) dσ‖.

Thus

‖
∫ w

0

[f(t+ s+ sn, σ, u)− f∗(t+ s, σ, u)] dσ‖

≤ LK2‖τi0‖‖u− xi0‖+MK2‖w − τi0‖‖u‖

+

∫ ‖τi0‖
0

‖f(t+ s+ sn, σ, xi0)− f∗(t+ s, σ, xi0)‖ dσ

+ LK2
‖w‖‖xi0 − u‖+MK2

‖w − τi0‖‖xi0‖

≤
∫ ‖τi0‖

0

‖f(t+ s+ sn, σ, xi0)− f∗(t+ s, σ, xi0)‖ dσ + 2(LK2
+MK2

)ε.



8 A. SADRATI, A. ZERTITI EJDE-2021/43

Now, by Minkowski’s inequality, the Hölder’s inequality, and (3.2), for each n > n0

and s ∈ [0, 1] with t+ s /∈ E we have[ ∫ t+1

t

(
sup

(w,u)∈K
‖
∫ w

0

(f(s+ sn, σ, u)− f∗(s, σ, u)) dσ‖
)p
ds
]1/p

=
[ ∫ 1

0

(
sup

(w,u)∈K
‖
∫ w

0

(f(t+ s+ sn, σ, u)− f∗(t+ s, σ, u)) dσ‖
)p
ds
]1/p

≤
m∑
i=1

‖τi‖
p−1
p

[ ∫ ‖τi‖
0

∫ 1

0

‖f(t+ s+ sn, σ, xi)− f∗(t+ s, σ, xi)‖pds dσ
]1/p

+ 2(LK2
+MK2

)ε

<

m∑
i=1

‖τi‖
p−1
p ‖τi‖

1
p
ε

m
+ 2(LK2

+MK2
)ε

≤ [‖K1‖+ 2(LK2
+MK2

)]ε.

Hence, we obtain

lim
n→∞

[ ∫ t+1

t

(
sup

(w,u)∈K
‖
∫ w

0

(f(s+ sn, σ, u)− f∗(s, σ, u)) dσ‖
)p
ds
]1/p

= 0,

for each t ∈ R. Analogously, one can show that

lim
n→∞

[ ∫ t+1

t

(
sup

(w,u)∈K
‖
∫ w

0

(f∗(s− sn, σ, u)− f(s, σ, u)) dσ‖
)p
ds
]1/p

= 0,

for each t ∈ R. The proof is complete. �

Proof of Theorem 3.1. Since τ, x ∈ BASp(R,Rn), f ∈ ASp,1(R × R+ × Rn+,Rn),
and (H1)(ii) holds, it is easy to show that Tx is bounded and Tx(·) ∈ Lploc(R,Rn+).
In addition, there exist x∗ and τ∗ ∈ Lploc(R,Rn) such that (2.1) holds, and f∗ :
R× R+ × Rn+ → Rn (as defined in Lemma 3.4) satisfies (2.2). Let

T ∗x∗(s) =

∫ τ∗(s)

0

f∗(s, σ, x∗(s− σ − l))ds.

Then we have[ ∫ t+1

t

‖Tx(s+ sn)− T ∗x∗(s)‖pds
]1/p

=
[ ∫ t+1

t

∥∥∫ τ(s+sn)

0

f(s+ sn, σ, x(s+ sn − σ − l)) dσ

−
∫ τ∗(s)

0

f∗(s, σ, x∗(s− σ − l)) dσ
∥∥pds]1/p

≤
[ ∫ t+1

t

∥∥∫ τ(s+sn)

0

[f(s+ sn, σ, x(s+ sn − σ − l))

− f∗(s, σ, x(s+ sn − σ − l))] dσ
∥∥pds]1/p

+
[ ∫ t+1

t

∥∥∫ τ∗(s)

0

[f∗(s, σ, x(s+ sn − σ − l))



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 9

− f∗(s, σ, x∗(s− σ − l))] dσ
∥∥pds]1/p

+
[ ∫ t+1

t

‖
∫ τ(s+sn)

τ∗(s)

f∗(s, σ, x(s+ sn − σ − l)) dσ‖pds
]1/p

≤
[ ∫ t+1

t

( sup
(w,u)∈K

‖
∫ w

0

(f(s+ sn, σ, u)− f∗(s, σ, u)) dσ‖)pds
]1/p

+ ‖τ∗‖
p−1
p
∞

[ ∫ t+1

t

∫ ‖τ∗‖∞
0

∥∥f∗(s, σ, x(s+ sn − σ − l))

− f∗(s, σ, x∗(s− σ − l))
∥∥p dσds]1/p

+MK2
‖x‖∞

[ ∫ t+1

t

‖τ(s+ sn)− τ∗(s)‖pds
]1/p

,

where K1 = {τ(s) : s ∈ R}, K2 = {x(s) : s ∈ R} and K = K1 ×K2.
Using Lemma 3.4, (2.2) and (2.1) we obtain

lim
n→+∞

[ ∫ t+1

t

‖Tx(s+ sn)− T ∗x∗(s)‖pds
]1/p

= 0.

Analogously we prove that limn→+∞
[ ∫ t+1

t
‖T ∗x∗(s−sn)−Tx(s)‖pds

]1/p
= 0. The

proof is complete. �

Now, we are ready to present our main results. In the sequel, we will consider
that the functions f and τ are defined as in system (1.1).

Theorem 3.5. Let f ∈ ASp,1(R×R+×Rn+,Rn+) be a function satisfying (H1) and
let τ ∈ BASp(R,Rn+). Assume that the following hypotheses hold:

(H2) There exist numbers r1, r2 ∈ R with r2 − r1 ≥ 1 and γ > 0 such that for
each compact subset K ⊂ Rn+ × Rn+,

inf
r∈[r1,r2], (w,u)∈K

‖
∫ w

0

f(r, σ, u) dσ‖ ≥ γ‖
∫ τ

0

f(s, σ, x) dσ‖,

for all (τ, x) ∈ K and s ∈ R.
(H3) There exists a function a : R× R+ → Rn such that

lim sup
‖u‖→0

f(s, σ, u)

‖u‖
= a(s, σ)

uniformly in (s, σ) ∈ R×R+, that is, for i = 1, . . . , n, there exists a function
ai : R× R+ → R such that

lim sup
‖u‖→0

fi(s, σ, u)

‖u‖
= ai(s, σ)

uniformly in (s, σ) ∈ R× R+, where (a1, a2, . . . , an) = a.
(H4) There exist a function b : R× R+ → Rn such that

lim inf
‖u‖→+∞

f(s, σ, u)

‖u‖
= b(s, σ)



10 A. SADRATI, A. ZERTITI EJDE-2021/43

uniformly in (s, σ) ∈ R×R+, that is, for i = 1, . . . , n, there exists a function
bi : R× R+ → R such that

lim inf
‖u‖→+∞

fi(s, σ, u)

‖u‖
= bi(s, σ)

uniformly in (s, σ) ∈ R× R+, where (b1, b2, . . . , bn) = b.

If

sup
r∈[r1,r2]

‖
∫ τ(r)

0

a(r, σ) dσ‖ < γ and inf
r∈[r1,r2]

‖
∫ τ(r)

0

b(r, σ) dσ‖ > 1

γ2
,

then system (1.1) has a nonzero positive solution x in BASp(R,Rn+). That is, x
not identically equal to zero and xi(t) ≥ 0, for all t ∈ R and all i = 1, . . . , n.

Proof. Let T : BASp(R,Rn+)→ BASp(R,Rn+) be the integral operator defined by

Tx(s) =

∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ, x ∈ P.

Consider the positive cone of BASp(R,Rn+) defined by

P = {x ∈ BASp(R,Rn+) : inf
r∈[r1,r2]

‖x(r)‖ ≥ γ‖x‖Sp}.

It is clear that x ∈ BASp(R,Rn+) is a solution of system (1.1) if and only if x is a
fixed point of the operator T in BASp(R,Rn+). We will prove that all assumptions
of Theorem 2.14 are satisfied. For the sake of convenience, we divide the proof into
three steps.

Step 1. We show that T (BASp(R,Rn+)) ⊂ P. For τ, x ∈ BASp(R,Rn+), denote by

K1 = τ(R), K2 = x(R) and K = K1 ×K2. Then, for all r ∈ [r1, r2],

‖Tx(r)‖ = ‖
∫ τ(r)

0

f(r, σ, x(r − σ − l)) dσ‖

≥ inf
r∈[r1,r2], (w,u)∈K

‖
∫ w

0

f(r, σ, u) dσ‖

=
[ ∫ t+1

t

(
inf

r∈[r1,r2], (w,u)∈K
‖
∫ w

0

f(r, σ, u) dσ‖
)p
ds
]1/p

≥ γ
[ ∫ t+1

t

‖
∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ‖pds
]1/p

,

for all s ∈ R and t ∈ R. Hence, for all r ∈ [r1, r2],

‖Tx(r)‖ ≥ γ‖Tx‖Sp .

Thus, infr∈[r1,r2] ‖Tx(r)‖ ≥ γ‖Tx‖Sp , which proves the assertion.

Step 2. We prove that T : P → P is completely continuous. Firstly, we claim
that T is a compact operator. That is, for every sequence {xk} ⊂ P with {xk}
is bounded in BASp(R,Rn+), the sequence {Txk} has a convergent subsequence
in BASp(R,Rn+). Indeed, since {xk} is bounded in BASp(R,Rn+), there exists a

constant M > 0 such that ‖xk‖∞ ≤ M , for all k = 0, 1, 2, . . . . In this case, there

exist a compact subset K ⊂ Rn+ such that xk(R) ⊂ K for all k. By using (H1)(ii),
for k = 0, 1, 2, . . . we have

‖Txk‖∞ = sup
s∈R
‖Txk(s)‖



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 11

= sup
s∈R
‖
∫ τ(s)

0

f(s, σ, xk(s− σ − l)) dσ‖

≤ sup
s∈R

∫ ‖τ‖∞
0

‖f(s, σ, xk(s− σ − l))‖ dσ

≤ sup
s∈R

∫ ‖τ‖∞
0

MK‖xk(s− σ − l)‖ dσ

≤ ‖τ‖∞MKM < +∞.

Therefore, {Txk} is uniformly bounded. Moreover, it is well known that if h ∈
Lp(Rn), 1 ≤ p < +∞, then limα→0

∫
Rn ‖h(s + α) − h(s)‖pds = 0. Hence, since

Txk ∈ BASp(R,Rn+) for all k, we obtain

lim
α→0

∫ t+1

t

‖Txk(s+ α)− Txk(s)‖pds = 0, ∀t ∈ R.

Thus, for all k = 0, 1, 2, . . . and all s ∈ R,

lim
α→0
‖Txk(s+ α)− Txk(s)‖ ≤ lim

α→0
sup
t∈Λ

[ ∫ t+1

t

‖Txk(θ + α)− Txk(θ)‖pdθ
]1/p

= 0,

where Λ is any compact subset in R, which implies that the sequence {Txk} of
continuous functions is equicontinuous.

Note that {Txk} is uniformly bounded and equicontinuous, this together with
the Ascoli’s theorem [19, p. 233-234 ] ensures that {Txk} is relatively compact in
uniform convergence on compacta. Thus {Txk} has a convergent subsequence with
respect to the topology of uniform convergence on compacta, say, {Txk′}. Hence,
there exists a continuous function x : R → Rn such that for each compact subset
Λ ⊂ R we have

lim
k′→∞

sup
s∈Λ
‖Txk′(s)− x(s)‖ = 0.

It follows that

lim
k′→∞

[ ∫ t+1

t

‖Txk′(θ)− x(θ)‖pdθ
]1/p

≤ lim
k′→∞

sup
s∈[t,t+1]

‖Txk′(s)− x(s)‖ = 0.

That means limk′→∞

[ ∫ t+1

t
‖Txk′(θ) − x(θ)‖pdθ

]1/p
= 0, for each t ∈ R. Then,

using [17, Lemma 2.7], one deduce that {Txk′} is a convergent subsequence of
{Txk} in BASp(R,Rn+).

On the other hand, let {xk} ⊂ P, x ∈ P such that limk→+∞ ‖xk − x‖Sp = 0 and

let Λ be a compact subset of Rn+ such that for all k, xk(R), x(R) ⊂ Λ. By using
(H1)(i), we have

‖Txk − Tx‖Sp

= sup
t∈R

[ ∫ t+1

t

‖
∫ τ(s)

0

[f(s, σ, xk(s− σ − l))− f(s, σ, x(s− σ − l))] dσ‖pds
]1/p

≤ sup
t∈R

[ ∫ t+1

t

(∫ ‖τ‖∞
0

‖f(s, σ, xk(s− σ − l))− f(s, σ, x(s− σ − l))‖ dσ
)p
ds
]1/p

≤ ‖τ‖
p−1
p
∞ LΛ sup

t∈R

[ ∫ t+1

t

∫ ‖τ‖∞
0

‖xk(s− σ − l)− x(s− σ − l)‖p dσds
]1/p



12 A. SADRATI, A. ZERTITI EJDE-2021/43

≤ ‖τ‖∞LΛ‖xk − x‖Sp .

This means that T is continuous. The proof of the assertion is complete.

Step 3. In this step, we show that (1) of Theorem 2.14 is satisfied. By (H3), for ev-

ery ε > 0 verifying supr∈[r1,r2] ‖
∫ τ(r)

0
(a(r, σ) + ε) dσ‖ ≤ γ, where ε = (ε, ε, . . . , ε) ∈

Rn, there exists δ > 0 such that

fi(s, σ, u) ≤ (ai(s, σ) + ε)‖u‖, i = 1, 2, . . . , n,

for all (s, σ) ∈ R× R+ and all u ∈ Rn+ with ‖u‖ ≤ δ.
We set Ω1 = {x ∈ P : ‖x‖Sp < δ}. Then, for x ∈ ∂Ω1 and t ∈ R we have

sup
t∈R

[ ∫ t+1

t

‖Tx(s)‖pds
]1/p

= sup
t∈R

[ ∫ t+1

t

‖
∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ‖pds
]1/p

≤ 1

γ
sup
t∈R

[ ∫ t+1

t

inf
r∈[r1,r2]
(w,u)∈K

‖
∫ w

0

f(r, σ, u) dσ‖pds
]1/p

=
1

γ
inf

r∈[r1,r2]
(w,u)∈K

‖
∫ w

0

f(r, σ, u) dσ‖.

Since ‖x‖Sp = δ, there exist r0 ∈ [r1, r2] such that ‖x(r0)‖ ≤ δ. Therefore

‖Tx‖Sp ≤ 1

γ
‖
∫ τ(r0)

0

f(r0, σ, x(r0)) dσ‖

≤ 1

γ
‖x(r0)‖‖

∫ τ(r0)

0

(a(r0, σ) + ε) dσ‖

≤ 1

γ
‖x(r0)‖ sup

r∈[r1,r2]

‖
∫ τ(r)

0

(a(r, σ) + ε) dσ‖

≤ ‖x(r0)‖ ≤ δ = ‖x‖Sp .

Inversely, by (H4), for every ε > 0 satisfying infr∈[r1,r2] ‖
∫ τ(r)

0
b(r, σ)− ε‖ dσ ≥ 1

γ2 ,

there exists M0 > 2δ such that

fi(s, σ, u) ≥ (bi(s, σ)− ε)‖u‖, i = 1, 2, . . . , n,

for all (s, σ) ∈ R× R+ and all u ∈ Rn+ with ‖u‖ ≥M0.
Let M = max{2δ,M0/γ}, and set Ω2 = {x ∈ P : ‖x‖Sp < M}. Then, for

x ∈ ∂Ω2 and r ∈ [r1, r2] we have

‖x(r)‖ ≥ inf
r∈[r1,r2]

‖x(r)‖ ≥ γ‖x‖Sp ≥M0.

It follows that for r ∈ [r1, r2],

‖
∫ τ(r)

0

f(r, σ, x(r − σ − l)) dσ‖ ≥ inf
r∈[r1,r2] ,(w,u)∈K

‖
∫ w

0

f(r, σ, u) dσ‖

≥ γ‖
∫ τ(r0)

0

f(r0, σ, x(r0)) dσ‖

≥ γ‖
∫ τ(r0)

0

(b(r0, σ)− ε) dσ‖‖x(r0)‖,



EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 13

for all r0 ∈ [r1, r2]. Hence

‖T (x)‖Sp = sup
t∈R

[ ∫ t+1

t

∥∥∫ τ(s)

0

f(s, σ, x(s− σ − l)) dσ
∥∥pds]1/p

≥ γ inf
r∈[r1,r2]

‖
∫ τ(r)

0

(b(r, σ)− ε) dσ‖ inf
r∈[r1,r2]

‖x(r)‖

≥ γ2 inf
r∈[r1,r2]

‖
∫ τ(r)

0

(b(r, σ)− ε) dσ‖‖x‖Sp

≥ ‖x‖Sp .

The proof is complete. �

Corollary 3.6. Let f ∈ ASp,1(R×R+×Rn+,Rn+) be a function satisfying (H1) and
let τ ∈ BASp(R,Rn+). Assume that (H2)–(H4) hold. In addition assume

(H5) The function τ = (τ1, . . . , τn) is such that τi(t) > 0, for all t ∈ R and all
i ∈ {1, . . . , n}.

(H6) f(s, σ, 0) = 0 for all (s, σ) ∈ R × R+, and for x = (x1, . . . , xn) ∈ Rn+, if
there is j ∈ {1, . . . , n} such that xj > 0 then fi(s, σ, x) > 0, for all i 6= j
and (s, σ) ∈ R× R+.

Then system (1.1) has a strictly positive solution x = (x1, . . . , xn) in BASp(R,Rn+).
That is, for each i ∈ {1, . . . , n}, xi(s) ≥ 0 for all s ∈ R and xi 6= 0.

Proof. We prove that if x is a solution as in the above theorem, then xi 6= 0 for
all i ∈ {1, . . . , n}. In fact, suppose that xj0 6= 0 for a j0 ∈ {1, . . . , n}, then there
exists s0 ∈ R such that xj0(s0) > 0 and consequently, for each i 6= j0, there exist
si ∈ R and σi ∈ [0, τi(si)] such that xj0(si − σi − l) > 0. Then by (H6) we have
fi(si, σi, x(si − σi − l)) > 0. Thus∫ τi(si)

0

fi(si, σ, x(si − σ − l)) dσ > 0.

This implies that xi(si) > 0 and therefore xi 6= 0 for all i ∈ {1, . . . , n}. �

Remark 3.7. System (1.1) with hypotheses (H5) and (H6) can be interpreted
as an epidemic model combining with population ecology. More precisely, it is a
model of n species x1, . . . , xn with Stepanov-like almost automorphic interaction
and infectious disease (assuming the n species are uniformly distributed in a given
geographical area). In this context, xi(t) is the population at time t of infectious
individuals in the species xi, function fi present the instantaneous rate of infection
in the species xi and τi(t) is the duration of infectivity in xi.

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EJDE-2021/43 POSITIVE STEPANOV-LIKE ALMOST AUTOMORPHIC SOLUTIONS 15

Abdellatif Sadrati

MSISI Laboratory, AM2CSI Group, Department of Mathematics, FST, Erracidia, Uni-

versity Moulay Ismäıl of Meknes, BP 509, Boutalamine, 52000, Errachidia, Morocco
Email address: abdo2sadrati@gmail.com

Abderrahim Zertiti
Department of Mathematics, Faculty of Sciences, University Abdelmalek Essaâdi, BP

2121, Tetouan, Morocco

Email address: abdzertiti@hotmail.fr


	1. Introduction
	2. Preliminaries
	3. Existence of positive Stepanov-Like almost automorphic solutions
	References