CUBO, A Mathematical Journal

Vol. 23, no. 01, pp. 145–159, April 2021

DOI: 10.4067/S0719-06462021000100145

Existence and attractivity results for ψ-Hilfer
hybrid fractional differential equations

Fatima Si bachir1

Säıd Abbas2

Maamar Benbachir3

Mouffak Benchohra4

Gaston M. N’Guérékata5

1 Laboratory of Mathematics and Applied

Sciences, University of Ghardaia, 47000,

Algeria.

sibachir.fatima@univ-ghardaia.dz.com

2 Department of Mathematics, University

of Säıda–Dr. Moulay Tahar, P.O. Box

138, EN-Nasr, 20000 Säıda, Algeria.

said.abbas@univ-saida.dz

3 Department of Mathematics, Saad

Dahlab Blida1, University of Blida,

Algeria.

mbenbachir2001@gmail.com

4 Laboratory of Mathematics, Djillali

Liabes University of Sidi Bel-Abbès, P.O.

Box 89, Sidi Bel-Abbès 22000, Algeria.

benchohra@yahoo.com

5 NEERLab, Department of

Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore

M.D. 21252, USA.

gaston.nguerekata@morgan.edu

ABSTRACT

In this work, we present some results on the existence of attractive

solutions of fractional differential equations of the ψ-Hilfer hybrid

type. The results on the existence of solutions are a consequence

of the Schauder fixed point theorem. Next, we prove that all

solutions are uniformly locally attractive.

RESUMEN

En este trabajo, presentamos algunos resultados sobre la existen-

cia de soluciones atractivas de ecuaciones diferenciales fraccionar-

ias de tipo ψ-Hilfer h́ıbridas. Los resultados de existencia de solu-

ciones son consecuencia del teorema de punto fijo de Schauder.

A continuación, probamos que todas las soluciones son uniforme-

mente localmente atractivas.

Keywords and Phrases: ψ-Hilfer fractional derivative; Schauder fixed-point Theorem; uniformly locally

attractive.

2020 AMS Mathematics Subject Classification: 26A33, 34A08.

Accepted: 01 March, 2021

Received: 12 November, 2020

©2021 F. Si bachir et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000100145
https://orcid.org/0000-0002-1947-9213
https://orcid.org/0000-0002-2518-8658
https://orcid.org/0000-0003-3519-1153
https://orcid.org/0000-0003-3063-9449
https://orcid.org/0000-0001-5765-7175


146 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

1 Introduction

The theory of derivatives and integrals to a real or complex order is none other than the fractional

theory which began in 1695 between G.A. de L’Hospital and G.W. Leibniz. The fractional integra-

tion and differentiation go back to Leibniz, Riemann, Liouville, Abel, Weyl, and Riesz [27]. Many

monographs to which the reader can refer such as Abbas et al. [1, 5, 6], Diethelm [13], Kilbas et al.

[17], Oldham et al. [22], Podlubny [23], Samko et al. [24], Zhou [32, 33], Zhou et al. [34] and the

works by Abbas and Benchohra [2], Lakshmikantham et al. [19, 20, 21]. Recently several works

have been done concerning hybrid fractional differential equations see [9, 12, 14, 26, 31], and the

references therein.

Functional ψ− fractional differential equations received a great importance in applied math-
ematics and other sciences, see [8, 16, 18, 25, 28, 29, 30], and the references therein.

Some interesting results on existence and attractivity have been obtained in [3, 4, 7]. In this

work, we are interested in the existence and attractivity of solutions for the following problem

D
λ,σ;ψ
0+

u(t)
v(t,u(t))

= w (t,u(t)) ; a.e. t ∈ R+,

(ψ(t) −ψ(0))1−ςu(t) |t=0= u0; u0 ∈ R,
(1.1)

where R+ := [0, +∞), 0 < λ < 1, 0 ≤ σ ≤ 1, ς = λ + σ(1−λ), HD
λ,σ;ψ
0+

is the ψ -Hilfer fractional

derivative of order λ and type σ, v : R+ ×R → R∗ and w : R+ ×R → R, are given functions.

Special cases:

• For σ = 0,ψ(t) = t,u0 = 0, we will get nonlinear hybrid FDEs of the form

RLDλ

0+

[
u(t)

v(t,u(t))

]
= w(t,u(t)), a.e. t ∈ R+,

u(0) = 0.

• For λ = 1,σ = 1,ψ(t) = t, we will get nonlinear integer order hybrid differential equations of the
form 


d
dt

[
u(t)

v(t,u(t))

]
= w(t,y(t)), a.e. t ∈ R+,

u(0) = u0 ∈ R.

For v = 1, we will get nonlinear ψ-Hilfer FDEs of the form

HD

λ,σ;ψ
0+

u(t) = w(t,y(t)), a.e. t ∈ R+,

(ψ(t) −ψ(0))1−ςu(t)
∣∣
t=0

= u0 ∈ R.

• For v = 1,σ = 0 (in this case ς = λ),ψ(t) = t, we will get nonlinear FDEs involving Riemann-
Liouville fractional derivative

RLDλ0+u(t) = w(t,y(t)), a.e. t ∈ R+.



CUBO
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Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 147

2 Preliminaries

Let ψ : [a,b] → R be an increasing differentiable function such that ψ′(t) 6= 0, for all t ∈ [a,b],
(−∞≤ a < b ≤ +∞). Define on [a,b], (0 < a < b < ∞) the weighted space

Cςψ[a,b] = {τ : (a,b] → R : (ψ(t) −ψ(a))ςτ(t) ∈ C[a,b]}, 0 ≤ ς < 1,

with the norm

‖τ‖Cς;ψ[a,b] = ‖(ψ(t) −ψ(a))
ςτ(t)‖C[a,b] = max{|(ψ(t) −ψ(a))

ςτ(t)| : t ∈ [a,b]} ,

where C([a,b]) denotes the Banach space of all real continuous functions on [a,b].

Let BC := BC(R+) be the Banach space of all bounded and continuous functions from R+
into R. By BCς := BCς(R+), we denote the weighted space of all bounded and continuous functions

defined by BCς = {φ : R+ → R : (ψ(t) −ψ(0))1−ςφ(t) ∈ BC}, with the norm

‖φ‖BCς := sup
t∈R+

∣∣(ψ(t) −ψ(0))1−ςφ(t)∣∣ .
Let us recall some definitions and properties of fractional calculus.

Definition 2.1. [17] The left-sided ψ-Riemann-Liouville fractional integral and fractional deriva-

tive of order λ, (n − 1 < λ < n) for an integrable function φ : [a,b] → R with respect to another
function ψ : [a,b] → R, that is an increasing differentiable function such that ψ′(t) 6= 0, for all
t ∈ [a,b], (−∞≤ a < b ≤ +∞), are respectively defined as follows:

I
λ;ψ
a+

φ(t) =
1

Γ(λ)

∫ t
a

ψ′(s)(ψ(t) −ψ(s))λ−1φ(s)ds,

and

D
λ;ψ
a+ φ(t) =

(
1

ψ′(t)

d

dt

)n
I
n−α;ψ
a+

φ(t) =
1

Γ(n−λ)

(
1

ψ′(t)

d

dt

)n ∫ t
a

ψ′(s)(ψ(t) −ψ(s))n−λ−1φ(s)ds,

where Γ(·) is the Euler gamma function defined by

Γ(δ) =

∫ ∞
0

e−ttδ−1dt, δ > 0.

Definition 2.2. [10] The left-sided ψ -Caputo fractional derivative of function χ ∈ Cn[a,b],
(n− 1 < λ < n) n = [α] + 1 with respect to another function ψ is defined by

cD
λ;ψ
a+

φ(t) = I
n−λ;ψ
a+

(
1

ψ′(t)

d

dt

)n
φ(t) =

1

Γ(n−λ)

∫ t
a

ψ′(s)(ψ(t) −ψ(s))n−λ−1φ[n]ψ (s)ds,

where φ
[n]
ψ (t) =

(
1

ψ′(t)
d
dt

)n
φ(t) and ψ defined as in Definition Q. Moreover, the ψ− Caputo frac-

tional derivative of function φ ∈ ACn[a,b] is determined as

cD
λ;ψ
a+

φ(t) = D
λ;ψ
a+


φ(t) −n−1∑

k=0

[
1

ψ′(t)
d
dt

]k
φ(a)

k!
(ψ(t) −ψ(a))k


 .



148 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

Definition 2.3. [29] Let n − 1 < λ < n,n ∈ N, with [a,b],−∞ ≤ a < b ≤ +∞, and
ψ ∈ Cn([a,b],R) a function such that ψ(t) is increasing and ψ′(t) 6= 0, for all t ∈ [a,b].
The ψ -Hilfer fractional derivative (left-sided) of function φ ∈ Cn([a,b],R) of order λ and type
σ ∈ [0, 1] is determined as

D
λ,σ;ψ
a+

φ(t) = I
σ(n−λ);ψ
a+

[
1

ψ′(t)

d

dt

]n
I
(1−σ)(n−λ);ψ
a+

φ(t), t > a.

In other way

D
λ,σ;ψ
a+

φ(t) = I
σ(n−λ);ψ
a+

D
γ;ψ
a+

φ(t), t > a,

where

D
γ;ψ
a+

φ(t) =

[
1

ψ′(t)

d

dt

]n
I
(1−σ)(n−λ);ψ
a+

φ(t).

In particular, the ψ -Hilfer fractional derivative of order λ ∈ (0, 1) and type σ ∈ [0, 1], can be
written in the following form

D
λ,σ;ψ
a+

φ(t) =
1

Γ(ς −λ)

∫ t
a

(ψ(t) −ψ(s))ς−λ−1Dγ;ψ
a+

φ(s)ds

= I
ς−λ;ψ
a+

D
ς;ψ
a+
φ(t),

where ς = λ + σ −λσ, and Dς;ψ
a+
φ(t) =

[
1

ψ′(t)
d
dt

]
I
1−ς;ψ
a+

φ(t).

Lemma 2.4. [29] Let λ > 0, 0 ≤ ς < 1 and φ ∈ L1(a,b). Then

I
λ;ψ
a+

I
σ;ψ
a+

φ(t) = I
λ+σ;ψ
a+

φ(t), a.e. t ∈ [a,b].

In particular (i) if φ ∈ Cς;ψ[a,b], then I
λ;ψ
a+ I

σ;ψ
a+ φ(t) = I

λ+σ;ψ
a+ φ(t), t ∈ (a,b].

(ii) If φ ∈ C[a,b], then Iλ;ψ
a+

I
σ;ψ
a+

φ(t) = I
λ+σ;ψ
a+

φ(t), t ∈ [a,b].

Lemma 2.5. [29] Let λ > 0, 0 ≤ σ ≤ 1 and 0 ≤ ς < 1. If h ∈ Cς;ψ[a,b] then

D
λ,σ;ψ
a+

I
λ;ψ
a+

φ(t) = φ(t), t ∈ (a,b].

If φ ∈ C1[a,b] then
D
λ,σ;ψ
a+ I

α;ψ
a+ φ(t) = φ(t), t ∈ [a,b].

Lemma 2.6. Let λ > 0, 0 ≤ ς < 1 and φ ∈ Cς;ψ[a,b]. If λ > ς, then I
λ;ψ
a+

φ ∈ C[a,b] and

I
λ;ψ
a+

φ(a) = lim
t→a+

I
λ;ψ
a+

φ(t) = 0.

Lemma 2.7. [29] Let φ ∈ Cn[a,b],n− 1 < λ < n, 0 ≤ σ ≤ 1, and ς = λ + σ −λσ. Then for all
t ∈ (a,b]

I
λ;ψ
a+

D
λ,σ;ψ
a+

φ(t) = φ(t) −
n∑
k=1

[ψ(t) −ψ(a)]ς−k

Γ(ς −k + 1)
φ
(n−k)
ψ I

(1−σ)(n−λ);ψ
a+

φ(a).

In particular, if 0 < λ < 1, we have

I
λ;ψ
a+

D
λ,σ;ψ
a+

φ(t) = φ(t) −
[ψ(t) −ψ(a)]ς−1

Γ(ς)
I
(1−σ)(1−λ);ψ
a+

φ(a).



CUBO
23, 1 (2021)

Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 149

Moreover, if φ ∈ C1−ς;ψ[a,b] and I
1−ς;ψ
a+

φ ∈ C11−ς;ψ[a,b] such that 0 < ς < 1. Then for all t ∈ (a,b]

I
ς;ψ
a+
D
ς;ψ
a+
φ(t) = φ(t) −

[ψ(t) −ψ(a)]γ−1

Γ(ς)
I
1−ς;ψ
a+

φ(a).

We deduce from the above lemma the following lemmas:

Lemma 2.8. [18] Let v ∈ C(Υ ×R,R∗); Υ := [0,d], d > 0, κ ∈ C1−ζ,ψ(Υ). Then the problem

D
λ,σ;ψ
0+

u(t)
v(t,u(t))

= κ(t),a.e. t ∈ Υ.

(ψ(t) −ψ(0))1−ςu(t) |t=0= u0, u0 ∈ R,

has a unique solution given by

u(t) = v(t,u(t))

{
u0

v(0,u(0))
(ψ(t) −ψ(0))ς−1 + Iλ;ψ

0+
κ(t)

}
.

Lemma 2.9. Let v ∈ C(Υ × R,R∗), w : Υ × R → R be a function such that w(·,u(·)) ∈ BCς for
any u ∈ BCς. Then the problem (1.1) is equivalent to the problem of obtaining the solutions of the
integral equation

u(t) = v(t,u(t))

{
u0

v(0,u(0))
(ψ(t) −ψ(0))ς−1 + Iλ;ψ

0+
w(·,u(·))(t)

}
.

Let ∅ 6= Λ ⊂ BC and let K : Λ → Λ. We consider the solution of the equation

(Ku)(t) = u(t). (2.1)

We introduce the concept of attractivity of solutions for equation (2.1).

Definition 2.10. Solutions of equation (2.1) are locally attractive if there exists a ball B (u0,µ)

in the space BC such that, for any solutions τ = τ(t) and ξ = ξ(t) of equations (2.1) that belong

to B (u0,µ) ∩ Λ, we can write

lim
t→∞

(τ(t) − ξ(t)) = 0. (2.2)

If the limit (2.2) is uniform with respect to B (u0,µ) ∩ Λ, then the solutions of equation (2.1)
are said to be uniformly locally attractive (or, equivalently, that the solutions of (2.1) are locally

asymptotically stable).

Lemma 2.11. [11] Let M ⊂ BC. Then M is relatively compact in BC if the following conditions
are satisfied:

(a) M is uniformly bounded in BC;

(b) the functions belonging to M are almost equicontinuous in R+, i.e., equicontinuous on every

compact set in R+;



150 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

(c) the functions from M are equiconvergent, i.e., given ε > 0, there exists L(ε) > 0 such that

∣∣∣u(t) − lim
t→∞

u(t)
∣∣∣ < ε,

for any t ≥ L(ε) and u ∈ M.

Theorem 2.12. (Schauder Fixed-Point Theorem [15]). Let F be a Banach space, let U be a

nonempty bounded convex and closed subset of F, and let K : U → U be a compact and continuous
map. Then K has at least one fixed point in U.

3 Existence and Attractivity Results

Definition 3.1. A measurable function u ∈ BCς is a solution of problem (1.1) if it verifies the
initial condition (ψ(t)−ψ(0))1−ςu(t) |t=0= u0 and the equation D

λ,σ;ψ
0+

u(t)
v(t,u(t))

= w (t,u(t)) on R+.

We will give the following hypotheses:

(H1) The function t 7→ w(t,u) is measurable on R+ for each u ∈ BCς, the function u 7→ w(t,u) is
continuous on BCς for a.e. t ∈ R+, and the function v is bounded such that u 7→ v(t,u) is
continuous.

(H2) There exists a continuous function T : R+ → R+ such that for a.e. t ∈ R+ and each u ∈ R,

|w(t,u)| ≤
T(t)

1 + |u|
,

and

lim
t→∞

(ψ(t) −ψ(0))1−ς
(
I
λ;ψ
0+

T
)

(t) = 0.

Set

T∗ = sup
t∈R+

(ψ(t) −ψ(0))1−ς
(
I
λ;ψ
0+

T
)

(t) < ∞.

Now we present a theorem on the existence and attractivity of solutions of the problem (1.1).

Theorem 3.2. Assume that the hypotheses (H1) and (H2) hold. Then the problem (1.1) has at

least one solution defined on R+ and the solutions of problem (1.1) are uniformly locally attractive.

Proof. Consider the operator K such that, for any u ∈ BCς,

(Ku)(t) = v(t,u(t))

{
u0

v(0,u(0))
(ψ(t) −ψ(0))ς−1 +

1

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds
}
.



CUBO
23, 1 (2021)

Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 151

Let L be a bound of the function v. For any u ∈ BCς, and for each t ∈ R+, we have∣∣∣∣(ψ(t) −ψ(0))1−ς(Ku)(t)
∣∣∣∣

≤|v(t,u(t))|
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + (ψ(t) −ψ(0))1−ςΓ(λ)
∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds
}

≤|v(t,u(t))|
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + (ψ(t) −ψ(0))1−ςΓ(λ)
∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds
}

≤L
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + T∗
}

:=R∗.

So

|K(u)‖BC ≤ R∗. (3.1)

Therefore, K(u) ∈ BCς. Since, the map K(u) is continuous on R+; for any u ∈ BCς, and K(BCς) ⊂
BCς, then the operator K maps BCς into itself. Furthermore, equation (3.1) implies that the

operator K transforms the ball

BR∗ := B(0,R∗) = {v ∈ BCς : ‖v‖BCς ≤ R∗}

into itself. From Lemma 2.9 the solution of problem (1.1) is reduced to finding the solutions of the

operator equation K(u) = u. We show that the operator K : BCς → BCς satisfies all assumptions
of Theorem 2.12. The proof is divided into several steps:

Step 1. K is continuous.

Let {un}n∈N be a sequence such that un → u in BR∗.

Then, for each t ∈ R+, we have∣∣((ψ(t) −ψ(0))1−ς (Kun) (t) − ((ψ(t) −ψ(0))1−ς(Ku)(t)∣∣
≤
∣∣∣∣v(t,un(t))

{
u0

v(0,u(0))
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds
}

−v(t,u(t))
{

u0
v(0,u(0))

+
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds
}∣∣∣∣

≤
∣∣∣∣v(t,un(t))

{
u0

v(0,u(0))
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds
}

−v(t,u(t))
{

u0
v(0,u(0))

+
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds
}

+ v(t,u(t))

{
u0

v(0,u(0))
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds
}

−v(t,u(t))
{

u0
v(0,u(0))

+
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds
}∣∣∣∣



152 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

≤
∣∣∣∣v(t,un(t)) −v(t,u(t))

∣∣∣∣
∣∣∣∣ u0v(0,u(0)) + (ψ(t) −ψ(0))

1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1

×w(s,un(s))ds
∣∣∣∣ + |v(t,u(t))|(ψ(t) −ψ(0))1−ςΓ(λ)

×
∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s)) −w(s,u(s))|ds.

Hence ∣∣(ψ(t) −ψ(0))1−ς (Kun) (t) − (ψ(t) −ψ(0))1−ς(Ku)(t)∣∣
≤
∣∣∣∣v(t,un(t)) −v(t,u(t))

∣∣∣∣
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s))|ds
}

+ L
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s)) −w(s,u(s))|ds. (3.2)

Case 1. If t ∈ [0,d], then, in view of the facts that un → u as n →∞, v and w are continuous,
by the Lebesgue dominated convergence theorem, from the equation (3.2), we have

‖K (un) −K (u)‖BCς → 0 as n →∞.

Case 2. If t ∈ (d,∞), then, from the hypotheses and (3.2), we have∣∣(ψ(t) −ψ(0))1−ς (Kun) (t) − (ψ(t) −ψ(0))1−ς(Ku)(t)∣∣
≤
∣∣∣∣v(t,un(t)) −v(t,u(t))

∣∣∣∣
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds
}

+ 2L
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds.

Then ∣∣(ψ(t) −ψ(0))1−ς (Kun) (t) − (ψ(t) −ψ(0))1−ς(Ku)(t)∣∣
≤
∣∣∣∣v(t,un(t)) −v(t,u(t))

∣∣∣∣
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + ((ψ(t) −ψ(0))1−ς (Iλ;ψ0+ T) (t)
}

+ 2L((ψ(t) −ψ(0))1−ς
(
I
λ;ψ
0+

T
)

(t). (3.3)

Since un → u as n →∞, v is continuous and (ψ(t)−ψ(0))1−ς
(
I
λ;ψ
0+

T
)

(t) → 0 as t →∞, it follows
from (3.3) that

‖K (un) −K(u)‖BCς → 0 as n →∞.

Step 2. L (BR∗) is uniformly bounded, and equicontinuous on every compact subset [0,d] of

R+, d > 0.



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Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 153

We have L (BR∗) ⊂ BR∗ and BR∗ is bounded, so L (BR∗) is uniformly bounded.

Next, for each t1, t2 ∈ [0,d], t1 < t2, and u ∈ BR∗, we have

∣∣(ψ(t2) −ψ(0))1−ς (Ku) (t2) − (ψ(t1) −ψ(0))1−ς(Ku)(t1)∣∣
≤
∣∣∣∣v(t2,u(t2))

{
u0

v(0,u(0))
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds
}

−v(t1,u(t1))
{

u0
v(0,u(0))

+
(ψ(t1) −ψ(0))1−ς

Γ(λ)

∫ t1
0

ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds
}∣∣∣∣

≤
∣∣∣∣v(t2,u(t2))

{
u0

v(0,u(0))
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds
}

−v(t1,u(t1))
{

u0
v(0,u(0))

+
(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds
}

+ v(t1,u(t1))

{
u0

v(0,u(0))
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds
}

−v(t1,u(t1))
{

u0
v(0,u(0))

+
(ψ(t1) −ψ(0))1−ς

Γ(λ)

∫ t1
0

ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds
}∣∣∣∣.

Thus ∣∣(ψ(t2) −ψ(0))1−ς (Ku) (t2) − (ψ(t1) −ψ(0))1−ς(Ku)(t1)∣∣
≤ |v(t2,u(t2)) −v(t1,u(t1))|

∣∣∣∣ u0v(0,u(0))
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds
∣∣∣∣

+ |v(t1,u(t1))|
∣∣∣∣(ψ(t2) −ψ(0))1−ςΓ(λ)

∫ t1
0

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds

+
(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
t1

ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds

−
(ψ(t1) −ψ(0))1−ς

Γ(λ)

∫ t1
0

ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds
∣∣∣∣.

Hence

∣∣(ψ(t2) −ψ(0))1−ς (Ku) (t2) − (ψ(t1) −ψ(0))1−ς(Ku)(t1)∣∣
≤ |v(t2,u(t2)) −v(t1,u(t1))|

(∣∣∣∣ u0v(0,u(0))
∣∣∣∣

+
(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1|w(s,u(s))|ds
)

+ L

(∫ t1
0

∣∣∣∣(ψ(t2) −ψ(0))1−ςΓ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1
−

(ψ(t1) −ψ(0))1−ς

Γ(λ)
ψ′(s)(ψ(t1) −ψ(s))λ−1

∣∣∣∣
|w(s,u(s))|ds +

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
t1

ψ′(s)(ψ(t2) −ψ(s))λ−1|w(s,u(s))|ds
)



154 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

≤ |v(t2,u(t2)) −v(t1,u(t1))|
(∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1T(s)ds
)

+ L

(∫ t1
0

∣∣∣∣(ψ(t2) −ψ(0))1−ςΓ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1
−

(ψ(t1) −ψ(0))1−ς

Γ(λ)
ψ′(s)(ψ(t1) −ψ(s))λ−1

∣∣∣∣
T(s)ds +

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
t1

ψ′(s)(ψ(t2) −ψ(s))λ−1T(s)ds
)
.

From the continuity of the functions T and v, by setting T∗ = supt∈[0,d] T(t), we obtain∣∣(ψ(t2) −ψ(0))1−ς (Ku) (t2) − (ψ(t1) −ψ(0))1−ς(Ku)(t1)∣∣
≤ |v(t2,u(t2)) −v(t1,u(t1))|

(∣∣∣∣ u0v(0,u(0))
∣∣∣∣ + T∗(ψ(t2) −ψ(0))1−ςΓ(λ)

∫ t2
0

ψ′(s)(ψ(t2) −ψ(s))λ−1ds
)

+ LT∗

(∫ t1
0

∣∣∣∣(ψ(t2) −ψ(0))1−ςΓ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1
−

(ψ(t1) −ψ(0))1−ς

Γ(λ)
ψ′(s)(ψ(t1) −ψ(s))λ−1

∣∣∣∣ds
+

(ψ(t2) −ψ(0))1−ς

Γ(λ)

∫ t2
t1

ψ′(s)(ψ(t2) −ψ(s))λ−1ds
)

≤ |v(t2,u(t2)) −v(t1,u(t1))|
(∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + T∗(ψ(t2) −ψ(0))1−ς+λΓ(λ + 1)
)

+ LT∗

(∫ t1
0

∣∣∣∣(ψ(t2) −ψ(0))1−ςΓ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1
−

(ψ(t1) −ψ(0))1−ς

Γ(λ)
ψ′(s)(ψ(t1) −ψ(s))λ−1

∣∣∣∣ds + (ψ(t2) −ψ(0))1−ςΓ(λ + 1) (ψ(t2) −ψ(t1))λ
)
.

As t1 → t2, the right-hand side of the inequality tends to zero.

Step 3. L (BR) is equiconvergent.

Let u ∈ BR∗. Then, for each t ∈ R+ we have∣∣∣∣(ψ(t) −ψ(0))1−ς(Ku)(t)|
∣∣∣∣ ≤ |v(t,u(t))|

{∣∣∣∣ u0v(0,u(0))
∣∣∣∣

+

∣∣∣∣(ψ(t) −ψ(0))1−ςΓ(λ)
∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds
∣∣∣∣
}

≤ |v(t,u(t))|
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ +
∣∣∣∣(ψ(t) −ψ(0))1−ςΓ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds
∣∣∣∣
}

≤ L
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + (ψ(t) −ψ(0))1−ς (Iλ;ψ0+ T) (t)
}
.

Since

(ψ(t) −ψ(0))1−ς
(
I
λ;ψ
0+

T
)

(t) → 0 as t → +∞,



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Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 155

we find

|(Ku)(t)| ≤ L
{∣∣∣∣ u0(ψ(t) −ψ(0))1−ςv(0,u(0))

∣∣∣∣ + (ψ(t) −ψ(0))
1−ς

(
I
λ;ψ
0+

T
)

(t)

(ψ(t) −ψ(0))1−ς

}
.

Hence,

|(Lu)(t) − (Lu)(+∞)|→ 0 as t → +∞,

in view of Lemma 2.11 as a consequence of Steps 1 − 4, we conclude that K : BR∗ → BR∗ is
compact and continuous. Applying the Theorem 2.12, we have that K has a fixed point u, which

is a solution of problem (1.1) on R+.

Step 4. The uniform local attractivity of solutions.

We assume that u∗ is a solution of problem (1.1) under the conditions of this theorem.

Set u ∈ B
(
u∗, 2L

{∣∣∣∣ u0v(0,u(0))
∣∣∣∣ + 2T∗

})
, we have

∣∣(ψ(t) −ψ(0))1−ς (Ku) (t) − (ψ(t) −ψ(0))1−ς(u∗)(t)∣∣
≤
∣∣(ψ(t) −ψ(0))1−ς (Ku) (t) − (ψ(t) −ψ(0))1−ς(Ku∗)(t)∣∣

≤ |v(t,u(t)) −v(t,u∗(t))|
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds
}

+ L
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds

≤ 2L
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + (ψ(t) −ψ(0))1−ςΓ(λ)
∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds
}

+ 2L
(ψ(t) −ψ(0))1−ς

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1T(s)ds

≤ 2L
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + 2T∗
}
.

Thus, we get

‖K(u) −u∗‖BCς ≤ 2L
{∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + 2T∗
}
.

So, we conclude that K is a continuous function such that

K

(
B

(
u∗, 2L

{∣∣∣∣ u0v(0,u(0))
∣∣∣∣ + 2T∗

}))
⊂ B

(
u∗, 2L

{∣∣∣∣ u0v(0,u(0))
∣∣∣∣ + 2T∗

})
.



156 F. Si bachir, S. Abbas, M. Benbachir, M. Benchohra & G. M. N’Guérékata CUBO
23, 1 (2021)

Moreover, if u is a solution of problem (1.1), then

|u(t) −u∗(t)| = |(Ku)(t) − (Ku∗) (t)|

≤ |v(t,u(t)) −v(t,u∗(t))|
{

(ψ(t) −ψ(0))ς−1
∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

1

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds
}

+
L

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds

≤ 2L
{

(ψ(t) −ψ(0))ς−1
∣∣∣∣ u0v(0,u(0))

∣∣∣∣
+

1

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds
}

+
L

Γ(λ)

∫ t
0

ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds

≤ 2L
{

(ψ(t) −ψ(0))ς−1
∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + 2(Iλ;ψ0+ T)(t)
}
.

Therefore,

|u(t) −u∗(t)| ≤ 2L
{

(ψ(t) −ψ(0))ς−1
∣∣∣∣ u0v(0,u(0))

∣∣∣∣ + 2 (ψ(t) −ψ(0))1−ς(Iλ;ψ0+ T)(t)(ψ(t) −ψ(0))1−ς
}
. (3.4)

By using (3.4) and the fact that

lim
t→∞

(ψ(t) −ψ(0))1−ς(Iλ;ψ
0+

T)(t) = 0,

we conclude

lim
t→∞

|u(t) −u∗(t)| = 0.

Consequently, all solutions of problem (1.1) are uniformly locally attractive.

4 An Example

As an application of our results, we consider the following problem for a ψ-Hilfer fractional differ-

ential equation 

D

1
2
, 1
2
;ψ

0+
u(t)

v(t,u(t))
= w (t,u(t)) ,a.e. t ∈ R+,

(ψ(t) −ψ(0))
1
4 u(t) |t=0= 1,

(4.1)

where ψ : [0, 1] → R with ψ(t) =
√
t + 3,

v(t,u) =
1

(1 + t)(1 + |u|)
,



w(t,u) =

β(ψ(t) −ψ(0))
−1
4 sin t

64(1 +
√
t)(1 + |u|)

, t ∈ (0,∞), u ∈ R,

w(0,u) = 0, u ∈ R,



CUBO
23, 1 (2021)

Existence and attractivity results for ψ-Hilfer hybrid fractional. . . 157

and

β =
9
√
π

16
.

Clearly, the function w is continuous. The hypothesis (H2) is satisfied with

T(t) =

β(ψ(t) −ψ(0))
−1
4 |sin t|

64(1 +
√
t)

, t ∈ (0,∞),

T(0) = 0.

In addition, we have

(ψ(t) −ψ(0))
1
4

(
I

1
2
;ψ

0+
T
)

(t) =
(ψ(t) −ψ(0))

1
4

Γ
(
1
2

) ∫ t
0

ψ′(τ)(ψ(t) −ψ(τ))
−1
2 T(τ)dτ

≤
1

4
(ψ(t) −ψ(0))

−1
4 → 0 as t →∞.

Simple computations show that all conditions of Theorem 3.2 are satisfied. Consequently,

our problem (4.1) has at least one solution defined on R+, and all solutions of this problem are

uniformly locally attractive.

5 Conclusion

In this paper, we provided some sufficient conditions ensuring the existence and the uniform locally

attractivity of solutions of some ψ-Hilfer fractional differential equations. The technique used is

based on Schauder’s fixed point theory theorem.

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	Introduction
	Preliminaries
	Existence and Attractivity Results
	An Example
	Conclusion