International Journal of Analysis and Applications

Volume 19, Number 2 (2021), 239-251

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-239

BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATION

IN SPECIAL BANACH SPACE

BEDDANI MOUSTAFA1,∗, HEDIA BENAOUDA2

1Department of Mathematics, Djillali Liabes University of Sidi Bel-Abbés, PO BOX 89 22000 Bel-Abbés,

Algeria

2Laboratory of Mathematics, University of Tiaret, PO BOX 78 14000 Tiaret, Algeria

∗Corresponding author: beddani2004@yahoo.fr

Abstract. This paper studies the existence of solutions of boundary value problem for fractional differential

equations on the half-line in a special Banach space. The main result is based on Mönch fixed point theorem

combining with a suitable measure of non-compactness, an example is given to illustrate our approach.

1. Introduction

Fractional differential equations play a very important role in describing some real world problems, for

example, in the description of hereditary properties of various materials and processes. They are also widely

applied in the mathematical modeling of processes in physics, chemistry, aerodynamics, electro-dynamics of

complex medium, polymer rheology, etc. Consequently, the fractional calculus and its applications in various

fields of science and engineering have received much attention and have developed very rapidly (cf. [18,20,23]

for instance).

Very recently, many research papers have appeared concerning the fractional differential equations in Banach

spaces, some of them investigated the existence results of solutions on finite intervals by classical tools from

Received October 16th, 2020; accepted November 11th, 2020; published February 24th, 2021.

2010 Mathematics Subject Classification. 34B15, 34B40, 26A33.

Key words and phrases. boundary value problem, measure of non-compactness, unbounded domain, special Banach space,

Mönch fixed point theorem, Riemann-Liouville fractional derivative.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

239

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-239


Int. J. Anal. Appl. 19 (2) (2021) 240

functional analysis; see, for example References. [1, 3, 6–8, 19, 21, 22].

In [4]. A. Arara and M. Benchohra studied the following problem


cDα
0+
y(t) = f(t,y(t)), t ∈ J = (0, +∞), 1 < α ≤ 2,

y(0) = y0, y is bounded on J,

where cDα
0+

is the Caputo fractional derivative of order α, f : J × R → R is a is a continuous function and

y0 ∈ R. The main approach is based on Schauder’s fixed point theorem. my results is to generalize the

previous work.

This article concerns the existence of solutions of boundary value problem for fractional differential equations

on unbounded interval. We consider the boundary value problem

(1.1) Dα0+y(t) = f(t,y(t)), t ∈ J = (0, +∞),

(1.2) I2−α
0+

y(0+) = y0,

(1.3) Dα−1
0+

y(∞) = y∞.

Dα
0+

denote Riemann-Liouville fractional derivative introduced, 1 < α ≤ 2. The operator I2−α
0+

denotes

the left-sided Riemann-Liouville fractional integral, the state y(·) takes value in a Banach space E, f :

(0,∞) ×E → E will be specified in later sections and (y0,y∞) ∈ E ×E.

This paper is organized in the following way. In Section 2, we give some general results and preliminaries

and in Section 3, we present existence results for the problem (1.1)-(1.3), using the Mönch’s fixed point

theorem combined with the technique of measure of noncompactness. Finally an illustrative example will be

presented in Section 4.

2. Preliminary results

In this section, we introduce some notation and technical results which are used throughout this paper.

Let I ⊂ J be a compact interval and denote by C(I,E) the Banach space of continuous functions y : I → E

with the usual norm

‖y‖∞ = sup{‖y(t)‖, t ∈ I}.

L1(J,E) the space of E-valued Bochner integrable functions on J with the norm

‖f‖L1 =
∫ +∞
0

‖f(t)‖dt.



Int. J. Anal. Appl. 19 (2) (2021) 241

We consider the space of functions

Cα([0,∞),E) = {y ∈ C(J,E) : lim
t→0+

t2−αy(t) exists and finite}.

For y ∈ Cα((0,∞),E), we define yα by

yα(t) =


 t

2−αy(t), t ∈ (0,∞),

limt−→0 t
2−αy(t), t = 0.

It is clear that yα ∈ C([0,∞),E).

We consider the following Banach space

Xα([0,∞),E) = {y ∈ Cα([0,∞),E) : lim
t→∞

t2−αy(t)

1 + tα
exists and finite}.

A norm in this space is given by

‖y‖α = sup
t∈J

t2−α‖y(t)‖
1 + tα

.

We begin with some definitions from the theory of fractional calculus. Let α > 0, n = dαe + 1 (the least

integer greater than or equal to α) and h ∈ C(J,E).

Definition 2.1. [18].

(1) The Riemann-Liouville fractional integral of the function h of order α is defined by

Iα0+h(t) = gα(t) ∗h(t) =
∫ t
0

gα(t−s)h(s)ds, t > 0,

where ∗ denotes convolution and gα(t) = tα−1/Γ(α).

(2) The Riemann-Liouville fractional derivative of the function h of order α is defined by

Dα0+h(t) =
dn

dtn
(gn−α(t) ∗h(t)),

for all t > 0. Where Γ is the gamma function.

For the existence of solutions for the problem (1.1)-(1.3), we need the following auxiliary lemmas.

Lemma 2.1. [26] Let α > 0, then the differential equation

Dα0+h(t) = 0,

has solutions h(t) = c1t
α−1 + c2t

α−2 + . . . + cnt
α−n, for some ci ∈ R, i = 1 . . .n, where n = [α] + 1.

Lemma 2.2. [26] Let α > 0, then

Iα0+D
α
0+h(t) = h(t) + c1t

α−1 + c2t
α−2 + . . . + cnt

α−n,

for some ci ∈ R, i = 0, . . . ,n, where n = [α] + 1.



Int. J. Anal. Appl. 19 (2) (2021) 242

Remark 2.1. For α > 0, k > −1, we have

Iα0+t
k =

Γ(k + 1)

Γ(α + k + 1)
tα+k and Dα0+t

k =
Γ(k + 1)

Γ(k −α + 1)
tk−α, t > 0,

giving in particular Dα0+t
α−m = 0, m = 1, . . . ,n, where n is the smallest integer greater than or equal to α.

Remark 2.2. If h is suitabe function (see for instance [18, 20, 23] ), we have the composition relations

Dα0+I
α
0+h(t) = h(t), α > 0 and D

α
0+I

k
0+h(t) = I

k−α
0+ h(t), k > α > 0, t > 0.

We note that γ,γC and γXα the Kuratowski noncompactness measure of bounded sets in the spaces

E,C(I,E) and Xα, respectively. As for the definition of the Kuratowski noncompactness measure, we refer

to references [5, 17]. The following properties of the Kuratowski measure of noncompactness and Mönch

fixed point theorem are needed for our discussion.

Lemma 2.3. If H ⊂ C(I,E) is bounded and equicontinuous, then γ(H(t)) is continuous on I and

γC(H) = max
t∈I

γ(H(t)), γ

(
{
∫
I

x(t)dt : x ∈ H}
)
≤
∫
I

γ(H(t))dt,

where H(t) = {x(t) : x ∈ H}.

Theorem 2.1. [2, 24] Let D be a bounded, closed and convex subset of a Banach space E such that 0 ∈ D

, and let N be a continuous mapping of D into itself. If the implication

V = convN(V ) or V = N(V ) ∪{0} =⇒ γ(V ) = 0,

holds for every subset V of D, then N has a fixed point.

3. Main result

We will need to introduce the following hypotheses which are assumed here after.

(H1) There exists a nonnegative functions a,b ∈ C(J,R+) such that

‖f(t,u)‖≤ a(t) + t2−αb(t)‖u‖ for all t ∈ J and u ∈ E,

where ∫ ∞
0

(1 + tα)b(t)dt < Γ(α),

∫ ∞
0

a(t)dt < ∞.

(H2) ∀t ∈ (0,L],∀x,y ∈ E : ‖f(t,x) −f(t,y)‖≤ t
2−α

1+tα
‖x−y‖.

(H3) There exists nonnegative function ` ∈ L1(J,R+) such that for each nonempty, bounded set Ω ⊂

Xα(J,E)

γ(f(t, Ω(t))) ≤ t2−α`(t)γ(Ω(t)), for all t ∈ J,

where ∫ ∞
0

(1 + tα)`(t)dt ≤ Γ(α).



Int. J. Anal. Appl. 19 (2) (2021) 243

Definition 3.1. A function y ∈ Xα([0, +∞)) is said to be solution of the problem (1.1)-(1.3) if y satisfies

the equation Dα
0+
y(t) = f(t,y(t)) and the conditions (1.2 − 1.3).

Lemma 3.1. Let 1 < α < 2 and let h : J → E be continuous. A function y is a solution of the fractional

integral equation

(3.1) y(t) =
1

Γ(α)
[y∞ −

∫ ∞
0

h(t)dt]tα−1 +
y0

Γ(α− 1)
tα−2 +

1

Γ(α)

∫ t
0

(t−s)α−1h(s)ds

if and only if y is solution of the problem

(3.2) Dα0+y(t) = h(t), t ∈ J = (0, +∞),

(3.3) I2−α
0+

y(0+) = y0,

(3.4) Dα−1
0+

y(∞) = y∞.

Proof. Assume that y satisfies the problem (3.2)-(3.4). We may apply Lemma 2.2 to reduce equation (3.2)

to an equivalent integral equation

(3.5) y(t) = c1t
α−1 + c2t

α−2 + Iα0+h(t),

for some c1, c2 ∈ R. Applying I2−α0+ to both side of (3.5), we have

I2−α
0+

y(t) = c1I
2−α
0+

tα−1 + c2I
2−α
0+

tα−2 + I2−α
0+

Iα0+h(t).

From Remark 2.1, we then get

I2−α
0+

y(t) =
c1Γ(α)

Γ(2)
t + c2Γ(α− 1) +

1

Γ(2)

∫ t
0

(t−s)h(s)ds.

As t −→ 0, we obtain

c2 =
y0

Γ(α− 1)
.

Applying Dα−1
0+

to both side of (3.5), we have

Dα−1
0+

y(t) = c1D
α−1
0+

tα−1 + c2D
α−1
0+

tα−2 + Dα−1
0+

Iα0+h(t).

From Remark 2.1 and Remark 2.2, we then get

Dα−1
0+

y(t) = c1Γ(α) +
1

Γ(1)

∫ t
0

h(s)ds.



Int. J. Anal. Appl. 19 (2) (2021) 244

Hence

c1 =
1

Γ(α)
(y∞ −

∫ ∞
0

h(t)dt).

Thus, we have

y(t) =
1

Γ(α)
[y∞ −

∫ ∞
0

h(t)dt]tα−1 +
y0

Γ(α− 1)
tα−2 +

1

Γ(α)

∫ t
0

(t−s)α−1h(s)ds.

Conversely. The proof is simple. �

Consider the operator N : Xα([0,∞),E) → Xα([0,∞),E) defined by

N(y)(t) =
y∞

Γ(α)
tα−1 +

y0
Γ(α− 1)

tα−2 −
1

Γ(α)

∫ t
0

[tα−1 − (t−s)α−1]f(s,y(s))ds

−
1

Γ(α)

∫ ∞
t

tα−1f(s,y(s))ds.

The following several lemmas present some properties of the operator N, which are necessary for the proof

of our main result.

Lemma 3.2. Suppose that conditions (H1) and (H2) are valid. Then the operator N is bounded and con-

tinuous.

Proof. For y ∈ Xα([0,∞),E), it is easy to deduce from (H1), and that Ny ∈ Xα(J,E). Furthermore, (H1)

guarantees that

t2−α‖N(y)(t)‖
1 + tα

≤
‖y∞‖
Γ(α)

+
‖y0‖

Γ(α− 1)
+

1

Γ(α)

∫ ∞
0

‖f(s,y(s))‖ds

≤
‖y∞‖
Γ(α)

+
‖y0‖

Γ(α− 1)
+
‖y‖α
Γ(α)

∫ ∞
0

(1 + tα)b(t)dt +
1

Γ(α)

∫ ∞
0

a(t)dt.

Hence, N : Xα(J,E) → Xα(J,E) is bounded. Next we prove that N is continuous. Let {yn}∞n=1 ⊂ Xα(J,E)

and y ∈ Xα(J,E) such that yn → y as n →∞. Then, {yn}∞n=1 is a bounded set of Xα(J,E), i.e. there exists

M > 0 such that ‖yn‖α ≤ M for n > 1. We also have by taking limit that ‖y‖α ≤ M. In view of condition

(H1), for any ε > 0, there exists L > 0 such that∫ ∞
L

a(t)dt <
Γ(α)

6
ε,

∫ ∞
L

(1 + tα)b(t)dt <
Γ(α)ε

6M
ε,

and there exists Ñ ∈ N such that, for all n ≥ Ñ, we have

‖f(s,yn(s)) − (s,y(s))‖ <
Γ(α)

3L
ε.

Therefore, for all t ∈ J and n > Ñ, we can obtain from

t2−α

1 + tα
‖N(yn)(t) −N(y)(t)‖≤

1

Γ(α)

∫ t
0

‖f(s,yn(s)) −f(s,y(s))‖ds

+
1

Γ(α)

∫ ∞
t

‖f(s,yn(s)) −f(s,y(s))‖ds.



Int. J. Anal. Appl. 19 (2) (2021) 245

If t ≤ L and n > Ñ, we can obtain from

t2−α

1 + tα
‖N(yn)(t) −N(y)(t)‖≤

1

Γ(α)

∫ t
0

‖f(s,yn(s)) −f(s,y(s))‖ds

+
1

Γ(α)

∫ ∞
t

‖f(s,yn(s)) −f(s,y(s))‖ds

≤
1

Γ(α)

∫ t
0

‖f(s,yn(s)) −f(s,y(s))‖ds

+
1

Γ(α)

[∫ L
t

‖f(s,yn(s)) −f(s,y(s))‖ds +
∫ ∞
L

‖f(s,yn(s)) −f(s,y(s))‖ds

]

≤
2

Γ(α)

∫ L
0

‖f(s,yn(s)) −f(s,y(s))‖ds +
2M

Γ(α)

∫ ∞
L

(1 + sα)b(s)ds +
2

Γ(α)

∫ ∞
L

a(s)ds

≤
ε

3
+
ε

3
+
ε

3
= ε.

The case when t > L and n > Ñ is treated similarly. Thus we conclude that

‖yn −y‖α → 0 as n →∞,

namely, N is continuous and the conclusion of the lemma follows. �

Lemma 3.3. Let condition (H1) be satisfied and B be a bounded subset of Xα(J,E). Then

(i)
t2−αN(B)(t)

1+tα
is equicontinuous on any compact interval of J.

(ii) For given ε > 0, there exists a constant N1 > 0 such that ‖
t
2−α
1 N(y)(t1)

1+tα1
− t

2−α
2 N(y)(t2)

1+tα2
‖ < ε for any

t1, t2 ≥ N1 and y(.) ∈ B.

Proof. We have

Ny(t) =
y∞ −

∫∞
0
f(t,y(t))dt

Γ(α)
tα−1 +

y0
Γ(α− 1)

tα−2 +
1

Γ(α)

∫ t
0

(t−s)α−1f(s,y(s))ds.

In view of condition (H1) and the boundedness of B, there exists M > 0 such that

(3.6)

∫ ∞
0

‖f(t,y(t))‖dt ≤ M for any y ∈ B.



Int. J. Anal. Appl. 19 (2) (2021) 246

In order to prove (i), let the constant r be such that ‖y‖α ≤ r for any y ∈ B, and without loss of generality,

let [a,b] ⊂ J be a compact interval and t1, t2 ∈ [a,b] with t1 < t2. Then

‖
t2−α1 N(y)(t1)

1 + tα1
−
t2−α2 N(y)(t2)

1 + tα2
‖≤
‖y∞‖ + M

Γ(α)
|

t1
1 + tα1

−
t2

1 + tα2
| +

‖y0‖
Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
1

Γ(α)

∥∥∥∥
∫ t1
0

(t1 −s)α−1f(s,y(s))ds−
∫ t2
0

(t2 −s)α−1f(s,y(s))ds
∥∥∥∥

≤
‖y∞‖ + M

Γ(α)
|

t1
1 + tα1

−
t2

1 + tα2
| +

‖y0‖
Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
1

Γ(α)

∫ t1
0

|(t2 −s)α−1 − (t1 −s)α−1|‖f(s,y(s))‖ds

+
1

Γ(α)

∫ t2
t1

(t2 −s)α−1‖f(s,y(s))‖ds

≤
‖y∞‖ + M

Γ(α)
|

t1
1 + tα1

−
t2

1 + tα2
| +

‖y0‖
Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
1

Γ(α)

∫ t1
0

|(t2 −s)α−1 − (t1 −s)α−1|a(s)ds

+
r

Γ(α)

∫ t1
0

|(t2 −s)α−1 − (t1 −s)α−1|(1 + sα)b(s)ds +
1

Γ(α)

∫ t2
t1

(t2 −s)α−1a(s)ds

+
r

Γ(α)

∫ t2
t1

(t2 −s)α−1(1 + sα)b(s)ds

≤
‖y∞‖ + M

Γ(α)
|

t1
1 + tα1

−
t2

1 + tα2
| +

‖y0‖
Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
a∗ + b∗r

Γ(α)

(∫ t1
0

(t2 −s)α−1 − (t1 −s)α−1ds
)

+
a∗ + b∗r

Γ(α)

∫ t2
t1

(t2 −s)α−1ds

+
2b∗r

Γ(α)

(∫ t2
0

(t2 −s)α−1sαds−
∫ t1
0

(t1 −s)α−1sαds
)

≤
‖y∞‖ + M

Γ(α)
|

t1
1 + tα1

−
t2

1 + tα2
| +

‖y0‖
Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
a∗ + b∗r

Γ(1 + α)
(tα2 − t

α
1 − (t2 − t1)

α) +
a∗ + b∗r

Γ(1 + α)
(t2 − t1)α +

2b∗rB(α,α + 1)
Γ(α)

(t2α2 − t
2α
1 ),

where a∗ = maxt∈[a,b] a(t) and b
∗ = maxt∈[a,b] b(t). As t2 → t1 the right-hand side of the above inequality

tends to zero. Then
t2−αN(B)(t)

1+tα
is equicontinuous on [a,b].

Next we verify assertion (ii). Let ε > 0, we heve

‖
t2−α1 N(y)(t1)

1 + tα1
−
t2−α2 N(y)(t2)

1 + tα2
‖≤
‖y∞‖ + M

Γ(α)

∣∣∣∣ t11 + tα1 − t21 + tα2
∣∣∣∣ + ‖y0‖Γ(α− 1)

∣∣∣∣ 11 + tα1 − 11 + tα2
∣∣∣∣

+
1

Γ(α)

∥∥∥∥
∫ t1
0

t2−α1 (t1 −s)
α−1

1 + tα1
f(s,y(s))ds−

∫ t2
0

t2−α2 (t2 −s)
α−1

1 + tα2
f(s,y(s))ds

∥∥∥∥ .



Int. J. Anal. Appl. 19 (2) (2021) 247

It is sufficient to prove that∥∥∥∥
∫ t1
0

t2−α1 (t1 −s)
α−1

1 + tα1
f(s,y(s))ds−

∫ t2
0

t2−α2 (t2 −s)
α−1

1 + tα2
f(s,y(s))ds

∥∥∥∥ ≤ ε.
Relation (3.6) yields that there exits N0 > 0 such that

(3.7)

∫ ∞
N0

‖f(t,y(t))‖dt ≤
ε

3
for any y ∈ B.

On the other hand, since limt→∞
t2−α(t−N0)α−1

1+tα
= 0, there exists N1 > N0 such that, for any t1, t2 ≥ N1 and

s ∈ [0,N0], we have

(3.8)

∣∣∣∣t2−α2 (t2 −s)α−11 + tα2 − t
2−α
1 (t1 −s)

α−1

1 + tα1

∣∣∣∣ < ε3M .
Now taking t1, t2 ≥ N1, from (3.7), (3.8) we can arrive at∥∥∥∥

∫ t1
0

t2−α1 (t1 −s)
α−1

1 + tα1
f(s,y(s))ds−

∫ t2
0

t2−α2 (t2 −s)
α−1

1 + tα2
f(s,y(s))ds

∥∥∥∥
≤
∫ N1
0

∣∣∣∣t2−α2 (t2 −s)α−11 + tα2 − t
2−α
1 (t1 −s)

α−1

1 + tα1

∣∣∣∣‖f(s,y(s))‖ds
+

∫ t1
N1

t2−α1 (t1 −s)
α−1

1 + tα1
‖f(s,y(s))‖ds +

∫ t2
N1

t2−α2 (t2 −s)
α−1

1 + tα2
‖f(s,y(s))‖ds

<
ε

3M

∫ ∞
0

‖f(s,y(s))‖ds + 2
∫ ∞
N1

‖f(s,y(s))‖ds < ε.

Therefore, we complete the proof of lemma 3.3. �

Lemma 3.4. [25] Suppose that condition (H1) holds and B is a bounded subset of Xα(J,E). Then

γXα(N(B)) = supt∈J γ
(
t2−αN(B)(t)

1+tα

)
.

Now we are in a position to give the main result of this work. Let

B = {y ∈ Xα([0,∞),E) : ‖y‖α ≤ R}.

Theorem 3.1. Suppose that conditions (H1), (H2) and (H3) are valid. If

(H4)

R >
‖y∞‖ + (α− 1)‖y0‖ +

∫∞
0
a(t)dt

Γ(α) −
∫∞
0

(1 + tα)b(t)dt
.

Then the problem (1.1)-(1.3) has at least one solution.

Proof. First we transform problem (1.1)-(1.3) into a fixed point problem. Consider the operator N :

Xα([0,∞),E) → Xα([0,∞),E) defined by

N(y)(t) =
y∞

Γ(α)
tα−1 +

y0
Γ(α− 1)

tα−2 −
1

Γ(α)

∫ t
0

[tα−1 − (t−s)α−1]f(s,y(s))ds

−
1

Γ(α)

∫ ∞
t

tα−1f(s,y(s))ds.



Int. J. Anal. Appl. 19 (2) (2021) 248

From Lemma 3.1, the fixed points of N are solutions to (1.1)-(1.3). We shall show that N satisfies the

assumptions of Mönch fixed point theorem (Theorem 2.1).

Then we can derive that N : B → B. Indeed, for any y ∈ B, by condition (H1) we get

‖
t2−αN(y)(t)

1 + tα
‖≤
‖y∞‖
Γ(α)

+
‖y0‖

Γ(α− 1)
+

1

Γ(α)

∫ ∞
0

‖f(t,y(t))‖dt

≤
(1

Γ(α)

(
‖y∞‖ + (α− 1)‖y0‖ +

∫ ∞
0

a(t)dt + R

∫ ∞
0

(1 + tα))b(t)dt

)
< R.

Hence, from (H4) we have ‖Ny‖α ≤ R, and we conclude that N : B → B. Clearly B is a bounded, convex

and closed subset of Xα([0,∞),E), together with Lemma 3.2 we know that N : B → B is continuous.

Finally we need to prove the following implication

V ⊂ conv{N(V ) ∪{0}} =⇒ γXα(V ) = 0, for any V ⊂ B.

Let V ⊂ B such that V ⊂ conv{N(V ) ∪{0}} and t ∈ J, we choose ξ > 0 and n > 0 such that ξ < t < n.

For each y ∈ V , we consider

Nξ,n(y)(t) =
y∞

Γ(α)
tα−1 +

y0
Γ(α− 1)

tα−2 +
1

Γ(α)

∫ t
ξ

[tα−1 − (t−s)α−1]f(s,y(s))ds

+
1

Γ(α)

∫ n
t

(t−s)α−1f(s,y(s))ds.

Then from (H1), we obtain that

t2−α

1 + tα
‖Nξ,n(y)(t) −N(y)(t)‖≤

1

Γ(α)

∫ ξ
0

‖f(t,y(t))‖dt +
1

Γ(α)

∫ ∞
n

‖f(t,y(t))‖dt

≤
1

Γ(α)

(∫ ξ
0

a(t)dt + R

∫ ξ
0

(1 + tα))b(t)dt +

∫ ∞
n

a(t)dt + R

∫ ∞
n

(1 + tα))b(t)dt

)
,

this shows that Hd

(
t2−αNξ,n(V )(t)

1+tα
,
t2−αN(V )(t)

1+tα

)
→ 0 as ξ → 0 and n → ∞, t ∈ J. Where Hd denotes the

Hausdorff metric in space E. By the prorerty of noncompactness mearure we get

(3.9) lim
ξ→0, n→∞

γ

(
t2−αNξ,n(V )(t)

1 + tα

)
= γ

(
t2−αN(V )(t)

1 + tα

)
.

From lemma 3.3, the set {t
2−αN(V )(t)

1+tα
}⊂ Xα([0,∞),E) is equicontinuous on any compact of J. By (H1), il

is easy to know that {f(.,y(.)) : y ∈ V} is equicontinuous on [ξ,n]. Moreover {f(.,y(.)) : y ∈ V} is bounded



Int. J. Anal. Appl. 19 (2) (2021) 249

on [ξ,n], by (H1). Using Lemma 2.3, Lemma 3.4 and (H3), we arrive at

γ

(
t2−αNξ,nV (t)

1 + tα

)
≤

1

Γ(α)

∫ n
ξ

(1 + tα)`(t)γ

(
t2−αV (t)

1 + tα

)
dt

≤
1

Γ(α)

∫ n
ξ

(1 + tα)`(t)γ

(
t2−αN(V )(t)

1 + tα

)
dt

≤
1

Γ(α)

∫ n
ξ

(1 + tα)`(t)γXα(N(V ))dt.

From (3.9), we know that

γ

(
t2−αN(V )(t)

1 + tα

)
≤

1

Γ(α)

∫ ∞
0

(1 + tα)`(t)γXα(N(V ))dt.

Thus

γXα(N(V )) ≤
1

Γ(α)

∫ ∞
0

(1 + tα)`(t)γXα(N(V ))dt.

Consequently, by condition (H3). We get γXα(N(V )) = 0; that is γXα(V ) = 0. From the theorem 2.1, we

conclude that N has a fixed point y ∈ B which is a solution of problem (1.1)-(1.3). �

4. Example

We consider the following problem.

(4.1) D
3
2 y(t) =

( √
tyn(t)

(1 + t
3
2 )e10t

+
2t)

(1 + t2)2

)∞
n=1

, t ∈ J = (0, +∞),

(4.2) I
1
2

0+
y(t) = y0,

(4.3) D
1
2

0+
y(∞) = y∞.

Let

E = {(y1, . . . ,yn, . . .) : sup |yn| < ∞},

with the norm ‖y‖ = supn |yn|, then E is a Banach space and problem (4.1)-(4.3) can be regarded as a

problem of the form (1.1)-(1.3), with

α =
3

2
and f(t,y(t)) = (f(t,y1(t)), . . . ,f(t,yn(t)), . . .),

where

f(t,yn(t)) =

√
tyn(t)

(1 + t
3
2 )e10t

+
2t

(1 + t2)2
,n ∈ N∗.



Int. J. Anal. Appl. 19 (2) (2021) 250

We shall verify the conditions (H1) − (H3). Evidently, f is continuous in J ×E and

‖f(t,y(t))‖≤
√
t

(1 + t
3
2 )e10t

‖y(t)‖ +
2t

(1 + t2)2
.

With the aid of simple computation we find that∫ ∞
0

e−5tdt =
1

10
< Γ(

3

2
) and

∫ ∞
0

2t

(1 + t2)2
dt = 1 < ∞.

Finally, we verify condition (H3). For any bounded set B ⊂ E, we have

f(t,B(t)) =

√
t

(1 + t
3
2 )e5t

B(t) + {
2t

(1 + t2)2
}.

Then

γ(f(t,B(t)) ≤
√
t

(1 + t
3
2 )e5t

γ(B(t)).

Since
∫∞
0
e−10tdt = 0.1 < Γ( 3

2
), we conclude that condition (H3) is satisfied. Therefore, Theorem 3.1 ensures

that problem (4.1)-(4.3) has a solution.

Acknowledgments: The authors would like to express their thanks to the editor and anonymous referees

for his/her comments that improved the quality of the paper.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

[1] R. P. Agarwal, B. Hedia and M. Beddani, Structure of solution sets for implulsive fractional differential equations,Fract.

Calc. Appl. Anal. 9 (2018), 15-36.

[2] R. P. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications , Cambridge Tracts in Mathematics,

141 , Cambridge University Press, Cambridge, 2001.

[3] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl.

Math. Lett. 23 (2010), 390-394.

[4] A. Arara, M. Benchohra, N. Hamidi and J.J. Nieto, Fractional order differential equations on an unbounded domain,

Nonlinear Anal., Theory Methods Appl. 72 (2010), 580-586.

[5] Z.B. Bai and H.S. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math.

Anal. Appl. 311 (2005), 495-505.

[6] K. Balachandran and J.Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear

Anal., Theory Methods Appl. 71 (2009), 4471-4475.

[7] K. Balachandran and J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in

Banach spaces, Nonlinear Anal., Theory Methods Appl. 72 (2010), 4587-4593.

[8] K. Balachandran, S. Kiruthika and J.J. Trujillo, Existence results for fractional impulsive integrodifferential equations in

Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1970-1977.

[9] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order and

nonlocal conditions, Nonlinear Anal., Theory Methods Appl. 71 (2009), 2391-2396.



Int. J. Anal. Appl. 19 (2) (2021) 251

[10] L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal

Cauchy problem, Selected problems of mathematics, 50th Anniv. Cracow Univ. Technol. Anniv. Issue 6, Cracow Univ.

Technol. Krakow, (1995), 25-33.

[11] L. Byszewski and V.Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract

Cauchy problem in a Banach space, Appl. Anal. 40 (1991), 11-19.

[12] G. Christopher, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Com-

puters Math. Appl. 61(2011), 191-202.

[13] W. Cheung, J. Ren, P.J.Y. Wong and D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems,

J. Math. Anal. Appl. 330 (2007), 900-915.

[14] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal.

Appl. 179 (1993), 630-637.

[15] A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., Theory Methods Appl.

33 (1998), 181-186.

[16] A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15-25.

[17] D.J. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers,

Dordrecht, 1996.

[18] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V.

Amsterdam, 2006.

[19] G. M. N’Guérékata, A Cauchy problem for some fractional abstract differential equation with non local conditions, Non-

linear Anal., Theory Methods Appl. 70 (2009), 1873-1876.

[20] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Academic Press,

New York, London, Toronto, 1999.

[21] H.A.H. Salem, On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional

order, Comput. Math. Appl. 59 (2010), 1278-1293.

[22] H.A.H. Salem, Multi-term fractional differential equation in reflexive Banach space, Math. Comput. Model. 49 (2009),

829-834.

[23] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and

Breach, Yverdon, 1993.

[24] S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Sem. Mat. Univ. Padova. 75

(1986), 1-14.

[25] X. Su, Solutions to boundary value problem of fractional order on unbounded domains in a Banach space, Nonlinear Anal.,

Theory Methods Appl. 74 (2011), 2844-2852.

[26] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ.

Equations, 2006 (2006), 36.


	1. Introduction
	2. Preliminary results
	3. Main result
	4. Example
	References