Thiramanus et al. Advances in Difference Equations ( 2016) 2016:83
DOI
10.1186/s13662-016-0813-7
R E S E A R C H Open Access
Positive solutions for Hadamard fractional
differential equations on infinite domain
Phollakr it Thiramanus
1
, Sotir is K Ntouyas
2,3
and Jessada Tariboon
1,4*
*
Correspondence:
jessada.t@sci.kmutnb.ac.th
1
Nonlinear Dynamic Analysis
Research Center, Department of
Mathematics, Faculty of Applied
Science, King M o ngkut’s University
of Technology North Bangkok,
Bangkok, 10800, Thailand
4
Centre of Excellence in
Mathematics, CHE, Sri Ayutthaya
Rd., Bangkok, 10400, Thailand
Full list of author information is
available at the end of the article
Abstract
This paper investigates the existence of nonnegative multiple solutions for nonlinear
fractional differential equations of Hadamard type, with nonlocal fractional integral
boundary conditions on an unbounded domain by means of Leggett-Williams and
Guo-Krasnoselskii’s fixed point theorems. Two examples are discussed for illustration
of the main work.
MSC: 34A08; 34A12; 34B15; 34B40
Keywords: Hadamard fractional derivative; Hadamard fractional integral; fractional
integral boundary conditions; fixed point theorems
1 Introduction
Fractional calculus has gained considerable attention from both theoretical and the ap-
plied points of view in recent years. There are numerous applications in a variety of fields
such as electrical networks, chemical physics, fluid flow, economics, signal and image pro-
cessing, viscoelasticity, porous media, aerodynamics, modeling for physical phenomena
exhibiting anomalous diffusion, and s o on. In contrast to integer-order differential and
integral op erators, fractional-order differential op erators are nonlocal in nature and pro-
vide the means to look into hereditary properties of several materials and processes. The
monographs [
–] are commonly cited for the theory of fractional derivatives and inte-
grals and applications to differential equations of fractional order. For more details and
examples, see [
–] and the references therein.
However, it has been noticed that most of the work on the topic is concerned with
Riemann-Liouville or Caputo type fractional differential equations. Besides these frac-
tional derivatives, another kind of fractional derivatives found in the literature is the
fractional derivative due to Hadamard introduced in [
], which differs from the
aforementioned derivatives in the sense that the kernel of the integral in the definition
of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed
description of Hadamard fractional derivative and integral can be found in [
, –].
Boundary value problems on infinite intervals appear often in applied mathematics and
physics, such as in unsteady flow of gas through a semi-infinite porous medium, the theory
of drain flows, etc. More examples and a collection of works on the existence of solutions
of boundary value problems on infinite intervals for differential, difference and integral
© 2016 Thiramanus et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Thiramanus et al. Advances in Difference Equations ( 2016) 2016:83 Page 2 of 18
equations may be found in the monographs [
, ]. For boundary value problems of frac-
tional order on infinite intervals we refer to [
–].
Zhao and Ge [
] studied the existence of unbounde d solutions for the following bo und-
ary value problem on the infinite interval:
D
α
u(t)+f
t, u(t)
=, <α ≤ , t ∈ [, ∞), (.)
u() = , lim
t →∞
D
α–
u(t)=βu(ξ), (.)
where D
α
denotes Riemann-Liouv ille fractional derivative of order α,and<β, ξ < ∞.
Zhang et al. [
] studied the existence of nonnegative solutions for the following bound-
ary value problem for fractional differential equations with nonlocal b oundary conditions
on unbounded domains:
D
α
u(t)+f
t, u(t)
=, <α ≤ , t ∈ [, ∞), (.)
I
α–
u() = , lim
t →∞
D
α–
u(t)=βI
α–
u(η), (.)
where D
α
denotes Riemann-Liouville f r actional derivative of order α, f ∈ C([, ∞) ×
R, R
+
)and<β, η < ∞. The Leray-Schauder nonlinear alternative is used.
Liang and Z hang [
] used a fixed-point theorem for operators on a cone, proved the
existence of positive solutions to the following fractional boundary value problem:
D
α
u(t)+f
t, u(t)
=, <α ≤ , t ∈ [, ∞), (.)
u() = u
′
() = , lim
t →∞
D
α–
u(t)=
m–
i=
β
i
u(ξ
i
), (.)
where D
α
denotes Riemann-Liouville f r actional derivative of order α, f ∈ C([, ∞) ×
R, R
+
), < ξ
< ξ
< ···< ξ
m–
< ∞, β
i
≥ , i =,,...,m –,with<
m–
i=
β
i
ξ
α–
i
< Ŵ(α).
Recently in [
] we studied a new class of boundary value problems on fractional differ-
ential equations with m-point Erdélyi-Kober fractional integral b oundary conditions on
an infinite interval of the form
D
α
u(t)+f
t, u(t)
=, <α ≤ , t ∈ (, ∞), (.)
u() = , D
α–
u(∞)=
m–
i=
β
i
I
γ
i
,δ
i
η
i
u(ξ
i
), (.)
where D
α
denotes the Riemann-Liouville fractional derivative of order α, I
γ
i
,δ
i
η
i
is the
Erdélyi-Kober fractional integral of order δ
i
>withη
i
>,γ
i
∈ R, i = ,,...,m –,
β
i
∈ R,andξ
i
∈ (, ∞), i =,,...,m – are given constants. We proved the existence and
uniqueness of unbounded solution of the boundary value problem (
.)-(.)byusingthe
Leray-Schauder nonlinear alternative and Banach contraction princ iple.
In this paper, we aim to investigate the existence criteria of positive solutions for frac-
tional differential e quations of Hadamard type, with integral boundary condition on infi-
nite intervals. Precisely, we consider the following boundary value problem for Hadamard
Thiramanus et al. Advances in Difference Equations ( 2016) 2016:83 Page 3 of 18
fractional differential equations:
D
α
u(t)+a(t)f
u(t)
=, <α ≤ , t ∈ (, ∞), (.)
u() = , D
α–
u(∞)=
m
i=
λ
i
I
β
i
u(η), (.)
where D
α
denotes the Hadamard fractional derivative of order α, η ∈ (, ∞), and I
β
i
is the
Hadamard fractional integral of order β
i
>,i =,,...,m,andλ
i
≥ , i =,,...,m are
given constants.
To the b est of our knowledge, there are no papers devoted to the study of positive so-
lutions for Hadamard fractional differential equations on infinite intervals; we fill the gap
in this paper. As we know, [, ∞) is noncompact. In the literature a special Banach space
were introduced. Unfortunately this Banach space is not suitable for Hadamard fractional
differential e q uations. In order to overcome this difficulty, a special Banach space is in-
troduced so that we can establish some inequ alities, which g uarantee that the functional s
defined on [, ∞) have better properties. Applying first the well-known Leggett-Williams
fixed point theorem, we obtain a new result on the existence of at least three distinct non-
negative solutions under some conditions. Next we prove the existence of at least one
positive solution by using Guo-Krasnoselskii’s fixed point theorem.
Therestofthepaperisorganizedasfollows:InSection
, we present some preliminaries
and lemmas that will be used to prove our main results. We also obtain the corresponding
Green’s function and some of its properties. The main result is for mulated and proved in
Sections
and . Especially in Section we prove the existence of at least three distinct
nonnegative solutions while, in Section
,weprovetheexistenceofatleastonepositive
solution. Examples illustrating our results are presented in Section
.
2Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [
]and
present preliminary results needed in our proofs later.
Definition . [
] The Hadamard fractional integral of order q for a function g is define d
as
I
q
g(t)=
Ŵ(q)
t
log
t
s
q–
g(s)
s
ds, q >,
provided the integral exists.
Definition . [
] The Hadamard derivative of fractional order q for a function g :
[, ∞) → R is defined as
D
q
g(t)=
Ŵ(n – q)
t
d
dt
n
t
log
t
s
n–q–
g(s)
s
ds, n –<q < n, n =[q]+,
where [q] denotes the integer part of the real number q and log(·)=log
e
(·).
Thiramanus et al. Advances in Difference Equations ( 2016) 2016:83 Page 4 of 18
Lemma . ([
], Property .) If a, α, β >then
D
α
a
log
t
a
β–
(x)=
Ŵ(β)
Ŵ(β – α)
log
x
a
β–α–
.
Lemma . [
, ] Let q >and x ∈ C[, ∞) ∩ L
[, ∞). Then the Hadamard fractional
differential equation D
q
x(t)=has the solutions
x(t)=
n
i=
c
i
(log t)
q–i
,
and the following formula holds:
I
q
D
q
x(t)=x(t)+
n
i=
c
i
(log t)
q–i
,
where c
i
∈ R, i =,,...,n, and n –<q < n.
Lemma . Let h ∈ C[, ∞) with <
∞
h(s)
ds
s
< ∞, and
= Ŵ(α)–
m
i=
λ
i
Ŵ(α)
Ŵ(α + β
i
)
(log η)
α+β
i
–
>. (.)
Then the unique solution of the following fractional differential equation:
D
α
u(t)+h(t)=, t ∈ (, ∞), α ∈(, ), (.)
subject to the boundary conditions
u() = , D
α–
u(∞)=
m
i=
λ
i
I
β
i
u(η), (.)
is given by the integral equation
u(t)=
∞
G(t, s)h(s)
ds
s
,(.)
where
G(t, s)=g(t, s)+
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
g
i
(η, s)(.)
and
g(t, s)=
Ŵ(α)
⎧
⎨
⎩
(log t)
α–
–(log
t
s
)
α–
,≤ s ≤ t < ∞,
(log t)
α–
,≤ t ≤ s < ∞,
(.)
g
i
(η, s)=
⎧
⎨
⎩
(log η)
α+β
i
–
–(log
η
s
)
α+β
i
–
,≤ s ≤ η < ∞,
(log η)
α+β
i
–
,≤ η ≤ s < ∞.
(.)
Thiramanus et al. Advances in Difference Equations ( 2016) 2016:83 Page 5 of 18
Proof Applying the Hadamard fractional integral of order α to both sides of (
.), we have
u(t)=c
(log t)
α–
+ c
(log t)
α–
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s
,(.)
where c
, c
∈ R.
The first condition of (
.)impliesc
= . Therefore,
u(t)=c
(log t)
α–
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s
.(.)
In accordance with Lemma
.,wehave
D
α–
u(t)=c
Ŵ(α)–
t
h(s)
ds
s
.
The second condition of (
.)leadsto
c
=
∞
h(s)
ds
s
–
m
i=
λ
i
Ŵ(α + β
i
)
η
log
η
s
α+β
i
–
h(s)
ds
s
,(.)
where is defined by (
.). Therefore, the unique solution of fractional boundary value
problem (
.)-(.)is
u(t)=
(log t)
α–
∞
h(s)
ds
s
–
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
η
log
η
s
α+β
i
–
h(s)
ds
s
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s
=
(log t)
α–
Ŵ(α)
Ŵ(α)–
m
i=
λ
i
Ŵ(α)
Ŵ(α+β
i
)
(log η)
α+β
i
–
∞
h(s)
Ŵ(α)
ds
s
–
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
η
log
η
s
α+β
i
–
h(s)
ds
s
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s
=
(log t)
α–
(Ŵ(α)–
m
i=
λ
i
Ŵ(α)
Ŵ(α+β
i
)
(log η)
α+β
i
–
+
m
i=
λ
i
Ŵ(α)
Ŵ(α+β
i
)
(log η)
α+β
i
–
)
Ŵ(α)–
m
i=
λ
i
Ŵ(α)
Ŵ(α+β
i
)
(log η)
α+β
i
–
×
∞
h(s)
Ŵ(α)
ds
s
–
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
η
log
η
s
α+β
i
–
h(s)
ds
s
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s
=(log t)
α–
∞
h(s)
Ŵ(α)
ds
s
+
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
(log η)
α+β
i
–
∞
h(s)
ds
s
–
m
i=
λ
i
(log t)
α–
Ŵ(α + β
i
)
η
log
η
s
α+β
i
–
h(s)
ds
s
–
Ŵ(α)
t
log
t
s
α–
h(s)
ds
s