Thiramanus et al. Journal of Inequalities and Applications 2014, 2014:398
http://www.journalofinequalitiesandapplications.com/content/2014/1/398
RESEARCH
Open Access
Integral inequalities with ‘maxima’ and their
applications to Hadamard type fractional
differential equations
Phollakrit Thiramanus1 , Jessada Tariboon1* and Sotiris K Ntouyas2
*
Correspondence:
[email protected]
1
Nonlinear Dynamic Analysis
Research Center, Department of
Mathematics, Faculty of Applied
Science, King Mongkut’s University
of Technology North Bangkok,
Bangkok, 10800, Thailand
Full list of author information is
available at the end of the article
Abstract
In this paper, some new integral inequalities with ‘maxima’ are established involving
Hadamard integral. Applications to Hadamard fractional diﬀerential equations with
‘maxima’ are also presented.
MSC: 26A33; 26D10; 26D15
Keywords: integral inequality; diﬀerential equations with ‘maxima’; fractional
diﬀerential equations
1 Introduction
It is well known that integral inequalities play a dominant role in the study of quantitative
properties of solutions of diﬀerential and integral equations [–]. Fractional inequalities
are important in studying the existence, uniqueness, and other properties of fractional differential equations. Recently many authors have studied integral inequalities on fractional
calculus using Riemann-Liouville and Caputo derivatives; see [–] and the references
therein. In [, ], the authors established some weakly singular integral inequalities of
Gronwall-Bellman type and also applied them in the qualitative analysis of solutions to
certain fractional diﬀerential equations of the Caputo type.
Another kind of fractional derivative that appears in the literature is the fractional
derivative due to Hadamard, introduced in  [], which diﬀers from the RiemannLiouville and Caputo derivatives in the sense that the kernel of the integral contains a logarithmic function of an arbitrary exponent. Details and properties of Hadamard fractional
derivative and integral can be found in [–]. Recently in the literature there appeared
some results on fractional integral inequalities using the Hadamard fractional integral; see
[–].
Let us recall here the deﬁnitions of Hadamard’s fractional integral and derivative [].
Deﬁnition . The Hadamard fractional integral of order α ∈ R+ of a function f (t), for all
t > , is deﬁned as
t

ds
t α–
α
(.)
f (s) ,
log
H J f (t) =
(α) +
s
s
where is the standard gamma function deﬁned by (α) =
integral exists, where log(·) = loge (·).
∞

e–s sα– ds, provided the
©2014 Thiramanus et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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Deﬁnition . The Hadamard fractional derivative of order α ∈ [n – , n), n ∈ Z+ , of a
function f (t) is given by
d n t

ds
t n–α–
t
f (s) .
log
H D f (t) =
+
(n – α) dt
s
s

α
(.)
Diﬀerential equations with ‘maxima’ are a special type of diﬀerential equations that contain the maximum of the unknown function over a previous interval. Several integral inequalities have been established in the case when the maxima of the unknown scalar function are involved in the integral; see [, ] and references cited therein.
Recently in [] some new types of integral inequalities on time scales with ‘maxima’
have been established, which can be used as a handy tool in the investigation of making
estimates for bounds of solutions of dynamic equations on time scales with ‘maxima’. In
this paper we establish some new integral inequalities with ‘maxima’ involving Hadamard’s
integral. The signiﬁcance of our work lies in the fact that ‘maxima’ are taken on intervals
[βt, t] which have non-constant lengths, where  < β < . Most papers take the ‘maxima’
on [t – h, t], where h >  is a given constant.
The paper is organized as follows: in Section  we recall some results from [] in the
special case T = R, used to prove our main results, which are presented in Section . In
Section  we give applications of our results for a Hadamard fractional diﬀerential equation with ‘maxima’.
2 Preliminaries
For convenience we let t >  throughout. The following results in Lemmas . and .
are obtained by reducing the time scale T = R, f (t) = g(t) ≡ , and a(t) = b(t) ≡  for all
t ∈ (t , T) in Theorems . and . ([], p. and p.), respectively.
Lemma . ([]) Let the following conditions be satisﬁed:
(H ) The functions p and q ∈ C((t , T), R+ ).
(H ) The function φ ∈ C([βt , T), R+ ) with maxs∈[βt ,t ] φ(s) > , where  < β < .
(H ) The function u ∈ C([βt , T), R+ ) and satisﬁes the inequalities
u(t) ≤ φ(t) +
t
p(s)u(s) + q(s) max u(ξ ) ds,
ξ ∈[βs,s]
t
u(t) ≤ φ(t),
t ∈ (t , T),
t ∈ [βt , t ].
Then
u(t) ≤ φ(t) + h(t) exp
t
p(s) + q(s) ds ,
t ∈ (t , T),
t
holds, where
t
h(t) = max φ(s) +
s∈[βt ,t ]
t
p(s)φ(s) + q(s) max φ(ξ ) ds,
ξ ∈[βs,s]
t ∈ (t , T).
By splitting the initial function φ into two functions, we deduce the following corollary.
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Corollary . Let the following conditions be satisﬁed:
(H ) The functions p, q, and v ∈ C((t , T), R+ ).
(H ) The function w ∈ C([βt , t ], R+ ) with maxs∈[βt ,t ] w(s) >  and w(t ) = v(t ), where
 < β < .
(H ) The function u ∈ C([βt , T), R+ ) and satisﬁes the inequalities
u(t) ≤ v(t) +
t
p(s)u(s) + q(s) max u(ξ ) ds,
ξ ∈[βs,s]
t
u(t) ≤ w(t),
t ∈ (t , T),
t ∈ [βt , t ].
Then
t
u(t) ≤ v(t) + h(t) exp
p(s) + q(s) ds ,
t ∈ (t , T),
t
holds, where
t
h(t) = max w(s) +
s∈[βt ,t ]
p(s)v(s) + q(s) max m(ξ ) ds,
ξ ∈[βs,s]
t
t ∈ (t , T),
with
m(t) =
⎧
⎨v(t),
t ∈ (t , T),
⎩w(t), t ∈ [βt , t ].
Lemma . ([]) Let the condition (H ) of Lemma . is satisﬁed. In addition, assume
that:
(H ) The function k ∈ C((t , T), (, ∞)) is nondecreasing.
(H ) The function φ ∈ C([βt , t ), R+ ), where  < β < .
(H ) The function u ∈ C([βt , T), R+ ) and satisﬁes the inequalities
u(t) ≤ k(t) +
t
p(s)u(s) + q(s) max u(ξ ) ds,
t
u(t) ≤ φ(t),
ξ ∈[βs,s]
t ∈ (t , T),
t ∈ [βt , t ].
Then
t
u(t) ≤ Nk(t) exp
p(s) + q(s) ds ,
t ∈ (t , T),
t
holds, where
maxs∈[βt ,t ] φ(s)
N = max ,
.
k(t )
The following lemma is a consequence of Jensen’s inequality, which can be found in [].
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Lemma . ([]) Let n ∈ N , and let x , . . . , xn be non-negative real numbers. Then for
σ > ,
n
σ
≤ nσ –
xi
i=
n
xσi .
i=
3 Main results
Theorem . Suppose that the following conditions are satisﬁed:
(A ) The functions p and r ∈ C((t , T), R+ ).
(A ) The function φ ∈ C([βt , t ], R+ ) with maxs∈[βt ,t ] φ(s) > , where  < β < .
(A ) The function u ∈ C([βt , T), R+ ) with
u(t) ≤ r(t) +
t
t
u(t) ≤ φ(t),
t
log
s
α–
p(s) max u(ξ )
ξ ∈[βs,s]
ds
,
s
t ∈ (t , T),
t ∈ [βt , t ],
(.)
(.)
where α > .
Then the following assertions hold:
(i) Suppose α >  , then

(α – ) t 

p (s) ds
,
u(t) ≤ t c r (t) + h (t) exp
t
t
t ∈ (t , T),
(.)
where
c = max t– , (βt )–
(.)
and
h (t) = c max φ  (s) +
s∈[βt ,t ]
×
c (α – )
t
t
t
p (s) max m (ξ ) ds,
ξ ∈[βs,s]
t ∈ (t , T),
(.)
with
m (t) =
⎧
⎨r(t),
t ∈ (t , T),
⎩φ(t), t ∈ [βt , t ].
(.)
In addition, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then
u(t) ≤
(α – ) t 
c N tr(t) exp
p (s) ds ,
t
t
t ∈ (t , T),
(.)
where
maxs∈[βt ,t ] φ  (s)
.
N = max ,
r (t )
(.)
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(ii) Suppose  < α ≤  , then

 b
((α  )) α t b
b
p (s) ds
,
u(t) ≤ t c r (t) + h (t) exp
t
t
t ∈ (t , T),
(.)
where b =  + α ,

c = max  α t–b , (βt )–b
(.)
and

h (t) = c max φ b (s) +
s∈[βt ,t ]
c ((α  )) α
t
t
×
t
pb (s) max mb (ξ ) ds,
ξ ∈[βs,s]
t ∈ (t , T).
(.)
Moreover, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then
 ((α  )) α t b

p (s) ds ,
u(t) ≤ (c N ) b tr(t) exp
bt
t
t ∈ (t , T),
(.)
where
maxs∈[βt ,t ] φ b (s)
.
N = max ,
rb (t )
(.)
Proof (i) α >  . For t ∈ (t , T), by using the Cauchy-Schwarz inequality in (.), we get
t u(t) ≤ r(t) +
t
t
×
t
t
log
s
α–

ds
 ds 
p (s) max u(ξ )
.
ξ ∈[βs,s]
s
(.)
It is easy to observe that
t
log
t
t
s
α–
ds = t
log tt

τ α– e–τ dτ < (α – )t.
(.)

Substituting (.) in (.), we obtain
u(t) ≤ r(t) + (α – )t

t

p (s) max u(ξ )
t
ξ ∈[βs,s]
 ds 
s
.
Applying Lemma . with n = , σ = , we get the estimate
u (t) ≤ r  (t) + (α – )t
t
t
 ds
p (s) max u(ξ )
,
ξ ∈[βs,s]
s
t ∈ (t , T).
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Setting v(t) = t – u (t), we have, for t ∈ (t , T),
v(t) ≤ t – r (t) +
(α – )
t
t
t
 ds
p (s) max u(ξ )
ξ ∈[βs,s]
s
(α – ) t 
p (s) max ξ – u (ξ ) ds
ξ ∈[βs,s]
t
t
t
(α – )
p (s) max v(ξ ) ds,
≤ c r (t) +
ξ ∈[βs,s]
t
t
≤ t– r (t) +
(.)
and for t ∈ [βt , t ],
v(t) ≤ t – φ  (t) ≤ (βt )– φ  (t) ≤ c φ  (t).
(.)
A suitable application of Corollary . for (.) and (.) leads to
(α – ) t 
p (s) ds ,
v(t) ≤ c r (t) + h (t) exp
t
t
t ∈ (t , T),
where c and h are deﬁned by (.) and (.), respectively. Therefore, we obtain the desired
bound in (.).
Now, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then, by Lemma . with (.)
and (.), it follows that
(α – ) t 
v(t) ≤ c N r (t) exp
p (s) ds ,
t
t

t ∈ (t , T),
where N is deﬁned by (.). Thus, we get the inequality in (.). This completes the proof
of the ﬁrst part.
(ii)  < α ≤  . Let a =  + α and b =  + α . It is obvious that a + b = . Using the Hölder
inequality in (.), for t ∈ (t , T), we obtain
u(t) ≤ r(t) +
t log
t
t
s
a(α–)
a t
ds
t
b ds b
pb (s) max u(ξ )
.
ξ ∈[βs,s]
sb
(.)
For the ﬁrst integral in (.), repeating the process to get (.), we have
t
t
t
log
s
a(α–)
ds <  – a( – α) t.
(.)
Obviously,  – a( – α) = α  >  and ( – a( – α)) ∈ R. Substituting (.) in (.), we get

u(t) ≤ r(t) + α  t a
t
b
p (s) max u(ξ )
t
ξ ∈[βs,s]
b ds b
sb
.
Applying Lemma . with n = , σ = b, we get the following estimate:
b ds
pb (s) max u(ξ )
ξ ∈[βs,s]
sb
t
b ds
 t b 
=  α rb (t) +  α  t α
p (s) max u(ξ )
, t ∈ (t , T).
ξ ∈[βs,s]
sb
t
b
ub (t) ≤ b– rb (t) + b– α  t a
t
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By taking v(t) = t –b ub (t), we have

v(t) ≤ c rb (t) +
((α  )) α
t
t
pb (s) max v(ξ ) ds,
t
ξ ∈[βs,s]
t ∈ (t , T)
(.)
and
v(t) ≤ c φ b (t),
t ∈ [βt , t ].
(.)
An application of Corollary . to (.) and (.) yields
 ((α  )) α t b
p (s) ds ,
v(t) ≤ c rb (t) + h (t) exp
t
t
t ∈ (t , T),
where c and h are deﬁned by (.) and (.), respectively. Thus, we get the required
inequality in (.).
Furthermore, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then, by applying Lemma . to (.) and (.), we get
 ((α  )) α t b
p (s) ds ,
v(t) ≤ c N r (t) exp
t
t
b
t ∈ (t , T),
where N is deﬁned by (.). Therefore, the desired inequality (.) is proved. This completes the proof.
Theorem . Suppose that the conditions (A ) and (A ) of Theorem . are satisﬁed. In
addition we assume that:
(A ) The function q ∈ C((t , T), R+ ).
(A ) The function u ∈ C([βt , T), R+ ) with
u(t) ≤ r(t) +
t
t
u(t) ≤ φ(t),
t
log
s
α– p(s)u(s) + q(s) max u(ξ )
ξ ∈[βs,s]
ds
s
,
t ∈ [βt , t ],
t ∈ (t , T),
(.)
(.)
where α > .
Then the following assertions hold:
(a) Suppose α >  , then

(α – ) t 


u(t) ≤ t c r (t) + h (t) exp
,
p (s) + q (s) ds
t
t
t ∈ (t , T),
(.)
where
c = max t– , (βt )–
(.)
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and
h (t) = c max φ  (s) +
s∈[βt ,t ]
×
t
t
c (α – )
t
p (s)r (s) + q (s) max m (ξ ) ds,
ξ ∈[βs,s]
t ∈ (t , T),
(.)
with m is deﬁned by (.).
Furthermore, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then
u(t) ≤
(α – ) t 
c N tr(t) exp
p (s) + q (s) ds ,
t
t
t ∈ (t , T),
(.)
where N is deﬁned by (.).
(b) Suppose  < α ≤  , then

 b
((α  )) α t b
b
b
,
u(t) ≤ t c r (t) + h (t) exp
p (s) + q (s) ds
t
t
t ∈ (t , T),
(.)
where b =  + α ,

c = max  α t–b , (βt )–b
(.)
and

h (t) = c max φ b (s) +
s∈[βt ,t ]
×
t
t
c ((α  )) α
t
pb (s)rb (s) + qb (s) max mb (ξ ) ds,
ξ ∈[βs,s]
t ∈ (t , T).
(.)
In addition, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then
 ((α  )) α t b

u(t) ≤ (c N ) b tr(t) exp
p (s) + qb (s) ds ,
bt
t
t ∈ (t , T),
(.)
where N is deﬁned by (.).
Proof (a) α >  . By using the Cauchy-Schwarz inequality in (.), for t ∈ (t , T), we have
 t

ds 
t α–


u(t) ≤ r(t) +
ds
p (s)u (s) 
log
s
s
t
t
α–  t
t  ds 
t

+
ds
q (s) max u(ξ )
log
ξ ∈[βs,s]
s
s
t
t
t


ds 
≤ r(t) + (α – )t 
p (s)u (s) 
s
t
t
 ds  +
q (s) max u(ξ )
.
ξ ∈[βs,s]
s
t
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By applying Lemma . with n = , σ = , we get
t
u (t) ≤ r  (t) + (α – )t
p (s)u (s)
t
t
q (s) max u(ξ )
+
 ds ξ ∈[βs,s]
t
s
,
ds
s
t ∈ (t , T).
Setting v(t) = t – u (t), we obtain
t
(α – )
p (s)v(s) ds
v(t) ≤ c r (t) +
t
t
t

+
q (s) max v(ξ ) ds , t ∈ (t , T)

(.)
ξ ∈[βs,s]
t
and
v(t) ≤ c φ  (t),
t ∈ [βt , t ].
(.)
Using Corollary . for (.) and (.), it follows that
(α – ) t 

p (s) + q (s) ds ,
v(t) ≤ c r (t) + h (t) exp
t
t

t ∈ (t , T),
where c and h are deﬁned by (.) and (.), respectively. Therefore, we get the desired
inequality in (.).
As a special case, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then by applying
Lemma . with (.) and (.), we have
(α – ) t 

p (s) + q (s) ds ,
v(t) ≤ c N r (t) exp
t
t

t ∈ (t , T),
where N is deﬁned by (.). Thus, we get the required inequality in (.). This completes
the proof of the ﬁrst part.
(b)  < α ≤  . Let a =  + α and b =  + α . Using the Hölder inequality in (.), for
t ∈ (t , T), we obtain
u(t) ≤ r(t) +
t t
t
log
s
a(α–)
a t
ds
t
 ds
p (s)u (s) b
s
b
b ds b
qb (s) max u(ξ )
ξ ∈[βs,s]
sb
t
t


ds b
≤ r(t) + α  t a
pb (s)ub (s) b
s
t
t
b ds b +
qb (s) max u(ξ )
.
ξ ∈[βs,s]
sb
t
+
t
b

b
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By applying Lemma . with n = , σ = b, we get


ub (t) ≤  α rb (t) +  α  t α
t
pb (s)ub (s)
t
t
qb (s) max u(ξ )
+
b ds sb
ξ ∈[βs,s]
t
,
ds
sb
t ∈ (t , T).
Taking v(t) = t –b ub (t), it follows that

v(t) ≤ c rb (t) +
t
+
((α  )) α
t
t
pb (s)v(s) ds
t
qb (s) max v(ξ ) ds ,
ξ ∈[βs,s]
t
t ∈ (t , T)
(.)
and
v(t) ≤ c φ b (t),
t ∈ [βt , t ].
(.)
Applying Corollary . for (.) and (.), we have the following estimate:
 ((α  )) α t b
b
p (s)d + q (s) ds ,
v(t) ≤ c r (t) + h (t) exp
t
t
b
t ∈ (t , T),
where c and h are deﬁned by (.) and (.), respectively. Hence, the result (.) is
proved.
As a special case, if r ∈ C((t , T), (, ∞)) is a nondecreasing function, then by using
Lemma . with (.) and (.), we get
 ((α  )) α t b
b
p (s)d + q (s) ds ,
v(t) ≤ c N r (t) exp
t
t
b
t ∈ (t , T),
where N is deﬁned by (.). Thus, the required inequality in (.) is proved. This completes the proof.
4 Applications to Hadamard fractional differential equations with ‘maxima’
In this section, the dependence of solutions on the orders with initial conditions and the
bound of solutions for the Hadamard fractional diﬀerential equations, are investigated.
We consider the following fractional diﬀerential equation with ‘maxima’:
y(t) = f t, y(t), max y(s) ,
HD
α
HD
α–k
s∈[βt,t]
y(t)|t=t+ = ηk ,
t ∈ I = (t , T),
k = , , . . . , n, n = –[–α],
(.)
(.)
and the initial function
y(t) = φ(t),
t ∈ [βt , t ],
(.)
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where H Dα represents the Hadamard fractional derivative of order α (α > ), f ∈ C(I ×
R × R, R), φ is a given continuous function on [βt , t ],  < β <  and ηk are constants.
The problem (.)-(.) describes a model of a fractional problem in real world phenomena in which often some parameters are involved. The values of these parameters can be
measured only up to certain degree of accuracy. Hence, the orders of fractional diﬀerential
equation α in (.) and the initial conditions α – k in (.) may be subject to some errors
either by necessity or for convenience. Thus, it is important to know how the solution of
(.)-(.) changes when the values of α and α – k are slightly altered.
Theorem . Let α >  and δ >  such that  ≤ n –  < α – δ < α ≤ n. Also let f : I × R ×
R → R be a continuous function satisfying the assumption:
(A ) There exist constants L , L >  such that |f (t, u , u )–f (t, v , v )| ≤ L |u –v |+L |u –
v |, for each t ∈ I and u , u , v , v ∈ R.
If y and z are the solutions of the initial value problems (.)-(.) and
z(t) = f t, z(t), max z(s) ,
HD
α–δ
HD
α–δ–k
s∈[βt,t]
z(t)|t=t+ = ηk ,
t ∈ I,
(.)
k = , , . . . , n, n = – –(α – δ) ,
(.)
with initial function
z(t) = φ(t),
t ∈ [βt , t ],
(.)
respectively, where ηk are constants and φ is a given continuous function on [βt , t ] such
that φ(t) ≡ φ(t) for all t ∈ [βt , t ], then the following estimates hold for t < t ≤ h < T:
(I) Suppose α – δ >  . Then for t ∈ I
z(t) – y(t) ≤ t c A (t) + h (t)


(α – δ – )(L + L )(t – t )
× exp
.

(α)t
(.)
(II) Suppose  < α – δ ≤  . Then for t ∈ I
z(t) – y(t) ≤ t c Ab (t) + h (t)


b
(((α – δ) )) α–δ (Lb + Lb )(t – t )
,
× exp
b
(α)t
where
n
n
ηj
ηj
t α–δ–j t α–j log
log
–
A(t) = (α – δ – j + )
t
(α – j + )
t
j=
j=

t α–δ

f –
+ log
t
(α – δ + ) (α – δ)(α) (.)
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t α–δ
t α 

log
log
f ,
–
(α – δ)(α)
t
(α + )
t f = sup f t, y(t), max y(s) ,
+ (.)
s∈[βt,t]
t ≤t≤h

,
α–δ
c = max t– , (βt )– ,

c = max  α–δ t–b , (βt )–b ,
b=+
 c (α – δ – )
h (t) = c max φ(s) – φ(s) +
s∈[βt ,t ]
 (α)t
t
L A (s) + L max m (ξ ) ds
t
ξ ∈[βs,s]
and
b
h (t) = c max φ(s) – φ(s)
s∈[βt ,t ]

+
c (((α – δ) )) α–δ
b (α)t
t
t
Lb Ab (s) + Lb max mb (ξ ) ds,
ξ ∈[βs,s]
with a continuous function m (t) is deﬁned by
⎧
⎨A(t),
t ∈ I,
m (t) =
⎩|φ(t) – φ(t)|, t ∈ [βt , t ].
Proof The solutions y and z of the initial value problems (.)-(.) and (.)-(.) satisfy
the following equations:
y(t) =
n
j=
t
ds
ηj
t α–j

t α– log
+
f s, y(s), max y(ξ )
log
ξ ∈[βs,s]
(α – j + )
t
(α) t
s
s
and
z(t) =
n
j=
ηj
(α – δ – j + )

+
(α – δ)
t
t
log
t
log
s
t
t
α–δ–j
α–δ– ds
,
f s, z(s), max z(ξ )
ξ ∈[βs,s]
s
respectively. So using the assumption (A ), it follows that
n
α–δ–j α–j n
ηj
η
t
t
j
z(t) – y(t) ≤ log
log
–
(α
–
δ
–
j
+
)
t
(α
–
j
+
)
t


j=
j=
t
ds

t α–δ– +
f s, z(s), max z(ξ )
log
ξ ∈[βs,s]
(α – δ) t
s
s
α–δ– t
ds t

f s, z(s), max z(ξ )
log
–
ξ ∈[βs,s]
(α) t
s
s t
ds

t α–δ– + f s, z(s), max z(ξ )
log
ξ ∈[βs,s]
(α) t
s
s
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t
ds t α–δ– 
log
f s, y(s), max y(ξ )
ξ ∈[βs,s]
(α) t
s
s t
ds

t α–δ– + log
f s, y(s), max y(ξ )
ξ ∈[βs,s]
(α) t
s
s
α– t
ds 
t
–
f s, y(s), max y(ξ )
log
ξ ∈[βs,s]
(α) t
s
s
t

t α–δ–
≤ A(t) +
log
(α) t
s
ds
× L z(s) – y(s) + L max z(ξ ) – max y(ξ )
ξ ∈[βs,s]
ξ ∈[βs,s]
s
α–δ–
t
t

log
≤ A(t) +
(α) t
s
ds
, t ∈ I,
× L z(s) – y(s) + L max z(ξ ) – y(ξ )
ξ ∈[βs,s]
s
–
where A(t) is deﬁned by (.), and
z(t) – y(t) = φ(t) – φ(t),
t ∈ [βt , t ].
Applying Theorem . yields the desired inequalities (.) and (.). This completes the
proof.
In the following theorem, we give the upper bounds of solution of the Hadamard fractional diﬀerential equation with ‘maxima’ and initial conditions (.)-(.).
Theorem . Assume that:
(A ) There exist functions μ, ν ∈ C(I, R+ ) such that for t ∈ I, u , u ∈ R,
f (t, u , u ) ≤ μ(t)|u | + ν(t)|u |.
(.)
If y is solution of the initial value problem (.)-(.) such that φ(t) ≡  for all t ∈ [βt , t ],
then the following estimates hold:
(III) Suppose α >  . Then for t ∈ I
n
y(t) ≤ t c
j=

|ηj |
t α–j
log
(α – j + )
t

(α – ) t 

+ h (t) exp
.
μ (s) + ν (s) ds
 (α)t
t
(.)
(IV) Suppose  < α ≤  . Then for t ∈ I
b(α–)
b
y(t) ≤ t c |η | log t
b (α)
t
b
 ((α  )) α t 

+ h (t) exp
,
μ (s) + ν (s) ds
b (α)t
t
(.)
Thiramanus et al. Journal of Inequalities and Applications 2014, 2014:398
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where b, c , c are deﬁned as in Theorem .,
c (α – )
 (α)t

t
n
|ηj |
s α–j



μ (s)
log
+ ν (s) max m (ξ ) ds
×
ξ ∈[βs,s]
(α – j + )
t
t
j=
h (t) = c max φ  (s) +
s∈[βt ,t ]
and
b
h (t) = c max φ(s)
s∈[βt ,t ]

+
c ((α  )) α
b (α)t
t
t
|η |b μb (s)
s b(α–)
b
b
log
+
ν
(s)
max
m
(ξ
)
ds,

ξ ∈[βs,s]
b (α)
t
with a continuous function m (t) deﬁned by
m (t) =
⎧
⎨ n
t ∈ I,
⎩|φ(t)|,
t ∈ [βt , t ].
|ηj |
t α–j
,
j= (α–j+) (log t )
Proof The solution y of the initial value problem (.)-(.) satisﬁes the following equations:
y(t) =
n
j=
+
ηj
t α–j
log
(α – j + )
t

(α)
y(t) = φ(t),
t
log
t
t
s
α– ds
,
f s, y(s), max y(ξ )
ξ ∈[βs,s]
s
t ∈ I,
t ∈ [βt , t ].
For α > , by using the assumption (A ), it follows that
n
y(t) ≤
j=
|ηj |
t α–j
log
(α – j + )
t
t
ds

t α– μ(s)y(s) + ν(s) max y(ξ )
,
log
ξ ∈[βs,s]
(α) t
s
s
y(t) = φ(t), t ∈ [βt , t ].
+
t ∈ I,
Hence Theorem . yields the estimate of the inequalities (.) and (.). This completes
the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the ﬁnal manuscript.
Thiramanus et al. Journal of Inequalities and Applications 2014, 2014:398
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Author details
1
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s
University of Technology North Bangkok, Bangkok, 10800, Thailand. 2 Department of Mathematics, University of Ioannina,
Ioannina, 451 10, Greece.
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz
University, Jeddah, Saudi Arabia.
Acknowledgements
The research of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
Received: 25 April 2014 Accepted: 30 September 2014 Published: 16 Oct 2014
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10.1186/1029-242X-2014-398
Cite this article as: Thiramanus et al.: Integral inequalities with ‘maxima’ and their applications to Hadamard type
fractional differential equations. Journal of Inequalities and Applications 2014, 2014:398
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