Some new nonlinear integral inequalities with weakly singular ... - Core

Liu and Meng Journal of Inequalities and Applications (2015) 2015:209 DOI 10.1186/s13660-015-0726-0

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Some new nonlinear integral inequalities with weakly singular kernel and their applications to FDEs Haidong Liu* and Fanwei Meng *

Correspondence: [email protected] School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P.R. China

Abstract In this paper, we investigate some nonlinear integral inequalities with weakly singular kernel which can be used as tools in deriving boundedness of the solutions of certain fractional differential equations and integral equations. Our results generalize and improve some results in the literature. Besides, we give some applications for some fractional differential equations involving the Riemann-Liouville derivative and the Caputo derivative, respectively. Keywords: integral inequalities; singular kernel; fractional differential equations; boundedness

1 Introduction In the study of the qualitative and quantitative properties of solutions of some fractional differential equations, many inequalities with singular kernels have been developed (for example, see [–] and the references therein). In these inequalities, Medveď [] discussed the following useful integral inequality:

t

u(t) ≤ a(t) +

(t – s)β– f (s)w u(s) ds,

t > ,



which came from the study of a global existence and an exponential decay result for a parabolic Cauchy problem by Henry []. Ma and Pečarić [] used the modification of Medveď’s method [] to study some new weakly singular integral inequality of Henry’s type: u (t) ≤ a(t) + b(t) p

t

t α – sα

β–

sγ – f (s)uq (s) ds,

t > ,



and used it to study the boundedness of certain fractional differential equations with the Caputo fractional derivatives and integral equations involving the Erdélyi-Kober fractional integrals. Recently, Ye and Gao [] studied a Henry-Gronwall type retarded integral inequalities:

t u(t) ≤ a(t) + t (t – s)β– b(s)u(s – r) ds, u(t) ≤ φ(t), t ∈ [t – r, t ),

t ∈ [t , T),

© 2015 Liu and Meng. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Liu and Meng Journal of Inequalities and Applications (2015) 2015:209

Page 2 of 17

and used it to obtain boundedness of a class of fractional differential equations. Very recently, Lin [] established some new weakly singular integral inequality of GronwallBellman type: u(t) ≤ a(t) +

n

t

(t – s)βi – ci (s)uγi (s) ds,

bi (t) t

i=

with the conditions that bi (t) (i = , , . . . , n) are bounded and monotonically increasing and used it to deal with the uniqueness of solutions for fractional differential equations. In this paper, we discuss the more general integral inequality with weakly singular kernel: u(t) ≤ a(t) +

l

+

t

(t – s)βj – gj (s)uγj (s) ds,

cj (t)

j=

(t – s)αi – f (s)u(s) ds t

i= n

t

bi (t)

t ∈ [t , T),

t

where the real constants αi >  (i = , , . . . , l), βj > ,  ≤ γj <  (j = , , . . . , n), and we have the following retarded integral inequality with weakly singular kernel: ⎧ t ⎪ + li= bi (t) t (t – s)αi – f (s)u(s – r) ds ⎨ u(t) ≤ a(t) t + nj= cj (t) t (t – s)βj – gj (s)uγj (s – r) ds, t ∈ [t , T), ⎪ ⎩ u(t) ≤ φ(t), t ∈ [t – r, t ). Our results not only generalize some integral inequalities that have been studied in [] and improve the results of [] by removing the conditions that cj (t) (j = , , . . . , n) are bounded and monotonically increasing but also provide a handy tool to derive the boundedness of the solutions of certain fractional differential equations and integral equations. Throughout the present paper, N denotes the set of the natural numbers; R denotes the set of the real numbers; R+ = [, +∞) is the subset of R; and C(D, E) denotes the class of all continuous functions defined on the set D with range in the set E.

2 Preliminaries and main results The following lemmas are useful in our main results. Lemma . (Jensen’s inequality) Let n ∈ N, and a , a , . . . , an ∈ R+ . Then, for r > ,

n i=

r ai

≤ nr–

n

ari .

i=

Lemma . [] Let c ≥ , x ≥ , and  ≤ λ < . Then, for any k > , cxλ ≤ kx + θ (c, k, λ) holds, where θ (c, k, λ) = ( – λ)λλ/(–λ) c/(–λ) k λ/(λ–) .


Page 3 of 17

Lemma . Let [t , T) ⊂ R (T ≤ ∞), a(t), b(t), f (t), ci (t), gi (t) ∈ C([t , T), R+ ) (i = , , . . . , n). If u(t) ∈ C([t , T), R+ ) and

t

u(t) ≤ a(t) + b(t)

f (s)u(s) ds +

n

t

t

gi (s)uγi (s) ds,

ci (t)

t ∈ [t , T),

()

t

i=

where the real constants  ≤ γi <  (i = , , . . . , n), then, for any positive functions ki (t) ∈ C([t , T), (, +∞)) (i = , , . . . , n),

t

u(t) ≤ A(t) exp B(t)

f (τ ) dτ +

n

t

t

Ci (t)

ki (τ ) dτ ,

t ∈ [t , T),

()

t

i=

where A(t) = max a(s) +

n

t ≤s≤t

t

Ci (t)

θ gi (s), ki (s), γi ds,

t

i=

Ci (t) = max ci (s),

B(t) = max b(s), t ≤s≤t

i = , , . . . , n.

t ≤s≤t

Proof From Lemma . and (), we get, for t ∈ [t , T),

t

u(t) ≤ a(t) + b(t)

f (s)u(s) ds + t

f (s)u(s) ds + t

= a(t) +

n

n i=

n

t

ci (t)

t

gi (s)uγi (s) ds t

ci (t)

t

ki (s)u(s) + θ gi (s), ki (s), γi ds

t

i=

θ gi (s), ki (s), γi ds + b(t)

t

f (s)u(s) ds

t

i=

+

ci (t)

i=

t

≤ a(t) + b(t)

n

t

t

ci (t)

ki (s)u(s) ds,

()

t

where ki (t) ∈ C([t , T), (, +∞)) (i = , , . . . , n) are any positive functions. Given any T ∈ (t , T), for t ∈ [t , T ], from (), we have

t

u(t) ≤ A(T ) + B(T )

f (s)u(s) ds + t

n i=

t

Ci (T )

ki (s)u(s) ds,

()

t

t where A(t) = maxt ≤s≤t a(s) + ni= Ci (t) t θ (gi (s), ki (s), γi ) ds, B(t) = maxt ≤s≤t b(s), Ci (t) = maxt ≤s≤t ci (s), i = , , . . . , n. Define z(t) by the right side of (), then z(t ) = A(T ), u(t) ≤ z(t), z(t) is nonnegative and nondecreasing, and z (t) = B(T )f (t)u(t) + = B(T )f (t) +

n i=

n i=

Ci (T )ki (t)u(t)

Ci (T )ki (t) u(t)


≤ B(T )f (t) +

n

Page 4 of 17

Ci (T )ki (t) z(t)

i=

= F(t)z(t),

() n

where F(t) = B(T )f (t) + have, for t ∈ [t , T ],

i= Ci (T )ki (t).

Based on a straightforward computation, we

t F(τ ) dτ , z(t) ≤ A(T ) exp

()

t

which implies that z(T ) ≤ A(T ) exp

T

F(τ ) dτ ,

()

t

and then we get u(T ) ≤ A(T ) exp

T

F(τ ) dτ

t T

= A(T ) exp

B(T )f (τ ) + t

n

Ci (T )ki (τ ) dτ

i=

T

= A(T ) exp B(T )

f (τ ) dτ + t

n i=

T

Ci (T )

ki (τ ) dτ .

t

By the arbitrariness of T ∈ (t , T), we obtain the inequality (). The proof is complete. Lemma . Let [t , T) ⊂ R (T ≤ ∞), a(t), b(t), f (t), ci (t), gi (t) ∈ C([t , T), R+ ) (i = , , . . . , n), φ(t) ∈ C([t – r, t ], R+ ), and a(t ) = φ(t ). If u(t) ∈ C([t – r, T), R+ ), and ⎧ t ⎪ + b(t) t f (s)u(s – r) ds ⎨ u(t) ≤ a(t) t + ni= ci (t) t gi (s)uγi (s – r) ds, ⎪ ⎩ u(t) ≤ φ(t), t ∈ [t – r, t ),

t ∈ [t , T),

()

where the real constants  ≤ γi <  (i = , , . . . , n). Then, for any positive functions ki (t) ∈ C([t , T), (, +∞)) (i = , , . . . , n), ⎧ t +r t +r ⎪ u(t) ≤ A(t) + [B(t) t f (s)φ(s – r) ds + ni= Ci (t) t ki (s)φ(s – r) ds] ⎪ ⎪ ⎪ t t ⎪ ⎪ · exp{B(t) t +r f (τ ) dτ + ni= Ci (t) t +r ki (τ ) dτ } ⎪ ⎪ t ⎪ t t ⎪ + B(t) t +r A(s – r)f (s) exp{B(t) s f (τ ) dτ + ni= Ci (t) s ki (τ ) dτ } ds ⎪ ⎨ t + ni= Ci (t) t +r A(s – r)ki (s) ⎪ t ⎪ t ⎪ · exp{B(t) s f (τ ) dτ + ni= Ci (t) s ki (τ ) dτ } ds, t ∈ [t + r, T), ⎪ ⎪ ⎪ t ⎪ ⎪ + b(t) t f (s)φ(s – r) ds ⎪ ⎪ u(t) ≤ a(t) ⎪ t ⎩ + ni= ci (t) t gi (s)φ γi (s – r) ds, t ∈ [t , t + r),

()


Page 5 of 17

where A(t) = a(t) +

n

t

ci (t)

θ gi (s), ki (s), γi ds,

t

i=

Ci (t) = max ci (s),

B(t) = max b(s), t ≤s≤t

i = , , . . . , n.

t ≤s≤t

Proof From (), for t ∈ [t , t + r), we can easily get

t

u(t) ≤ a(t) + b(t)

f (s)φ(s – r) ds + t

n

t

gi (s)φ γi (s – r) ds.

ci (t)

()

t

i=

From Lemma . and (), for t ∈ [t + r, T) and any positive functions ki (t) ∈ C([t , T), (, +∞)) (i = , , . . . , n), we obtain the following inequality:

t

u(t) ≤ a(t) + b(t)

f (s)u(s – r) ds + t

f (s)u(s – r) ds + t

= a(t) +

n

n

n

t

t

gi (s)uγi (s – r) ds

ci (t) t

ci (t)

t

ki (s)u(s – r) + θ gi (s), ki (s), γi ds

t

i=

t

t

θ gi (s), ki (s), γi ds + b(t)

ci (t)

i=

+

i=

t

≤ a(t) + b(t)

n

f (s)u(s – r) ds t

t

ci (t)

ki (s)u(s – r) ds t

i=

t

= A(t) + b(t)

f (s)u(s – r) ds + t

n

t

ci (t)

ki (s)u(s – r) ds,

()

t

i=

t where A(t) = a(t) + ni= ci (t) t θ (gi (s), ki (s), γi ) ds. Given any T ∈ (t + r, T), for t ∈ [t + r, T ], from (), we obtain u(t) ≤ A(t) + B(T )

t

f (s)u(s – r) ds + t

n

t

Ci (T )

ki (s)u(s – r) ds,

()

t

i=

where B(t) = maxt ≤s≤t b(s), Ci (t) = maxt ≤s≤t ci (s), i = , , . . . , n. Let

t

z(t) = B(T )

f (s)u(s – r) ds + t

n

t

Ci (T )

ki (s)u(s – r) ds, t

i=

then z(t ) = , u(t) ≤ A(t) + z(t), z(t) is nonnegative and nondecreasing and z (t) = B(T )f (t)u(t – r) +

n

Ci (T )ki (t)u(t – r)

i= n Ci (T )ki (t) A(t – r) + z(t – r) ≤ B(T )f (t) A(t – r) + z(t – r) + i=

()


= B(T )f (t) +

n

≤ B(T )f (t) +

n

Ci (T )ki (t) A(t – r) + B(T )f (t) +

i=

Page 6 of 17

n

Ci (T )ki (t) z(t – r)

i=

Ci (T )ki (t) A(t – r) + B(T )f (t) +

i=

n

Ci (T )ki (t) z(t)

i=

= P(t)A(t – r) + P(t)z(t), where P(t) = B(T )f (t) + (), we have

t +r

z(t) ≤

n

i= Ci (T )ki (t).

P(s)φ(s – r) ds · exp

t +r

P(τ ) dτ

s

t +r

= B(T )

f (s)φ(s – r) ds +

t

f (τ ) dτ + t +r

n

t +r

Ci (T )

ki (s)φ(s – r) ds t

i=

· exp B(T )

t

Ci (T )

ki (τ ) dτ t +r

i=

t

+ B(T )

A(s – r)f (s) t +r

t

· exp B(T )

f (τ ) dτ +

n

s n

n

t

+

t

t A(s – r)P(s) exp P(τ ) dτ ds

t

+

Based on a straightforward computation, from

t +r

t

()

t

Ci (T )

ki (τ ) dτ ds s

i= t

Ci (T )

A(s – r)ki (s) t +r

i=

t

· exp B(T )

f (τ ) dτ +

n

s

t

Ci (T )

ki (τ ) dτ ds,

()

s

i=

then

t +r

z(T ) ≤ B(T )

f (s)φ(s – r) ds + t

T

f (τ ) dτ + t +r

Ci (T )

ki (s)φ(s – r) ds t

T

Ci (T )

ki (τ ) dτ t +r

i=

A(s – r)f (s) t +r

T

· exp B(T )

f (τ ) dτ + s

n

n

t +r

T

+ B(T )

+

i=

· exp B(T )

n

T

Ci (T )

ki (τ ) dτ ds

s

i= T

Ci (T )

A(s – r)ki (s) t +r

i=

n

T

· exp B(T )

f (τ ) dτ + s

n i=

Ci (T )

T

ki (τ ) dτ ds.

s

()


Page 7 of 17

By the arbitrariness of T ∈ (t + r, T), we obtain the inequality (). The proof is complete. Remark . Assume that for Lemma ., if b(t) = , ci (t) = , i = , , . . . , n, then we can obtain Lemma . in []. Theorem . Let [t , T) ⊂ R (T ≤ ∞), a(t), f (t), bi (t), cj (t), gj (t) ∈ C([t , T), R+ ) (i = , , . . . , l; j = , , . . . , n). If u(t) ∈ C([t , T), R+ ) and u(t) ≤ a(t) +

l

(t – s)αi – f (s)u(s) ds t

i=

+

n

t

bi (t)

t

j=

t ∈ [t , T),

(t – s)βj – gj (s)uγj (s) ds,

cj (t)

()

t

where the real constants αi >  (i = , , . . . , l) and βj > ,  ≤ γj <  (j = , , . . . , n). Then, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , . . . , n), the following assertions hold: (i) Suppose that αi >  (i = , , . . . , l), βj >  (j = , , . . . , n), then n Bj (t) t B(t) t F(τ ) dτ + kj (τ ) dτ , u(t) ≤ A(t) exp t +  t  t j=

t ∈ [t , T),

()

where A(t) = max H(s) + t ≤s≤t

n

t

Bj (t)

θ Gj (s), kj (s), γj ds,

j=

Bj (t) = max Kj (s),

ncj (t) (βj – )

Kj (t) =

t ≤s≤t

Gj (t) = gj (t)Mj (t),

βj – (γj –)t

Mj (t) = e K(t) = l

B(t) = max K(s), t ≤s≤t

H(t) = e–t a (t),

t

,

 

()

j = , , . . . , n,

l b (t)et (αi – ) i

i=

(ii) Suppose that  < αi ≤

,

αi –

(i = , , . . . , l) or  < βj ≤

 

F(t) = f  (t).

,

(j = , , . . . , n), then

n Bj (t) t B(t) t /q u(t) ≤ A (t) exp t + ki (τ ) dτ , F(τ ) dτ + q t q t j=

t ∈ [t , T),

()

where (s) + A(t) = max H t ≤s≤t

n j=

Bj (t)

t


t

j (s), Bj (t) = max K

j (t) = (n)(q–) cqj (t) K

q j (t), Gj (t) = gj (t)M

j (t) = eq(γj –)t , M

t ≤s≤t

(t) = q– aq (t)e–qt , H

( – p( – βi )) p–p(–βj )

j = , , . . . , n,

pq , ()


B(t) = max K(s),

l

= (l)(q–) K(t)

t ≤s≤t

Page 8 of 17

ept ( – p( – αi )) p–p(–αi )

q bi (t)

i=

 q=+ , θ

p =  + θ,

pq ,

F(t) = f q (t),

θ = min{αi , βj , i = , , . . . , l; j = , , . . . , n}.

Proof (i) For t ∈ [t , T), from (), using the Cauchy-Schwarz inequality and a simple computation, l

u(t) ≤ a(t) +

bi (t)

(t – s)

n

αi – s

(t – s)βj – es ds



αi

–s 

f (s)e

u (s) ds

t

n  cj (t)et (βj – )  βj

j=







t

i



+

gj (s)e–s uγj (s) ds

t

i=

u (s) ds

t

 l b (t)et (αi – ) 

< a(t) +

–s 

f (s)e

t

j=



t

t

cj (t)



t

e ds

t

i=

+



t

t

t

gi (s)e–s uγj (s) ds

 .

()

Using Lemma ., we obtain u (t) ≤ a (t) + l

l b (t)et (αi – ) αi –

i=

+n

n

t

i

cj (t)et (βj – ) βj –



j=

f  (s)e–s u (s) ds t

t

t

gj (s)e–s uγj (s) ds.

()

Let w(t) = [e–t u(t)] , we obtain

t

w(t) ≤ H(t) + K(t)

F(s)w(s) ds + t

n

t

Gj (s)wγj (s) ds,

Kj (t) t

j=

where H(t), F(t), Kj (t), and Gj (t) (j = , , . . . , n) are defined in (). Using Lemma ., we get, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , . . . , n), t t n F(τ ) dτ + Bj (t) kj (τ ) dτ , () w(t) ≤ A(t) exp B(t) t

t

j=

where A(t), B(t), and Bj (t) (j = , , . . . , n) are defined in (). From the definition of w(t), we get (). (ii) For t ∈ [t , T), by the hypothesis, we get p + q = . Using the Hölder inequality and a simple computation, we obtain u(t) ≤ a(t) +

l

bi (t)

+

j=

(t – s)

pαi –p ps

p

(t – s) t

q

–qs q

f (s)e

u (s) ds

t

t

cj (t)

q

t

e ds

t

i= n

t

pβj –p ps

p

t

e ds t

q gj (s)e–qs uqγj (s) ds

q


≤ a(t) +

l i=

+

n

ept ( – p( – αi )) bi (t) p–p(–αi )

cj (t)

ept ( – p( – βj ))

Page 9 of 17

p

–qs q

q

f (s)e

u (s) ds

t

p

p–p(–βj )

j=

q

t

t t

q

gj (s)e–qs uqγj (s) ds

q .

Obviously,  – p( – αi ) =  – ( + θ )( – αi ) ≥  – ( + αi )( – αi ) = αi > , i = , , . . . , l,  – p( – βj ) =  – ( + θ )( – βj ) ≥  – ( + βj )( – βj ) = βj > , j = , , . . . , n. Using Lemma ., we obtain u (t) ≤  q

q– q

a (t) + (l)

(q–)

l

q bi (t)

i=

+ (n)(q–)

n

q

cj (t)

j=

ept ( – p( – αi )) p–p(–αi )

ept ( – p( – βj )) –p(–βj )

p

pq

t

t

pq

t

f q (s)e–qs uq (s) ds

t

q

gj (s)e–qs uqγj (s) ds.

()

Let w(t) = [e–t u(t)]q , we get (t) + K (t) w(t) ≤ H

t

F (s)w(s) ds +

t

n

j (t) K

t

Gj (s)wγj (s) ds,

t

j=

j (t) (j = , , . . . , n), F(t), and Gj (t) (j = , , . . . , n) are defined in (). Using where H(t), K Lemma ., we have, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , . . . , n), w(t) ≤ A(t) exp B(t)

t

t

F(τ ) dτ +

n j=

Bj (t)

t

kj (τ ) dτ ,

()

t

where A(t), B(t), and Bj (t) (j = , , . . . , n) are defined in (). From the definition of w(t), we get (). The proof is complete. Theorem . Let [t , T) ⊂ R (T ≤ ∞), a(t), f (t), bi (t), cj (t), gj (t) ∈ C([t , T), R+ ) (i = , , . . . , l; j = , , . . . , n). φ(t) ∈ C([t – r, t ], R+ ) and a(t ) = φ(t ). If u(t) ∈ C([t – r, T), R+ ), and ⎧ t ⎪ + li= bi (t) t (t – s)αi – f (s)u(s – r) ds ⎨ u(t) ≤ a(t) t () + nj= cj (t) t (t – s)βj – gj (s)uγj (s – r) ds, t ∈ [t , T), ⎪ ⎩ u(t) ≤ φ(t), t ∈ [t – r, t ), where the real constants αi >  (i = , , . . . , l), βj > ,  ≤ γj <  (j = , , . . . , n). Then, for any positive functions ki (t) ∈ C([t , T), (, +∞)) (i = , , . . . , n), the following assertions hold: (i) Suppose that αi >  (i = , , . . . , l), βj >  (j = , , . . . , n), then ⎧ t +r t +r ⎪ u(t) ≤ et {A(t) + [B(t) t F(s)φ (s – r) ds + nj= Bj (t) t kj (s)φ (s – r) ds] ⎪ ⎪ ⎪ t t ⎪ ⎪ · exp{B(t) t +r F(τ ) dτ + nj= Bj (t) t +r kj (τ ) dτ } ⎪ ⎪ t t t ⎪ ⎪ + B(t) t +r A(s – r)F(s) exp{B(t) s F(τ ) dτ + nj= Bj (t) s kj (τ ) dτ } ds ⎪ ⎨ t + nj= Bj (t) t +r A(s – r)kj (s) ⎪ t ⎪ t ⎪ · exp{B(t) s F(τ ) dτ + nj= Bj (t) s kj (τ ) dτ } ds}/ , t ∈ [t + r, T), ⎪ ⎪ ⎪ t ⎪ ⎪ + li= bi (t) t (t – s)αi – f (s)φ(s – r) ds ⎪ ⎪ u(t) ≤ a(t) ⎪ t ⎩ + nj= cj (t) t (t – s)βj – gj (s)φ γj (s – r) ds, t ∈ [t , t + r),

()


Page 10 of 17

where A(t) = H(t) +

n

t

Kj (t)


t

j=

nci (t)e–γj r (βj – )

H(t) = e–t a (t),

Kj (t) =

Gj (t) = gj (t)Mj (t),

Mj (t) = e(γj –)t , K(t) = l

B(t) = max K(s), t ≤s≤t

βj –

j = , , . . . , n,

l b (t)et (αi – ) i

αi –

i=

F(t) = f  (t),

φ (t) = e–t φ  (t),

(ii) Suppose that  < αi ≤

t ≤s≤t

 

()

,

Bj (t) = max Kj (s),

(i = , , . . . , l) or  < βj ≤

 

,

j = , , . . . , n.

(j = , , . . . , n) then

⎧ t +r t +r A(t) + [ B(t) t F(s)φ (s – r) ds + nj= Bj (t) t kj (s)φ (s – r) ds] u(t) ≤ et { ⎪ ⎪ ⎪ t t ⎪ ⎪ · exp{ B(t) t +r F(τ ) dτ + nj= Bj (t) t +r kj (τ ) dτ } ⎪ ⎪ ⎪ t t t ⎪ ⎪ ⎪ F (τ ) dτ + nj= Bj (t) s kj (τ ) dτ } ds A(s – r) F (s) exp{ B(t) s + B(t) ⎨ n t +r t + j= Bj (t) t +r A(s – r)kj (s) ⎪ t t ⎪ ⎪ ⎪ Bj (t) s kj (τ ) dτ } ds}/q , t ∈ [t + r, T), · exp{B(t) s F (τ ) dτ + nj= ⎪ ⎪ ⎪ ⎪ u(t) ≤ a(t) + l bi (t) t (t – s)αi – f (s)φ(s – r) ds ⎪ i= ⎪ ⎪ t t ⎩ + nj= cj (t) t (t – s)βj – gj (s)φ γj (s – r) ds, t ∈ [t , t + r),

()

where n

(s) + A(t) = max H t ≤s≤t

Bj (t)

t


t

j=

j (s), Bj (t) = max K

(t) = q– e–qt aq (t), H

t ≤s≤t

j (t) = (n)(q–) cqj (t)e–qγj r K q j (t), Gj (t) = gj (t)M

( – p( – βj ))

= (l)(q–) K(t)

t ≤s≤t

,

p–p(–βj )

j (t) = eq(γj –)t , M

B(t) = max K(s),

pq

l

j = , , . . . , n, q

bi (t)

i=

φ (t) = e–qt φ q (t),

p =  + θ,

j = , , . . . , n,

ept ( – p( – αi )) p–p(–αi )

 q=+ , θ

θ = min{αi , βj , i = , , . . . , l; j = , , . . . , n},

F (t) = f q (t).

Proof From (), for t ∈ [t , t + r), we can easily get u(t) ≤ a(t) +

l

+

j=

(t – s)αi – f (s)φ(s – r) ds t

i= n

t

bi (t)

t

(t – s)βj – gj (s)φ γj (s – r) ds.

cj (t) t

() pq ,


Page 11 of 17

(i) For t ∈ [t , T), from a proof procedure similar to (i) of Theorem ., we obtain u (t) ≤ a (t) + l 



l b (t)et (αi – ) αi –

i=

+n

t

i

f  (s)e–s u (s – r) ds t

n

cj (t)et (βj – )

j=

βj –

t

t

gj (s)e–s uγj (s – r) ds.

()

Let w(t) = [e–t u(t)] , we obtain

t

w(t) ≤ H(t) + K(t) +

n

F(s)w(s – r) ds

t t

t ∈ [t , T)

Gj (s)wγj (s – r) ds,

Kj (t)

()

t

j=

and w(t) ≤ φ (t),

t ∈ [t – r, t ),

where H(t), K(t), F(t), φ (t), and Kj (t), Gj (t) (j = , , . . . , n) are defined in (). Using Lemma ., we get, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , . . . , n),

w(t) ≤ A(t) + B(t)

t +r

F(s)φ (s – r) ds +

n

t

· exp B(t)

t

F(τ ) dτ +

n

+

kj (τ ) dτ

A(s – r)F(s) exp B(t) t +r

n

t

F(τ ) dτ + s

t

Bj (t)

n

Bj (t)

t

F(τ ) dτ +

t

kj (τ ) dτ ds s

j=

A(s – r)kj (s) exp B(t) t +r

j=

t t +r

+ B(t)

kj (s)φ (s – r) ds t

Bj (t)

j=

t

t +r

Bj (t)

j=

t +r

n

s

j=

Bj (t)

t

kj (τ ) dτ ds, s

t ∈ [t + r, T), where A(t), B(t), and Bj (t) (i = , , . . . , n) are defined in (). From the definition of w(t), we get (). (ii) For t ∈ [t , T), from the similar proof procedure of (ii) of Theorem ., we have u (t) ≤  q

q– q

a (t) + (l)

·

(q–)

l

q bi (t)

i=

ept ( – p( – αi )) p–p(–αi )

pq

t

f q (s)e–qs uq (s – r) ds t

+ (n)

(q–)

n j=

q cj (t)

ept ( – p( – βj )) p–p(–βj )

pq

t

t

q

gj (s)e–qs uqγj (s – r) ds.

()


Page 12 of 17

Let w(t) = [e–t u(t)]q , we get

(t) + K (t) w(t) ≤ H

t

F (s)w(s – r) ds +

n

t

j (t) K

t

Gj (s)wγj (s – r) ds,

t ∈ [t , T)

t

j=

and w(t) ≤ φ (t),

t ∈ [t – r, t ),

(t), j (t) and where H F (t), φ (t), K Gj (t) (j = , , . . . , n) are defined in (). Using Lemma ., we get, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , . . . , n), w(t) ≤ A(t) + B(t)

t +r

F (s)φ (s – r) ds +

n

t

· exp B(t)

t

F(τ ) dτ +

+ B(t)

n

Bj (t)

A(s – r) F (s) exp B(t)

t +r

+

n

Bj (t)

kj (τ ) dτ

t

F (τ ) dτ +

n

Bj (t)

t

kj (τ ) dτ ds s

j=

t t n F (τ ) dτ + Bj (t) A(s – r)kj (s) exp B(t) kj (τ ) dτ ds,

t t +r

j=

t

s

kj (s)φ (s – r) ds

t +r

j=

t

t +r

t

j=

t +r

Bj (t)

s

j=

s

t ∈ [t + r, T), where A(t), B(t), and Bj (t) (j = , , . . . , n) are defined in (). From the definition of w(t), we get (). The proof is complete.

3 Applications In this section, we present some applications in studying the boundedness of the solutions of certain fractional differential equations with a Riemann-Liouville fractional derivative and a Caputo fractional derivative, respectively. First, we consider some fractional differential equations with the Riemann-Liouville fractional derivative. The Riemann-Liouville fractional order derivative and fractional integral are defined as follows. Definition  [] For any  < β < , the βth Riemann-Liouville fractional order derivative of a function f : [, +∞) → R is defined by β

DR f (t) =

d  ( – β) dt

t

(t – s)–β f (s) ds, 

where is the gamma function. Definition  [] The βth Riemann-Liouville fractional order integral of a function f : [, +∞) → R is defined by β IR f (t) =

where β > .

 (β)

t

(t – s)β– f (s) ds, 


Page 13 of 17

Let us consider the initial value problems for fractional differential equations in the following form: n

β DRi y(t) = f t, y(t) ,

()

i= n

–βi

IR

y(t)|t= = δ,

()

i=

where  < β < β < · · · < βn < , t ∈ [, T). The solution y(t) of the initial value problem (IVP) ()-() can be written as (see [])

y(t) =

β IR n f t, y(t) –

=

+

n– i=

β –βi

IR n

y(t) + δ

i=

 (βn )

n–

t

t βn – (βn )

(t – s)βn – f s, y(s) ds



 (βn – βi )

t

(t – s)βn –βi – y(s) ds + δ 

t βn – . (βn )

()

The following theorem deals with the boundedness of the solutions of the initial value problem ()-(). Theorem . Suppose that the function f ∈ C([, T) × R, R) satisfies f (t, y) ≤ h (t)|y|γ + h (t)|y|γ + h (t)|y|γ ,

(t, y) ∈ [, T) × R,

()

where  < γ , γ , γ <  are constants independent of t, y in R, h (t), h (t), and h (t) are nonnegative continuous functions defined on [, T). If y(t) is any solution of the initial value problem ()-(), then, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , , ), the following assertions hold: (i) Suppose that βn – βi >  (i = , , . . . , n – ), βn >  , then  t( + B(t)) Bj (t) t u(t) ≤ A(t) exp kj (τ ) dτ , +    j=

t ∈ [, T),

where

A(t) = max H(s) + ≤s≤t

Bj (t) =

 j=

 (βn – ) , βn –  (βn )

B(t) = (n – )

Bj (t)

t




Gj (t) = hj (t)e(γj –)t ,

n– et (βn – βi – ) i=

H(t) =

βn –βi –  (βn – βi )

.

j = , , ,

δ  e–t t (βn –) ,  (βn )

()


(ii) Suppose that βn – βi ≤

 

Page 14 of 17

(i = , , . . . , n – ), or  < βn ≤  , then

 Bj (t) t t(q + B(t)) /q + u(t) ≤ A (t) exp ki (τ ) dτ , q q  j=

t ∈ [, T),

()

where (s) + A(t) = max H



≤s≤t

Bj (t) = ()

Bj (t)

(t) = H

|δ|q q– e–qt t q(βn –) , q (βn )

q ( – p( – βn )) p  , q (βn ) p–p(–βn )

q Gj (t) = hj (t)eq(γj –)t ,

j = , , ,

n– = (n – ) (q–) K(t) i=

(s), B(t) = max K ≤s≤t

ept ( – p( – αi )) q (βn – βi ) p–p(–αi ) 

 q=+ , θ

p =  + θ,




j=

(q–)

t

pq ,

θ = min{βn – βi , βn , i = , , . . . , n – }.

Proof From () and () we have y(t) ≤

 (βn ) +

n– i=

≤ |δ| +

t

(t – s)βn – f s, y(s) ds



 (βn – βi )

 t βn – + (βn ) (βn )

n– i=

 (βn – βi )

t 

t βn – (t – s)βn –βi – y(s) ds + |δ| (βn ) t

γ γ γ (t – s)βn – h (s)y(s)  + h (s)y(s)  + h (s)y(s)  ds



t

(t – s)βn –βi – y(s) ds.



As an application of Theorem ., we obtain the inequalities () and (). This process completes the proof of Theorem .. Next, we consider some fractional differential equations with the Caputo fractional derivative. The Caputo fractional order derivative is defined as follows. Definition  [] The Caputo fractional derivative of order α (n –  < α < n, n is a positive integer) of a continuous function f : R+ → R is given by Dαt f (x) :=

 (n – α)

x

(x – t)–α+n– f (n) (t) dt. t

Let us consider the initial value problems for fractional differential equations with delay in the following form: β Dt y(t) = f t, y(t – r) ,

t ∈ [t , T),

()


Page 15 of 17

with the given initial condition t ∈ [t – r, t ],

y(t) = φ(t),

()

where f ∈ C([t , T) × R, R) and φ is a given continuously differentiable function on [t – r, t ] up to order n (n = –[–β]), and we denote φ k (t ) = bk , k = , , , . . . , n – . Theorem . Suppose that f (t, y) ≤ h (t)|y| + h (t)|y|γ + h (t)|y|γ ,

(t, y) ∈ [t , T) × R,

()

where  < γ , γ <  are constants independent of t, y in R, h (t), h (t), and h (t) are nonnegative continuous functions defined on [t , T). If y(t) is any solution of the initial value problem ()-(), then, for any positive functions kj (t) ∈ C([t , T), (, +∞)) (j = , ), the following assertions hold: (i) Suppose that β >  , then

t +r

u(t) ≤ et A(t) + B(t)

F(s)φ (s – r) ds +



t

· exp B(t)

t

F(τ ) dτ +

t

Bj (t)

kj (τ ) dτ t +r

j=

A(s – r)F(s)

t +r

· exp B(t)

t

F(τ ) dτ + s



kj (s)φ (s – r) ds t

t

+ B(t)

+



t +r

Bj (t)

j=

t +r



t

Bj (t)

kj (τ ) dτ ds s

j= t

Bj (t)

A(s – r)kj (s) t +r

j=

· exp B(t)

t

F(τ ) dτ + s



Bj (t)

t

kj (τ ) dτ ds

/ ,

t ∈ [t + r, T),

s

j=

where

A(t) = H(t) +



H(t) = e

n– bk k=

Bj (t) = Kj (t) = B(t) =


t

j=

–t

t

Kj (t)

k!

 (t – t )

k

e–γj r (β – ) , β–  (β)

et (β – ) , β–  (β)

, Gj (t) = hj (t)e(γj –)t ,

F(t) = h (t),

j = , ,

φ (t) = e–t φ  (t).

()


Page 16 of 17

(ii) Suppose that  < β ≤  , then u(t) ≤ e A(t) + B(t) t

t +r

F (s)φ (s – r) ds +

t

· exp B(t)

t

t

F (τ ) dτ +



Bj (t)

t +r

kj (s)φ (s – r) ds t

t

kj (τ ) dτ t +r

j=

A(s – r) F (s)

t

F (τ ) dτ +

s 

t +r

· exp B(t)

+

Bj (t)

j=

t +r

+ B(t)



Bj (t)



Bj (t)

t

kj (τ ) dτ ds s

j= t

A(s – r)kj (s)

t +r

j=

· exp B(t)

t

F (τ ) dτ +

s



Bj (t)

t

/q

kj (τ ) dτ ds

,

t ∈ [t + r, T),

()

s

j=

where (s) + A(t) = max H



t ≤s≤t

(t) = q– e–qt H

Bj (t)

t


t

j=

n– bk k=

k!

q (t – t )

k

,

q q– e–qγj r ( – p( – β)) p , q (β) p–p(–β) q q– ept ( – p( – β)) p B(t) = q , (β) p–p(–β)

q Gj (t) = gj (t)eq(γj –)t ,

Bj (t) =

p =  + β,

q=+

 , β

j = , ,

φ (t) = e–qt φ q (t),

q F (t) = h (t).

Proof The solution y(t) of the initial value problem ()-() can be written as (see [])

y(t) =

n–

bk k= k! (t

u(t) = φ(t),

– t )k +

 (β)

t ∈ [t – r, t ].

t

t (t

– s)β f (s, y(s)) ds,

t ∈ [t , T),

So t n–  |bk | k y(t) ≤ (t – t ) + (t – s)β k! (β) t k= γ γ · h (s)y(s) + h (s)y(s)  + h (s)y(s)  ds,

t ∈ [t , T).

As an application of Theorem ., we obtain the inequalities () and (). This process completes the proof of Theorem ..


Page 17 of 17

Competing interests The authors declare that there is no conflict of interests regarding the publication of this paper. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The authors would like to thank the referee for his/her careful reading and kind suggestions. This research was supported by the National Natural Science Foundations of China (Nos. 11171178, 11271225), the Science and Technology Project of High Schools of Shandong Province (No. J14LI09). Received: 3 February 2015 Accepted: 22 May 2015 References 1. Ma, QH, Yang, EH: Estimations on solutions of some weakly singular Volterra integral inequalities. Acta Math. Appl. Sin. 25, 505-515 (2002) 2. Grinshpan, AZ: Weighted integral and integro-differential inequalities. Adv. Appl. Math. 41, 227-246 (2008) 3. Medved’, M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214, 349-366 (1997) 4. Medved’, M: Singular integral inequalities and stability of semi-linear parabolic equations. Arch. Math. 24, 183-190 (1998) 5. Kirane, M, Tatar, N-E: Global existence and stability of some semilinear problems. Arch. Math. 36, 33-44 (2000) 6. Medved’, M: Nonlinear singular integral inequalities for functions of two and n independent variables. J. Inequal. Appl. 5, 287-308 (2000) 7. Tatar, N-E: Exponential decay for a semilinear problem with memory. Arab J. Math. Sci. 7(1), 29-45 (2001) 8. Tatar, N-E: On an integral inequality with a kernel singular in time and space. J. Inequal. Pure Appl. Math. 4(4), Article 82 (2003) 9. Ye, HP, Gao, JM, Ding, YS: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075-1081 (2007) 10. Ma, QH, Peˇcarić, J: Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J. Math. Anal. Appl. 341, 894-905 (2008) 11. Ma, QH, Peˇcarić, J, Zhang, JM: Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems. Comput. Math. Appl. 61, 3258-3267 (2011) 12. Bainov, D, Simeonov, P: Integral Inequalities and Applications. Mathematics and Its Applications, vol. 57. Kluwer Academic, Dordrecht (1992) 13. Jiang, FC, Meng, FW: Explicit bounds on some new nonlinear integral inequality with delay. J. Comput. Appl. Math. 205, 479-486 (2007) 14. Ferreira, RAC, Torres, DFM: Generalized retarded integral inequalities. Appl. Math. Lett. 22, 876-881 (2009) 15. Feng, Q, Meng, F: Some new Gronwall-type inequalities arising in the research of fractional differential equations. J. Inequal. Appl. 2013, Article ID 429 (2013) 16. Shao, J, Meng, FW: Gronwall-Bellman type inequalities and their applications to fractional differential equations. Abstr. Appl. Anal. 2013, Article ID 217641 (2013) 17. Ye, HP, Gao, JM: Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay. Appl. Math. Comput. 218, 4152-4160 (2011) 18. Lin, SY: New results for generalized Gronwall inequalities and their applications. Abstr. Appl. Anal. 2014, Article ID 168594 (2014) 19. Henry, D: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., vol. 840. Springer, Berlin (1981) 20. Sun, YG: Nonlinear dynamical integral inequalities in two independent variables and their applications. Discrete Dyn. Nat. Soc. 2011, Article ID 320794 (2011) 21. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)