Zheng et al. Journal of Inequalities and Applications (2016) 2016:7 DOI 10.1186/s13660-015-0943-6
RESEARCH
Open Access
Some new generalized retarded inequalities for discontinuous functions and their applications Zhaowen Zheng1* , Xin Gao1 and Jing Shao2 *
Correspondence:
[email protected] 1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P.R. China Full list of author information is available at the end of the article
Abstract In this paper, some new generalized retarded inequalities for discontinuous functions are discussed, which are effective in dealing with the qualitative theory of some impulsive differential equations and impulsive integral equations. Compared with some existing integral inequalities, these estimations can be used as tools in the study of differential-integral equations with impulsive conditions. Keywords: retarded differential-integral equation; global existence; estimation; impulsive equation
1 Introduction In analyzing the impulsive phenomenon of a physical system governed by certain differential and integral equations, one often needs some kinds of inequalities, such as Gronwalllike inequalities; these inequalities and their various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential and integral equations (see [–] and references therein). In [], Lipovan studied the inequality with delay (b(t) ≤ t, b(t) → ∞ as t → ∞)
t
u(t) ≤ c +
f (s)w u(s) ds +
b(t)
g(s)w u(s) ds,
t < t < t ,
b(t )
t
in [], Agarwal et al. investigated the retarded Gronwall-like inequality
u(t) ≤ a(t) +
n i=
bi (t)
fi (t, s)wi u(s) ds,
bi (t )
in , Borysenko [] obtained the explicit bound to the unknown function of the following integral inequality with impulsive effect: u(t) ≤ a(t) +
t
f (s)u(s) ds + t
αi ur (ti – ),
t
© 2015 Zheng et al. 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.
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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in , Iovane [] studied the following integral inequalities:
t
u(t) ≤ a(t) +
f (s)u λ(s) ds + αi ur (ti – ),
t
t
t
u(t) ≤ a(t) + q(t)
f (s)u α(s) ds +
t
+
t
s
f (s) t
r αi u (ti – ) ,
g(t)u τ (t) dt ds
t
t
in , Wang and Li [] gave the upper bound of solutions for the nonlinear inequality p p–q
vp (t) ≤ A (t) +
t
f (s)vq τ (s) ds + αi vq (ti – ),
t
t
in , Yan [] considered the following inequality:
t
u(t) ≤ a(t) +
f (t, s)u α(s) ds +
t
+ q(t)
t
t
s
f (t, s)
g(s, λ)u τ (λ) dλ ds
t
αi ur (ti – ),
t
and gave an upper bound estimation. Because of the fundamental importance, over the years, many generalizations and analogous results have been established. However, the bounds given on such inequalities are not directly applicable in the study of some complicated retarded inequalities for discontinuous functions. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of retarded nonlinear differential and integral equations. So in this paper, the following new integral inequalities are presented:
u(t) ≤ a(t) +
N
t
L gi (s)u φi (s) ds +
t
i=
j=
p p – q i= N
up (t) ≤ a (t) + +
t
t
s
bj (s) t
()
t
L gi (s)uq φi (s) ds +
t
cj (θ )u wj (θ ) dθ ds,
j=
t
s
bj (s) t
cj (θ )uq wj (θ ) dθ ds
t
βi uq (ti – ),
()
t
up (t) ≤ a (t) + q (t)
N i=
+
L j=
gi (t, s)uq φi (s) ds
t
t
t
t
s
bj (t, s) t
cj (s, θ )uq wj (θ ) dθ ds + q (t) βi uq (ti – ).
()
t
We give the explicit upper bounds estimation of unknown function of these new inequalities, some applications of these inequalities in impulsive differential equations are also involved.
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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2 Main results We consider the inequality () first. Theorem . Suppose that for t ∈ R and t ≤ t < ∞, the functions u(t), a(t), and gi (t), bj (t), cj (t) ( ≤ i ≤ N , ≤ j ≤ L) are positive and continuous functions on [t , ∞), and cj (t) are nondecreasing functions on [t , ∞). Moreover, φi (t), wj (t) are continuous functions on [t , ∞) and t ≤ φi (t) ≤ t, t ≤ wj (t) ≤ t for ≤ i ≤ N and ≤ j ≤ L. Then the inequality () implies that t s t Q (s) ds Y (s) exp – Q (τ ) dτ ds , u(t) ≤ a(t) + exp t
t
()
t
where Q (t) = L +
N
L
gi (t) +
i=
bj (t)cj (t),
j=
and cj (t) = max{cj (t), }. Proof Let a(t)+z (t) denote the function on the right-hand side of inequality (). Obviously z (t) is a positive and increasing function, and it satisfies u(t) ≤ a(t) + z (t), t N L dz (t) gi (t)u φi (t) + bj (t) cj (θ )u wj (θ ) dθ = dt t i= j= ≤
N
gi (t) a φi (t) + z φi (t)
i=
+
L
t
bj (t)
cj (θ ) a wj (θ ) + z wj (θ ) dθ .
()
t
j=
Let Y (t) =
N
L gi (t)a φi (t) + bj (t)
i=
gi (t)z (t) +
L
i=
Obviously,
dz (t) dt
z (t) ≤
j=
t
cj (θ )a wj (θ ) dθ ,
()
t
j=
N
z (t) =
t
bj (t)
cj (θ )z wj (θ ) dθ .
()
t
≤ Y (t) + z (t). Let cj (t) = max{cj (t), }. We can obtain
N
gi (t) +
i=
L
bj (t)cj (t)
j=
Let z (t) = z (t) +
L j=
t t
z wj (θ ) dθ ,
z (t) +
L j=
t
t
z wj (θ ) dθ .
()
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we note that z (t) is a positive and nondecreasing function on I with z (t ) = , and z (t) ≤ z (t), which satisfies dz (t) dz (t) = + z wj (t) dt dt j= L
≤ Y (t) +
N
L
gi (t) +
i=
≤ L+
N
bj (t)cj (t)
z (t) +
j=
gi (t) +
i=
L
L z wj (t) j=
bj (t)cj (t) z (t) + Y (t)
j=
= Q (t)z (t) + Y (t).
()
Consider the initial value problem of the differential equation
dz (t) dt
= Q (t)z (t) + Y (t), z (t ) = .
()
The solution of equation () is
t
z (t) = exp
t
Q (s) ds t
s Y (s) exp – Q (τ ) dτ ds .
t
()
t
Then by comparison of the differential inequality, we have z (t) ≤ z (t), so t t s u(t) ≤ a(t) + exp Q (s) ds Y (s) exp – Q (τ ) dτ ds . t
t
()
t
This completes the proof. Now, we consider the inequality ().
Theorem . Suppose that gi (t), bj (t), φi (t), wj (t) are defined as those in Theorem ., p > q > , t < t < t < · · · , βi ≥ , a (t) is continuous and nondecreasing function on [t , t ) and a (t) ≥ . u(t) is a piecewise continuous nonnegative function on [t , ∞) with only the first discontinuous points ti , i = , , . . . . Then, for all t ∈ Ik and Ik = [tk– , tk ], we obtain q–
u(t) ≤ τk (t),
()
where
τk (t) = ak– (t) exp
pq
t
Q (s) ds tk–
p q + ak– (t) p–q × +L – p q s p–q t q – Q (τ ) dτ ds × exp – , p tk– tk–
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
p ak (t) = a (t) + p – q j= k
+
k j=
L
tj
N
tj
Page 5 of 14
gi (s)τj φi (s) ds
tj– i=
s
bm (s)
k cm (θ )τj wm (θ ) dθ ds + βj τj (tj ),
t
tj– m=
t ∈ Ik .
j=
Proof Let V = uq . The inequality () is equivalent to p V (t) ≤ a (t) + p – q i= N
p q
+
L
t
gi (s)V φi (s) ds
t
cj (θ )V wj (θ ) dθ ds
s
bj (s) t
j=
+
t
t
∀t ∈ [, ∞).
βi uq (ti – ),
()
t
Let Ii = [ti– , ti ], i = , , . . . . First, we consider the following inequality on I : p V (t) ≤ a (t) + p – q i= N
p q
+
L
t
t
gi (s)V φi (s) ds
t
cj (θ )V wj (θ ) dθ ds.
s
bj (s) t
j=
()
t
For T ∈ I and t ∈ [t , T), let p p – q i= N
Y (t) = a (T) +
+
L j=
t
s
bj (s)
t
t
gi (s)V φi (s) ds
t
cj (θ )V wj (θ ) dθ ds.
()
t q p
p
Obviously Y (t) ≥ , and V q (t) ≤ Y (t), so V (t) ≤ Y (t), and p dY (t) = gi (t)V φi (t) + bj (t) dt p – q i= j= N
L
q p gi (t)Yp (t) + bj (t) p – q i= j=
N
≤
≤
L
p gi (t) + bj (t)cj (t) p – q i= j= N
L
Let Y (t) = Y (t) +
L j=
t
t
q p Y wj (θ ) dθ .
t
cj (θ )V wj (θ ) dθ
t t
t
q cj (θ )Yp wj (θ ) dθ
Y (t) +
L j=
t t
q p
Y wj (θ ) dθ .
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Then Y (t) ≤ Y (t), and differentiating Y (t) implies dY (t) dY (t) pq = + Y wj (t) dt dt j= L
≤
N L q p p gi (t) + bj (t)cj (t) Y (t) + LY (t). p – q i= j=
()
Let p gi (t) + bj (t)cj (t). p – q i= j= N
Q (t) =
Considering p–q p
dY (t) dt
L
–q p
q p
q
– p
= Q (t)Y (t) + LY (t), we obtain Y (t) dYdt (t) = Q (t)Y
(t) + L. Denote
p p–q
R(t) = Y (t), we have Y (t) = R (t). Furthermore, ⎧ ⎨ dR(t) = (Q (t)R(t) + L)( – q ), dt
⎩ R(t ) = a
p–q p
p
()
(T).
Then we get t q Q (s) ds R(t) = exp – p t t s p–q p–q q p Q (τ ) dτ ds . exp – × a (T) + L – p t p t
()
Then t Q (s) ds Y (t) = a (T) exp t
p p s p–q t q p–q q × +L – a (T) Q (τ ) dτ ds exp – . – p p t t
() p
By comparison of the differential inequality, we have Y (t) ≤ Y (t). Moreover, V q (t) ≤ q p
Y (t) implies V (t) ≤ Y (t), and this inequality is equivalent to t pq Q (s) ds V (t) ≤ a (T) exp
t q p s p–q t q p–q q a (T) Q (τ ) dτ ds × +L – exp – . – p p t t
()
Letting t = T, where T is a positive constant chosen arbitrarily, we get V (T) ≤ a (T) exp
pq
T
Q (s) ds
t q p s p–q T q p–q q a (T) Q (τ ) dτ ds × +L – exp – . – p p t t
()
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Obviously,
t pq V (t) ≤ a (t) exp Q (s) ds t
q p s p–q t q q p–q – a (t) Q (τ ) dτ ds × +L – exp – p p t t
= τ (t).
()
For all t ∈ I , we can obtain the following estimation by () and (): p p – q i= N
p
V q (t) ≤ a (t) +
+
L
t
s
cj (θ )V wj (θ ) dθ ds + β V (t – )
t
p p – q i= N
≤ a (t) +
gi (s)V φi (s) ds
t
t
bj (s) t
j=
t
L gi (s)τ φi (s) ds +
t
j=
p + β τ (t ) + p – q i= N
+
L
t t
j=
+
p p – q i=
L j=
t
t
s
bj (s) t
cj (θ )τ wj (θ ) dθ ds
t
gi (s)V φi (s) ds
t
cj (θ )V wj (θ ) dθ ds
t N
= a (t) +
s
bj (s)
t
gi (s)V φi (s) ds
t s
bj (s) t
t
cj (θ )V wj (θ ) dθ ds.
()
t
Since it has the same style as (), we can use the same ways to obtain the estimation as (). Therefore t pq V (t) ≤ a (t) exp Q (s) ds t
q p s p–q t q q p–q – a (t) Q (τ ) dτ ds × +L – exp – . p p t t
()
Let τ (t) denote the function of the right-hand side of (), which is a positive and nondecreasing function on I . Using mathematical induction, ∀k ∈ Z, when ∀t ∈ Ik , the estimation is obtained. We have V (t) ≤ ak– (t) exp
pq
t
Q (s) ds
tk–
q p s p–q t q q p–q – a (t) Q (τ ) dτ ds × +L – exp – . p k– p tk– tk–
This completes the proof.
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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We consider the inequality () now. Theorem . Suppose φi (t), wj (t), a (t), p, q are defined as those in Theorem .. gi (t, s), bj (t, s), cj (t, s) are nondecreasing functions with their two variables. q (t), q (t) are continuous and nondecreasing functions on [t , ∞) and positive on [t , ∞) and u(t) is a piecewise continuous nonnegative function on [t , ∞) with only the first discontinuous points ti , i = , , . . . , and satisfying (). Then, for all t ∈ Ik , q–
u(t) ≤ Rk (t),
()
where Rk (t) = ak– (t)q(t) exp
t
˜ (s) + B˜ (s) ds Q
pq
tk– q s p–q t q q ˜ (τ ) + B˜ (τ ) dτ ds × + – exp , – Q p tk– tk– p k tk Ri (φi (s)) ak– (t) = a (t)q(t) + ds gi (t, s) a (s) i= tk–
+
L
N i=
B˜ (t) =
L j=
tk–
j=
˜ (t) = Q
tk
bj (t, s) a (s)
k Ri (ti – ) , cj (s, θ )Rj wj (θ ) dθ ds + βi a(ti – ) tk– i=
s
()
q
[q(φi (t))a (φi (t))] p gi (t, s) , a (t) q
[q(wj (t))a (wj (t))] p . bj (t, s)cj (t, s) a (t)
Proof Let v = uq , so the inequality () is equivalent to p q
v (t) ≤ a (t) + q (t)
N
L j=
t
s
bj (t, s)
t
gi (t, s)v φi (s) ds
t
i=
+
t
cj (s, θ )v wj (θ ) dθ ds + q (t) βi v(ti – ).
t
()
t
p
Note that w(t) = v q (t), then from (), we get w(t) ≤ + q (t) a (t) i= N
+
t
gi (t, s) t
v(φi (s)) ds a (s)
s L t bj (t, s) cj (s, θ )v wj (θ ) dθ ds a (s) t j= t
+ q (t)
t
βi
v(ti – ) . a (ti – )
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Moreover, with the assumption that q(t) = max{q (t), q (t)} + , we see N t w(t) v(φi (s)) ≤ q(t) + ds gi (t, s) a (t) a (s) i= t
s L t bj (t, s) v(ti – ) . cj (s, θ )v wj (θ ) dθ ds + βi + a (s) a (ti – ) t t
()
i
Let N
(t) = +
i=
+
gi (t, s)
t
L t t
j=
t
bj (t, s) a (s)
v(φi (s)) ds a (s)
s
t
v(ti – ) . cj (s, θ )v wj (θ ) dθ ds + βi a (ti – ) t
()
i
Then (t) is a positive and nondecreasing function on I with (t ) = and w(t) ≤ q(t) (t), w(t) ≤ q(t) (t)a (t), a (t)
()
q
so v(t) ≤ (q(t) (t)a (t)) p . Applying () to (), we obtain (t) ≤ +
N i=
+
t
t
q
[q(φi (s))a (φi (s))] p pq gi (t, s) (s) ds a (s)
s L t q q bj (t, s) cj (s, θ ) q wj (θ ) a wj (θ ) p p (θ ) dθ ds a (s) t j= t q
[q(ti – )a (ti – ) (ti – )] p + . βi a (ti – ) t
()
i
Let Ii = [ti– , ti ), first, we consider the condition under which, for all t in [t , t ), we have (t) ≤ +
N i=
q
t
gi (t, s)
t
[q(φi (s))a (φi (s))] p pq (s) ds a (s)
s L t pq q bj (t, s) p cj (s, θ ) q wj (θ ) a wj (θ ) (θ ) dθ ds. + a (s) t j= t
()
For all t ∈ [t , T), where T ∈ I , we get (t) ≤ +
N i=
+
t
t
q
[q(φi (s))a (φi (s))] p pq gi (T, s) (s) ds a (s)
s L t q q bj (T, s) cj (s, θ ) q wj (θ ) a wj (θ ) p p (θ ) dθ ds. a (s) t j= t
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Let (t) denote the function on the right-hand side of (), which is a positive and nondecreasing function on I with (t ) = , and (t) ≤ (t). For all t ∈ [t , T), differentiating (t), q
[q(φi (t))a (φi (t))] p pq d (t) gi (T, t) ≤ (t) dt a (t) i= N
+
L bj (T, t) j=
a (t)
t
q q cj (s, θ ) q wj (θ ) a wj (θ ) p p (θ ) dθ .
()
t
Let Q(t) =
N
q
gi (T, t)
i=
B(t) =
L j=
[q(φi (t))a (φi (t))] p , a (t) q
[q(wj (t))a (wj (t))] p . bj (T, t)cj (T, t) a (t)
Since cj , q, a are nondecreasing functions, we can estimate () further to obtain q d (t) ≤ Q(t) p (t) + B(t) dt
t
q
p (θ ) dθ .
()
t
Moreover, we can get t q pq d (t) p ≤ Q(t) + B(t) (t) + (θ ) dθ dt t t q p (θ ) dθ . ≤ Q(t) + B(t) (t) +
()
t
t q Let (t) = (t) + t p (θ ) dθ . We see that (t) satisfies (t) ≤ (t), and differentiating (t), we can obtain q d (t) d (t) = + p (t) dt dt q ≤ Q(t) + B(t) (t) + p (t).
()
Consider
q p
d (t) dt
= (Q(t) + B(t)) (t) + (t), (t ) = .
()
Since () is a Bernoulli equation, we compute it to obtain t (t) = exp Q(s) + B(s) ds t
p t p–q s q q × + – Q(τ ) + B(τ ) dτ ds exp – – . p t p t
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Then by comparison of the differential inequality, we have (t) ≤ (t). Therefore, pq t v(t) ≤ a (t)q(t) exp Q(s) + B(s) ds t
q p
× + –
q s p–q q exp – – . Q(τ ) + B(τ ) dτ ds p t
t t
()
˜ (t) and B˜ (t), we By taking t = T in the above inequality, and noticing the definitions of Q get t pq ˜ (s) + B˜ (s) ds v(t) ≤ a (t)q(t) exp Q t q t p–q t q ˜ q exp × + – – . – Q (s) + B˜ (s) ds p p t t
()
Let R (t) denote the function on the right-hand side of (). When t ∈ I , we obtain p q
v (t) ≤ a (t)q(t) +
N
+
j=
t
t
+ a (t)q(t)
bj (t, s) a (s)
s t
N i=
gi (t, s)
t
i= L
t
R (φi (s)) ds a (s)
k Ri (ti – ) cj (s, θ )R wj (θ ) dθ ds + βi a(ti – ) i=
v( φi (s)) gi (t, s) ds a (s)
t
t
s L t bj (t, s) + cj (s, θ )v wj (θ ) dθ ds . a (s) t j= t
()
Let a (t) = a (t)q(t) +
N i=
t
gi (t, s) t
R (φi (s)) ds a (s)
s L t bj (t, s) cj (s, θ )R wj (θ ) dθ ds + a (s) t j= t +
k i=
Ri (ti – ) . βi a(ti – )
()
Obviously, a (t) ≥ a (t). Since q(t) ≥ , we can go further to obtain p q
v (t) ≤ a (t)q(t) +
N i=
t
t
v( φi (s)) gi (t, s) ds a (s)
s L t bj (t, s) cj (s, θ )v wj (θ ) dθ ds . + a (s) t j= t
()
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Since () has the same style as (), we can use the same solution to deal with it, finally the estimation of the unknown function in the inequality () is obtained. We have v(t) ≤ ak– (t)q(t) exp
t
˜ (s) + B˜ (s) ds Q
q p
tk–
q × + – p
q p–q s q ˜ ˜ exp – – . Q (τ ) + B (τ ) dτ ds p tk– tk–
t
()
Let Rk (t) denote the function on the right-hand side, and ak– (t) = a (t)q(t) +
k
tk
gi (t, s) tk–
i=
Ri (φi (s)) ds a (s)
s L tk k bj (t, s) Ri (ti – ) + cj (s, θ )Rj wj (θ ) dθ ds + βi . a (s) a(ti – ) tk– j= tk– i=
()
So we obtain q–
u(t) ≤ Rk (t).
This proves Theorem ..
3 Applications In this section we will apply our Theorem . and Theorem . to discuss the following differential-integral equation and retarded differential equation for discontinuous functions, respectively. We present the following propositions. Proposition . Consider the following equation: t t dx(t) = H(t, x(φ (t)), . . . , x(φN (t)), K(s, x(w (s))) ds, . . . , K(s, x(wL (s))) ds), dt x() = x , ∀t ∈ I = [, ∞),
()
where the function K is in C(R × R, R+ ) and φi (t) ≤ t, wj (t) ≤ t, for t > , H satisfies the following condition: t t H t, u , u , . . . , uN , K(s, v ) ds, . . . , K(s, vL ) ds
≤
N
gi (t)ui +
i=
L
bj (t)
t
cj (θ )vj dθ ,
()
j=
where gi (t), bj (t), cj (t), wj (t) are defined as in Theorem .. If
t
Q (s) ds < ∞,
t
t
s Y (s) exp – Q (τ ) dτ ds < ∞.
t
t
Then all the solutions of equation () exist on I and for all t in I = [, ∞), and they satisfy the following estimate: t t s x(t) ≤ x + exp Q (s) ds Y (s) exp – Q (τ ) dτ ds . () t
t
t
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
Page 13 of 14
Proof Integrating both sides of equation () from to t, we get
t
x(t) = x +
s H s, x φ (s) , . . . , x φN (s) , K τ , x w (τ ) dτ , . . . ,
s
K τ , x wL (τ ) dτ ds.
()
Using the conditions () and (), we can obtain N x(t) ≤ x +
t
L gi (s)x φi (s) ds +
t
i=
j=
t
s
bj (s) t
cj (θ )x wj (θ ) dθ ds.
()
t
Applying Theorem . to (), we can obtain the estimate.
Proposition . Consider the impulsive differential system ⎧ dxp (t) t t ⎪ ⎨ dt = H(t, x(φ (t)), . . . , x(φN (t)), K(s, x(w (s))) ds, . . . , K(s, x(wL (s))) ds), (x)|t=t = βi xq (ti – ), ⎪ ⎩ x() = c, ∀t ∈ I = [, ∞),
()
where φi (t), wj (t) are defined as in Theorem . and the function K is in C(R × R, R+ ). Furthermore, H satisfies t t H t, x φ (t) , . . . , x φN (t) , K s, x w (s) ds, . . . , K s, x w (s) ds L
≤
N
gi (t)xq φi (t) +
i=
L
s
bj (s)
cj (t)xq wj (t) dt.
()
j=
Then all the solutions of equation () exist on I and satisfy |x(t)| ≤ τk (t) for all t ∈ Ik , where τk (t) is defined as in Theorem .. Proof Integrating () we obtain p x (t) ≤ cp +
s H s, x φ (s) , . . . , x φN (s) , K τ , x w (τ ) dτ , . . . ,
t
s
+
K τ , x wL (τ ) dτ ds
βi uq (ti – ),
∀t ∈ I.
()
t
Furthermore, we get N p x (t) ≤ cp +
L j=
gi (s)xq φi (s) ds
i=
+
t
t
bj (s)
s
cj (θ )xq wj (θ ) dθ ds + βi uq (ti – ).
Then we use Theorem . to obtain the estimation.
()
t
Zheng et al. Journal of Inequalities and Applications (2016) 2016:7
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Competing interests The authors declare that they have no competing interests. Authors’ contributions ZZ came up with the main ideas and helped to draft the manuscript. XG proved the main theorems. JS revised the paper. All authors read and approved the final manuscript. Author details 1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P.R. China. 2 Department of Mathematics, Jining University, Qufu 273155, P.R. China. Acknowledgements The authors sincerely thank the referees for their constructive suggestions and corrections. This project is supported by the NNSF of China (Grants 11171178 and 11271225), NSF of Shandong Province (ZR2015PA005). Received: 30 June 2015 Accepted: 13 December 2015 References 1. Lipovan, O: A retarded Gronwall-like inequality and its applications. J. Math. Anal. Appl. 252, 389-401 (2000). doi:10.1006/jmaa.2000.7085 2. Agarwal, RP, Deng, S, Zhang, W: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 165, 599-612 (2005). doi:10.1016/j.amc.2004.04.067 3. Borysenko, DS: About one integral inequality for piece-wise continuous functions. In: Proceedings of 10th International Kravchuk Conference, Kyiv, p. 323 (2004) 4. Iovane, G: Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions. Nonlinear Anal. 66(2), 498-508 (2007) 5. Wang, W-S, Li, Z: A new nonlinear retarded integral inequality and its application. J. Sichuan Normal Univ. Nat. Sci. 35, 180-183 (2012) (in Chinese) 6. Yan, Y: Some new Gronwall-Bellman type impulsive integral inequality and its application. J. Sichuan Normal Univ. Nat. Sci. 36, 603-608 (2013) (in Chinese) 7. Gronwall, TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 292-296 (1919). doi:10.2307/1967124 8. Bellman, R: The stability of solutions of linear differential equations. Duke Math. J. 10, 643-647 (1943). doi:10.1215/S0012-7094-43-01059-2 9. Wang, W: Some new generalized retarded nonlinear integral inequalities with iterated integrals and its applications. J. Inequal. Appl. 2012, 31 (2012) 10. Abdeldaim, A, Yakout, M: On some new integral inequalities of Gronwall-Bellman-Pachpatte type. Appl. Math. Comput. 217, 7887-7889 (2011). doi:10.1016/j.amc.2011.02.093 11. Agarwal, RP, Kim, YH, Sen, SK: New retarded integral inequalities with application. J. Inequal. Appl. 2008, Article ID 908784 (2008) 12. Liu, C, Li, Z: A new class of integral inequality for discontinuous function and its application. Pure Math. 3, 4-8 (2013) (in Chinese)
