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Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations
Journal of Inequalities and Applications volume 2011, Article number: 29 (2011)
Abstract
In this paper, some new nonlinear delay integral inequalities on time scales are established, which provide a handy tool in the research of boundedness of unknown functions in delay dynamic equations on time scales. The established results generalize some of the results in Lipovan [J. Math. Anal. Appl. 322, 349358 (2006)], Pachpatte [J. Math. Anal. Appl. 251, 736751 (2000)], Li [Comput. Math. Appl. 59, 19291936 (2010)], and Sun [J. Math. Anal. Appl. 301, 265275 (2005)].
MSC 2010: 26E70; 26D15; 26D10.
1 Introduction
In the 1980s, Hilger initiated the concept of time scales [1], which is used as a theory capable to contain both difference and differential calculus in a consistent way. Since then, many authors have expounded on various aspects of the theory of dynamic equations on time scales. For example [2–10], and the references therein. In these investigations, integral inequalities on time scales have been paid much attention by many authors, and a lot of integral inequalities on time scales have been established (see [5–10] and the references therein), which are designed to unify continuous and discrete analysis, and play an important role in the research of boundedness, uniqueness, stability of solutions of dynamic equations on time scales. But to our knowledge, delay integral inequalities on time scales have been paid little attention so far in the literature. Recent results in this direction include the works of Li [11] and Ma [12].
Our aim in this paper is to establish some new nonlinear delay integral inequalities on time scales, which are generalizations of some known continuous inequalities and discrete inequalities in the literature. Also, we will present some applications for the established results, in which we will use the present inequalities to derive new bounds for unknown functions in certain delay dynamic equations on time scales.
At first, we will give some preliminaries on time scales and some universal symbols for further use. Throughout this paper, R denotes the set of real numbers and R_{+} = [0, ∞), while Z denotes the set of integers. For two given sets G, H, we denote the set of maps from G to H by (G, H).
A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, T denotes an arbitrary time scale. On T, we define the forward and backward jump operators σ ∈ (T, T), and ρ ∈ (T, T) such that σ(t) = inf{s ∈ T, s > t}, ρ(t) = sup{s ∈ T, s < t}.
Definition 1.1: A point t ∈ T is said to be leftdense if ρ(t) = t and t ≠ inf T, rightdense if σ(t) = t and t ≠ sup T, leftscattered if ρ(t) < t and rightscattered if σ(t) > t.
Definition 1.2: The set T ^{κ} is defined to be T if T does not have a leftscattered maximum, otherwise it is T without the leftscattered maximum.
Definition 1.3: A function f ∈ (T, R) is called rdcontinuous if it is continuous at rightdense points and if the leftsided limits exist at leftdense points, while f is called regressive if 1 + μ(t)f(t) ≠ 0, where μ(t) = σ(t)  t. C_{ rd } denotes the set of rdcontinuous functions, while denotes the set of all regressive and rdcontinuous functions, and .
Definition 1.4: For some t ∈ T ^{κ}, and a function f ∈ (T, R), the delta derivative of f at t is denoted by f^{Δ}(t) (provided it exists) with the property such that for every ε > 0, there exists a neighborhood of t satisfying
Remark 1.1: If T = R, then f^{Δ}(t) becomes the usual derivative f'(t), while f^{Δ}(t) = f(t + 1)  f(t) if T = Z, which represents the forward difference.
Definition 1.5: If F^{Δ}(t) = f(t), t ∈ T ^{κ}, then F is called an antiderivative of f, and the Cauchy integral of f is defined by
The following two theorem include some important properties for delta derivative on time scales.
Theorem 1.1 [[13], Theorem 2.2]: If a, b, c ∈ T, α ∈ R, and f, g ∈ C_{ rd }, then

(i)
,

(ii)
,

(iii)
,

(iv)
,

(v)
,

(vi)
if f(t) ≥ 0 for all a ≤ t ≤ b, then .
For more details about the calculus of time scales, we advise to refer to [14].
2 Main results
In the rest of this paper, for the sake of convenience, we denote T_{0} = [t_{0}, ∞) ∩T, and always assume T_{0} ⊂ T ^{κ}.
Lemma 2.1 [15]: Assume that a ≥ 0, p ≥ q ≥ 0, and p ≠ 0, then for any K > 0
Lemma 2.2: Suppose u, a ∈ C_{ rd }, , m ≥ 0, and a is nondecreasing. Then,
implies
where e_{ m }(t, t_{0}) is the unique solution of the following equation
Proof: From [[16], Theorem 5.6], we have , t ∈ T_{0}. Since a(t) is nondecreasing on T_{0}, then . On the other hand, from [[14], Theorem 2.39 and 2.36 (i)], we have . Combining the above information, we can obtain the desired inequality.
Theorem 2.1: Suppose u, a, b, f ∈ C_{ rd }(T_{0}, R_{+}), and a, b are nondecreasing. ω ∈ C(R_{+}, R_{+}) is nondecreasing. τ ∈ (T_{0}, T), τ (t) ≤ t, ∞ < α = inf{τ(t), t ∈ T_{0}} ≤ t_{0}, ϕ ∈ C_{ rd }([α, t_{0}] ∩T, R_{+}). p > 0 is a constant. If u(t) satisfies, the following integral inequality
with the initial condition
then
where G is an increasing bijective function, and
Proof: Let T ∈ T_{0} be fixed, and
Then considering a, b are nondecreasing, we have
Furthermore, for t ∈ [t_{0}, T ] ∩T, if τ(t) ≥ t_{0}, considering τ (t) ≤ t, then τ_{ i }(t) ∈ [t_{0}, T ] ∩T, and from (6) we obtain
If τ(t) ≤ t_{0}, from (2) we obtain
So from (7) and (8), we always have
Moreover,
that is,
On the other hand, for t ∈ [t_{0}, T ] ∩T, if σ(t) > t, then
If σ(t) = t, then
where ξ lies between v(s) and v(t). So we always have .
Using the statements above, we deduce that
Replacing t with s in the inequality above, and an integration with respect to s from t_{0} to t yields
where G is defined in (4).
Considering G is increasing, and v(t_{0}) = a(T ), it follows that
Combining (6) and (12), we get
Taking t = T in (12), yields
Since T ∈ T_{0} is selected arbitrarily, then substituting T with t in (13) yields the desired inequality (3).
Remark 2.1: Since T is an arbitrary time scale, then if we take T for some peculiar cases in Theorem 2.1, then we can obtain some corollaries immediately. Especially, if T = R, t_{0} = 0, then Theorem 2.1 reduces to [[17], Theorem 2.2], which is the continuous result. However, if we take T = Z, we obtain the discrete result, which is given in the following corollary.
Corollary 2.1: Suppose T = Z, n_{0} ∈ Z, and Z_{0} = [n_{0}, ∞) ∩ Z. u, a, b, f ∈ (Z_{0}, R_{+}), and a, b are decreasing on Z_{0}. τ ∈ (Z_{0}, Z), τ (n) ≤ n, ∞ < α = inf{τ(n), n ∈ Z_{0}} ≤ n_{0}, ϕ ∈ C_{ rd }([α, n_{0}] ∩ Z, R_{+}). ω is defined the same as in Theorem 2.1. If for n ∈ Z_{0}, u(n) satisfies
with the initial condition
then
In Theorem 2.1, if we change the conditions for a, b, ω p; then, we can obtain another bound for the function u(t).
Theorem 2.2: Suppose u, a, b, f ∈ C_{ rd }(T_{0}, R_{+}), ω ∈ C(R_{+}, R_{+}) is nondecreasing, subadditive, and submultiplicative, that is, for ∀α ≥ 0, β ≥ 0 we always have ω(α + β) ≤ ω(α) + ω (β) and ω(αβ) ≤ ω(α)ω(β). τ, α, ϕ are the same as in Theorem 2.1. If u(t) satisfies the inequality (1) with the initial condition (2), then for ∀K > 0, we have
where is an increasing bijective function, and
Proof: Let
Then,
Similar to the process of (7)(9), we have
Considering ω is nondecreasing, subadditive, and submultiplicative, Combining (16), (18), and Lemma 2.1, we obtain
where A(t) is defined in (15).
Let T be fixed in T_{0}, and t ∈ [t_{0}, T] ⋂ T. Denote
Considering A(t) is nondecreasing, then we have
Furthermore,
Similar to Theorem 2.1, we have
Substituting t with s in (22), and an integration with respect to s from t_{0} to t yields
which is followed by
Combining (17), (21), and (23), we obtain
Taking t = T in (24), yields
Since T is selected from T_{0} arbitrarily, then substituting T with t in (25), we can obtain the desired inequality (14).
Remark 2.2: Theorem 2.2 unifies some known results in the literature. If we take T = R, t_{0} = 0, τ(t) = t, K = 1, then Theorem 2.2 reduces to [[18], Theorem 2(b3)], which is one case of continuous inequality. If we take T = Z, t_{0} = 0, τ(t) = t, K = 1, then Theorem 2.2 reduces to [[18], Theorem 4(d3)], which is the discrete analysis of [[18], Theorem 2(b3)].
Now we present a more general result than Theorem 2.1. We study the following delay integral inequality on time scales.
where u, a, b, f, g, h ∈ Crd(T_{0}, R_{+}), ω ∈ C(R_{+}, R_{+}), and a, b, ω are nondecreasing, η ∈ C(R_{+}, R_{+}) is increasing, τ_{ i } ∈ (T_{0}, T) with τ_{ i }(t) ≤ t, i = 1, 2, and ∞ < α = inf{min{τ_{ i }(t), i = 1, 2}, t ∈ T_{0}} ≤ t_{0}.
Theorem 2.3: Define a bijective function such that , with . If is increasing, and for t ∈ T_{0}, u(t) satisfies the inequality (26) with the initial condition
where ϕ ∈ C_{ rd }([α, t_{0}] ⋂ T, R_{+}), then
Proof: Let the right side of (26) be v(t), then
For t ∈ T_{0}, if τ_{ i }(t) ≥ t_{0}, considering τ_{ i }(t) ≤ t, then τ_{ i }(t) ∈ T_{0}, and from (29), we have
If τ_{ i }(t) ≤ t_{0}, from (27), we obtain
So from (30) and (31), we always have
Furthermore, considering η is increasing, we get that
Fix a T ∈ T_{0}, and let t ∈ [t_{0}, T] ⋂ T. Define
Since a, b are nondecreasing on T_{0}, it follows that
On the other hand,
Similar to Theorem 2.1, we have
Replacing t with s, and an integration for (36) with respect to s from t_{0} to t yields
Since c(t_{0}) = a(T), and G is increasing, it follows that
Combining (29), (35), (38), we have
Taking t = T in (39), yields
Since T ∈ T_{0} is selected arbitrarily, then substituting T with t in (40) yields the desired inequality (28).
Remark 2.3: If we take η(u) = u^{p}, g(t) ≡ 0, then Theorem 2.3 reduces to Theorem 2.1.
Next, we consider the delay integral inequality of the following form.
where u, f, g, h, a, τ_{ i }, i = 1, 2 are the same as in Theorem 2.3, m ∈ C(R_{+}, R_{+}), p > 0 is a constant, ω ∈ C(R_{+}, R_{+}) is nondecreasing, and ω is submultitative, that is, ω(αβ) ≤ ω(α)ω(β) holds for ∀α ≥ 0, β ≥ 0.
Theorem 2.4: Suppose G ∈ (R_{+}, R) is an increasing bijective function defined as in Theorem 2.1. If u(t) satisfies, the inequality (41) with the initial condition
then
Proof: Let the right side of (41) be v(t). Then,
and similar to the process of (30)(32) we have
Furthermore,
A suitable application of Lemma 2.2 to (46) yields
Fix a T ∈ T_{0}, and let t ∈ [t_{0}, T] ⋂ T. Define
Then,
and
Similar to Theorem 2.1, we have
An integration for (50) from t_{0} to t yields
Considering G is increasing and , it follows
Combining (44), (49), and (51), we have
Taking t = T in (52), yields
Since T ∈ T_{0} is selected arbitrarily, after substituting T with t in (53), we obtain the desired inequality (43).
Remark 2.4: If we take ω(u) = u, τ_{1}(t) = t, h(t) ≡ 0, then Theorem 2.4 reduces to [[11], Theorem 3]. If we take m(t) = f(t) = h(t) ≡ 0, then Theorem 2.4 reduces to Theorem 2.1 with slight difference.
Finally, we consider the following integral inequality on time scales.
where u, f, g, ω, τ_{1}, τ_{2} are the same as in Theorem 2.3, p, q, C are constants, and p > q > 0, C > 0.
Theorem 2.5: If u(t) satisfies (54) with the initial condition (42), then
where , H are two increasing bijective functions, and
Proof: Let the right side of (54) be v(t). Then,
and similar to the process of (30)(32) we have
Furthermore,
Similar to Theorem 2.1, we have
An integration for (59) from t_{0} to t yields
Considering is increasing, and v(t_{0}) = C, then (60) implies
Given a fixed number T in T_{0}, and t ∈ [t_{0}, T]. Let
Then,
and furthermore,
that is,
Integrating (64) from t_{0} to t yields
Since H is increasing, and , then (65) implies
Combining (57), (63), and (66), we obtain
Taking t = T in (67), and since T is an arbitrary number in T_{0}, then the desired inequality can be obtained after substituting T with t.
Remark 2.5: If we take T = R, τ_{1}(t) = τ_{2}(t), then we can obtain a new bound of for the unknown continuous function u(t), which is different from the result using the method in [[19], Theorem 2.1].
Remark 2.6: If we take T = R in Theorem 2.32.4, or take T = Z in Theorem 2.32.5, then immediately we obtain a number of corollaries on continuous and discrete analysis, which are omitted here.
3 Applications
In this section, we will present some applications for the established results above. Some new bounds for solutions of certain dynamic equations on time scales will be derived in the following examples.
Example 1: Consider the delay dynamic integral equation on time scales
with the initial condition
where u ∈ C_{ rd }(T_{0}, R), C = u^{p}(t_{0}), p is a positive number with p ≥ 1, τ, α are defined as in Theorem 2.1, ϕ ∈ C_{ rd }([α, t_{0}] ⋂ T, R).
Theorem 3.1 Suppose, u(t) is a solution of (68) and assumes F(t, u) ≤ f(t)u, where f ∈ C_{ rd }(T_{0}, R_{+}), then we have
where
Proof: From (68), we obtain
Let ω ∈ C(R_{+}, R_{+}), and ω(v) = v. Then, (72) can be rewritten as
A suitable application of Theorem 2.1 to (73) yields the desired inequality.
Remark 3.1: In the proof for Theorem 3.1, if we apply Theorem 2.2 instead of Theorem 2.1 to (73), then we obtain another bound for u(t) as follows.
where K > 0 ia an arbitrary constant, and
Example 2: Consider the following delay dynamic differential equation on time scales
with the initial condition
where u ∈ C_{ rd }(T_{0}, R), C = u^{p}(t_{0}), p is a positive number with p ≥ 1, α, τ_{ i }, i = 1, 2 are defined as in Theorem 2.3, ϕ ∈ C_{ rd }([α, t_{0}] ⋂ T, R).
Theorem 3.2: Suppose u(t) is a solution of (76), and assume F(t, u, v) ≤ f(t)u + v, M(t, u) ≤ h(t)u, where f, h ∈ C_{ rd }(T_{0}, R_{+}), then have
where G is defined as in Theorem 3.1.
Proof: The equivalent integral form of (75)(76) can be denoted by
Then,
where ω ∈ C (R_{+}, R_{+}), and ω(u) = u.
A suitable application of Theorem 2.3 to (79) yields the desired inequality.
4 Conclusions
In this paper, some new integral inequalities on time scales have been established. As one can see through the present examples, the established results are useful in dealing with the boundedness of solutions of certain delay dynamic equations on time scales. Finally, we note that the process of Theorem 2.12.5 can be applied to establish delay integral inequalities with two independent variables on time scales.
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Acknowledgements
This work is supported by National Natural Science Foundation of China (11026047 and 10571110), Natural Science Foundation of Shandong Province (ZR2009AM011, ZR2010AQ026, and ZR2010AZ003) (China) and Specialized Research Fund for the Doctoral Program of Higher Education (20103705 110003)(China). The authors thank the referees very much for their careful comments and valuable suggestions on this paper.
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The authors declare that they have no competing interests.
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QF carried out the main part of this article. All authors read and approved the final manuscript.
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Feng, Q., Meng, F., Zhang, Y. et al. Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J Inequal Appl 2011, 29 (2011). https://doi.org/10.1186/1029242X201129
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Keywords
 delay integral inequality
 time scales
 dynamic equation
 bound