# Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations

- Qinghua Feng
^{1, 2}Email author, - Fanwei Meng
^{1}, - Yaoming Zhang
^{2}, - Bin Zheng
^{2}and - Jinchuan Zhou
^{2}

**2011**:29

https://doi.org/10.1186/1029-242X-2011-29

© Feng et al; licensee Springer. 2011

**Received: **16 February 2011

**Accepted: **5 August 2011

**Published: **5 August 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**, 349-358 (2006)], Pachpatte [J. Math. Anal. Appl. **251**, 736-751 (2000)], Li [Comput. Math. Appl. **59**, 1929-1936 (2010)], and Sun [J. Math. Anal. Appl. **301**, 265-275 (2005)].

**MSC 2010**: 26E70; 26D15; 26D10.

## Keywords

## 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 left-dense if *ρ*(*t*) = *t* and *t ≠* inf **T**, right-dense if *σ*(*t*) = *t* and *t ≠* sup **T**, left-scattered if *ρ*(*t*) *< t* and right-scattered if *σ*(*t*) > *t*.

**Definition 1.2**: The set **T** ^{
κ
} is defined to be **T** if **T** does not have a left-scattered maximum, otherwise it is **T** without the left-scattered maximum.

**Definition 1.3**: A function *f* ∈ (**T**, **R**) is called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at left-dense 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 rd-continuous functions, while
denotes the set of all regressive and rd-continuous 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)

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
}.

**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

*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

where ξ lies between *v*(*s*) and *v*(*t*). So we always have
.

*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).

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

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

*ω*is nondecreasing, subadditive, and submultiplicative, Combining (16), (18), and Lemma 2.1, we obtain

where *A*(*t*) is defined in (15).

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)].

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

*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

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.

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

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.

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.

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 2.3-2.4, or take **T** = **Z** in Theorem 2.3-2.5, 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.

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

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 *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.

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 2.1-2.5 can be applied to establish delay integral inequalities with two independent variables on time scales.

## Declarations

### 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.

## Authors’ Affiliations

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