Open Access

Nonlinear integral inequalities on time scales with ‘maxima’

  • Phollakrit Thiramanus1,
  • Jessada Tariboon1Email author and
  • Sotiris K Ntouyas2
Journal of Inequalities and Applications20142014:255

https://doi.org/10.1186/1029-242X-2014-255

Received: 26 November 2013

Accepted: 13 June 2014

Published: 22 July 2014

Abstract

In this paper, some new types of nonlinear integral inequalities on time scales with ‘maxima’, which provide explicit bounds on unknown functions, are established. The importance of these integral inequalities is given by their wide applications in qualitative investigations of differential equations with ‘maxima’. An example is also presented to illustrate our results.

MSC:34A40, 26D15, 39A13.

Keywords

nonlinear integral inequalitytime scalesmaximadynamic equationdifferential equations with ‘maxima’

1 Introduction

The theory of time scales (closed subsets of ) was created by Hilger [1] in order to unify continuous and discrete analysis and in order to extend those theories to other kinds of the so-called dynamic equations. Many authors have expounded on various aspects of the theory of dynamic equations on time scales. We refer the reader to the monograph [2] and the references cited therein. Also, a few papers studied the theory of dynamic inequalities on time scales; see, for example, [317].

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 maxima of the unknown scalar function is involved in the integral; see [1821] and the references cited therein.

Recently in [22] we initiated the study of integral inequalities on time scales with ‘maxima’, where some new integral inequalities were established. The significance of our work in [22] lies in the fact that ‘maxima’ are taken on intervals [ β t , t ] which have non-constant length, where 0 < β < 1 . Most papers take the ‘maxima’ on [ t h , t ] , where h > 0 is a given constant.

In this paper we continue the study of [22] and investigate some nonlinear dynamic integral inequalities on time scales with ‘maxima’. This paper is organized as follows. In Section 2 we give some preliminary results with respect to the calculus on time scales. In Section 3 we deal with our nonlinear dynamic inequalities on time scales with ‘maxima’. In Section 4 we give an example to illustrate our main results.

2 Preliminaries

In this section, we list the following well-known definitions and some lemmas which can be found in [2] and the references therein.

Definition 2.1 A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from .

The forward and backward jump operators σ , ρ : T T and the graininess μ : T R + are defined, respectively, by
σ ( t ) : = inf { s T | s > t } , ρ ( t ) : = sup { s T | s < t } , μ ( t ) : = σ ( t ) t

for all t T . If σ ( t ) > t , t is said to be right scattered, and if ρ ( t ) < t , t is said to be left scattered; if σ ( t ) = t , t is said to be right dense, and if ρ ( t ) = t , t is said to be left dense. If has a right-scattered minimum m, define T k = T { m } ; otherwise set T k = T . If has a left-scattered maximum M, define T k = T { M } ; otherwise set T k = T .

Definition 2.2 A function f : T R is rd-continuous (rd-continuous is short for right-dense continuous) provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions f : T R will be denoted by C rd ( T ) = C rd ( T , R ) .

Definition 2.3 For f : T R and t T k , the delta derivative of f at the point t is defined to be the number f ( t ) (provided it exists) with the property that for each ε > 0 , there is a neighborhood U of t such that
| f ( σ ( t ) ) f ( s ) f ( t ) ( σ ( t ) s ) | ε | σ ( t ) s |

for all s U .

Definition 2.4 For a function f : T R (the range of f may be actually replaced by a Banach space), the (delta) derivative is defined at point t by
f ( t ) = f ( σ ( t ) ) f ( t ) σ ( t ) t ,
if f is continuous at t and t is right scattered. If t is not right scattered, then the derivative is defined by
f ( t ) = lim s t f ( σ ( t ) ) f ( s ) σ ( t ) s = lim s t f ( t ) f ( s ) t s ,

provided this limit exists.

Definition 2.5 If F ( t ) = f ( t ) , then we define the delta integral by
a t f ( s ) Δ s = F ( t ) F ( a ) .

Lemma 2.1 ([2])

Assume that ν : T R is strictly increasing and T ˜ : = ν ( T ) is a time scale. If f : T R is an rd-continuous function and ν is differentiable with rd-continuous derivative, then for a , b T ,
a b f ( t ) ν ( t ) Δ t = ν ( a ) ν ( b ) ( f ν 1 ) ( s ) Δ ˜ s .

Lemma 2.2 ([23])

Assume that a 0 , p q 0 , and p 0 . Then
a q p ( q p k q p p a + p q p k q p ) for any  k > 0 .

3 Main results

For convenience of notation, we let throughout t 0 T , t 0 0 , T 0 = [ t 0 , ) T and an interval [ γ , η ] T = [ γ , η ] T . In addition, for a strictly increasing function α : T R , T ˜ = α ( T ) is a time scale such that T ˜ T . For f C rd ( T , R ) , we define a notation of the composition of two functions on time scales by
f ( γ ) α 1 ( s ) = f ( α 1 ( s ) ) , γ T , s T ˜ .
Example 3.1 Let f ( t ) = 5 t 2 for t T : N 0 1 2 = { n : n N 0 } and α ( t ) = t 2 for t T . Then we have α 1 ( t ) = t for t T ˜ = N 0 and
f ( γ ) α 1 ( s ) = ( 5 γ 2 ) s = 5 s , s T ˜ .
Theorem 3.1 Let the following conditions be satisfied:
  1. (i)

    The function α C rd ( T 0 , R + ) is strictly increasing.

     
  2. (ii)

    The functions a, b, p and q C rd ( T 0 , R + ) .

     
  3. (iii)

    The function ϕ C rd ( [ β τ , t 0 ] T , R + ) , where 0 < β < 1 and τ = min { t 0 , α ( t 0 ) } .

     
  4. (iv)

    The function h C ( R + , ( 0 , ) ) is increasing.

     
  5. (v)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    u ( t ) k + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s u ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] u ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.1)
     
u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T ,
(3.2)

where k 0 .

Then, for all t T 0 satisfying
H ( M ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H 1 ) ,
we have
u ( t ) H 1 ( H ( M ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) ,
(3.3)
where
M = max { k , max s [ β τ , t 0 ] T ϕ ( s ) }
(3.4)
and
H ( x ) = x 0 x 1 h ( r ) d r , x 0 > 0 ,
(3.5)

which H ( ) = , and H 1 is the inverse of H.

Proof We define a function z : [ β τ , ) T R + by
z ( t ) = { M + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , M , t [ β τ , t 0 ] T ,

where M is defined by (3.4). Note that the function z ( t ) is nondecreasing.

It follows that the inequality
u ( t ) z ( t ) , t [ β τ , ) T
holds. Therefore, for t T 0 and s [ t 0 , t ] T , we have
max ξ [ β s , s ] T u ( ξ ) max ξ [ β s , s ] T z ( ξ ) = z ( s ) .
For t T 0 and s [ α ( t 0 ) , α ( t ) ] T ˜ , we have
h ( max ξ [ β γ , γ ] T u ( ξ ) ) α 1 ( s ) h ( max ξ [ β γ , γ ] T z ( ξ ) ) α 1 ( s ) = h ( max ξ [ β α 1 ( s ) , α 1 ( s ) ] T z ( ξ ) ) = h ( z ( α 1 ( s ) ) ) = h ( z ( γ ) ) α 1 ( s ) .
Then, from the definition of z ( t ) and the above analysis, we get for t T 0 that
z ( t ) M + t 0 t [ p ( s ) h ( z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T z ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T z ( ξ ) ) ] α 1 ( s ) Δ ˜ s M + t 0 t [ p ( s ) h ( z ( s ) ) + q ( s ) h ( z ( s ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( z ( γ ) ) + b ( γ ) h ( z ( γ ) ) ] α 1 ( s ) Δ ˜ s = M + t 0 t [ p ( s ) h ( z ( s ) ) + q ( s ) h ( z ( s ) ) ] Δ s + t 0 t [ a ( s ) h ( z ( s ) ) + b ( s ) h ( z ( s ) ) ] α ( s ) Δ s = M + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] h ( z ( s ) ) Δ s .
(3.6)
From inequality (3.6) we have
z ( t ) [ p ( t ) + q ( t ) + a ( t ) α ( t ) + b ( t ) α ( t ) ] h ( z ( t ) ) ,
which implies
z ( t ) h ( z ( t ) ) p ( t ) + q ( t ) + a ( t ) α ( t ) + b ( t ) α ( t ) .
(3.7)
On the other hand, for t T 0 , if σ ( t ) > t , then
[ H ( z ( t ) ) ] = H ( z ( σ ( t ) ) ) H ( z ( t ) ) σ ( t ) t = 1 σ ( t ) t z ( t ) z ( σ ( t ) ) 1 h ( r ) d r z ( σ ( t ) ) z ( t ) σ ( t ) t 1 h ( z ( t ) ) = z ( t ) h ( z ( t ) ) .
(3.8)
If σ ( t ) = t , then
[ H ( z ( t ) ) ] = lim s t H ( z ( t ) ) H ( z ( s ) ) t s = lim s t 1 t s z ( s ) z ( t ) 1 h ( r ) d s = lim s t z ( t ) z ( s ) t s 1 h ( ω ) = z ( t ) h ( z ( t ) ) ,
(3.9)
where ω lies between z ( s ) and z ( t ) . Hence from (3.8) and (3.9) we have
[ H ( z ( t ) ) ] z ( t ) h ( z ( t ) ) .
(3.10)
Combining (3.7) and (3.10), we get
[ H ( z ( t ) ) ] p ( t ) + q ( t ) + a ( t ) α ( t ) + b ( t ) α ( t ) .
An integration for the above inequality with respect to t from t 0 to t yields
H ( z ( t ) ) H ( M ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s .
Since H 1 is an increasing function, we obtain
z ( t ) H 1 ( H ( M ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.3). This completes the proof. □

We introduce the following classes of functions in connection with the nonlinearity of the considered integral inequality.

Definition 3.1 ([24])

We will say that a function h C ( R + , R + ) is from class Φ if the following conditions are satisfied:
  1. (i)

    h is a nondecreasing function;

     
  2. (ii)

    h ( x ) > 0 for x > 0 ;

     
  3. (iii)

    h ( t x ) t h ( x ) for 0 t 1 , x 0 ;

     
  4. (iv)

    1 d x h ( x ) = .

     

Definition 3.2 ([24])

We will say that a function h C ( R + , R + ) is from class Ω if the following conditions are satisfied:
  1. (i)

    h is a nondecreasing function;

     
  2. (ii)

    h ( x ) > 0 for x > 0 ;

     
  3. (iii)

    h ( t x ) t h ( x ) for 0 t 1 , x 0 ;

     
  4. (iv)

    h ( x + y ) h ( x ) + h ( y ) for x , y 0 ;

     
  5. (v)

    1 d x h ( x ) = .

     

Note that the functions h ( x ) = x and h ( x ) = x are from class Ω.

In the case when in place of the constant k involved in Theorem 3.1 we have a function k ( t ) , we obtain the following result using functions from class Φ.

Theorem 3.2 Let the following conditions be satisfied:
  1. (i)

    The conditions (i)-(iii) of Theorem  3.1 are satisfied.

     
  2. (ii)

    The function h C ( R + , R + ) and h Φ .

     
  3. (iii)

    The function k C rd ( T 0 , [ 1 , ) ) is nondecreasing.

     
  4. (iv)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    u ( t ) k ( t ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s u ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] u ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.11)
    u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
    (3.12)
     
Then, for all t T 0 satisfying
H ( N ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H 1 ) ,
we have
u ( t ) k ( t ) H 1 ( H ( N ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) ,
(3.13)
where
N = max { 1 , max s [ β τ , t 0 ] T ϕ ( s ) k ( t 0 ) } ,
(3.14)

and H ( x ) is defined by (3.5).

Proof From inequality (3.11) we obtain for t T 0
u ( t ) k ( t ) 1 + t 0 t [ p ( s ) h ( u ( s ) ) k ( t ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) k ( t ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) k ( t ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) k ( t ) ] α 1 ( s ) Δ ˜ s .
(3.15)
Let us define functions k : [ β τ , ) T R + and w : [ β τ , ) T R + by
k ( t ) = { k ( t ) , t T 0 , k ( t 0 ) , t [ β τ , t 0 ] T , w ( t ) = u ( t ) k ( t ) , t [ β τ , ) T .

Note that the function k ( t ) is nondecreasing on t [ β τ , ) T .

By conditions (ii) and (iii) of Theorem 3.2, it follows that h ( u ( s ) ) k ( t ) h ( u ( s ) k ( t ) ) for t T 0 and s [ t 0 , t ] T . From the monotonicity of k ( t ) and α ( t ) , we get for t T 0 and s [ t 0 , t ] T that
max ξ [ β s , s ] T u ( ξ ) k ( t ) max ξ [ β s , s ] T u ( ξ ) k ( s ) = max ξ [ β s , s ] T u ( ξ ) k ( s ) max ξ [ β s , s ] T u ( ξ ) k ( ξ ) .
(3.16)
For t T 0 and s [ α ( t 0 ) , α ( t ) ] T ˜ , we have
max ξ [ β γ , γ ] T u ( ξ ) α 1 ( s ) k ( t ) = max ξ [ β α 1 ( s ) , α 1 ( s ) ] T u ( ξ ) k ( t ) max ξ [ β α 1 ( s ) , α 1 ( s ) ] T u ( ξ ) k ( α 1 ( s ) ) = max ξ [ β α 1 ( s ) , α 1 ( s ) ] T u ( ξ ) k ( α 1 ( s ) ) max ξ [ β α 1 ( s ) , α 1 ( s ) ] T u ( ξ ) k ( ξ ) = max ξ [ β γ , γ ] T u ( ξ ) k ( ξ ) α 1 ( s ) .
(3.17)
From inequalities (3.15), (3.16) and (3.17) and the definition of w ( t ) , we have
w ( t ) 1 + t 0 t [ p ( s ) h ( w ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T w ( ξ ) ) ] Δ s w ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( w ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T w ( ξ ) ) ] w ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.18)
w ( t ) ϕ ( t ) k ( t 0 ) , t [ β τ , t 0 ] T .
(3.19)
Using Theorem 3.1 for (3.18) and (3.19), we get
w ( t ) H 1 ( H ( N ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.13). This completes the proof. □

In the case when the function k ( t ) involved in the right part of inequality (3.11) is not a monotonic function, we obtain the following result.

Theorem 3.3 Let the following conditions be satisfied:
  1. (i)

    The conditions (i)-(ii) of Theorem  3.1 are satisfied.

     
  2. (ii)

    The function ϕ C rd ( [ β τ , ) T , R + ) with max s [ β τ , t 0 ] T ϕ ( s ) > 0 , where 0 < β < 1 and τ = min { t 0 , α ( t 0 ) } .

     
  3. (iii)

    The function h C ( R + , R + ) and h Ω .

     
  4. (iv)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    u ( t ) ϕ ( t ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s u ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] u ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.20)
    u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
    (3.21)
     
Then, for all t T 0 satisfying
H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H 1 ) ,
we have
u ( t ) ϕ ( t ) + f ( t ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) ,
(3.22)
where H ( x ) is defined by (3.5) and
f ( t ) = max s [ β τ , t 0 ] T ϕ ( s ) + t 0 t [ p ( s ) h ( ϕ ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T ϕ ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( ϕ ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T ϕ ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 .
(3.23)
Proof Let us define a function z : [ β τ , ) T R + by
z ( t ) = { t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , 0 , t [ β τ , t 0 ] T .
(3.24)
Therefore,
u ( t ) ϕ ( t ) + z ( t ) , t [ β τ , ) T .
(3.25)
From the definition of the function z ( t ) , it follows that
z ( t ) t 0 t { p ( s ) h ( ϕ ( s ) + z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T ϕ ( ξ ) + max ξ [ β s , s ] T z ( ξ ) ) } Δ s z ( t ) + α ( t 0 ) α ( t ) { a ( γ ) h ( ϕ ( γ ) + z ( γ ) ) z ( t ) + b ( γ ) h ( max ξ [ β γ , γ ] T ϕ ( ξ ) + max ξ [ β γ , γ ] T z ( ξ ) ) } α 1 ( s ) Δ ˜ s z ( t ) f ( t ) + t 0 t { p ( s ) h ( z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T z ( ξ ) ) } Δ s z ( t ) + α ( t 0 ) α ( t ) { a ( γ ) h ( z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T z ( ξ ) ) } α 1 ( s ) Δ ˜ s , t T 0 ,
(3.26)
z ( t ) ϕ ( t ) , t [ β τ , t 0 ] T ,
(3.27)

where the function f ( t ) is defined in (3.23).

Since the function f ( t ) : T 0 ( 0 , ) is nondecreasing and f ( t 0 ) = max s [ β τ , t 0 ] T ϕ ( s ) , by using Theorem 3.2 for (3.26) and (3.27), we get
z ( t ) f ( t ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.22). This completes the proof. □

Now we will consider an inequality in which the unknown function into the left part is presented in a power.

Theorem 3.4 Let the following conditions be fulfilled:
  1. (i)

    The conditions (i)-(iii) of Theorem  3.1 and (iii) of Theorem  3.3 are satisfied.

     
  2. (ii)
    The function k C rd ( T 0 , ( 0 , ) ) is nondecreasing and the following inequality
    L : = max s [ β τ , t 0 ] T ϕ ( s ) k ( t 0 ) n , n > 1
    (3.28)
     
holds.
  1. (iii)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    u n ( t ) k ( t ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s u n ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] u n ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.29)
    u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
    (3.30)
     
Then, for all t T 0 satisfying
H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H 1 ) ,
we have
u ( t ) 1 n c 1 n n k ( t ) + n 1 n c 1 n + ( L + g ( t ) ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) ,
(3.31)
where
g ( t ) = t 0 t [ p ( s ) h ( w ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T w ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( w ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T w ( ξ ) ) ] α 1 ( s ) Δ ˜ s ,
(3.32)
with
w ( t ) = { 1 n c 1 n n k ( t ) + n 1 n c 1 n , t T 0 , 1 n c 1 n n k ( t 0 ) + n 1 n c 1 n , t [ β τ , t 0 ] T
(3.33)

for any constant c 1 .

Proof Define a function z : [ β τ , ) T R + by
z ( t ) = { t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , 0 , t [ β τ , t 0 ] T .
(3.34)
It follows from inequality (3.29) for t T 0 that
u ( t ) [ k ( t ) + z ( t ) ] 1 n .
Using Lemma 2.2, for any c 1 , we obtain
u ( t ) 1 n c 1 n n [ k ( t ) + z ( t ) ] + n 1 n c 1 n = 1 n c 1 n n k ( t ) + n 1 n c 1 n + 1 n c 1 n n z ( t ) = w ( t ) + 1 n c 1 n n z ( t ) , t T 0 .
(3.35)
From inequality (3.28) and applying Lemma 2.2, for any c 1 , we have
k ( t 0 ) n 1 n c 1 n n k ( t 0 ) + n 1 n c 1 n .
(3.36)
Indeed, by using inequality (3.36), we have for t [ β τ , t 0 ] T
u ( t ) ϕ ( t ) ϕ ( t ) + 1 n c 1 n n z ( t ) w ( t ) + 1 n c 1 n n z ( t ) ,
(3.37)

where w ( t ) is defined by (3.33).

Now we define a nondecreasing function v : T 0 ( 0 , ) by v ( t ) = L + g ( t ) , where L and g ( t ) are defined by (3.28) and (3.32), respectively.

From the definition of the function z ( t ) , it follows that
z ( t ) t 0 t { p ( s ) h ( w ( s ) + 1 n c 1 n n z ( s ) ) z ( t ) + q ( s ) h ( max ξ [ β s , s ] T w ( ξ ) + 1 n c 1 n n max ξ [ β s , s ] T z ( ξ ) ) } Δ s z ( t ) + α ( t 0 ) α ( t ) { a ( γ ) h ( w ( γ ) + 1 n c 1 n n z ( γ ) ) z ( t ) + b ( γ ) h ( max ξ [ β γ , γ ] T w ( ξ ) + 1 n c 1 n n max ξ [ β γ , γ ] T z ( ξ ) ) } α 1 ( s ) Δ ˜ s z ( t ) v ( t ) + t 0 t [ p ( s ) h ( 1 n c 1 n n z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T 1 n c 1 n n z ( ξ ) ) ] Δ s z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( 1 n c 1 n n z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T 1 n c 1 n n z ( ξ ) ) ] z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.38)
z ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
(3.39)
From inequalities (3.38) and (3.39), we get for c 1 , n > 1
1 n c 1 n n z ( t ) v ( t ) + t 0 t [ p ( s ) h ( 1 n c 1 n n z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T 1 n c 1 n n z ( ξ ) ) ] Δ s 1 n c 1 n n z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( 1 n c 1 n n z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T 1 n c 1 n n z ( ξ ) ) ] 1 n c 1 n n z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.40)
1 n c 1 n n z ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
(3.41)
Applying Theorem 3.2 for (3.40) and (3.41), we obtain
1 n c 1 n n z ( t ) v ( t ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.31). This completes the proof. □

Next we will consider an inequality which has powers on both sizes.

Theorem 3.5 Let the following conditions be fulfilled:
  1. (i)

    The conditions (i)-(iii) of Theorem  3.1 and (iii) if Theorem  3.3 are satisfied.

     
  2. (ii)
    The function k C rd ( T 0 , ( 0 , ) ) is nondecreasing and the following inequality
    K : = max s [ β τ , t 0 ] T { ϕ ε ( s ) , ϕ l ( s ) } m n c m n n k ( t 0 ) + n 1 n c m n
    (3.42)
     
holds for any constant c 1 and n m l δ ε > 1 .
  1. (iii)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    u n ( t ) k ( t ) + t 0 t [ p ( s ) h ( u m ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u l ( ξ ) ) ] Δ s u n ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u δ ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ε ( ξ ) ) ] u n ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.43)
    u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
    (3.44)
     
Then, for all t T 0 satisfying
H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H 1 ) ,
we have
u ( t ) 1 n c 1 n n k ( t ) + n 1 n c 1 n + 1 m c 1 m n ( K + λ ( t ) ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) ,
(3.45)
where
λ ( t ) = t 0 t [ p ( s ) h ( w ¯ ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T w ¯ ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( w ¯ ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T w ¯ ( ξ ) ) ] α 1 ( s ) Δ ˜ s ,
(3.46)
with
w ¯ ( t ) = { m n c m n n k ( t ) + n 1 n c m n , t T 0 , m n c m n n k ( t 0 ) + n 1 n c m n , t [ β τ , t 0 ] T .
(3.47)
Proof We define a function z : [ β τ , ) T R + by
z ( t ) = { t 0 t [ p ( s ) h ( u m ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u l ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u δ ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ε ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , 0 , t [ β τ , t 0 ] T .
(3.48)
From inequality (3.43) we have for t T 0
u ( t ) [ k ( t ) + z ( t ) ] 1 n , u l ( t ) [ k ( t ) + z ( t ) ] l n , u m ( t ) [ k ( t ) + z ( t ) ] m n , u δ ( t ) [ k ( t ) + z ( t ) ] δ n , u ε ( t ) [ k ( t ) + z ( t ) ] ε n .
By using Lemma 2.2, for any c 1 , we obtain
u ( t ) 1 n c 1 n n k ( t ) + n 1 n c 1 n + 1 n c 1 n n z ( t ) , t T 0 ,
(3.49)
u ε ( t ) ε n c ε n n k ( t ) + n ε n c ε n + ε n c ε n n z ( t ) u ε ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t T 0 ,
(3.50)
u δ ( t ) δ n c δ n n k ( t ) + n δ n c δ n + δ n c δ n n z ( t ) u δ ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t T 0 ,
(3.51)
u l ( t ) l n c l n n k ( t ) + n l n c l n + l n c l n n z ( t ) u l ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t T 0 ,
(3.52)
u m ( t ) m n c m n n k ( t ) + n m n c m n + m n c m n n z ( t ) u m ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t T 0 .
(3.53)
Moreover, we have
u ε ( t ) ϕ ε ( t ) ϕ ε ( t ) + m n c m n n z ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t [ β τ , t 0 ] T
(3.54)
and
u l ( t ) ϕ l ( t ) ϕ l ( t ) + m n c m n n z ( t ) w ¯ ( t ) + m n c m n n z ( t ) , t [ β τ , t 0 ] T ,
(3.55)
where w ¯ ( t ) is defined by (3.47). From the definition of the function z ( t ) , it follows that
z ( t ) t 0 t { p ( s ) h ( w ¯ ( s ) + m n c m n n z ( s ) ) z ( t ) + q ( s ) h ( max ξ [ β s , s ] T w ¯ ( ξ ) + max ξ [ β s , s ] T m n c m n n z ( ξ ) ) } Δ s z ( t ) + α ( t 0 ) α ( t ) { a ( γ ) h ( w ¯ ( γ ) + m n c m n n z ( γ ) ) z ( t ) + b ( γ ) h ( max ξ [ β γ , γ ] T w ¯ ( ξ ) + max ξ [ β γ , γ ] T m n c m n n z ( ξ ) ) } α 1 ( s ) Δ ˜ s z ( t ) ρ ( t ) + t 0 t [ p ( s ) h ( m n c m n n z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T m n c m n n z ( ξ ) ) ] Δ s z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( m n c m n n z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T m n c m n n z ( ξ ) ) ] z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.56)
z ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
(3.57)
From inequalities (3.56) and (3.57), we have
m n c m n n z ( t ) ρ ( t ) + t 0 t [ p ( s ) h ( m n c m n n z ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T m n c m n n z ( ξ ) ) ] Δ s m n c m n n z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( m n c m n n z ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T m n c m n n z ( ξ ) ) ] m n c m n n z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.58)
m n c m n n z ( t ) ϕ ( t ) , t [ β τ , t 0 ] T ,
(3.59)

where a nondecreasing function ρ ( t ) : T 0 ( 0 , ) is defined by ρ ( t ) : = K + λ ( t ) , where K and λ ( t ) are defined in (3.42) and (3.46), respectively.

Applying Theorem 3.2 for (3.58) and (3.59), we obtain
m n c m n n z ( t ) ρ ( t ) H 1 ( H ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.45). This completes the proof. □

In the case when the unknown function is involved nonlinearly in the left part of the inequality, we obtain the following result.

Theorem 3.6 Let the following conditions be fulfilled:
  1. (i)

    The conditions (i)-(iv) of Theorem  3.1 are satisfied.

     
  2. (ii)

    The function Ψ C ( R + , R + ) is strictly increasing, lim t Ψ ( t ) = .

     
  3. (iii)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    Ψ ( u ( t ) ) k + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s Ψ ( u ( t ) ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] Ψ ( u ( t ) ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.60)
     
u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T ,
(3.61)

where k 0 .

Then, for all t T 0 satisfying
H ˜ ( Ψ ( P ) ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H ˜ 1 )
and
H ˜ 1 ( H ˜ ( Ψ ( P ) ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) Dom ( Ψ 1 ) ,
we have
u ( t ) Ψ 1 { H ˜ 1 ( H ˜ ( Ψ ( P ) ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) } ,
(3.62)
where
P = max { Ψ 1 ( k ) , max s [ β τ , t 0 ] T ϕ ( s ) }
(3.63)
and
H ˜ ( x ) = x 0 x 1 h ( Ψ 1 ( r ) ) d r , x 0 > 0 ,
(3.64)

where H ˜ ( ) = , and H ˜ 1 is the inverse of H ˜ .

Proof Define a function z : [ β τ , ) T R + by
z ( t ) = { Ψ ( P ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , Ψ ( P ) , t [ β τ , t 0 ] T ,

where P is defined by (3.63). Note that the function z ( t ) is nondecreasing.

It follows that the inequality
u ( t ) Ψ 1 ( z ( t ) ) , t [ β τ , ) T
holds. Therefore, for t T 0 and s [ t 0 , t ] T , we have
max ξ [ β s , s ] T u ( ξ ) max ξ [ β s , s ] T Ψ 1 ( z ( ξ ) ) = Ψ 1 ( z ( s ) ) .
For t T 0 and s [ α ( t 0 ) , α ( t ) ] T ˜ , we have
h ( max ξ [ β γ , γ ] T u ( ξ ) ) α 1 ( s ) h ( max ξ [ β γ , γ ] T Ψ 1 ( z ( ξ ) ) ) α 1 ( s ) = h ( max ξ [ β α 1 ( s ) , α 1 ( s ) ] T Ψ 1 ( z ( ξ ) ) ) = h ( Ψ 1 ( z ( α 1 ( s ) ) ) ) = h ( Ψ 1 ( z ( γ ) ) ) α 1 ( s ) .
Then, from the definition of z ( t ) and the above analysis, we get for t T 0 that
z ( t ) Ψ ( P ) + t 0 t [ p ( s ) h ( Ψ 1 ( z ( s ) ) ) + q ( s ) h ( max ξ [ β s , s ] T Ψ 1 ( z ( ξ ) ) ) ] Δ s z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( Ψ 1 ( z ( γ ) ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T Ψ 1 ( z ( ξ ) ) ) ] z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.65)
z ( t ) Ψ ( P ) , t [ β τ , t 0 ] T .
(3.66)
According to Theorem 3.1, from inequalities (3.65) and (3.66), we have
z ( t ) H ˜ 1 ( H ˜ ( Ψ ( P ) ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.62). This completes the proof. □

In the case when in place of the constant k involved in Theorem 3.6 we have a function k ( t ) , we obtain the following result.

Theorem 3.7 Let the following conditions be fulfilled:
  1. (i)

    The conditions (i)-(iii) of Theorem  3.1, (ii) of Theorem  3.2 and (ii) of Theorem  3.6 are satisfied.

     
  2. (ii)

    The function k C rd ( T 0 , [ 1 , ) ) is nondecreasing and the inequality Q = max s [ β τ , t 0 ] T ϕ ( s ) Ψ 1 ( k ( t 0 ) ) holds.

     
  3. (iii)
    The function u C rd ( [ β τ , ) T , R + ) and satisfies the inequalities
    Ψ ( u ( t ) ) k ( t ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s Ψ ( u ( t ) ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] Ψ ( u ( t ) ) α 1 ( s ) Δ ˜ s , t T 0 ,
    (3.67)
    u ( t ) ϕ ( t ) , t [ β τ , t 0 ] T .
    (3.68)
     
Then, for all t T 0 satisfying
H ˜ ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s Dom ( H ˜ 1 )
and
k ( t ) H ˜ 1 ( H ˜ ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) Dom ( Ψ 1 ) ,
we have
u ( t ) Ψ 1 { k ( t ) H ˜ 1 ( H ˜ ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) } ,
(3.69)

where H ˜ ( x ) is defined by (3.64).

Proof Define a function z : [ β τ , ) T R + by
z ( t ) = { k ( t ) + t 0 t [ p ( s ) h ( u ( s ) ) + q ( s ) h ( max ξ [ β s , s ] T u ( ξ ) ) ] Δ s + α ( t 0 ) α ( t ) [ a ( γ ) h ( u ( γ ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T u ( ξ ) ) ] α 1 ( s ) Δ ˜ s , t T 0 , k ( t 0 ) , t [ β τ , t 0 ] T .
Note that the function z ( t ) is nondecreasing. It follows that the inequality
u ( t ) Ψ 1 ( z ( t ) ) , t [ β τ , ) T
holds. Therefore, for t T 0 and s [ t 0 , t ] T , we have
max ξ [ β s , s ] T u ( ξ ) max ξ [ β s , s ] T Ψ 1 ( z ( ξ ) ) = Ψ 1 ( z ( s ) ) .
For t T 0 and s [ α ( t 0 ) , α ( t ) ] T ˜ , we have
h ( max ξ [ β γ , γ ] T u ( ξ ) ) α 1 ( s ) h ( max ξ [ β γ , γ ] T Ψ 1 ( z ( ξ ) ) ) α 1 ( s ) = h ( max ξ [ β α 1 ( s ) , α 1 ( s ) ] T Ψ 1 ( z ( ξ ) ) ) = h ( Ψ 1 ( z ( α 1 ( s ) ) ) ) = h ( Ψ 1 ( z ( γ ) ) ) α 1 ( s ) .
Then, from the definition of z ( t ) and the above analysis, we get for t T 0 that
z ( t ) k ( t ) + t 0 t [ p ( s ) h ( Ψ 1 ( z ( s ) ) ) + q ( s ) h ( max ξ [ β s , s ] T Ψ 1 ( z ( ξ ) ) ) ] Δ s z ( t ) + α ( t 0 ) α ( t ) [ a ( γ ) h ( Ψ 1 ( z ( γ ) ) ) + b ( γ ) h ( max ξ [ β γ , γ ] T Ψ 1 ( z ( ξ ) ) ) ] z ( t ) α 1 ( s ) Δ ˜ s , t T 0 ,
(3.70)
z ( t ) k ( t 0 ) , t [ β τ , t 0 ] T .
(3.71)
According to Theorem 3.2, from inequalities (3.70) and (3.71), we have
z ( t ) k ( t ) H ˜ 1 ( H ˜ ( 1 ) + t 0 t [ p ( s ) + q ( s ) + a ( s ) α ( s ) + b ( s ) α ( s ) ] Δ s ) , t T 0 ,

which results in (3.69). This completes the proof. □

4 An application

In this section, in order to illustrate our results, we consider the following first-order dynamic equation with ‘maxima’:
x ( t ) = F ( t , x ( t ) , max s [ β t , t ] T x ( s ) ) , t T 0 ,
(4.1)
and initial condition
x ( t ) = ϕ ( t ) , t [ β τ , t 0 ] T ,
(4.2)

where F C rd ( T 0 × R × R , R ) , ϕ C rd ( [ β t 0 , t 0 ] T , R ) , 0 < β < 1 , τ is a constant such that β τ t 0 .

Corollary 4.1 Assume that:

(H1) There exists a strictly increasing function α C rd ( T 0 , R + ) such that α ( T ) = T ˜ is a time scale and min { t 0 , α ( t 0 ) } = τ .

(H2) There exist functions A , B , C , D , α C rd ( T 0 , R + ) and an integer p > 1 such that for t T 0 , u , v R ,
| F ( t , u , v ) | ( A ( t ) + B ( t ) α ( t ) ) | u | p + ( C ( t ) + D ( t ) α ( t ) ) | v | p .
(4.3)
Then the solution x ( t ) of IVP (4.1)-(4.2) satisfies the following inequality:
| x ( t ) | { M p 1 p + p 1 p t 0 t [ A ( s ) + C ( s ) + B ( s ) α ( s ) + D ( s ) α ( s ) ] Δ s } p p 1 , t T 0 ,
(4.4)
where
M = max s [ β τ , t 0 ] T | ϕ ( s ) | .
Proof It is easy to see that the solution x ( t ) of IVP (4.1)-(4.2) satisfies the following equation:
x ( t ) = ϕ ( t 0 ) + t 0 t F ( s , x ( s ) , max ξ [ β s , s ] T x ( ξ ) ) Δ s .
(4.5)
Using the assumption (H2), it follows from (4.5) that
| x ( t ) | | ϕ ( t 0 ) | + t 0 t | F ( s , x ( s ) , max ξ [ β s , s ] T x ( ξ ) ) | Δ s | ϕ ( t 0 ) | + t 0 t [ ( A ( s ) + B ( s ) α ( s ) ) | x ( s ) | p + ( C ( s ) + D ( s ) α ( s ) ) | max ξ [ β s , s ] T x ( ξ ) | p ] Δ s | ϕ ( t 0 ) | + t 0 t [ A ( s ) | x ( s ) | p + C ( s ) max ξ [ β s , s ] T | x ( ξ ) | p ] Δ s + t 0 t [ B ( s ) | x ( s ) | p + D ( s ) max ξ [ β s , s ] T | x ( ξ ) | p ] α ( s ) Δ s = | ϕ ( t 0 ) | + t 0 t [ A ( s ) | x ( s ) | p + C ( s ) max ξ [ β s , s ] T | x ( ξ ) | p ] Δ s + α ( t 0 ) α ( t ) [ B ( γ ) | x ( γ ) | p + D ( γ ) max ξ [ β γ , γ ] T | x ( ξ ) | p ] α 1 ( s ) Δ ˜ s .
(4.6)
Hence Corollary 4.1 yields the estimate
| x ( t ) | { M p 1 p + p 1 p t 0 t [ A ( s ) + C ( s ) + B ( s ) α ( s ) + D ( s ) α ( s ) ] Δ s } p p 1 , t T 0 .
(4.7)

Inequality (4.7) gives the bound on the solution x ( t ) of IVP (4.1)-(4.2). □

Example 4.1 Consider the following first-order dynamic equation with ‘maxima’ on time scale T = { 2 n : n Z } { 0 } ( stands for the integer set):
{ x ( t ) = 1 2 tan 1 ( ( 2 + 8 t 2 ) x ( t ) 3 ) x ( t ) = + 2 sin ( ( e 2 t + 4 cos 2 ( π t ) ) max s [ 1 16 t , t ] T x ( s ) 3 ) , t T 0 , x ( t ) = 3 , t [ 1 8 , 2 ] T ,
(4.8)

where T 0 = [ 2 , ) T .

Here ϕ ( t ) = 3 , β = 1 / 16 , p = 3 , F ( t , x ( t ) , max s [ β t , t ] T x ( s ) ) = ( tan 1 ( ( 2 + 8 t 2 ) × x ( t ) 3 ) ) / 2 + 2 sin ( ( e 2 t + 4 cos 2 ( π t ) ) max s [ 1 16 t , t ] T x ( s ) 3 ) , t 0 = 2 , τ = 2 .

By choosing α ( t ) = 4 t , we can show that α ( T ) = T ˜ T and min { t 0 , α ( t 0 ) } = 2 . Clearly,
| F ( t , x ( t ) , max s [ β t , t ] T x ( s ) ) | = | 1 2 tan 1 ( ( 2 + 8 t 2 ) x ( t ) 3 ) + 2 sin ( ( e 2 t + 4 cos 2 ( π t ) ) max s [ 1 16 t , t ] T x ( s ) 3 ) | ( 1 + 4 t 2 ) | x ( t ) | 3 + ( 2 e 2 t + 8 cos 2 ( π t ) ) | max s [ 1 16 t , t ] T x ( s ) | 3
and
max s [ ( 1 / ( 8 ) ) , ( 2 ) ] T | ϕ ( s ) | = 3 .
On the other hand, we have α ( t ) = 4 . Set A ( t ) = 1 , B ( t ) = t 2 , C ( t ) = 2 e 2 t and D ( t ) = 2 cos 2 ( π t ) . Hence, Corollary 4.1 yields the estimate
| x ( t ) | { 3 2 3 + 2 3 2 t [ 1 + 2 e 2 s + 4 s 2 + 8 cos 2 ( π s ) ] Δ s } 3 2 , t T 0 .

Authors’ information

Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Declarations

Acknowledgements

The research of P Thiramanus and J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

Authors’ Affiliations

(1)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok
(2)
Department of Mathematics, University of Ioannina

References

  1. Hilger S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 18–56. 10.1007/BF03323153MathSciNetView ArticleGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Basel; 2001.View ArticleGoogle Scholar
  3. Feng Q, Meng F, Zhang Y, Zhou J, Zheng B: Some delay integral inequalities on time scales and their applications in the theory of dynamic equations. Abstr. Appl. Anal. 2012., 2012: Article ID 538247Google Scholar
  4. Feng Q, Meng F, Zheng B: Gronwall-Bellman type nonlinear delay integral inequalities on time scales. J. Math. Anal. Appl. 2011, 382: 772–784. 10.1016/j.jmaa.2011.04.077MathSciNetView ArticleGoogle Scholar
  5. Li W-N: Explicit bounds for some special integral inequalities on time scales. Results Math. 2010, 58: 317–328. 10.1007/s00025-010-0040-6MathSciNetView ArticleGoogle Scholar
  6. Li W-N: Some delay integral inequalities on time scales. Comput. Math. Appl. 2010, 59: 1929–1936. 10.1016/j.camwa.2009.11.006MathSciNetView ArticleGoogle Scholar
  7. Li W-N: Bounds for certain new integral inequalities on time scales. Adv. Differ. Equ. 2009., 2009: Article ID 484185Google Scholar
  8. Li W-N, Sheng W: Some nonlinear integral inequalities on time scales. J. Inequal. Appl. 2007., 2007: Article ID 70465Google Scholar
  9. Sun Y: Some new integral inequalities on time scales. Math. Inequal. Appl. 2012, 15: 331–341.MathSciNetGoogle Scholar
  10. Sun Y: Some sublinear dynamic integral inequalities on time scales. J. Inequal. Appl. 2010., 2010: Article ID 983052Google Scholar
  11. Wang T, Xu R: Bounds for some new integral inequalities with delay on time scales. J. Math. Inequal. 2012, 6: 355–366.MathSciNetView ArticleGoogle Scholar
  12. Xu R, Meng F, Song C: On some integral inequalities on time scales and their applications. J. Inequal. Appl. 2010., 2010: Article ID 464976Google Scholar
  13. Feng Q, Zheng B: Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications. Appl. Math. Comput. 2012, 218: 7880–7892. 10.1016/j.amc.2012.02.006MathSciNetView ArticleGoogle Scholar
  14. Feng Q, Meng F, Zhang Y, Zheng B, Zhou J: Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J. Inequal. Appl. 2011., 2011: Article ID 29Google Scholar
  15. Zheng B, Feng Q, Meng F, Zhang Y: Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales. J. Inequal. Appl. 2012., 2012: Article ID 201Google Scholar
  16. Zheng B, Zhang Y, Feng Q: Some new delay integral inequalities in two independent variables on time scales. J. Appl. Math. 2011., 2011: Article ID 659563Google Scholar
  17. Agarwal R, Bohner M, Peterson A: Inequalities on time scales: a survey. Math. Inequal. Appl. 2001, 4: 535–557.MathSciNetGoogle Scholar
  18. Bainov D, Hristova S Pure and Applied Mathematics. In Differential Equations with Maxima. Chapman & Hall/CRC, New York; 2011.Google Scholar
  19. Hristova S, Stefanova K: Some integral inequalities with maximum of the unknown functions. Adv. Dyn. Syst. Appl. 2011, 6: 57–69.MathSciNetGoogle Scholar
  20. Hristova S, Stefanova K: Linear integral inequalities involving maxima of the unknown scalar functions. J. Math. Inequal. 2010, 4: 523–535.MathSciNetView ArticleGoogle Scholar
  21. Hristova, S, Stefanova, K: Nonlinear Bihari type integral inequalities with maxima. REMIA (2010)Google Scholar
  22. Tariboon J, Thiramanus P, Ntouyas SK: Dynamic integral inequalities on time scales with ‘maxima’. J. Inequal. Appl. 2013., 2013: Article ID 564Google Scholar
  23. Jiang F, Meng F: Explicit bounds on some new nonlinear integral inequalities with delay. J. Comput. Appl. Math. 2007, 205: 479–486. 10.1016/j.cam.2006.05.038MathSciNetView ArticleGoogle Scholar
  24. Henderson J, Hristova S: Nonlinear integral inequalities involving maxima of unknown scalar functions. Math. Comput. Model. 2011, 53: 871–882. 10.1016/j.mcm.2010.10.024MathSciNetView ArticleGoogle Scholar

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© Thiramanus et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.