ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 71-77, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
LANGENHOP’S INEQUALITY AND APPLICATIONS
FOR DYNAMIC EQUATIONS
B. KAYMAKÇALAN AND A. ZAFER
Abstract. A Langenhop-type inequality is given for dynamic equations on time scales. This result is further
employed to obtain lower bounds for solutions of certain dynamic equations. As an application, usage of the derived
Langenhop’s inequality in determining the oscillatory behavior of a damped second order delay dynamic equation is
illustrated. The results obtained are important in the qualitative sense.
Key words. Langenhop inequality, time scale, lower bounds, oscillation
AMS subject classifications. 26D20, 34A40, 34K11
1. Introduction. There is no doubt that the Gronwall inequality and its generalization,
the Bihari inequality in continuous and discrete cases, have been very powerful tools in
studying the qualitative behavior of differential and difference equations. These inequalities
have been applied very successfully to investigate the global existence, uniqueness, stability,
boundedness and other properties of solutions of various nonlinear differential and difference
equations. Langenhop-type inequalities have also been used quite successfully in studying
the qualitative behavior of differential equations [9, 10] and difference equations [11, 12].
2. Preliminaries. Time scale calculus which unifies continuous and discrete analysis
was first introduced by Stefan Hilger [3]. Later, the theory developed very rapidly, see the
monographs [2, 6] and the references cited therein. Here we shall only provide some basic
facts on time scales extracted from [2].
A time scale is an arbitrary nonempty closed subset of the real numbers . The most
well-known examples are
and
. An interval
in is defined to be the set
. Other types of intervals are defined similarly. To define a continuity
on a time scale we need the concept of forward and backward jump operators,
given by
!"$#%
'& ( )+*-, /. $ .102 and & ( )4365 /. 7 .182 )=0E*-,B . A ( point C (F0E
is
together with the convention that )+*-,:9;<)=365> and )43/5?9@A
D
(
'
&
'
&
&
called
right-scattered,
right-dense,
left-scattered,
left-dense,
if
,
,
is then defined by GH& ( E'& (JIK . The,
& ( is satisfied, respectively.
The
graininess
at
O if has a left-scattered maximum O , and as otherwise.
set ML is defined as MN
= >L
D
2.1. A function PEQR#S is called T -differentiable at a point
ifX there
exists a real number U and for a given V there is a neighborhood W of such that
PY&Z'& ([(\I PY& .](I UF&^'& (I.(X8 V X '& (ID._X for( all .1 W . The number U is denoted>by Pa`& ( ( .
As usual, P is said to be differentiable
on &^ if it is differentiable
at
every point
&^ ( L .
(
(
(
(
'
i
g
h
M
(
I
It can be shown that Pb`& cPdZ& if e , and P`D& cT=PY&
fePY&
PY& if
;j . In fact, it is possible to show that
.](JI (
( (YI (
P ` & ( ilnk-*-m) o PY& .?IPY& if GH& ( Epq P ` & ( PY&^''& & (YIPY& if GH& (r0 pq
s
EFINITION
Received February 28, 2004. Accepted for publication September, 2004. Recommended by A. Ruffing.
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460
(billur@georgiasouthern.edu).
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
(zafer@metu.edu.tr).
71
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72
B. KAYMAKÇALAN AND A. ZAFER
P tEHu
#v ( are T ( -differentiable, then so are P g t , P I t , Pwt , Paxt . Moreover,
(
(
a
(
g
PY&^'& yPY& GH& P`& holds on any arbitrary time scale.
D
2.2. A function Pai#z is called rd-continuous, if it is continuous at
every
right-dense
point and if the left-sided limit exists at every left-dense point. A function
tb& { ( |$}~E# is called rd-continuous if tb& [{Q& ( ( |€# is so. (
The set of all rd-continuous functions defined on is denoted by ƒ‚„&Z . The space
( of
functions that are differentiable with rd-continuous derivatives is denoted by >‚†… „ &^ . It is
noteworthy to mention that every rd-continuous function P has an antiderivative
in the
( yPY‡ & .( As
continuous
case,
a
function
is
called
an
antiderivative
of
on
if
holds
for
‡
P
ˆ
‡
`
&
M ƒLŠ‰
all
hFg GH& ( PY& (!c
Πp for
D
2.3. A function PjQ#‹ is called regressive if
D
all
  M&Z L ( .. The set of all rd-continuous and regressive functions defined on is denoted by
In this study, we also employ a notion of a generalized exponential function on an arbitrary time scale. The definition below is extracted from
[2].
 and ! . The unique solution
D
2.4 (Exponential function).
Let
Ž
(
‘
/

(
h
of the initial value problem {`<‘
called the generalized exponential
( ŽQ& { , {J & is out
‘( function and is denoted by ’“"Œ & ”• I1h . Let – ‘]. ( It turns
that if 2i , then ’6—Š& M
h
g
(
š
o
w
™
œ
o
[
›

š
o
™
œ
o
›
’ —|˜ , and if ž and – , then ’6—& & – . ‘/(
For an extensive list of properties and detailed discussion on ’“q&n”-
we refer to [2].
and if
EFINITION
EFINITION
EFINITION
Another useful tool is a variation of parameters formula for first order linear nonhomogeneous dynamic equations which can be stated as follows.
and
. Let
T HEOREM 2.5 (Theorem 2.77, [2]). Suppose that
and
. Then the unique solution of
,
is given by
Ž (   ( P  '‚†„&^ ( ‘4 (
g
Ÿq`$žŽQ& Ÿ PY& Ÿb& Ÿ
o
Ÿb& ( E’\“"& ‘/( Ÿ Mg$ o › ’\“"& ‘ ['&^¡ ( ( PY&œ¡ ( Tˆ¡_š‰
In this set up,
o  (( (
(

Ÿ “ E’ “ & oœ› ’ “ & ['&^¡ PY&^¡ Tˆ¡
is becomes a particular solution of the dynamic equation
Ÿ ` žŽQ& ( Ÿ g PY& ( ‰
o h
(
(
(
(


In case ŽQ& – and PY& ’ — & , it follows that Ÿ “ i’ — & o › hMg –qGH&^¡ ( — Tˆo ¡q‰
As special cases, this
expression results in the well-known particular solutions Ÿ¢“£ ’
if
€E and Ÿ/“¤ & hMg – ( o if €E .
3. Langenhop inequality. In 1960, C. E. Langenhop [8] proved a version of the following theorem when ju . Recently, the same theorem was proved by Zafer [12] in the
case when 2R . It is also noteworthy to mention that in [11], Theorem 2.3.2 is given as a
discrete version of Langenhop’s inequality. Here we unify the previous results by use of time
scale methods.
 . Suppose that tH¥#v is nondecreasing, tb&œ¦ ( is rdT
3.1. (§Let
0
p for ¦ 0 p . Let ¦a#¨ and ©€a#¨'ª be rd-continuous.
continuous, and tb&œ¦
If
o
(3.1)
¦Y& (>« ¦Y& .](YIž l ©&^¬ ( tb&œ¦Y&œ¬ ( ( Tˆ¬/ .1 ‘ [­?®
Ÿ HEOREM
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73
LANGENHOP INEQUALITY AND APPLICATIONS
then
where
(3.2)
If
¯
¦Y& (>«2¯ & .( .¤ ‘  ­?
is the maximal solution of
¯ ` I ©& ( tb& ¯F( ¯ & . .]( ¦Y& .]( ‰
tb&œ¦ ( ¦ , then
¦J& (>« ¦Y& ]. ( ’ ™ ° & .]( .1 ‘ [­?®‰
> &  ­? and for .1  define
Proof. Fix
o
Ÿb& .]( E¦J& (ag l ©&^¬ ( t&^¦J&^¬ ( ( Tˆ¬/‰
(3.3)
It is clear that
Ÿ ` & .](ag ©& ]. ( t &^¦J& .([( Ep
(3.4)
and
Ÿb& ( E¦Y& ( ‰
Moreover, from (3.1) and (3.3) we have
Ÿb& .](>« ¦J& . ( 1. ‘ š‰
(3.5)
Using (3.5) in (3.4), we obtain
Ÿ ` & (>«iI ©& ( bt &œŸb& ( ( ‰
In view of the theory developed in [5] for maximal-minimal solutions on time scales and by
comparison with (3.2), (see also [4, 7]), we have
Ÿb& (>«2¯ & .]( .  ­?š‰
(3.6)
. tends
žj the
Employing (3.5) and (3.6), we obtain the desired conclusion.
It is remarkable that the conclusion of the theorem remains valid in the limit as
to , but if is fixed as in (3.1), then as was shown by Langenhop for the case
estimate is no longer true.
‘
.
n
±4|€#'² and PY& [±& ([( be rd-continuous.
(4.1)
± ` jPY& [± ( >«$‘
XX
Let us assume that for some norm in ³1² , which we shall denote by ‰ , the function P
satisfies
X PY& [± (\X 2©& ( tb& X ± X ( >«2‘ ´±  ² (4.2)
4. Bounds on the norm of solutions. Let
We shall consider the first order system
where
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B. KAYMAKÇALAN AND A. ZAFER
h?I GH& ( © 0 p
¦ « p , and strictly positive for ¦ 0 p .
X w± & \( X X ±& .](Xg o ©&^¬ ( tb& X ±w&^¬ (X ( T§¬
l
C© |ž#Yª
t&^¦ (
is rd-continuous,
;
(a)
(b)
is rd-continuous and nondecreasing for
It follows from (4.1) and (4.2) that
(4.3)
and
(4.4)
. > ‘  † µ (
X ±w& (\X_«X ±& .](X/Iž o ©&^¬ ( bt & X ±w&^¬ (X ( T§¬
l
.
for all
The main results of this section are as follows.
T HEOREM 4.1. If
is solution of (4.1) such that
±& (
X ±w& (\X ¯ & (
(4.5)
and
(4.6)
for all
and
¯·
>«$‘ , where ¯
, then
X ±w& (\X_«2¯?· & (
is a minimal solution of
¯ ` ©& ( t& ¯ & ([( ¯ &  ( X ¶ X
is a maximal solution of
T HEOREM
(4.7)
±& n( j¶
¯ ` I ©& ( t& ¯ & ([( ¯ & ‘/( X ¶ X ‰
(
(
>«$n ,
4.2. Let tb&œ¦ ¦ . If ±& is solution of (4.1), then for all
X ¶ X ’ ™° &  ( X ±w& (\X X ¶ X ’ ° &  ( ‰
Upper bounds in (4.5) and (4.7) are obtained from (4.3) by applying Bihari and Gronwall
type inequalities on time Scales, (see [2, 7]), respectively. Lower bounds in (4.6) and (4.7),
however, are new and follow from (4.4) on using Theorem 3.1.
5. Oscillation of damped second order delay dynamic equations. We shall consider
the oscillatory behavior of solutions of second order delay equations of the form
(·
¸
(
7
g
¹
(
`
{
&
{ `'` & & { ` & (¸g { ` &^'& ( ( g QŽ & ( {Q&ºtb& ([( º»DEpq‰
(5.1)
n (
We restrict our attention to solutions of (5.1) which exist
on some ray †µ , the interval
(
being understood in time scale sense. A solution {Q& is called oscillatory if it is neither
eventually positive nor eventually negative.
With regards to (5.1) the following conditions are assumed to hold:
(a)
is differentiable, nondecreasing,
and
;
(b)
and
are rd-continuous, there is an rd-continuous function
such that
and
for all
.
¼t |€# ª
ŽžO j#¾ ª
|€#À ª
¹ w j#¿
?h I G OÁ0 p
tb& (
¹ &( O
&(
k-*•) ^o m¤½ tb& ( iµ
> pqµ (
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LANGENHOP INEQUALITY AND APPLICATIONS
(c)
k•*-) o^m¤½Ã & ­ M( 8 µ
­ pqµ ( , where
o
à & [­ ( ÄÅ ’ ™ ² & . ­ ( T .¢Æ
·
(
¸
(
7
g
¹
(
Ÿ ` & & Ÿb& (¸
Ÿbg & Ÿb&^ '& ( ( 8 p
75
for any fixed
(d) The inequality
(5.2)
has no oscillatory solution.
Note that if
then (d) holds without imposing any condition at all. Indeed, if
, then
which means that
cannot have a zero
larger than .
L EMMA 5.1. Let (a)–(d) hold, and
be an eventually positive solution of (5.1). Then
there is a
such that
is of constant sign for
and
<Ç
ŸqdZ& (!8IB¹ & ( Ÿ& ( x6É7Áp
Ÿb& (
{J& (
$
«
‘

(
M ­:†µ (
­
{`D&
(5.3)
{Q& (r«jI à &®µE ( { ` & ( M ­:†µ ( ‰
(
such that
that {Q& is eventually
positive. Clearly,
there exist
{Q& (rProof.
0 p andSuppose
[
(
r
(
0
(
(
otherwise,
{Q&ºtb& p on … †µ . Furthermore, {`& is not oscillatory; since,
(
(>8
(d). We may assume that {b`D&
Ÿ={w`& ([(1is 8 an oscillatory( solution of (5.2), contradicting
(
p , {`D&Z'& p on ­:†µ . The case where (Y{g `& is eventually
positive can be handled in
{w`&Z'& ( ( and integrating over ­B we see
a similar manner. Multiplying (5.1) by {w`&
Ÿb& ( È
p
that
o { ` & .](¸g { ` ^& '& .]([( Ê{ `'` & ]. ( T .rg$ o ¹ & .]( { ` & . ( · T 1. 0 qp Ä
Ä
from which we get
·«
· €I o O .]( .]( · .
(
(
{ ` & { ` &œ­ Ä & { ` & T ‰
Hence, by the Langenhop inequality,
·
·
`
{ & ( « { ` &^­ ( ’ ™ ² & [­ ( ‰
Clearly, this inequality results in
{ ` & ( $ { ` &^­ ( Å ’ ™ ² & ­ ( ‰
Integrating the above inequality over ­B , we have
{Q& ( ${J&œ­ (Qg { ` &^­ ( à & ­ (
and hence
{Q& (?«yI { ` & ( à &šµ[­ ( ‰
T HEOREM 5.2. In addition to (a)–(d) assume that
(5.4)
h?I GH& ( ¹ & ( à (J&šµg '& ( ( ([( 0 pŠ
à ®& µE à &šµ'&
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B. KAYMAKÇALAN AND A. ZAFER
½ ŽJ& ( T yµ
(5.5)
(( ½ ŽQ& ( à &šµ ( à ( &šµ'& ([( Ë Ã ®& µE ( ( g à š& µ' ([& (‹
µE
¬_&
¬"&
¬_&Z'& Ì T y
where ¬ is a solution of
((
(5.7)
¬ ` ¹ & ( à &®µE à (¸&®µEg [à '&®& µE['& ( ( ¬/͉
and
(5.6)
Then every solution of (5.1) is oscillatory.
{J& (
{Q& (
{ ` &(
Proof. We may assume that
is eventually positive, since a similar argument holds
when
is eventually negative. By (d)
is either eventually positive or eventually
negative.
on
for some large . From (5.1) we have
Case 1.
{w`& (r0 p
†µ (
( (
¸
(
g
(
{ `'` & JŽ & {J&ºtb& Ê»§$pŠ
(5.8)
0 p,
Integrating (5.8) over we see that for some constant Î
o .]( .
Y
(
I
¸
(
g
{ ` & { ` & … Î oZÏ ŽQ& T pq
(8
¬"& ( h { ` & p
µ (
which clearly contradicts (5.5).
Case 2.
on
for some large . Let be a solution of (5.7) satisfying
. In view of (5.4) we see that the function is positive for all
. In fact, it can
be expressed in terms of the exponential function. It follows from (5.1) that
¬
¬
?«
&œ¬]{ ` ( ` E{ ` Ë ¬ ` I ¬ Í ¹ { ` w{g ` { ` Í Ì I ¬ Í ŽQ
{Q&œt& ( ( » ‰
Employing (5.3) leads to
&œ¬]{ ` ( ` 2{ ` Ë ¬ ` I ¬ Í ¹ { x à { g Í x { Ã Í x Ã Í Ì I ¬ Í ŽQ
{Q&ºtb& ([( » (
from which, on using the nonincreasing nature of {Q& as well as (5.7), we have
hxÃ
(
I
¹
&œ¬]{ ` ` 2{ ` Ë ¬ ` ¬ Í h x à gh x Ã Í Ì I ¬ Í ŽQ
{Q&œt& ( ( » I ¬ Í ŽB{ » and so
I& &œ¬]{ ` ( ™ · ( ` &œ¬]{w` ( ` Ë h g h Ì
¬]{ ` ¬ Í { ` Í ¬]h{ ` ¬ Í h{ ` Í
«iI ŽB{ » Ë g
¬]{ ` { ` Í ¬]{ ` ¬ Í { ` Í Ì ‰
Employing (5.3), it is not difficult to see from the last inequality that
· ( « Ž à ·· Ã Í g Ž Ã Ã Í ·
I
(
™
(5.9)
& &œ¬]{ ` `
¬
¬]¬ Í ‰
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· «2 to and using I œ& ¬"& ( {`1& ([( ™ · 8 p , we have
Integrating (5.9) from
o Ž à ·… · Ã Í g Ž Ã Ã Í · r8 · ( · ( ( ™ ·
oœÐ Ë ¬
¬]¬ Í Ì T &œ¬"& { ` & LANGENHOP INEQUALITY AND APPLICATIONS
{Q&ºtb& ([( »
which obviously contradicts (5.6).
It is possible to prove a similar theorem when the term
in (5.1) is replaced
by
, where
. In this situation, one has to be careful though, with
tedious calculations coming into picture that arise from the complicated nature of the time
scale calculus. Since our focus was to illustrate an application of the Langenhop inequality
on time scales, for simplicity we have only considered a particular case, namely
.
X {Q&œt& ( (\X Ñ ™ …\{Q&œt& ( (
Ò 0eh
ÒEÓ
REFERENCES
[1] R. P. A GARWAL , Difference Equations and Inequalities, Marcel Dekker Inc., New York, 1992.
[2] M. B OHNER AND A. P ETERSON , Dynamic Equations on Time Scales, An Introduction with Applications,
Birkhäuser, Boston, 2001.
[3] S. H ILGER , Analysis on measure chains–a unified approach to continuous and discrete calculus, Results
Math., (1990), pp. 18–56.
[4] B. K AYMAKÇALAN , Existence and comparison results for dynamic systems on time scale, J. Math. Anal.
Appl., 172 (1993), pp. 243-255.
[5] B. K AYMAKÇALAN , Monotone Iterative Method for Dynamic Systems on Time Scales, Dynam. Systems
Appl., 2 (1993), pp. 213-220.
[6] B. K AYMAKÇALAN , V. L AKSMIKANTHAM , AND S. S IVASUNDARAM , Dynamical Systems on Measure
Chains, Kluwer Academic Publishers, Boston, 1996.
[7] B. K AYMAKÇALAN , S. A. Ö ZG ÜN AND A. Z AFER , Asymptotic behavior of higher-order nonlinear eqautions on time scales, Comput. Math. Appl., 36 (1990), pp. 299–306.
[8] C. E. L ANGENHOP , Bounds on the norm of a solution of a general differential equation, Proc. Amer. Math.
Soc., 11 (1960), pp. 795–799.
[9] M. N AITO , Oscillation theorems for a damped nonlinear differential equation, Proc. Japan Acad., 50 (1974),
pp. 104–108.
[10] H. O NOSE , An oscillation theorem for a damped nonlinear differential equation, Mathematica (Cluj), 17
(1975), pp. 203–207.
[11] B. G. PACHPATTE , Inequalities for Finite Difference Equations, Marcel Dekker, New York, 2002.
[12] A. Z AFER , Applications of Langenhop inequality to difference equations: Lower bounds and oscillation,
Appl. Math. E-Notes, 3 (2003), pp. 80–87.