SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
The Pure and Applied Mathematics. 2015. Nov, 22(4): 315-331
• Received : March 09, 2015
• Accepted : September 07, 2015
• Published : November 30, 2015
PDF
e-PUB
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
YOUNG JIN, KIM

Abstract
The purpose of this paper is to obtain Opial-type inequalities that are useful to study various qualitative properties of certain differential equations involving impulses. After we obtain some Opial-type inequalities, we apply our results to certain differential equations involving impulses.
Keywords
1. INTRODUCTION
Opial-type inequalities are very useful to study various qualitative properties of differential equations. For a good reference of the work on such inequalities together with various applications, we recommend the monograph [1] . In this paper we obtain some Opial-type inequalities that involve Stieltjes derivatives which are applicable to differential equations with impulses. Differential equations involving impulses arise in various real world phenomena, we refer to the monograph [8] .
2. PRELIMINARIES
To obtain our results in this paper we need some preliminaries.
Let R be the set of all real numbers. Assume that [ a , b ] ⊂ R is a bounded interval. A function f : [ a , b ] → R is called regulated on [ a , b ] if both
PPT Slide
Lager Image
exist for every point s ∈ [ a , b ), t ∈ ( a , b ], respectively. Let G ([ a , b ]) be the set of all regulated functions on [ a , b ]. For f G ([ a , b ]) we define f ( a −) = f ( a ); f ( b +) = f ( b ). For convenience we define
PPT Slide
Lager Image
Remark 2.1. Let f G ([ a , b ]). Since both f ( s +) and f ( s −) exist for every s ∈ [ a , b ] it is obvious that f is bounded on [ a , b ], and since f is the uniform limit of step functions, f is Borel measurable (see [3, Theorem 3.1.]).
For a closed interval I = [ c , d ], we define f ( I ) = f ( d ) − f ( c ).
A tagged interval ( τ , [ c , d ]) in [ a , b ] consists of an interval [ c , d ] ⊂ [ a , b ] and a point τ ⊂ [ c , d ].
Let Ii = [ ci , di ] ⊂ [ a , b ], i = 1, …, m . We say that the intervals Ii are pairwise non-overlapping if
PPT Slide
Lager Image
for i j where int ( I ) denotes the interior of an interval I .
A finite collection {( τi , Ii ) : i = 1, 2, …, m } of pairwise non-overlapping tagged intervals is called a tagged partition of [ a , b ] if
PPT Slide
Lager Image
. A positive function δ on [ a , b ] is called a gauge on [ a , b ].
From now on we use notation
PPT Slide
Lager Image
.
Definition 2.2 ( [6 , 9] ) . Let δ be a gauge on [ a , b ]. A tagged partition
PPT Slide
Lager Image
of [ a , b ] is said to be δ fine if for every
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
Moreover if a δ −fine partition P satisfies the implications
PPT Slide
Lager Image
then it is called a δ *−fine partition of [ a , b ].
The following lemma implies that for a gauge δ on [ a , b ] there exists a δ *−fine partition of [ a , b ]. This also implies the existence of a δ − partition of [ a , b ].
Lemma 2.3 ( [6] ) . Let δ be a gauge on [ a , b ] and a dense subset ­ Ω⊂ ( a , b ) be given . Then there exists a δ*−fine partition
PPT Slide
Lager Image
of [ a , b ] such that ti ∈ Ω for
PPT Slide
Lager Image
We now give a formal definition of two types of the Kurzweil integrals.
Definition 2.4 ( [6 , 9] ) . Assume that f , g : [ a , b ] → R are given. We say that f d g is Kurzweil integrable (or shortly, K- integrable ) on [ a , b ] and v R is its integral if for every ε > 0 there exists a gauge δ on [ a , b ] such that
PPT Slide
Lager Image
provided
PPT Slide
Lager Image
is a δ −fine tagged partition of [ a , b ]. In this case we define
PPT Slide
Lager Image
(or, shortly,
PPT Slide
Lager Image
).
If, in the above definition, δ −fine is replaced by δ *−fine, then we say that f d g is Kurzweil* integrable (or, shortly, K*- integrable ) on [ a , b ] and we define
PPT Slide
Lager Image
.
Remark 2.5. By the above definition it is obvious that K-integrability implies K*-integrability.
The following results are needed in this paper. For other properties of the K-integrals, see, e.g., [2 , 7 , 9 , 10] .
In this paper BV ([ a , b ]) denotes the set of all functions that are of bounded variation on [ a , b ].
Theorem 2.6 ([ 11 , 2.15. Theorem]) . Assume that f G ([ a , b ]) and g BV ([ a , b ]). Then both f d g and g d f are K-integrable on [ a , b ] and
PPT Slide
Lager Image
Remark 2.7. In the above theorem, the sum Σ t∈[a,b] f ( t g ( t )−Δ + f ( t + g ( t )] is actually a countable sum because every regulated function has only countable dis-continuities.
Theorem 2.8 ([ 10 , p. 40, 4.25. Theorem]) . Let h BV ([ a , b ]), g : [ a , b ] → R and f : [ a , b ] → R . If the integral
PPT Slide
Lager Image
g dh exists and f is bounded on [ a , b ], then the integral
PPT Slide
Lager Image
exists if and only if the integral
PPT Slide
Lager Image
exists and in this case we have
PPT Slide
Lager Image
Theorem 2.9 ([ 10 , p. 34, 4.13. Corollary]) . Assume that f G ([ a , b ]) and g BV ([ a , b ]). Then we have for every t ∈ [ a , b ]
PPT Slide
Lager Image
The following result is the Hölder’s inequality for K-integral. In this paper we frequently use this inequality.
Theorem 2.10. (Hölder’s inequality) Assume that f , g G ([ a , b ]) and h is a nondecreasing function defined on [ a , b ]. Let
PPT Slide
Lager Image
. Then we have
PPT Slide
Lager Image
Proof . The proof of this theorem is very similar to the proof of the classical Hölder’s inequality. So we omit the proof.   ☐
3. STIELTJES DERIVATIVES
In this section we state the results in [4 , 5] that are essential to obtain our main results.
Throughout this section, we assume that f G ([ a , b ]) and g is a nondecreasing function on [ a , b ].
We say that the function g is not locally constant at t ∈ ( a , b ) if there exists η > 0 such that g is not constant on ( t ε , t + ε ) for every 0 < ε < η . We also say that the function g is not locally constant at a and b , respectively if there exist η , η * > 0 such that g is not constant on [ a , a + ε ), ( b ε *, b ] respectively, for every ε ∈ (0, η ), ε *∈ (0, η* ).
Definition 3.1 ( [4] ) . If g is not locally constant at t ∈ ( a , b ), we define
PPT Slide
Lager Image
provided that the limit exists.
If g is not locally constant at t = a and t = b respectively, we define
PPT Slide
Lager Image
respectively, provided that the limits exist. Frequently we use
PPT Slide
Lager Image
PPT Slide
Lager Image
.
If both f and g are constant on some neighborhood of t , then we define
PPT Slide
Lager Image
.
Remark 3.2. It is obvious that if g is not continuous at t then
PPT Slide
Lager Image
exists. Thus if
PPT Slide
Lager Image
does not exist then g is continuous at t .
PPT Slide
Lager Image
is called a Stieltjes derivative of f with respect to g .
Theorem 3.3 ( [4] ) . Assume that if g is not locally constant at t ∈ [ a , b ]. If f is continuous at t or g is not continuous at t; then we have
PPT Slide
Lager Image
K*-integrals recover Stieltjes derivatives.
Theorem 3.4 ( [4] ) . Assume that if g is constant on some neighborhood of t then there is a neighborhood of t where both f and g are constant. Suppose that
PPT Slide
Lager Image
exists at every t ∈ [ a , b ] − { c 1 , c 2 , …}, where f is continuous at every t ∈ { c 1 , c 2 , …}. Then we have
PPT Slide
Lager Image
Lemma 3.5 ( [4] ) . Assume that if g is constant on some neighborhood of t then there is a neighborhood of t such that both f 1 and f 2 are constant there. If both
PPT Slide
Lager Image
and
PPT Slide
Lager Image
exist and f 1 , f 2 G ([ a , b ]), t hen we have
PPT Slide
Lager Image
Similarly to the Riemann integral we have the following integration by parts formula.
Theorem 3.6. (Integration by Parts) Assume that functions f, g, h G ([ a , b ]) are all left-continuous and h is nondecreasing. Suppose that both
PPT Slide
Lager Image
and
PPT Slide
Lager Image
exist for every t ∈ [ a , b ] and
PPT Slide
Lager Image
. Then we have
PPT Slide
Lager Image
Proof . By Theorem 2.8 and Theorem 3.4 we have
PPT Slide
Lager Image
So by Theorem 2.6 we get
PPT Slide
Lager Image
This completes the proof.   ☐
Let
PPT Slide
Lager Image
The Heaviside function Hτ : R → {0, 1} is defined by
PPT Slide
Lager Image
Using the Heaviside function Hτ , we define function ϕ : [ a , b ] → R by
PPT Slide
Lager Image
Remark 3.7. It is obvious that the function ϕ is strictly increasing and of bounded variation on [ a , b ], and left-continuous on [ a , b ].
Lemma 3.8 ( [5] ) . Assume that f G ([ a , b ]) and f' ( t ) exists for t tk ,
PPT Slide
Lager Image
Then we have
PPT Slide
Lager Image
4. OPIAL-TYPE INTEGRAL INEQUALITIES INVOLVING STIELTJES DERIVATIVES
In this section we obtain some Opial-type integral inequalities involving Stieltjes derivatives. The Opial-type inequalities have many interesting applications in the theory of differential equations(see, e.g., [1] ).
Throughout this paper we always assume that
PPT Slide
Lager Image
and that a function α : [ a , b ] → R is strictly increasing on [ a , b ], and continuous at t tk , and Δ α ( tk ) ≠ 0; for every
PPT Slide
Lager Image
.
Remark 4.1. Note that strictly increasing implies nondecreasing, and a nondecreasing function is regulated.
Let PC ([ a , b ]) = { u G ([ a , b ]) : u is continuous at every t tk ,
PPT Slide
Lager Image
}.
From now on we always assume that u ,
PPT Slide
Lager Image
, and we define
PPT Slide
Lager Image
The following result is an Opial-type inequality with Stieltjes derivatives.
Theorem 4.2. Assume that u ( a ) = u ( b ) = 0. If both u and α are left-continuous on [ a , b ], then we have
PPT Slide
Lager Image
where Kα = inf h∈[a, b] max{ α ( h ) − α ( a ), α ( b ) − α ( h )}.
Proof . Let for t ∈[ a , b ],
PPT Slide
Lager Image
By Theorem 2.9, the functions, y and z are left-continuous on [ a , b ]. Also by Theorem 3.3, we have
PPT Slide
Lager Image
and we have by Theorem 3.4 and u ( a ) = u ( b ) = 0
PPT Slide
Lager Image
for t ∈ [ a , b ]. So by Theorem 3.4, Lemma 3.5, and using Hölder’s inequality, we get
PPT Slide
Lager Image
and similarly we obtain
PPT Slide
Lager Image
So we have
PPT Slide
Lager Image
The proof is complete.   ☐
A slightly more general result is as follows.
Theorem 4.3. Assume that u ( b ) = 0. If both u and α are left-continuous on [ a , b ], then we have
PPT Slide
Lager Image
Proof . From (4.2) we have
PPT Slide
Lager Image
So we get
PPT Slide
Lager Image
This gives (4.3). The proof is complete.   ☐
More generally we have the following result.
Theorem 4.4. Let p ≥ 0, q ≥ 1, r ≥ 0, m ≥ 1 be real numbers and let f PC ([ a , b ]) be a positive function on [ a , b ] with inf s∈[a, b] f ( s ) > 0. Assume that both functions u and α are left-continuous on [ a , b ]. If u ( b ) = 0, then we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
,
PPT Slide
Lager Image
for m ≠ 1, and
PPT Slide
Lager Image
for m = 1.
Proof . Let for t ∈ [ a , b ],
PPT Slide
Lager Image
Then by Theorem 3.4, | u ( t )| ≤ z ( t ) and by Theorem 2.9, z is left-continuous, and non-increasing on [ a , b ].
If t tk ,
PPT Slide
Lager Image
, Then by Theorem 3.3,
PPT Slide
Lager Image
exists, and by Theorem 2.9, z is continuous at t . Using the Mean Value Theorem and by the definition of the Stieltjes derivatives, if z is not locally constant at t , then we have,
PPT Slide
Lager Image
If z is constant on some neighborhood of t , then since
PPT Slide
Lager Image
, the above equality is also true. If t = tk ,
PPT Slide
Lager Image
, since z is non-increasing on [ a , b ], and
PPT Slide
Lager Image
and by the Mean Value Theorem, and by the definition of the Stieltjes derivatives, we have,
PPT Slide
Lager Image
where z ( tk +) ≤ ω z ( tk ) = z( tk −): Thus we have
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
. Then by hypotheses, β is strictly increasing on [ a , b ].
Since
PPT Slide
Lager Image
we have by Theorem 3.4 and (4.5), since z ( b ) = 0 and
PPT Slide
Lager Image
,
PPT Slide
Lager Image
Using Hölder’s inequality with indices m ,
PPT Slide
Lager Image
, we have
PPT Slide
Lager Image
Integrating (4.6) on [ a , b ] and using Hölder’s inequality with indices q ,
PPT Slide
Lager Image
, and considering
PPT Slide
Lager Image
by Theorem 2.8, we get
PPT Slide
Lager Image
If
PPT Slide
Lager Image
, then
PPT Slide
Lager Image
is obviously true, otherwise, dividing both sides of (4.7) by
PPT Slide
Lager Image
and then taking the q th power on both sides of the resulting inequality we get also (4.8).
Using the Hölder’s inequality with indice
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, we have, by (4.8),
PPT Slide
Lager Image
Using Hölder’s inequality with indices
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, we get by (4.9)
PPT Slide
Lager Image
If
PPT Slide
Lager Image
, then the inequality
PPT Slide
Lager Image
is obviously true, otherwise, dividing both sides of (4.10) by
PPT Slide
Lager Image
and then taking the
PPT Slide
Lager Image
power on both sides of the resulting inequality we get also (4.11). Since | u | ≤ z and
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
This gives (4.4). The proof is complete.   ☐
5. SOME APPLICATIONS TO CERTAIN DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
In this section we always assume that both functions u and u' are left- continuous on [ a , b ], and that α = ϕ (see (3.2)). Consider the following impulsive differential equation: for
PPT Slide
Lager Image
,
PPT Slide
Lager Image
where q 1 PC ([ a , b ]): Now we define
PPT Slide
Lager Image
Since by Lemma 3.8 for
PPT Slide
Lager Image
PPT Slide
Lager Image
the equation (5.1) implies the following equation:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
We need the following result.
Lemma 5.1. If the function u satisfies the equation (5.1) and c ∈ [ a , b ], then we have
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . In the proof, we frequently use Lemma 3.8,
PPT Slide
Lager Image
.
PPT Slide
Lager Image
, and
PPT Slide
Lager Image
,
PPT Slide
Lager Image
.
PPT Slide
Lager Image
This gives (5.4). And
PPT Slide
Lager Image
This gives (5.5). And
PPT Slide
Lager Image
This gives (5.6). Also
PPT Slide
Lager Image
This gives (5.7). The proof is complete.   ☐
Theorem 5.2. Assume that u satisfies the equation (5.1) and u' ( a ) = 0, u ( a ) ≠ 0. If we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, then u ( t ) ≠ 0 for every t ∈ [ a , b ].
Proof . Assume that there is a number c ∈ ( a , b ] with u ( c ) = 0. Then multiplying both sides of (5.2) by u and integrating we have
PPT Slide
Lager Image
Using Theorem 3.3, Lemma 3.5 and Theorem 3.6, and u ( c ) = Q ( a ) = 0, we get, since, by Theorem 2.9 and Remark 3.7, Q is left-continuous on [ a , b ]; and Δ α ( tk ) = Δ + α ( tk ) = 1, q ( tk ) = 0,
PPT Slide
Lager Image
,
PPT Slide
Lager Image
Since both u and u' are left-continuous
PPT Slide
Lager Image
By Lemma 3.8 and Lemma 5.1, we get, since u ( c ) = u' ( a ) = 0,
PPT Slide
Lager Image
By (5.9), (5.10) and (5.11), we have
PPT Slide
Lager Image
Hence by Theorem 4.3 and Lemma 5.1, we get
PPT Slide
Lager Image
If
PPT Slide
Lager Image
then, since
PPT Slide
Lager Image
,
PPT Slide
Lager Image
and u' α ( tk ) = u ( tk +) − u ( tk ) = 0. This implies that u is a constant on [ a , c ]. So u ( c ) = u ( a ) ≠0. But this is a contradiction to u ( c ) = 0. Hence we conclude that
PPT Slide
Lager Image
.
In (5.12), canceling
PPT Slide
Lager Image
, we get a contradiction to (5.8). This completes the proof.   ☐
In the following result we apply Theorem 4.4.
Theorem 5.3. Let q PC ([ a , b ]) and let α = ϕ (see (3.2)). If u PC ([ a , b ]) is left-continuous and a nontrivial solution of the following equation:
PPT Slide
Lager Image
then we have
PPT Slide
Lager Image
Proof . Substituting f ≡ 1, p = 0, q = 1, r = 0 into Theorem 4.4, then we have
PPT Slide
Lager Image
So we have
PPT Slide
Lager Image
Canceling
PPT Slide
Lager Image
, we get (5.13).   ☐
References
Agarwal R.P. , Pang P.Y.H. 1995 Opial inequalities with applications in differential anddifference equations Kluwer Academic Publishers Dordrecht
Henstock R. 1988 Lectures on the theory of integration World Scientific Singapore
Hönig C.S. 1973 Volterra Stieltjes-integral equations North Holand and American Elsevier, Mathematics Studies Amsterdam and New York
Kim Y.J. (2011) Stieltjes derivatives and its applications to integral inequalities of Stieltjes type J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (1) 63 - 78
Kim Y.J. (2014) Stieltjes derivative method for integral inequalities with impulses J. Korean Soc. Math. Educ.Ser. B: Pure Appl. Math. 21 (1) 61 - 75
Krejčí P. , Kurzweil J. (2002) A nonexistence result for the Kurzweil integral Math. Bohem. 127 571 - 580
Pfeffer W.F. 1993 The Riemann approach to integration: local geometric theory Cambridge University Press
Samoilenko A.M. , Perestyuk N.A. 1995 Impulsive differential equations World Scientific Singapore
Schwabik Š. 1992 Generalized ordinary differential equations World Scientific Singapore
Schwabik Š. , Tvrdý M. , Vejvoda O. 1979 Differntial and integral equations: boundary value problems and adjoints Academia and D. Reidel, Praha and Dordrecht
Tvrdý M. (1989) Regulated functions and the Perron-Stieltjes integral Časopis pešt. mat. 114 (2) 187 - 209