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SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 315-331
Copyright © 2015, Korean Society of Mathematical Education
  • Received : March 09, 2015
  • Accepted : September 07, 2015
  • Published : November 30, 2015
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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
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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
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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
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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
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. A positive function δ on [ a , b ] is called a gauge on [ a , b ].
From now on we use notation
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.
Definition 2.2 ( [6 , 9] ) . Let δ be a gauge on [ a , b ]. A tagged partition
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of [ a , b ] is said to be δ fine if for every
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we have
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Moreover if a δ −fine partition P satisfies the implications
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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
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of [ a , b ] such that ti ∈ Ω for
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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
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provided
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is a δ −fine tagged partition of [ a , b ]. In this case we define
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(or, shortly,
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).
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
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.
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
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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
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g dh exists and f is bounded on [ a , b ], then the integral
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exists if and only if the integral
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exists and in this case we have
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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 ]
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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
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. Then we have
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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
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provided that the limit exists.
If g is not locally constant at t = a and t = b respectively, we define
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respectively, provided that the limits exist. Frequently we use
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instead of
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.
If both f and g are constant on some neighborhood of t , then we define
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.
Remark 3.2. It is obvious that if g is not continuous at t then
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exists. Thus if
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does not exist then g is continuous at t .
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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
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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
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exists at every t ∈ [ a , b ] − { c 1 , c 2 , …}, where f is continuous at every t ∈ { c 1 , c 2 , …}. Then we have
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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
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and
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exist and f 1 , f 2 G ([ a , b ]), t hen we have
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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
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and
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exist for every t ∈ [ a , b ] and
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. Then we have
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Proof . By Theorem 2.8 and Theorem 3.4 we have
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So by Theorem 2.6 we get
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This completes the proof.   ☐
Let
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The Heaviside function Hτ : R → {0, 1} is defined by
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Using the Heaviside function Hτ , we define function ϕ : [ a , b ] → R by
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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 ,
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Then we have
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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
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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
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.
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 ,
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}.
From now on we always assume that u ,
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, and we define
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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
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where Kα = inf h∈[a, b] max{ α ( h ) − α ( a ), α ( b ) − α ( h )}.
Proof . Let for t ∈[ a , b ],
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By Theorem 2.9, the functions, y and z are left-continuous on [ a , b ]. Also by Theorem 3.3, we have
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and we have by Theorem 3.4 and u ( a ) = u ( b ) = 0
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for t ∈ [ a , b ]. So by Theorem 3.4, Lemma 3.5, and using Hölder’s inequality, we get
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and similarly we obtain
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So we have
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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
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Proof . From (4.2) we have
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So we get
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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
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where
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,
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for m ≠ 1, and
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for m = 1.
Proof . Let for t ∈ [ a , b ],
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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 ,
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, Then by Theorem 3.3,
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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,
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If z is constant on some neighborhood of t , then since
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, the above equality is also true. If t = tk ,
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, since z is non-increasing on [ a , b ], and
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and by the Mean Value Theorem, and by the definition of the Stieltjes derivatives, we have,
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where z ( tk +) ≤ ω z ( tk ) = z( tk −): Thus we have
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Let
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. Then by hypotheses, β is strictly increasing on [ a , b ].
Since
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we have by Theorem 3.4 and (4.5), since z ( b ) = 0 and
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,
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Using Hölder’s inequality with indices m ,
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, we have
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Integrating (4.6) on [ a , b ] and using Hölder’s inequality with indices q ,
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, and considering
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by Theorem 2.8, we get
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If
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, then
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is obviously true, otherwise, dividing both sides of (4.7) by
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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
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,
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, we have, by (4.8),
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Using Hölder’s inequality with indices
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,
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, we get by (4.9)
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If
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, then the inequality
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is obviously true, otherwise, dividing both sides of (4.10) by
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and then taking the
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power on both sides of the resulting inequality we get also (4.11). Since | u | ≤ z and
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we have
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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
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,
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where q 1 PC ([ a , b ]): Now we define
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Since by Lemma 3.8 for
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the equation (5.1) implies the following equation:
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where
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We need the following result.
Lemma 5.1. If the function u satisfies the equation (5.1) and c ∈ [ a , b ], then we have
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Proof . In the proof, we frequently use Lemma 3.8,
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.
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, and
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,
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.
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This gives (5.4). And
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This gives (5.5). And
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This gives (5.6). Also
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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
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where
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, 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
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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,
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,
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Since both u and u' are left-continuous
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By Lemma 3.8 and Lemma 5.1, we get, since u ( c ) = u' ( a ) = 0,
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By (5.9), (5.10) and (5.11), we have
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Hence by Theorem 4.3 and Lemma 5.1, we get
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If
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then, since
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,
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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
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.
In (5.12), canceling
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, 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:
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then we have
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Proof . Substituting f ≡ 1, p = 0, q = 1, r = 0 into Theorem 4.4, then we have
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So we have
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Canceling
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, we get (5.13).   ☐
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