Abstract
The paper deals with the statedependent impulsive problem
where , , f fulfils the Carathéodory conditions on , the impulse function is continuous on , the barrier function γ has a continuous first derivative on some subset of ℝ and ℓ is a linear bounded functional which is defined on the Banach space of leftcontinuous regulated functions on equipped with the supnorm. The functional ℓ is represented by means of the KurzweilStieltjes integral and covers all linear boundary conditions for solutions of firstorder differential equations subject to statedependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time.
MSC: 34B37, 34B15.
Keywords:
firstorder ODE; statedependent impulses; transversality conditions; general linear boundary conditions; existence; KurzweilStieltjes integral1 Introduction
The investigation of impulsive differential equations has a long history; see, e.g., the monographs [13]. Most papers dealing with impulsive differential equations subject to boundary conditions focus their attention on impulses at fixed moments. But this is a very particular case of a more complicated case with statedependent impulses. Boundary value problems with statedependent impulses, where difficulties with an operator representation appear (cf. Remark 6.2), are substantially less developed. We refer to the papers [46] and [7] which are devoted to periodic problems, and for problems with other boundary conditions, see [8,9] or [1012].
Here, in our paper, we present an approach leading to a new existence principle for impulsive boundary value problems. This approach is applicable to each linear boundary condition which is considered with some firstorder differential equation subject to statedependent impulses. The important step is a proof of a transversality (Remark 2.3 and Lemmas 5.1 and 5.2), which makes possible a construction of a continuous operator (Section 6) whose fixed point leads to a solution of our original impulsive problem (Section 7).
Notation
• is the set of real functions continuous on M.
• is the set of real functions absolutely continuous on M.
• is the set of real functions Lebesgue integrable on .
• is the set of real functions essentially bounded on .
• is the set of real functions with bounded variation on .
• is the set of real leftcontinuous regulated functions on , that is, if and only if , and for each and each ,
• is the set of functions such that
(iii) for each compact set , there exists satisfying
• The set equipped with the norm
is a Banach space.
• Since , we equip the sets and with the norm and get also Banach spaces (cf.[13]). Then (1.2) can be written as
and
• is the Banach space of functions such that and , where the norm is given by
2 Formulation of problem
We investigate the solvability of the nonlinear differential equation
subject to the statedependent impulse condition
and the general linear boundary condition
Here we assume that
and is a linear bounded functional.
Definition 2.1 A function is a solution of problem (2.1), (2.2) if
• there exists a unique such that ;
• the restrictions and are absolutely continuous;
• z satisfies equation (2.1) for a.e. .
Definition 2.2 A graph of a function is called a barrierγ.
Remark 2.3 Let be the set of all solutions of problem (2.1), (2.2). According to Definition 2.1, each function satisfies a transversality property, which means that the graph of z crosses a barrier γ at a unique point , where the impulse acts on z. After that (for ) the graph of z lies on the right of the barrier γ. This transversality property follows from transversality conditions (cf. (4.5), (4.6)) and it is proved in Section 5.
Assume that and . Then there exists a unique such that for and can occur. Therefore different functions from can have their discontinuities at different points from . Our aim in this paper is to prove the existence of a solution of problem (2.1), (2.2) satisfying the general linear boundary condition (2.3). To do this, we need a suitable linear space containing . Due to statedependent impulses, the Banach space of piecewise continuous functions on with the supnorm cannot be used here. Therefore we choose the Banach space . Clearly, by (1.1), . The operator ℓ in the general linear boundary condition (2.3) can be written uniquely in the form
where , and is the KurzweilStieltjes integral (cf.[14], Theorem 3.8). Representation (2.5) is correct on , because for each the integral exists. Its definition and properties can be found in [15] (see PerronStieltjes integral based on the work of Kurzweil).
Definition 2.4 A function is a solution of problem (2.1)(2.3) if z is a solution of problem (2.1), (2.2) and fulfils (2.3).
3 Green’s function
For further investigation, we will need a linear homogeneous problem corresponding to problem (2.1)(2.3). Such problem has the form
because the impulse in (2.2) disappears if . We will also work with the nonhomogeneous equation
Definition 3.1 A solution of problem (3.3), (3.2) is a function satisfying equation (3.3) for a.e. and fulfilling condition (3.2).
Remark 3.2 If x is a solution of problem (3.3), (3.2), then x belongs to , and consequently condition (3.2) can be written in the form (cf. (2.5))
where , and the Lebesgue integral is used.
Definition 3.3 A function is the Green’s function of problem (3.1), (3.2) if
(i) for any , the restrictions , are solutions of equation (3.1) and , where ;
fulfils condition (3.4).
Lemma 3.4Letℓbe from (2.5) withand.
(i) if and only if there exists the Green’s functionGof problem (3.1), (3.2) which has the form
(ii) if and only if there exists a unique solutionxof problem (3.3), (3.4), which has a form of (3.5) withGfrom (3.6).
Proof Clearly, G given by (3.6) fulfils (i) and (ii) of Definition 3.3 if and only if . A general solution of equation (3.3) is , where . By (3.4),
The equation
has a unique solution c if and only if . Then a unique solution x of problem (3.3), (3.4) is written as
□
Lemma 3.5LetGbe the Green’s function of problem (3.1), (3.2), whereℓis from (2.5) and. Then, for each, the functionbelongs toand
Consequently, the function belongs to . This yields that the integral exists for each . Note that since is not continuous on , formula (3.4) cannot be used for in place of x. Instead, we use the properties of the KurzweilStieltjes integral which justify the following computation
Hence, by (2.5), we get
□
Example 3.6 Consider a solution x of problem (3.3), (3.2), where ℓ has a form of the twopoint boundary condition
We will show that ℓ can be expressed in a form of (3.4). If , then k and v can be found from the equality
and hence , . In addition, if , then (cf. (3.6))
Example 3.7 Consider a solution x of problem (3.3), (3.2), where ℓ has a form of the multipoint boundary condition
Here . If , then k and v of (3.4) can be found from the equality
Assume that v is a piecewise constant rightcontinuous function on , that is,
Consequently,
and further (cf. (3.6))
Example 3.8 Consider a solution x of problem (3.3), (3.2), where ℓ has a form of the integral condition
where . If , then k and v of (3.4) can be found from the equality
Let us put
Then
and (3.11) gives , . Consequently,
Similarly, if
In both cases, G is written as
4 Assumptions
An existence result for problem (2.1)(2.3) will be proved in the next sections under the basic assumption (2.4) and the following additional assumptions imposed on f, ℓ, and γ.
(i) Boundedness of f
(ii) Boundedness of
(iii) Boundedness of γ
(iv) Properties of ℓ
(v) Transversality conditions
where h is from (4.1) and , are from (4.3).
Remark 4.1
(a) Boundedness of f and can be replaced by more general conditions, for example, growth or sign ones, if the method of a priori estimates is used. See, e.g., [16,17].
(b) Continuity of v on is necessary for the construction of a continuous operator in Section 6. Note that then we need in Example 3.7.
(c) Clearly, if f is continuous on , then f fulfils (4.7).
(d) Let there exist , and , , such that
for a.e. and all . Then f fulfils (4.7). An example of such a function f is
5 Transversality
Consider , and define a set ℬ by
The following two lemmas for functions from ℬ are the modifications of lemmas in [10] and provide the transversality (cf. Remark 2.3) which will be essential for operator constructions in Section 6.
Lemma 5.1Letγsatisfy (2.4), (4.3) and (4.5). Then, for each, there exists a uniquesuch that
Proof Let us take an arbitrary and denote
Then, by (2.4) and (5.1), we see that and
Consequently, there exists at least one zero of σ in . Let be a zero of σ. By virtue of (4.5) and (5.1), we get, for , ,
That is,
Hence τ is a unique zero of σ, and (4.3) yields . □
Due to Lemma 5.1, we can define a functional by
where τ fulfils (5.2).
Lemma 5.2Letγsatisfy (2.4), (4.3) and (4.5). Then the functionalis continuous.
Proof Let us choose a sequence which is convergent in . Then
So, by virtue of (1.5) and (5.5),
By Lemma 5.1,
We need to prove that
Conditions (2.4), (1.5) and (5.6) yield
Let us take an arbitrary . By (5.3) and (5.9) we can find , and such that , for each . By Lemma 5.1 and the continuity of , we see that for , and (5.8) follows. □
6 Fixed point problem
In this section we assume that
and we construct a fixed point problem whose solvability leads to a solution of problem (2.1)(2.3). To this aim, having the set ℬ from (5.1), we define a set Ω by
and for , we define a function as follows. We set, for a.e. ,
where is defined by (5.4) and the point is uniquely determined due to Lemma 5.1. By (4.1)
Now, we can define an operator by , where
Here the functionals and are defined such that the functions and are continuous at the point . Therefore
Differentiating (6.5) and using (3.6) and (6.3), we get
This together with (4.1) yields
Since (cf. (4.4)), we see that (6.4)(6.6), (3.6), (4.1) and (4.2) give
Due to (6.8)(6.10), we see that , , and the operator ℱ is defined well.
Lemma 6.1Assume that (6.1) holds and that Ω and ℱ are given by (6.2) and (6.5), (6.6), respectively. Then the operator ℱ is compact on.
ProofStep 1. We show that ℱ is continuous on . Choose a sequence
which is convergent in , that is, (cf. (1.5)) there exists such that
Lemma 5.1 and Lemma 5.2 yield
where is defined by (5.4). Denote
We will prove that
By (4.7), (6.8), (6.11) and (6.13),
Using (4.1), we get
Since
the Lebesgue dominated convergence theorem and (6.16) give
Using (6.13) and (6.5), we get
The continuity and boundedness of , and v (cf. Lemma 5.2, (2.4), (4.2), (4.4) and (6.12)) imply
wherefrom, by the boundedness of G and (6.17),
Using (6.13) and integrating (6.8), we get
and, due to (6.15) and (6.18), we arrive at
Similarly, we derive
Properties (6.15), (6.19) and (6.20) yield (6.14).
Step 2. We show that the set is relatively compact in . Choose an arbitrary sequence
We need to prove that there exists a convergent subsequence. Clearly, there exists such that
Choose . By (5.1) and (6.2), it holds
for , . Therefore, the ArzelàAscoli theorem yields that there exists a subsequence
which converges in . Consequently, for each , there exists such that for each ,
Similarly as in Step 1, we prove (cf. (6.15), (6.19), (6.20))
which gives by (1.5) that is convergent in . □
Remark 6.2 If there exists such that for , then problem (2.1)(2.3) has an impulse at fixed time and a standard operator , acting on the space of piecewise continuous functions on and having the form
can be used instead of the operator ℱ from (6.5), (6.6). But this is not possible if γ is not constant on . The reason is that then an impulse is realized at a statedependent point , and with τ instead of should be investigated on the space . But if we write a statedependent τ instead of a fixed in (6.21), loses its continuity on , which we show in the next example.
Example 6.3 Let , and ℓ be from (2.5) with , and . Consider the functions
Clearly, uniformly on and hence
For , denote and . Assume that the barrier γ is given by the linear function on ℝ and the impulse function for , . Then
and, according to (6.21), we have for
Consequently,
due to (3.6). Hence and we have also , and is not continuous on .
Lemma 6.1 results in the following theorem.
Theorem 6.4Assume that (6.1) holds and that the set Ω is given by (6.2), where
Further, let the operator ℱ be given by (6.5), (6.6). Then ℱ has a fixed point in.
Proof By Lemma 6.1, ℱ is compact on . Due to (5.1), (6.2), (6.5), (6.6), (6.10) and (6.22),
Therefore, the Schauder fixed point theorem yields a fixed point of ℱ in . □
7 Main result
The main result, which is contained in Theorem 7.1, guarantees the solvability of problem (2.1)(2.3) provided the data functions f, and γ are bounded (cf. (4.1)(4.3)). As it is mentioned in Remark 4.1, Theorem 7.1 serves as an existence principle which, in combination with the method of a priori estimates, can lead to more general existence results for unbounded f and and concrete boundary conditions.
Theorem 7.1Assume that (6.1) and (6.22) hold. Then there exists a solutionzof problem (2.1)(2.3) such that
Proof By Theorem 6.4, there exists which is a fixed point of the operator ℱ defined in (6.5) and (6.6). This means that
where G, , , , are given by (3.6), (5.4), (6.3), (6.7), respectively. Recall that is a unique point in satisfying
Differentiating (7.2), (7.3) and using (3.6) and (6.3), we get for a.e. , , and consequently
By virtue of (7.2)(7.5), we have
Let us show that is a unique solution of the equation
in . According to (7.4) and (7.5), it suffices to prove
Since , we have (cf. (5.1) and (6.2))
Assume that the first condition in (4.6) is fulfilled. Then , for . Put
By (7.6), , and since γ is nonincreasing, we have
due to (7.4). Using (4.5), we derive for
So, (7.8) is valid. If the second condition in (4.6) is fulfilled, we use the dual arguments.
Finally, let us check that . By (7.2)(7.6) and (3.6), we have
Put
Then, according to (iii) of Definition 3.3 and Remark 3.2, we get . Further, using (3.7) from Lemma 3.5, we arrive at . Consequently, due to (2.5), (7.9) and (7.10), results in
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
This research was supported by the grant Matematické modely, PrF_2013_013. The authors thank the referees for suggestions which improved the paper.
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