The paper deals with the state-dependent 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 left-continuous regulated functions on equipped with the sup-norm. The functional ℓ is represented by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of first-order differential equations subject to state-dependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time.
MSC: 34B37, 34B15.
Keywords:first-order ODE; state-dependent impulses; transversality conditions; general linear boundary conditions; existence; Kurzweil-Stieltjes integral
The investigation of impulsive differential equations has a long history; see, e.g., the monographs [1-3]. 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 state-dependent impulses. Boundary value problems with state-dependent impulses, where difficulties with an operator representation appear (cf. Remark 6.2), are substantially less developed. We refer to the papers [4-6] and  which are devoted to periodic problems, and for problems with other boundary conditions, see [8,9] or [10-12].
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 first-order differential equation subject to state-dependent 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).
is a Banach space.
• Since , we equip the sets and with the norm and get also Banach spaces (cf.). Then (1.2) can be written as
2 Formulation of problem
We investigate the solvability of the nonlinear differential equation
subject to the state-dependent impulse condition
and the general linear boundary condition
Here we assume that
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 state-dependent impulses, the Banach space of piece-wise continuous functions on with the sup-norm 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 Kurzweil-Stieltjes integral (cf., Theorem 3.8). Representation (2.5) is correct on , because for each the integral exists. Its definition and properties can be found in  (see Perron-Stieltjes integral based on the work of Kurzweil).
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
fulfils condition (3.4).
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 Kurzweil-Stieltjes 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 two-point boundary condition
Example 3.7 Consider a solution x of problem (3.3), (3.2), where ℓ has a form of the multi-point boundary condition
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
Let us put
In both cases, G is written as
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
The following two lemmas for functions from ℬ are the modifications of lemmas in  and provide the transversality (cf. Remark 2.3) which will be essential for operator constructions in Section 6.
where τ fulfils (5.2).
Lemma 5.2Letγsatisfy (2.4), (4.3) and (4.5). Then the functionalis continuous.
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
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
Differentiating (6.5) and using (3.6) and (6.3), we get
This together with (4.1) yields
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
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).
Similarly as in Step 1, we prove (cf. (6.15), (6.19), (6.20))
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 piece-wise 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 state-dependent point , and with τ instead of should be investigated on the space . But if we write a state-dependent τ instead of a fixed in (6.21), loses its continuity on , which we show in the next example.
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
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
By virtue of (7.2)-(7.5), we have
So, (7.8) is valid. If the second condition in (4.6) is fulfilled, we use the dual arguments.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.
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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