First-order nonlinear differential equations with state-dependent impulses
© Rachůnek and Rachůnková; licensee Springer 2013
Received: 13 May 2013
Accepted: 13 August 2013
Published: 28 August 2013
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.
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).
Let , , .
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 left-continuous regulated functions on , that is, if and only if , and for each and each ,(1.1)
is the set of functions such that
is measurable for all ,
is continuous for a.e. ,
- (iii)for each compact set , there exists satisfying
The set equipped with the norm(1.2)
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(1.3)
is the Banach space of functions such that and , where the norm is given by(1.5)
is the characteristic function of a set A, where .
2 Formulation of problem
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 barrier γ.
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.
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).
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
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).
where , and the Lebesgue integral is used.
for any , the restrictions , are solutions of equation (3.1) and , where ;
for any ;
- (iii)for any , the function(3.5)
fulfils condition (3.4).
- (i)if and only if there exists the Green’s function G of problem (3.1), (3.2) which has the form(3.6)
if and only if there exists a unique solution x of problem (3.3), (3.4), which has a form of (3.5) with G from (3.6).
- (i)Boundedness of f(4.1)
- (iii)Boundedness of γ(4.3)
- (iv)Properties of ℓ(4.4)
- (v)Transversality conditions(4.5)
- (vi)-continuity of f(4.7)
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.
Clearly, if f is continuous on , then f fulfils (4.7).
- (d)Let there exist , and , , such that
where , , , .
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.
In addition .
Hence τ is a unique zero of σ, and (4.3) yields . □
where τ fulfils (5.2).
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
Due to (6.8)-(6.10), we see that , , and the operator ℱ is defined well.
Lemma 6.1 Assume 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 .
Properties (6.15), (6.19) and (6.20) yield (6.14).
which gives by (1.5) that is convergent in . □
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.
due to (3.6). Hence and we have also , and is not continuous on .
Lemma 6.1 results in the following theorem.
Further, let the operator ℱ be given by (6.5), (6.6). Then ℱ has a fixed point in .
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.
So, (7.8) is valid. If the second condition in (4.6) is fulfilled, we use the dual arguments.
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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