Research

# First-order nonlinear differential equations with state-dependent impulses

Lukáš Rachůnek and Irena Rachůnková*

Author Affiliations

Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic

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Boundary Value Problems 2013, 2013:195  doi:10.1186/1687-2770-2013-195

 Received: 13 May 2013 Accepted: 13 August 2013 Published: 28 August 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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 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 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

### 1 Introduction

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 [7] 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).

#### Notation

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

(i) is measurable for all ,

(ii) 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.[13]). Then (1.2) can be written as

(1.3)

and

(1.4)

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

We investigate the solvability of the nonlinear differential equation

(2.1)

subject to the state-dependent impulse condition

(2.2)

and the general linear boundary condition

(2.3)

Here we assume that

(2.4)

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 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 in the general linear boundary condition (2.3) can be written uniquely in the form

(2.5)

where , and is the Kurzweil-Stieltjes 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 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

For further investigation, we will need a linear homogeneous problem corresponding to problem (2.1)-(2.3). Such problem has the form

(3.1)

(3.2)

because the impulse in (2.2) disappears if . We will also work with the non-homogeneous equation

(3.3)

where .

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))

(3.4)

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 ;

(ii) for any ;

(iii) for any , the function

(3.5)

fulfils condition (3.4).

Lemma 3.4Letbe from (2.5) withand.

(i) if and only if there exists the Green’s functionGof problem (3.1), (3.2) which has the form

(3.6)

(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), whereis from (2.5) and. Then, for each, the functionbelongs toand

(3.7)

Proof Choose . By (3.6),

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 has a form of the two-point boundary condition

(3.8)

We will show that can be expressed in a form of (3.4). If , then k and v can be found from the equality

Assuming that , we get

and hence , . In addition, if , then (cf. (3.6))

Example 3.7 Consider a solution x of problem (3.3), (3.2), where has a form of the multi-point boundary condition

(3.9)

Here . If , then k and v of (3.4) can be found from the equality

(3.10)

Assume that v is a piece-wise constant right-continuous function on , that is,

where , . By (3.10), we get

Consequently,

To summarize, if , then

and further (cf. (3.6))

Example 3.8 Consider a solution x of problem (3.3), (3.2), where has a form of the integral condition

where . If , then k and v of (3.4) can be found from the equality

(3.11)

Let us put

Then

and (3.11) gives , . Consequently,

Similarly, if

and , we derive

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, , and γ.

(i) Boundedness of f

(4.1)

(ii) Boundedness of

(4.2)

(iii) Boundedness of γ

(4.3)

(iv) Properties of

(4.4)

(v) Transversality conditions

(4.5)

(4.6)

where h is from (4.1) and , are from (4.3).

(vi) -continuity of f

(4.7)

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

where , , , .

### 5 Transversality

Consider , and define a set ℬ by

(5.1)

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

(5.2)

Proof Let us take an arbitrary and denote

Then, by (2.4) and (5.1), we see that and

Since , condition (4.3) gives

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,

(5.3)

Hence τ is a unique zero of σ, and (4.3) yields . □

Due to Lemma 5.1, we can define a functional by

(5.4)

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

(5.5)

and there exists such that

(5.6)

So, by virtue of (1.5) and (5.5),

We see that . For , define

By Lemma 5.1,

(5.7)

We need to prove that

(5.8)

Conditions (2.4), (1.5) and (5.6) yield

(5.9)

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

(6.1)

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

(6.2)

and for , we define a function as follows. We set, for a.e. ,

(6.3)

where is defined by (5.4) and the point is uniquely determined due to Lemma 5.1. By (4.1)

(6.4)

Now, we can define an operator by , where

(6.5)

(6.6)

Here the functionals and are defined such that the functions and are continuous at the point . Therefore

(6.7)

Differentiating (6.5) and using (3.6) and (6.3), we get

(6.8)

This together with (4.1) yields

(6.9)

Since (cf. (4.4)), we see that (6.4)-(6.6), (3.6), (4.1) and (4.2) give

(6.10)

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 andare given by (6.2) and (6.5), (6.6), respectively. Then the operatoris 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

(6.11)

Lemma 5.1 and Lemma 5.2 yield

(6.12)

where is defined by (5.4). Denote

(6.13)

We will prove that

(6.14)

By (4.7), (6.8), (6.11) and (6.13),

(6.15)

Using (4.1), we get

(6.16)

Since

the Lebesgue dominated convergence theorem and (6.16) give

(6.17)

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),

(6.18)

Using (6.13) and integrating (6.8), we get

and, due to (6.15) and (6.18), we arrive at

(6.19)

Similarly, we derive

(6.20)

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 Arzelà-Ascoli 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 piece-wise continuous functions on and having the form

(6.21)

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.

Example 6.3 Let , and 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

(6.22)

Further, let the operatorbe given by (6.5), (6.6). Thenhas 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

(7.1)

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

(7.2)

(7.3)

where G, , , , are given by (3.6), (5.4), (6.3), (6.7), respectively. Recall that is a unique point in satisfying

(7.4)

For , define a function z by

(7.5)

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

(7.6)

Let us show that is a unique solution of the equation

(7.7)

in . According to (7.4) and (7.5), it suffices to prove

(7.8)

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 non-increasing, 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

(7.9)

Put

(7.10)

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.

### References

1. Bainov, D, Simeonov, P: Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow (1993)

2. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

3. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific, Singapore (1995)

4. Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl.. 204, 65–73 (1996). PubMed Abstract | Publisher Full Text

5. Belley, J, Virgilio, M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl.. 306, 646–662 (2005). Publisher Full Text

6. Belley, J, Virgilio, M: Periodic Liénard-type delay equations with state-dependent impulses. Nonlinear Anal. TMA. 64, 568–589 (2006). Publisher Full Text

7. Frigon, M, O’Regan, D: First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl.. 233, 730–739 (1999). Publisher Full Text

8. Benchohra, M, Graef, JR, Ntouyas, SK, Ouahab, A: Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and variable times. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal.. 12, 383–396 (2005)

9. Frigon, M, O’Regan, D: Second order Sturm-Liouville BVP’s with impulses at variable times. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal.. 8, 149–159 (2001)

10. Rachůnková, I, Tomeček, J: A new approach to BVPs with state-dependent impulses. Bound. Value Probl.. 2013, (2013) Article ID 22

11. Rachůnková, I, Tomeček, J: Second order BVPs with state-dependent impulses via lower and upper functions. Cent. Eur. J. Math. (to appear)

12. Rachůnková, I, Tomeček, J: Existence principle for BVPs with state-dependent impulses. Topol. Methods Nonlinear Anal. (submitted)

13. Tvrdý, M: Linear integral equations in the space of regulated functions. Math. Bohem.. 123, 177–212 (1998)

14. Tvrdý, M: Regulated functions and the Perron-Stieltjes integral. Čas. Pěst. Mat.. 114, 187–209 (1989)

15. Schwabik, Š, Tvrdý, M, Vejvoda, O: Differential and Integral Equations, Academia, Prague (1979)

16. Rachůnková, I, Tomeček, J: Singular Dirichlet problem for ordinary differential equation with impulses. Nonlinear Anal. TMA. 65, 210–229 (2006). Publisher Full Text

17. Rachůnková, I, Tomeček, J: Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions. J. Math. Anal. Appl.. 292, 525–539 (2004). Publisher Full Text