The paper provides an existence principle for a general boundary value problem of the form , a.e. , , , with the state-dependent impulses , where the impulse points t are determined as solutions of the equations , , . Here, , , the functions , , are Lebesgue integrable on and satisfies the Carathéodory conditions on . The impulse functions , , , and the barrier functions , , are continuous on and , respectively. The functionals , , are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on vector functions. Provided the data functions h and are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier at the unique impulse point are presented, and consequently the existence of a solution to the problem is proved.
MSC: 34B37, 34B10, 34B15.
Keywords:nonlinear higher-order ODE; state-dependent impulses; general linear boundary conditions; transversality conditions; fixed point
In this paper we are interested in the nonlinear ordinary differential equation of the nth-order () with state-dependent impulses and general linear boundary conditions on the interval . Studies of real-life problems with state-dependent impulses can be found e.g. in [1-6]. Here we consider the equation
subject to the impulse conditions
and the linear boundary conditions
In what follows we use this notation. Let . By we denote the set of all matrices of the type with real valued coefficients. Let denote the transpose of . Let be the set of all n-dimensional column vectors , where , , and . By we denote the set of all mappings with continuous components. By , , , , , , we denote the sets of all mappings whose components are, respectively, essentially bounded functions, Lebesgue integrable functions, left-continuous regulated functions, absolutely continuous functions, functions with bounded variation and functions with continuous derivatives of the kth order on the interval . By we denote the set of all functions satisfying the Carathéodory conditions on the set . Finally, by we denote the characteristic function of the set .
is a linear space, and equipped with the sup-norm it is a Banach space (see [, Theorem 3.6]). In particular, we set
In this paper we provide sufficient conditions for the solvability of problem (1)-(3). This problem is a generalization of problems studied in the papers [8-10] which are devoted to the second-order differential equation. Other types of initial or boundary value problems for the first- or second-order differential equations with state-dependent impulses can be found in [11-19]. To get the existence results for problem (1)-(3), we exploit the paper  with fixed-time impulsive problems.
Here we assume that
Remark 1 The integral in formula (4) is the Kurzweil-Stieltjes integral, whose definition and properties can be found in . The fact that each linear bounded functional on can be written uniquely in the form described in (4) is proved in . See also .
Now let us define a solution of problem (1)-(3).
2 Problem with impulses at fixed times
In the paper  we have found an operator representation to the special type of problem (1)-(3) having impulses at fixed times. This is the case that the barrier functions in (2) are constant functions, i.e. there exist satisfying such that
Note that boundary value problems for higher-order differential equations with impulses at fixed times have been studied for example in [23-31] and for delay higher-order impulsive equations in [32,33].
Let us summarize the results of the paper  according to our needs. Assume that the linear homogeneous problem
Remark 3 In order to state one of the main results of  we introduce the set of all functions u continuous on the intervals , with from (5), having their derivatives continuously extendable onto these intervals. This set is denoted by . For we define
Equipped with the standard addition, scalar multiplication, and with the norm
Theorem 4 [, Theorem 17]
Remark 5 Let us note that the row vector
Remark 6 Let us put
Then the operator ℋ in Theorem 4 can be written as
Therefore a (unique) solution of problem (17) has the form
Remark 7 Under the assumption (9) we are allowed using (11) to define the functions
for . Let us stress that , as well as , do not depend on the choice of fundamental system , but only on the data of problem (6). The functions possess crucial properties for our approach. From their definition it follows that for each
3 Transversality conditions
The most results for differential equations with state-dependent impulses concern initial value problems. Theorems about the existence, uniqueness or extension of solutions of initial value problems, and about intersections of such solutions with barriers can be found for example in [, Chapter 5].
A different approach has to be used when boundary value problems with state-dependent impulses are discussed and boundary conditions are imposed on a solution anywhere in the interval including unknown points of impulses. This is the case of problem (1)-(3).
Our approach is based on the existence of a fixed point of an operator ℱ in some set (cf. Lemma 12), where is a ball defined in (28). In order to get a fixed point, we need to prove for functions of assertions about their transversality through barriers. Such assertions are contained in Lemmas 9 and 10 and it is important that they are valid for all functions in and not only for solutions of problem (1), (2).
Remark 8 Having the lemmas about the transversality, we will prove in Section 4 the existence of a solution u of problem (1)-(3), which has the following property:
Now, we are ready to formulate the following transversality conditions:
Let us define the set
From (25), (26), (28), and the Mean Value Theorem we obtain
which is a contradiction.
which completes the proof. □
Our next step is to define an appropriate operator representation of the BVP with state-dependent impulses. The first idea would be a direct exploitation of the operator ℋ from Theorem 4, putting in place of . This is not possible for many reasons. First, each acts on the space of functions having continuous derivatives - but we need functions having p discontinuities. Even if we would overcome this difficulty we arrive at a problem of choosing an appropriate Banach space on which ℋ would be acting. According to Remark 8, we search a solution u of problem (1)-(3), which has its jumps (together with ) at the points , (see (31)). In general, these points are different for different solutions. Consequently, such solutions have to be searched in the Banach space . But then there is a difficulty with the continuity of such operator. In fact the operator ℋ from (16) having in place of is not continuous in the space (cf. Remark 6.2 and Example 6.3 in ).
Therefore, we choose the way here, which we have developed in our joint papers [8-10]. The main idea of our approach lies in representing the solution u of problem (1)-(3) by an ordered -tuple as follows:
Consequently, we work with the space
equipped with the norm
It is well known that X is a Banach space.
4 Main results
Let us turn our attention to problem (1)-(3) with state-dependent impulses under the assumptions (4) and (9). In our approach we will make use of the tools introduced in the previous sections.
In addition we assume
Remark 11 Let us note that the constants from (35) do not depend on the choice of the fundamental system of solutions , but only on the coefficients of the differential equation (1) and on the operators from (3) (and, of course, on the constants ).
Now, we are ready to construct a convenient operator for a representation of problem (1)-(3). Let us choose its domain as the closure of the set
Let us compare (16) for the operator ℋ with (37) for the operator ℱ. The first term in (16) expresses a solution of homogeneous boundary value problem without impulses. This term is decomposed in (37) on subintervals which depend on the choice of -tuple . The second term in (16) caused (according to the discontinuity of functions ) needed impulses of solutions of the fixed-time impulsive problem (13)-(15). We significantly modify this term in (37) in such a way that, instead of discontinuous functions which have jumps at the points , we use smooth functions , defined in (18). Due to this modification the operator ℱ maps one tuple of smooth functions onto another tuple of smooth functions , and we will be able to prove the compactness of ℱ on .
In the next lemma we arrive at a justification of our definition.
Proof Let ℬ be defined by (28) and . Further, let be such that . For each , we have , and hence by Lemma 9 and (31), there exists a unique solution of the equation . Due to (24), the inequalities are valid and u can be defined by (33). We will prove that u is a fixed point of the operator ℋ from Theorem 4, taking the space from Remark 3 with
Of course we have
These facts imply that
for . Consequently, by virtue of (16) and Theorem 4, u is a solution of problem (13)-(15). Clearly u fulfils equation (1) a.e. on and satisfies the boundary conditions (3). In addition, since u fulfils the impulse conditions (14) with , where , , we see that u also fulfils the state-dependent impulse conditions (2). According to Remark 8, it remains to prove that are the only instants at which the function u crosses the barriers , respectively. To this aim, due to (24) and (33), it suffices to prove that
Let us denote
From Lemma 9 it follows that ψ is decreasing. So, by virtue of (38), it suffices to prove that
Using (33), (2), and (27), we have
which yields (39). This completes the proof. □
Proof The last term in (37) is the same as in (16) for the compact operator ℋ. Therefore it suffices to prove the compactness of the operator ℱ on for . To do it we can use the same arguments as in the proof of Lemma 6 in , where the second-order state-dependent impulsive problem is investigated. In particular, the compactness of ℱ on is a consequence of the following properties of functions and functionals contained in (37):
• the first term in (37) can be written in the form
For the application of the Schauder Fixed Point Theorem it remains to prove that
Theorem 14Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. Then there exists at least one solution to problem (1)-(3) satisfying (22).
Proof The assertion follows directly from Lemma 12 and Lemma 13. □
Remark 15 The existence result from Theorem 14 can be extended to unbounded functions h and by means of the method of a priori estimates. This can be found for the special case in .
The authors declare that they have no competing interests.
The authors contributed equally to the manuscript and read and approved the final draft.
The authors were supported by the grant IGA_PrF_2014028. The authors sincerely thank the anonymous referees for their valuable comments and suggestions.
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