Abstract
The paper provides an existence principle for a general boundary value problem of the form , a.e. , , , with the statedependent 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 leftcontinuous regulated (i.e. having finite onesided 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 higherorder ODE; statedependent impulses; general linear boundary conditions; transversality conditions; fixed point1 Introduction
In this paper we are interested in the nonlinear ordinary differential equation of the nthorder () with statedependent impulses and general linear boundary conditions on the interval . Studies of reallife problems with statedependent impulses can be found e.g. in [16]. 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 ndimensional 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, leftcontinuous 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 .
Note that a mapping is leftcontinuous regulated on if for each and each there exist finite limits
is a linear space, and equipped with the supnorm it is a Banach space (see [[7], Theorem 3.6]). In particular, we set
A function satisfies the Carathéodory conditions on if
• for each compact set there exists a function such that for a.e. and each .
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 [810] which are devoted to the secondorder differential equation. Other types of initial or boundary value problems for the first or secondorder differential equations with statedependent impulses can be found in [1119]. To get the existence results for problem (1)(3), we exploit the paper [20] with fixedtime impulsive problems.
Here we assume that
Remark 1 The integral in formula (4) is the KurzweilStieltjes integral, whose definition and properties can be found in [21]. The fact that each linear bounded functional on can be written uniquely in the form described in (4) is proved in [22]. See also [20].
Now let us define a solution of problem (1)(3).
Definition 2 A function is said to be a solution of problem (1)(3) if u satisfies (1) for a.e. and fulfils conditions (2) and (3).
2 Problem with impulses at fixed times
In the paper [20] 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
In this case, each solution of the problem crosses ith barrier at same time instant for .
Note that boundary value problems for higherorder differential equations with impulses at fixed times have been studied for example in [2331] and for delay higherorder impulsive equations in [32,33].
Let us summarize the results of the paper [20] according to our needs. Assume that the linear homogeneous problem
has only the trivial solution. Let be a fundamental system of solutions of the differential equation from (6), W be their Wronski matrix and w its first row, i.e.
Denote
From [[20], Lemma 8] (see also Chapter 3 in [34]) it follows that the unique solvability of (6) is equivalent to the condition
Further assume (9), consider , , from (4), and denote
If we denote by and elements of the matrices H and , respectively, that is,
we can define functions , , as
For each fixed the functions , , will be understood as rightcontinuous extensions at and leftcontinuous extensions at and . In this way the Green’s function of problem (6) is built (cf. Remark 6).
Remark 3 In order to state one of the main results of [20] 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
Now we are ready to state the operator representation theorem for the problem with impulses at fixed times which has the form
Theorem 4 [[20], Theorem 17]
Let (4), (9) hold, and letW, w, and, be defined in (7), (8), and (12). Thenis a fixed point of an operatordefined by
, if and only ifuis a solution of problem (13)(15). Moreover, the operator ℋ is completely continuous.
Remark 5 Let us note that the row vector
does not depend on the choice of a fundamental system of solutions , but only on the data of problem (6).
Remark 6 Let us put
and
Then the operator ℋ in Theorem 4 can be written as
Theorem 4 implies that u is a fixed point of if and only if u is a solution of the problem
Therefore a (unique) solution of problem (17) has the form
and consequently is the Green’s function of (6).
Remark 7 Under the assumption (9) we are allowed using (11) to define the functions
for , . Obviously, due to (12),
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
for , , . Moreover, for each , , , there exists a constant such that
This follows from the definition of (), from the fact and from the boundedness of the matrices and H (cf. (7), (10) and (11)).
3 Transversality conditions
The most results for differential equations with statedependent 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 [[35], Chapter 5].
A different approach has to be used when boundary value problems with statedependent 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:
Consider real numbers , , and denote
Now, we are ready to formulate the following transversality conditions:
Let us define the set
Our current goal is to find a continuous functional for , which maps each function u from to some time instant of (2).
Lemma 9Let, , , , , be real numbers satisfying (26), and letand ℬ be given by (23) and (28), respectively. Finally, assume that, , satisfy (24), (25), and choose. Then the function
is continuous and decreasing onand it has a unique root in the interval, i.e. there exists a unique solution of the equation
is valid. This together with the fact that σ is continuous shows that σ has at least one root in . Now, we will prove that σ is decreasing, by a contradiction. Let , be such that
i.e.
From (25), (26), (28), and the Mean Value Theorem we obtain
which is a contradiction.
According to Lemma 9, we can define a functional by
where is a solution of (30), i.e. a unique root of the function σ from Lemma 9, for . □
Lemma 10Let the assumptions of Lemma 9 be satisfied. The functionals, , are continuous.
Let us choose and prove that as . We denote
From Lemma 9 it follows that are the unique roots of the functions
and these functions are strictly decreasing. Let , be such that . Then and . According to (32) we see that uniformly on , in particular and as . These facts imply that
From the continuity of and the Intermediate Value Theorem it follows that
which completes the proof. □
Our next step is to define an appropriate operator representation of the BVP with statedependent 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 [36]).
Therefore, we choose the way here, which we have developed in our joint papers [810]. 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 statedependent 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
Consider from (3), w from (7) and from (8), and denote
and
for , where are constants from (21).
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
where ℬ is defined in (28) with from (36).
Now, we have to modify the operator ℋ from Theorem 4 using and instead of the Green’s functions , that is, we define an operator by with
and W, w, , , , , and are from (7), (12), (18), and (35), respectively.
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 fixedtime 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.
Lemma 12Let assumptions (4), (9), (23)(27), (34)(36) be satisfied. Ifis a fixed point of the operator ℱ, then the functionudefined by (33) is a solution of problem (1)(3) satisfying (22).
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
Denote
and choose , . Then, according to (33), we have
Of course we have
Let be such that . Then and therefore (19) gives
Let be such that (such k exists only if ). Then and therefore we get by (19)
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 statedependent 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
Choose an arbitrary and consider σ from (29). Since u fulfils (2), we have
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. □
Lemma 13Let assumptions (4), (9), (23)(27), (34)(36) be satisfied. Then the operator ℱ from (37) has a fixed point in.
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 [9], where the secondorder statedependent 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
• are continuous on (due to Lemma 10),
• , satisfy (20), satisfies (19),
For the application of the Schauder Fixed Point Theorem it remains to prove that
Let for some . Then, by (21), (34), (35), and (37), we have
and so . We have proved (40), and consequently there exists at least one fixed point of ℱ in . □
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 [10].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to the manuscript and read and approved the final draft.
Acknowledgements
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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