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
Consider from (3), w from (7) and from (8), and denote
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
for , , where
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 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.
Lemma 12 Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. If is a fixed point of the operator ℱ, then the function u defined 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
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 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
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 13 Let 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 , 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):
where for , , ,
are continuous on (due to Lemma 10),
, satisfy (20), satisfies (19),
are continuous on .
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
for , , . From (36) we get
and so . We have proved (40), and consequently there exists at least one fixed point of ℱ in . □
Theorem 14 Let 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 .