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
(34)
Consider from (3), w from (7) and from (8), and denote
(35)
and
(36)
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
(37)
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
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 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
(38)
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 [9], 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 [10].