In this section, we present the main result of this paper. We prove the existence of at least one solution to nonlinear problem (1), (2), and (3) lying in an admissible bounding set. To achieve this, we turn our impulsive boundary value problem into an equivalent nonlinear equation and use Leray-Schauder degree theory. We compute the degree using three homotopies, the Leray index theorem and the multiplication theorem.
The first homotopy involves , where
The second and third homotopies are constructed using one-parameter families of systems of ordinary differential equations.
We construct our first family of systems of differential equations using defined below.
Let Φ, , M, and K be given in Lemma 1, Remark 1, and Definition 2, respectively. Let
(14)
for , where
(15)
(16)
and ε is given below. Firstly, we consider where M is given in Definition 2. For case , see Remark 5. Let
(17)
Remark 4
-
(i)
It follows that has a continuous extension to .
-
(ii)
It follows from Remark 1 that , , and L are well defined when where M is given in Definition 2.
For , we define by
(18)
where and f are given in (14) and (1), respectively.
We consider the system
(19)
Lemma 2 Let be an admissible bounding set for (1) and assume that f satisfies the Hartman-Nagumo condition and that is given by (18). Then for ,
where Φ is given in Lemma 1, is given in (17) and is given by (5) and , are nonnegative numbers.
If x is a solution of (19) with , then where N is given in Lemma 1.
Moreover, if , , , and , then .
Proof It follows from (17) that
for . Since f satisfies the Hartman-Nagumo condition, thus for , and is given in (10), it is easy to see that
Since , it follows from (17) that
If x is a solution of (19) with , it follows from Lemma 1 that where N is given in Lemma 1.
From Definition 1(iii), if , , , and , then . Since f satisfies the Hartman-Nagumo condition, so . It follows from (15) that
If from (14), then and
If from (14), then . It follows from (16) and (7) that
Thus
□
Now we construct the second one-parameter family of systems of ordinary differential equations.
For , we define by
(20)
where , , L are given in (14), (15), and (16), respectively.
We consider the system
(21)
Lemma 3 Assume that is an admissible bounding set for (1) and that is defined in (20). Then, for ,
where is given in (10).
If x is a solution of (21) with , then , where N is given in Lemma 1.
Moreover, if , , , and , then .
Proof Clearly,
for all , where is given in (10). From the proof of Lemma 2, for all , . It follows from (17) and (8) that
Thus
If x is a solution of (21) with , then it follows from Lemma 1 that , where N is given in Lemma 1.
From the proof of Lemma 2, if , , , , then . It follows from (15) and (16), respectively, that
where is given in Remark 1(i). Therefore,
□
Remark 5 If , where M is given in Definition 2, we do not need to choose and in (14). We set
for . Moreover, we do not need the second one-parameter family of systems of ordinary differential equations based on to construct our homotopy.
For and , let be Green’s function for (1) restricted to together with the homogeneous boundary conditions , thus
(22)
For , let
(23)
Using the above two families of systems of ordinary differential equations, we can homotopy the original problem (1), (2), and (3) to the following solvable system of ordinary differential equations subject to Picard boundary conditions and impulses.
(24)
(25)
where , L are given in (15) and (16), respectively. Then (24) and (25) have a solution of the form
(26)
for and , where we have identified V with .
We show that for all .
Lemma 4 Assume that is an admissible bounding set for (1) and is given by (26). Then for . Moreover, , where N is given in Lemma 1.
Proof Suppose for some . Set . Since is a solution of (24) and (25), it follows that , , and so and . Therefore . So has a local maximum at and . Hence , where ε is given below. But it follows from (15) and (16) that for all and hence
a contradiction. Thus for . Since is a solution of (24) and (21) is (24) when , it follows from Lemma 3 that , where N is given in Lemma 1. □
Now we present our main result.
Theorem 1 Assume that is an admissible bounding set for (1) and that f satisfies the Hartman-Nagumo condition. Suppose that the boundary conditions (2) and impulses (3) are compatible with Ω. Then there is at least one solution of problem (1), (2), and (3) such that for .
Proof Now for . First consider the case that all are strongly compatible with Ω.
Choose such that for all . It follows from Remark 1(ii) that , where , , for and all . Let , L, and be given in (15), (16), and (17), respectively. Let where M is given in Definition 2.
Let
and let , where and is given in (9) and N is given in Lemma 1.
Following [4], we interpret to mean for and set . Let , where is a strongly inwardly pointing vector field on for each k. Let , where is given in (11), for all .
Let be given in (18). For all , let
(27)
where is given in (22). Define
(28)
where we identify x and .
Consider the solutions of
(29)
where
and is given in (23) for all .
From (18) and (28), problem (1), (2), and (3) has a solution x satisfying if and only if is a solution of (29) in since and for in that case.
To show that (29) has a solution, we use Leray-Schauder degree theory.
Define for by
where
for , is given in (20).
Now is completely continuous since T is completely continuous. We show that either there is a solution to our problem or the above functions define homotopies.
It is easy to see that is a solution of (1), (2), and (3) with if
when . Now if there is a solution of (30) with for , then and is the required solution, so we assume there is no solution on ∂ Σ. We show that is a homotopy for the Leray-Schauder degree on Σ at 0, that is, there are no solutions of (30) for . We argue by contradiction and assume that there is a solution of (30) with and . From the definition of , x is a solution of (1) such that
for . Suppose . Assume . Since , then so that . Since is a strongly inwardly pointing vector field on for each k and , thus
Thus as g is strongly compatible with Ω. Thus , a contradiction. Similarly, the other cases lead to a contradiction, so . Suppose . By the choice of N, for all k. Assume that for some . Then . Since for , it follows that r attains a local maximum at . Thus . However, , a contradiction. Thus for any , .
Suppose that has a solution . From the definition of , x is a solution of (19) with . Since on ∂△, it follows that . Suppose . By the choice of N, for all k. Assume that for some . Then . Since for , it follows that r attains a local maximum at . Thus . However, by Lemma 2. Since and for all , so for . Moreover, from Remark 1(i), is convex for all , it follows that and for all . Thus . Therefore for any , .
Suppose that has a solution . From the definition of , x is a solution of (21) with and for all and . Since on ∂△, so . Suppose . By the choice of N, for all k. Assume that for some . Then . Since for , it follows that r attains a local maximum at . Thus . However, by Lemma 3, so . Thus for any , .
Therefore are homotopies for . For all and , by the homotopy invariance of the Leray-Schauder degree, we have
In particular,
where T is defined in (28) and W is given by
for , where and are given in (23) and (22), respectively. Moreover, since is the solution of (26), using the Leray index theorem, Theorem 8.10 in [23], it is easy to show
where is an open ball in . Thus there is a solution of and is a solution of (29). By the above argument, x is the required solution of (1), (2), and (3).
Suppose now that for is compatible with . Then there is a sequence strongly compatible with and converging uniformly to on compact subsets of for . Let be the corresponding solution. By compactness, there is a subsequence of converging in to the desired solution of integral equation (29), and hence the differential equation, satisfying the boundary conditions and impulses. □
Remark 6
-
(i)
It is easy to see from the above proof that we can weaken our assumptions as follows. We assume that and look for solutions . Moreover, we may assume that and has an extension .
-
(ii)
We can vary the assumptions on our admissible bounding sets. It is easy to see from the proof that instead of assuming that for some constants , , and , it suffices to assume that for and . Indeed, we can still recover our existence result by an approximation argument if we can weaken this further to for and . We apply our Theorem using noting that satisfies (6) and . Since solutions with satisfy , we obtain derivative bounds independent of ε. Since on , strongly compatible boundary conditions on Ω will be strongly compatible on for sufficiently small. Letting ε approach 0 and choosing a subsequence if necessary, converges to a solution of our problem.