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Systems of differential equations with implicit impulses and fully nonlinear boundary conditions
Boundary Value Problems volume 2013, Article number: 240 (2013)
Abstract
We show that systems of second-order ordinary differential equations, , subject to compatible nonlinear boundary conditions and impulses, have a solution x such that lies in an admissible bounding subset of when f satisfies a Hartman-Nagumo growth bound with respect to . We reformulate the problem as a system of nonlinear equations and apply Leray-Schauder degree theory. We compute the degree by homotopying to a new system of nonlinear equations based on the simpler system of ordinary differential equations, , subject to Picard boundary conditions and impulses and using the Leray index theorem. Our proof is simpler than earlier existence proofs involving nonlinear boundary conditions without impulses and requires weak assumptions on f.
MSC:34A37, 34A34, 34B15.
1 Introduction
Let , the natural numbers,
and for . We call Q a division of the interval .
We consider the system of second-order ordinary differential equations
subject to very general nonlinear boundary conditions of the form
and very general nonlinear implicit impulses of the form
where
satisfies has an extension to and
for . Our fully nonlinear boundary conditions (2) include the Picard, periodic, and Neumann boundary conditions as special cases. We establish a general existence result for solutions lying in an admissible bounding set for the system of ordinary differential equations (1) satisfying boundary conditions (2) and impulses (3).
Our result is closely related to those of Thompson [1] and of Kongson et al. [2]. In [1] and [2], the authors established existence results for systems of second-order ordinary differential equations in more general bounding sets and subject to general boundary conditions (2) but not subject to impulses. Moreover, the proof in [1] is incomplete as it fails to establish the required derivative bounds; these appear to require more assumptions on the Hartman-Nagumo growth bound than we assume here. Although our bounding sets are more restrictive than those in [2], our proof is much simpler than theirs. In particular, the ideas introduced in our proof offer a fresh starting point for further work aimed at identifying the natural and most general concept of a bounding set and with this the natural and most general existence results possible for system (1) subject to nonlinear boundary conditions (2).
Earlier works on boundary value problems homotopies the original problem (1), plus nonlinear boundary conditions (2), to plus the Picard boundary conditions; see, for example, [3]. This requires f to be redefined for outside the admissible bounding set in such a way that solutions to the associated boundary value problem lie in the admissible bounding set. This in turn imposes restrictive assumptions on f and the associated bounding set. A key to our new idea is the observation that it suffices to homotopy our associated system of nonlinear equations to a new system of nonlinear equations associated with the simpler system subject to Picard boundary conditions and impulses. This is uniquely solvable with the solution lying in the admissible bounding set. We use the Leray index theorem and the multiplication theorem to show that the degree of the associated nonlinear equation is not zero. Using our homotopies, we do not need to redefine the system outside the admissible bounding set. In the current work, we require the bounding set to be , where and is strongly convex as a function of x (see Remark 1(i)).
A further motivation for our work comes from the paper by Cabada and Thompson [4] for a single equation with impulses. Recently, many papers devoted to the study of boundary value problems for nonlinear differential equations with impulses have appeared because of their wide applicability and associated rich theory. In the literature one can find different kinds of existence results for first-order [5, 6], second-order [7–9], and higher-order [10, 11] ordinary differential equations with periodic boundary conditions and impulses. In addition, some existence results for first-order impulsive differential equation with nonlinear boundary conditions can be found in [12–15]. In the papers [4, 16, 17], the ϕ-Laplacian and φ-Laplacian equations with impulses are considered.
This paper is organized as follows. In Section 2, we introduce the notation and definitions that we use in this paper. We give the definition of compatible boundary conditions and introduce our definition of compatible impulses in Section 3. In Section 4, we present the Nagumo-type condition that we use in our existence result to a priori bound the derivative of solutions. Section 5 is principally devoted to our main result where we prove that there are solutions to (1), (2), and (3) lying in an admissible bounding set. In Section 6, we present an example.
2 Notation and definitions
In this section, we present the notation, definitions, and assumptions that we use to obtain a priori bounds on solutions.
Let H denote finite or infinite dimensional Hilbert spaces. For a bounded subset V of H, let denote its interior, ∂V its boundary and its closure. For a bounded subset U of and , let denote its t-cross section and denote the boundary of in . Thus . Let denote the curved boundary of U, so excludes the sets and from ∂U. For , denotes the absolute value of x. For and , denotes the transpose of x while denotes the scalar product of x and y. Let ℐ denote the identity on H so for all x. If X is a Banach space and , then denotes the space of m-times continuously differentiable functions from A to X with a finite norm. In the case of continuous functions, we omit the m, while in the case of real-valued functions, we omit the X.
Let be an interval. For , let denote the partial derivative with respect to t, denote the gradient, and denote the matrix of second-order partial derivatives of r with respect to x.
The norm on is given by
where denotes the k th derivative of u. By abuse of notation, we abbreviated to . Further we will abbreviate to when the meaning is clear from the context.
For , let and for , let for . To simplify statements of results, set and for , where and are the appropriate one-sided derivatives.
In order to define the concept of solution for our problem, we consider the following sets. Let
All our limits are assumed to be -valued when they exist. Thus, for , exists for , . Note that is defined in the obvious way. Thus we may identify with , where , for all . By abuse of notation, we will denote by x where the meaning is clear from the context. Further we define a norm on by
If A is a bounded open subset of H, , where , has compact closure and , then denotes the Leray-Schauder degree of on A at p. In the special case that and , , is the Brouwer degree.
By a solution x we mean a function satisfying (1) for all , (2) and (3).
We look for solutions to problem (1) together with the fully nonlinear boundary conditions (2) and impulses (3) in the following admissible bounding set which provides a priori bounds on solutions to (1).
Definition 1 Let be a bounded set and . We call an admissible bounding set for (1) if it has the following properties:
-
(i)
There is such that
-
(a)
can be uniquely extended to for all ;
-
(b)
;
-
(c)
for some constants , all and ;
-
(a)
-
(ii)
There is such that , where for all ;
-
(iii)
If , , and , then
where
(4)(5) -
(iv)
for all and some constant .
Remark 1
-
(i)
A function is strongly convex iff for some constants ,
(6)for (see Part 4 in [18]). If satisfies (6), then r is uniformly convex, see Appendix B.1. in [19]. Moreover, satisfies (6) when iff r is convex (see Appendix B.1. in [19]). From the definition of convex function, it is easy to see that
is a convex set for .
It follows from Definition 1(i)(c) that for ,
(7)where and .
-
(ii)
It follows from Definition 1(i)(a), (ii) and (iv) that
(8)for all and some .
Set
for , where and . Let
We assume that f satisfies the following conditions.
Definition 2 Let be an admissible bounding set for (1). We say that f satisfies the Hartman-Nagumo condition on Ω if:
-
(i)
has an extension to ;
-
(ii)
for all , where
-
(iii)
for all , where M and K are nonnegative constants and is given by (5).
Remark 2 If conditions (ii) and (iii) above are satisfied, a solution x of (1) with satisfies the Hartman-Nagumo inequality (see the second paragraph on p.702 in [20]).
3 Compatibility
Following [1], we give the definition of compatible boundary conditions and introduce the definition of compatible impulses. These are simple, degree-based relationships between the boundary conditions, the impulses, and the associated admissible bounding set. For more information on compatibility of boundary conditions, we refer the reader to [1, 21], and [[4], Definition 14].
Definition 3 For , we call the vector field strongly inwardly pointing on if for all ,
where , , are given in (9). We call the vector field strongly inwardly pointing on if for all ,
where () is the extension to of and is given by (4). From (9), ; . For , we call inwardly pointing on if the above inequalities are weak.
In what follows, where there is a strongly inwardly pointing vector field on for all , then is defined by
for all , .
The following definition is a variant of Definition 2.5 given in [1].
Definition 4 Let and . We say is strongly compatible with Ω if
for all such that
and/or
and
for any strongly inwardly pointing vector field on .
For , we say that is compatible with Ω if there is a sequence strongly compatible with Ω and converging uniformly to on compact subsets of .
4 Nagumo-type conditions
In the literature, there are many variants of the ‘Nagumo condition’ which are used to establish a priori bounds on the derivative of bounded solutions.
We use the following variant of Lemma 4.1 in [2].
Lemma 1 Let satisfy
and r be given in Definition 1(i). Let x be a solution of(1) satisfying . Assume that
for such that and , where , , and K are nonnegative constants and is given by (5). Then there exists such that .
Proof Since r is given in Definition 1(i), then when , where is given in (10). Thus the proof of Lemma 5.2 of Hartman [22] carries over to our case on , and it follows that for . Thus for , where . □
Remark 3 The function satisfies (13).
5 The main result
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
for , where
and ε is given below. Firstly, we consider where M is given in Definition 2. For case , see Remark 5. Let
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
where and f are given in (14) and (1), respectively.
We consider the system
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
where , , L are given in (14), (15), and (16), respectively.
We consider the system
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
For , let
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.
where , L are given in (15) and (16), respectively. Then (24) and (25) have a solution of the form
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
where is given in (22). Define
where we identify x and .
Consider the solutions of
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.
6 Example
In this section we present an example to illustrate the power of our existence result. This example is modeled on that in [2] and we have added impulses.
Example 1 Let and and consider the problem
for , where w is a bounded continuous function. Let , where
Let for all , and let the Sturm-Liouville boundary conditions be given by
Let the impulses be given by
where .
To see that is an admissible bounding set, first we note and
for . If , and , then Kongson et al. [2] proved that for . We prove for .
Now for all since , . Moreover, it is not difficult to show that
for . Thus for .
It is not difficult to prove that f satisfies the Hartman-Nagumo condition. Some details are similar to those in the analysis of Example 1 given in Kongson et al. [2].
To show that the impulses given in (33) are compatible with Ω, let be a strongly inwardly pointing vector field on . Then
Let be given by
for so that the boundary conditions (33) are given by . Therefore
and so
for . Thus
is a homotopy for the Brouwer degree and
Therefore, the impulses are strongly compatible with Ω and hence compatible. Using a similar proof, we can show that the boundary conditions given in (32) are strongly compatible with Ω and hence compatible.
Therefore our impulsive boundary value problem satisfies the conditions of Remark 6 (ii) and therefore has a solution with for all .
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The first author thanks the University of Queensland for University of Queensland International Scholarship (UQI) and University of Queensland Research Scholarship (UQRS).
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Song, Y., Thompson, B. Systems of differential equations with implicit impulses and fully nonlinear boundary conditions. Bound Value Probl 2013, 240 (2013). https://doi.org/10.1186/1687-2770-2013-240
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DOI: https://doi.org/10.1186/1687-2770-2013-240