Systems of differential equations with implicit impulses and fully nonlinear boundary conditions
© Song and Thompson; licensee Springer. 2013
Received: 16 July 2013
Accepted: 16 October 2013
Published: 11 November 2013
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.
and for . We call Q a division of the interval .
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  and of Kongson et al. . In  and , 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  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 , 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, . 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  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.
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.
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).
- (i)There is such that
can be uniquely extended to for all ;
for some constants , all and ;
There is such that , where for all ;
- (iii)If , , and , thenwhere(4)(5)
for all and some constant .
- (i)A function is strongly convex iff for some constants ,(6)for (see Part 4 in ). If satisfies (6), then r is uniformly convex, see Appendix B.1. in . Moreover, satisfies (6) when iff r is convex (see Appendix B.1. in ). 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 .
We assume that f satisfies the following conditions.
has an extension to ;
- (ii)for all , where
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 ).
Following , 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 [, Definition 14].
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.
for all , .
The following definition is a variant of Definition 2.5 given in .
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 .
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  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 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.
It follows that has a continuous extension to .
It follows from Remark 1 that , , and L are well defined when where M is given in Definition 2.
where and f are given in (14) and (1), respectively.
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 .
If x is a solution of (19) with , it follows from Lemma 1 that where N is given in Lemma 1.
Now we construct the second one-parameter family of systems of ordinary differential equations.
where , , L are given in (14), (15), and (16), respectively.
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 .
If x is a solution of (21) with , then it follows from Lemma 1 that , where N is given in Lemma 1.
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 , 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.
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.
and let , where and is given in (9) and N is given in Lemma 1.
Following , 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 .
where we identify x and .
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.
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.
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 , .
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. □
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 .
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.
In this section we present an example to illustrate the power of our existence result. This example is modeled on that in  and we have added impulses.
for . If , and , then Kongson et al.  proved that for . We prove for .
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. .
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 .
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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