The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse
Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space . Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions.
MSC: 34B37, 34B15.
Keywords:impulsive differential equation; state-dependent impulses; Dirichlet problem; second-order ODE
Most papers in the literature on impulsive boundary value problems concern the case with fixed moments of impulsive effects. Papers dealing with state-dependent impulses, called also impulses at variable times, focus their attention on initial value problems or periodic problems. Such papers investigate the existence, stability or asymptotic properties of solutions of initial value problems [4-8] or solvability of autonomous periodic problems [9,10] and nonautonomous ones [11-15]. We can also find papers investigating other boundary value problems with state-dependent impulses through some initial value problems for multi-valued maps [16,17].
In this paper we provide a new approach to boundary value problems with state-dependent impulses based on a construction of proper sets and operators and the topological degree arguments. Unlike previous existing results, our approach enables us to find simple existence conditions for data functions and it can be used for other regular (and also singular) problems. We demonstrate it on the second-order Dirichlet boundary value problem with one state-dependent impulse
where we assume
Under assumptions (4)-(8), we prove the solvability of problem (1)-(3). In particular, we transform problem (1)-(3) to a fixed point problem for a proper operator in the space . This approach can be also used for other types of boundary conditions and it can be easily extended to more impulses.
Here, we denote by the set of all continuous functions on the interval J, by the set of all functions having continuous derivatives on the interval J and by the set of all Lebesgue integrable functions on J. For a compact interval J, we consider the linear space of functions from or equipped, respectively, with the norms
We say that is a solution of problem (1)-(3), if z is continuous on , there exists unique such that , and have absolutely continuous first derivatives, z satisfies equation (1) for a.e. and fulfills conditions (2), (3).
In this section we assume that (4)-(8) are fulfilled. We introduce sets and operators corresponding to problem (1)-(3) and prove their properties which are needed for an application of the Leray-Schauder degree theory. Let us consider K of (7) and define the set
Therefore, by (13), we get
is a solution of problem (1)-(3).
and by Lemma 1,
Further, we get
3 Main result
Here, using the Leray-Schauder degree theory, we prove our main result about the solvability of problem (1)-(3). To this end, we will need the following lemma on a priori estimates.
which implies, due to (6) and (8),
This inequality together with (7) implies
which is a contradiction.
Theorem 6Assume (4)-(8). Then the operator ℱ has a fixed point in Ω.
and consequently the equation
has a solution in Ω. This solution is a fixed point of the operator ℱ. □
Theorem 7Assume (4)-(8). Then problem (1)-(3) has a solutionzsuch that
Proof From Theorem 6 it follows that there exists a fixed point of the operator ℱ. Lemma 4 yields that the function z defined in (17) (with ) is a solution of problem (1)-(3). Estimates (25) follow from (17) and from the definitions of Ω and (cf. (12) and (8)). □
Remark 8 Let us note that assumption (7) follows from the condition
In this section we demonstrate that Theorem 7 can be applied to sublinear, linear and superlinear problems.
Example 9 (Sublinear problem)
Let us consider problem (1)-(3) with
that is, f and I are sublinear in x. Then assumptions (5) and (6) are valid for
Remark 8 yields that condition (7) is satisfied for any sufficiently large K. In particular, let us put
we can check that conditions (8) are satisfied in both cases. Therefore, by Theorem 7, the corresponding problem (1)-(3) has at least one solution.
Note that (27) shows that γ need not be monotonous.
Example 10 (Linear problem)
Let us consider problem (1)-(3) with f and I having the linear behavior in x and put
Then assumptions (5) and (6) are valid for
Theorem 7 can be applied, due to Remark 8, under the additional assumption
If (28) holds, then for any sufficiently large K, condition (7) is satisfied. By (8), we have , and problem (1)-(3) has a solution for any γ satisfying (8). Consequently, if γ is given by (26) or (27), problem (1)-(3) is solvable.
Example 11 (Superlinear problem) Let us consider problem (1)-(3) with f and I superlinear in x. Put, for example,
Then assumptions (5) and (6) are valid for
satisfies (30) as well. Put, for example, , . Then we get that for inequality (30) holds. Consequently, (8) gives and the corresponding problem (1)-(3) is solvable for any γ satisfying (8). In particular, γ given by (26) or (27) can be considered in this case as well.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Dedicated to Jean Mawhin on the occasion of his 70th birthday.
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by the grant Matematické modely a struktury, PrF_ 2012_ 017.
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