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First order periodic problem at resonance with nonlinear impulses
Boundary Value Problems volume 2014, Article number: 186 (2014)
We consider the nonlinear first order periodic problem with nonlinear impulses. We apply the Schaeffer fixed point theorem and prove the existence results under Landesman-Lazer type sufficient conditions. We formulate also necessary conditions in some special cases. The impulses can be viewed as a control which compensates the influence of external forces and vice versa.
MSC: 34A37, 34B37, 34F15, 47H11.
In this paper we deal with a nonlinear boundary value problem of the first order,
At prescribed points in the interval the solution is subject to the impulses which depend on an actual value of the solution. These impulses can be interpreted as a control of the external forces represented by f in order to get existence or nonexistence a periodic solution of (1). Vice versa, the external forces can be interpreted as a control of given impulses.
Impulsive problems have attracted the attention of mathematicians for several decades. Let us mention the classical books –. The first order boundary value problems with impulses were studied recently in many research articles such as –. Optimal control of space trajectories with applications to maneuvers of spacecraft or satellite constellation are considered in papers –.
According to the best of our knowledge, this paper is the first one to treat the periodic first order problem at resonance with external forces and impulses satisfying the Landesman-Lazer type conditions (see (3)-(7) below and compare with conditions in ). Similar conditions for Dirichlet problems at resonance, but for second order semilinear and quasilinear equations (involving the p-Laplacian), are formulated in the authors’ papers  and , respectively. The authors apply the topological methods in  and variational methods in  to prove the existence results. In this paper we rely on the Schaeffer fixed point theorem, which is based on the topological degree argument.
In order to formulate the main results precisely, we need some notation.
Let , , and set . We denote by the Banach space of all bounded and continuous functions defined on equipped with the usual supremum norm on . We also let
equipped with the norm
Then is a Banach space.
We assume that , is such that , the limits
Let , , . We consider the following nonlinear impulse conditions:
Our main results concern the resonance case:
We also assume that f is bounded on and for the limits
exist. We admit and but we require to be sublinear at ±∞, that is, we assume
In the following theorem the mutual connection among and plays the key role.
Let the following two inequalities hold:
In particular, for the linear equation we have the following necessary and sufficient condition.
Letfor all, and
then (7) is also a necessary condition for the existence of a solution. A similar assertion holds true if we switchandin the above inequalities.
We give the following simple examples, which illustrate our main results.
Let us consider the linear periodic problem
It is well known that (9) has a solution if and only if
The impulse condition
corresponds to , and
Similarly, applying Corollary 1, we obtain an existence result for the linear problem (9) with impulse condition
if and only if
Let us consider the nonlinear periodic problem
with impulses (2). In this case, we have and
Let us consider the nonlinear periodic problem (1) with impulses (2). Assume for at least one , otherwise and, similarly, for at least one and otherwise. Then the impulse problem (1), (2) has at least one solution.
The above examples illustrate that suitably chosen impulses may control the forcing term in order to get existence or nonexistence of periodic solutions.
2 Some elementary facts
For the reader’s convenience we provide a survey of well-known facts about the linear periodic problem and constant impulses:
(Nonresonance case) Periodic problem(16)
has a unique solution for all f if and only if .
This solution is expressed in the following form:
In particular, implies .
(Nonresonance case with impulses) If , the solution of (16) on with constant impulses(17)
is given by
(Resonance case) If , then (16) has a solution if and only if
The solution (if it exists) is then not unique. It is given by
where is arbitrary.
The solution (if it exists) is not unique. It is given by
where is arbitrary.
The properties (i)-(iv) follow from the variation of constants formula.
We define operator as follows:
The operator F is compact. Indeed, let be a bounded set. Then it follows from the definition of F that the family of functions is equicontinuous in and for each the set is bounded in ℝ. It follows from Arzela-Ascoli’s theorem (see , Theorem 1.2.13]) that is relatively compact in and hence also in X. Since F is clearly continuous, compactness of F follows.
The following a priori estimate is the key to the proof of Theorem 1.
There existssuch that for allandsatisfyingwe have.
Assume via contradiction that there exist , () such that and
This is equivalent to
Set . Then ; , satisfy
This is equivalent to the integral equation
Since , , uniformly with respect to , , , by similar argument as above, the family is equicontinuous in and for each the set is bounded. It follows from Arzela-Ascoli’s theorem that is relatively compact in and hence also in X.
Passing to subsequences, if necessary, we may assume that there exist and such that in X, . Taking the limit for in (20) we get
If then (21) has only a trivial solution (see (i)). But this contradicts ! Hence, and solves
where is such that .
Consider first the case in J. Since in X (i.e., uniformly in J) we have as uniformly in J. Multiply the equation in (19) by , integrate over J and perform the integration by parts in the first integral to get
we arrive at
Recall that uniformly in J. In particular, we have also , . These facts together with (22), boundedness of f, and the Lebesgue dominated convergence theorem yield
a contradiction with (3).
Similarly, if in J, we arrive at a contradiction with (4). □
Compactness of F, Lemma 1 and the Schaeffer fixed point theorem (see , Example 5.8.4]) imply that (1), (2) has a solution provided that (3) and (4) hold true. Choosing at the beginning of the proof, we get the existence result provided that (5) and (6) hold.
The proof of Theorem 1 is finished.
Integrating by parts the first integral we get
Evaluating the sum on the left-hand side as above, we arrive at
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This research was supported by the Grant 13-00863S of the Grant Agency of Czech Republic and by the European Regional Development Fund (ERDF), project ‘NTIS - New Technologies for the Information Society’, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.
The authors declare that they have no competing interests.
Both authors read and approved the final manuscript.
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Drábek, P., Langerová, M. First order periodic problem at resonance with nonlinear impulses. Bound Value Probl 2014, 186 (2014). https://doi.org/10.1186/s13661-014-0186-3
- impulsive differential equations
- Landesman-Lazer conditions
- topological degree theory
- Schaeffer fixed point theorem