- Open Access
First order periodic problem at resonance with nonlinear impulses
© Drábek and Langerová; licensee Springer 2014
- Received: 2 May 2014
- Accepted: 19 July 2014
- Published: 26 September 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.
- impulsive differential equations
- Landesman-Lazer conditions
- topological degree theory
- Schaeffer fixed point theorem
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.
Then is a Banach space.
In the following theorem the mutual connection among and plays the key role.
In particular, for the linear equation we have the following necessary and sufficient condition.
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 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.
- (i)(Nonresonance case) Periodic problem(16)
has a unique solution for all f if and only if .
- (ii)(Nonresonance case with impulses) If , the solution of (16) on with constant impulses(17)
- (iii)(Resonance case) If , then (16) has a solution if and only if
where is arbitrary.
The properties (i)-(iv) follow from the variation of constants formula.
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.
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.
where is such that .
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.
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.
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