# First order periodic problem at resonance with nonlinear impulses

- Pavel Drábek
^{1, 2}and - Martina Langerová
^{2}Email author

**2014**:186

https://doi.org/10.1186/s13661-014-0186-3

© Drábek and Langerová; licensee Springer 2014

**Received: **2 May 2014

**Accepted: **19 July 2014

**Published: **26 September 2014

## Abstract

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.

## Keywords

## 1 Introduction

At prescribed points in the interval $(0,T)$ 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 [1]–[4]. The first order boundary value problems with impulses were studied recently in many research articles such as [5]–[12]. Optimal control of space trajectories with applications to maneuvers of spacecraft or satellite constellation are considered in papers [13]–[16].

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 [17]). 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 [18] and [19], respectively. The authors apply the topological methods in [18] and variational methods in [19] 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 $(X,\parallel \cdot \parallel )$ is a Banach space.

exist.

By a solution of (1), (2) we understand a function $u\in X$ such that ${u}^{\prime}\in X$ and (1), (2) hold true.

*resonance case*:

*f*is bounded on $J\times \mathbb{R}$ and for $t\in J$ the limits

In the following theorem the mutual connection among ${f}_{\pm}(t)$ and ${I}_{j}(\pm \mathrm{\infty})$ plays the key role.

### Theorem 1

*Let the following two inequalities hold*:

*or*,

*alternatively*,

*Then the nonlinear impulsive problem* (1), (2) *has at least one solution*.

In particular, for the linear equation we have the following necessary and sufficient condition.

### Corollary 1

*Let*$f(t,s)=f(t)\in X$

*for all*$s\in \mathbb{R}$,

*and*

*Then the impulsive problem*(1), (2)

*has at least one solution*.

*Moreover*,

*if for all*$s\in \mathbb{R}$, $j=1,2,\dots ,p$,

*we have*

*then* (7) *is also a necessary condition for the existence of a solution*. *A similar assertion holds true if we switch*${I}_{j}(-\mathrm{\infty})$*and*${I}_{j}(+\mathrm{\infty})$*in the above inequalities*.

We give the following simple examples, which illustrate our main results.

### Example 1

Comparison of (10) and (12) shows that the problem with impulses has a solution for a wider class of external forces *f* than the original problem without impulses.

Comparison of (10) and (14) shows that the sets of external forces *f* for which the problem with impulse (13) and the problem without impulse (13) have a solution have an empty intersection.

### Example 2

### Example 3

Let us consider the nonlinear periodic problem (1) with impulses (2). Assume ${I}_{j}(+\mathrm{\infty})=+\mathrm{\infty}$ for at least one $j\in \{1,\dots ,p\}$, $|{I}_{j}(+\mathrm{\infty})|<\mathrm{\infty}$ otherwise and, similarly, ${I}_{k}(-\mathrm{\infty})=-\mathrm{\infty}$ for at least one $k\in \{1,\dots ,p\}$ and $|{I}_{k}(-\mathrm{\infty})|<-\mathrm{\infty}$ 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

- (i)(Nonresonance case) Periodic problem$\begin{array}{r}{u}^{\prime}(t)+a(t)u(t)=f(t),\phantom{\rule{1em}{0ex}}t\in J,\\ u(0)=u(T),\end{array}$(16)

has a unique solution for all *f* if and only if ${\int}_{0}^{T}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \ne 0$.

- (ii)(Nonresonance case with impulses) If ${\int}_{0}^{T}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \ne 0$, the solution of (16) on ${J}^{\prime}$ with constant impulses$u\left({{t}_{j}}^{+}\right)=u({t}_{j})+{\theta}_{j},\phantom{\rule{1em}{0ex}}j=1,2,\dots ,p,$(17)

- (iii)(Resonance case) If ${\int}_{0}^{T}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau =0$, then (16) has a solution if and only if${\int}_{0}^{T}f(t){e}^{{\int}_{0}^{t}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t=0.$

- (iv)(Resonance case with impulses) If ${\int}_{0}^{T}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau =0$, then (16) on ${J}^{\prime}$ with impulses (17) has a solution if and only if${\int}_{0}^{T}f(t){e}^{{\int}_{0}^{t}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\sum _{i=1}^{p}{e}^{{\int}_{0}^{{t}_{j}}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}{\theta}_{j}=0.$

where $c\in \mathbb{R}$ is arbitrary.

The properties (i)-(iv) follow from the variation of constants formula.

## 3 Proofs

*u*is a fixed point of

*F*,

*i.e.*,

The operator *F* is compact. Indeed, let $M\subset X$ be a bounded set. Then it follows from the definition of *F* that the family of functions $F(M)$ is equicontinuous in $BC({J}^{\prime})$ and for each $t\in {J}^{\prime}$ the set $\{u(t):u\in F(M)\}$ is bounded in ℝ. It follows from Arzela-Ascoli’s theorem (see [20], Theorem 1.2.13]) that $F(M)$ is relatively compact in $BC({J}^{\prime})$ 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.

### Lemma 1

*There exists*$K>0$*such that for all*$\lambda \in [0,1]$*and*$u\in X$*satisfying*$u=\lambda F(u)$*we have*$\parallel u\parallel <K$.

### Proof

Since $|{v}_{n}(s)|\le 1$, $\frac{f(s,{u}_{n}(s))}{\parallel {u}_{n}\parallel}\to 0$, uniformly with respect to $s\in [0,T]$, $\frac{{I}_{j}({u}_{n}({t}_{j}))}{\parallel {u}_{n}\parallel}\to 0$, $j=1,2,\dots ,p$, by similar argument as above, the family $\{{v}_{n}\}$ is equicontinuous in $BC({J}^{\prime})$ and for each $t\in {J}^{\prime}$ the set $\{{v}_{n}(t)\}$ is bounded. It follows from Arzela-Ascoli’s theorem that $\{{v}_{n}\}$ is relatively compact in $BC({J}^{\prime})$ and hence also in *X*.

*X*, ${\lambda}_{n}\to \lambda $. Taking the limit for $n\to \mathrm{\infty}$ in (20) we get

*i.e.*,

where $c>0$ is such that $\parallel v\parallel =1$.

*J*. Since ${v}_{n}\to v$ in

*X*(

*i.e.*, uniformly in

*J*) we have ${u}_{n}(t)\to +\mathrm{\infty}$ as $n\to \mathrm{\infty}$ uniformly in

*J*. Multiply the equation in (19) by ${e}^{{\int}_{0}^{t}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}$, integrate over

*J*and perform the integration by parts in the first integral to get Since we arrive at

*J*. In particular, we have also ${u}_{n}({t}_{j})\to +\mathrm{\infty}$, $j=1,2,\dots ,p$. These facts together with (22), boundedness of

*f*, and the Lebesgue dominated convergence theorem yield

a contradiction with (3).

Similarly, if $v(t)=-c\cdot {e}^{-{\int}_{0}^{t}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}<0$ in *J*, we arrive at a contradiction with (4). □

Compactness of *F*, Lemma 1 and the Schaeffer fixed point theorem (see [20], Example 5.8.4]) imply that (1), (2) has a solution provided that (3) and (4) hold true. Choosing $\delta <0$ at the beginning of the proof, we get the existence result provided that (5) and (6) hold.

The proof of Theorem 1 is finished.

Let $f=f(t,s)$ be independent of *s*, that is, $f(t,s)=f(t)$, $t\in J$, $s\in \mathbb{R}$. Then $f(t)={f}_{+}(t)={f}_{-}(t)$ and (3), (4) reduces to (7). Hence the linear problem (16) with nonlinear impulses (2) has a solution provided that (7) holds.

*u*be a solution of (16), (2). Multiply the equation in (16) by ${e}^{{\int}_{0}^{t}a(\tau )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau}$ and integrate over

*J*with respect to

*t*:

Inequality (7) follows immediately from (23) provided that (8) holds true. The proof of Corollary 1 is finished.

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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