# Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem

## Abstract

The paper provides an existence principle for a general boundary value problem of the form ${\sum }_{j=0}^{n}{a}_{j}\left(t\right){u}^{\left(j\right)}\left(t\right)=h\left(t,u\left(t\right),\dots ,{u}^{\left(n-1\right)}\left(t\right)\right)$, a.e. $t\in \left[a,b\right]\subset \mathbb{R}$, ${\ell }_{k}\left(u,{u}^{\prime },\dots ,{u}^{\left(n-1\right)}\right)={c}_{k}$, $k=1,\dots ,n$, with the state-dependent impulses ${u}^{\left(j\right)}\left(t+\right)-{u}^{\left(j\right)}\left(t-\right)={J}_{ij}\left(u\left(t-\right),{u}^{\prime }\left(t-\right),\dots ,{u}^{\left(n-1\right)}\left(t-\right)\right)$, where the impulse points t are determined as solutions of the equations $t={\gamma }_{i}\left(u\left(t-\right),{u}^{\prime }\left(t-\right),\dots ,{u}^{\left(n-2\right)}\left(t-\right)\right)$, $i=1,\dots ,p$, $j=0,\dots ,n-1$. Here, $n,p\in \mathbb{N}$, ${c}_{1},\dots ,{c}_{n}\in \mathbb{R}$, the functions ${a}_{j}/{a}_{n}$, $j=0,\dots ,n-1$, are Lebesgue integrable on $\left[a,b\right]$ and $h/{a}_{n}$ satisfies the Carathéodory conditions on $\left[a,b\right]×{\mathbb{R}}^{n}$. The impulse functions ${J}_{ij}$, $i=1,\dots ,p$, $j=0,\dots ,n-1$, and the barrier functions ${\gamma }_{i}$, $i=1,\dots ,p$, are continuous on ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{n-1}$, respectively. The functionals ${\ell }_{k}$, $k=1,\dots ,n$, are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on $\left[a,b\right]$ vector functions. Provided the data functions h and ${J}_{ij}$ are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier ${\gamma }_{i}$ at the unique impulse point ${\tau }_{i}$ are presented, and consequently the existence of a solution to the problem is proved.

MSC:34B37, 34B10, 34B15.

## 1 Introduction

In this paper we are interested in the nonlinear ordinary differential equation of the n th-order ($n\ge 2$) with state-dependent impulses and general linear boundary conditions on the interval $\left[a,b\right]\subset \mathbb{R}$. Studies of real-life problems with state-dependent impulses can be found e.g. in [16]. Here we consider the equation

(1)

subject to the impulse conditions

(2)

and the linear boundary conditions

${\ell }_{k}\left(u,{u}^{\prime },\dots ,{u}^{\left(n-1\right)}\right)={c}_{k},\phantom{\rule{1em}{0ex}}k=1,\dots ,n.$
(3)

In what follows we use this notation. Let $k,m,n\in \mathbb{N}$. By ${\mathbb{R}}^{m×n}$ we denote the set of all matrices of the type $m×n$ with real valued coefficients. Let ${A}^{T}$ denote the transpose of $A\in {\mathbb{R}}^{m×n}$. Let ${\mathbb{R}}^{n}={\mathbb{R}}^{n×1}$ be the set of all n-dimensional column vectors $c={\left({c}_{1},\dots ,{c}_{n}\right)}^{T}$, where ${c}_{i}\in \mathbb{R}$, $i=1,\dots ,n$, and $\mathbb{R}={\mathbb{R}}^{1×1}$. By $\mathbb{C}\left({\mathbb{R}}^{n};{\mathbb{R}}^{m}\right)$ we denote the set of all mappings $x:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ with continuous components. By ${\mathbb{L}}^{\mathrm{\infty }}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, ${\mathbb{L}}^{1}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, ${\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, $\mathbb{AC}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, $\mathbb{BV}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, ${\mathbb{C}}^{k}\left(\left[a,b\right];{\mathbb{R}}^{m×n}\right)$, we denote the sets of all mappings $x:\left[a,b\right]\to {\mathbb{R}}^{m×n}$ whose components are, respectively, essentially bounded functions, Lebesgue integrable functions, left-continuous regulated functions, absolutely continuous functions, functions with bounded variation and functions with continuous derivatives of the k th order on the interval $\left[a,b\right]$. By $Car\left(\left[a,b\right]×{\mathbb{R}}^{n};\mathbb{R}\right)$ we denote the set of all functions $f:\left[a,b\right]×{\mathbb{R}}^{n}\to \mathbb{R}$ satisfying the Carathéodory conditions on the set $\left[a,b\right]×{\mathbb{R}}^{n}$. Finally, by ${\chi }_{M}$ we denote the characteristic function of the set $M\subset \mathbb{R}$.

Note that a mapping $u:\left[a,b\right]\to {\mathbb{R}}^{n}$ is left-continuous regulated on $\left[a,b\right]$ if for each $t\in \left(a,b\right]$ and each $s\in \left[a,b\right)$ there exist finite limits

$u\left(t\right)=u\left(t-\right)=\underset{\tau \to t-}{lim}u\left(\tau \right),\phantom{\rule{2em}{0ex}}u\left(s+\right)=\underset{\tau \to s+}{lim}u\left(\tau \right).$

${\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{n}\right)$ is a linear space, and equipped with the sup-norm ${\parallel \cdot \parallel }_{\mathrm{\infty }}$ it is a Banach space (see [[7], Theorem 3.6]). In particular, we set

A function $f:\left[a,b\right]×{\mathbb{R}}^{n}\to \mathbb{R}$ satisfies the Carathéodory conditions on $\left[a,b\right]×{\mathbb{R}}^{n}$ if

• $f\left(\cdot ,x\right):\left[a,b\right]\to \mathbb{R}$ is measurable for all $x\in {\mathbb{R}}^{n}$,

• $f\left(t,\cdot \right):{\mathbb{R}}^{n}\to \mathbb{R}$ is continuous for a.e. $t\in \left[a,b\right]$,

• for each compact set $K\subset {\mathbb{R}}^{n}$ there exists a function ${m}_{K}\in {\mathbb{L}}^{1}\left(\left[a,b\right];\mathbb{R}\right)$ such that $|f\left(t,x\right)|\le {m}_{K}\left(t\right)$ for a.e. $t\in \left[a,b\right]$ and each $x\in K$.

In this paper we provide sufficient conditions for the solvability of problem (1)-(3). This problem is a generalization of problems studied in the papers [810] which are devoted to the second-order differential equation. Other types of initial or boundary value problems for the first- or second-order differential equations with state-dependent impulses can be found in [1119]. To get the existence results for problem (1)-(3), we exploit the paper [20] with fixed-time impulsive problems.

Here we assume that

(4)

Remark 1 The integral in formula (4) is the Kurzweil-Stieltjes integral, whose definition and properties can be found in [21]. The fact that each linear bounded functional on ${\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{n×1}\right)$ can be written uniquely in the form described in (4) is proved in [22]. See also [20].

Now let us define a solution of problem (1)-(3).

Definition 2 A function $u\in {\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{n}\right)$ is said to be a solution of problem (1)-(3) if u satisfies (1) for a.e. $t\in \left[a,b\right]$ and fulfils conditions (2) and (3).

## 2 Problem with impulses at fixed times

In the paper [20] we have found an operator representation to the special type of problem (1)-(3) having impulses at fixed times. This is the case that the barrier functions ${\gamma }_{i}$ in (2) are constant functions, i.e. there exist ${t}_{1},\dots ,{t}_{p}\in \mathbb{R}$ satisfying $a<{t}_{1}<\cdots <{t}_{p} such that

(5)

In this case, each solution of the problem crosses i th barrier at same time instant ${\tau }_{i}={t}_{i}$ for $i=1,\dots ,p$.

Note that boundary value problems for higher-order differential equations with impulses at fixed times have been studied for example in [2331] and for delay higher-order impulsive equations in [32, 33].

Let us summarize the results of the paper [20] according to our needs. Assume that the linear homogeneous problem

(6)

has only the trivial solution. Let $\left\{{\stackrel{˜}{u}}_{1},\dots ,{\stackrel{˜}{u}}_{n}\right\}$ be a fundamental system of solutions of the differential equation from (6), W be their Wronski matrix and w its first row, i.e.

$W\left(t\right)=\left(\begin{array}{ccc}{\stackrel{˜}{u}}_{1}\left(t\right)& \cdots & {\stackrel{˜}{u}}_{n}\left(t\right)\\ {\stackrel{˜}{u}}_{1}^{\prime }\left(t\right)& \cdots & {\stackrel{˜}{u}}_{n}^{\prime }\left(t\right)\\ {\stackrel{˜}{u}}_{1}^{\left(n-1\right)}\left(t\right)& \cdots & {\stackrel{˜}{u}}_{n}^{\left(n-1\right)}\left(t\right)\end{array}\right),\phantom{\rule{2em}{0ex}}w\left(t\right)=\left({\stackrel{˜}{u}}_{1}\left(t\right),\dots ,{\stackrel{˜}{u}}_{n}\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \left[a,b\right].$
(7)

Denote

$\ell \left(W\right)={\left({\ell }_{i}\left({\stackrel{˜}{u}}_{j},{\stackrel{˜}{u}}_{j}^{\prime },\dots ,{\stackrel{˜}{u}}_{j}^{\left(n-1\right)}\right)\right)}_{i,j=1}^{n}.$
(8)

From [[20], Lemma 8] (see also Chapter 3 in [34]) it follows that the unique solvability of (6) is equivalent to the condition

$det\ell \left(W\right)\ne 0.$
(9)

Further assume (9), consider ${V}_{j}$, $j=1,\dots ,n$, from (4), and denote

$V\left(t\right)=\left(\begin{array}{c}{V}_{1}\left(t\right)\\ {V}_{2}\left(t\right)\\ \cdots \\ {V}_{n}\left(t\right)\end{array}\right),\phantom{\rule{2em}{0ex}}A\left(t\right)=\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ 0& 0& 0& \cdots & 1\\ -\frac{{a}_{0}\left(t\right)}{{a}_{n}\left(t\right)}& -\frac{{a}_{1}\left(t\right)}{{a}_{n}\left(t\right)}& -\frac{{a}_{2}\left(t\right)}{{a}_{n}\left(t\right)}& \cdots & -\frac{{a}_{n-1}\left(t\right)}{{a}_{n}\left(t\right)}\end{array}\right),$

$t\in \left[a,b\right]$ and

$H\left(\tau \right)=-{\left[\ell \left(W\right)\right]}^{-1}\left({\int }_{\tau }^{b}V\left(s\right)A\left(s\right)W\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\cdot {W}^{-1}\left(\tau \right)+V\left(\tau \right)\right),\phantom{\rule{1em}{0ex}}\tau \in \left[a,b\right].$
(10)

If we denote by ${H}_{ij}$ and ${\omega }_{ij}$ elements of the matrices H and ${W}^{-1}$, respectively, that is,

$H\left(\tau \right)={\left({H}_{ij}\left(\tau \right)\right)}_{i,j=1}^{n},\phantom{\rule{2em}{0ex}}{W}^{-1}\left(\tau \right)={\left({\omega }_{ij}\left(\tau \right)\right)}_{i,j=1}^{n},$
(11)

we can define functions ${g}_{j}$, $j=1,\dots ,n$, as

${g}_{j}\left(t,\tau \right)=\sum _{k=1}^{n}{\stackrel{˜}{u}}_{k}\left(t\right)\left({H}_{kj}\left(\tau \right)+{\chi }_{\left(\tau ,b\right]}\left(t\right){\omega }_{kj}\left(\tau \right)\right),\phantom{\rule{1em}{0ex}}t,\tau \in \left[a,b\right].$
(12)

For each fixed $\tau \in \left[a,b\right]$ the functions $\frac{{\partial }^{k}{g}_{j}\left(t,\tau \right)}{\partial {\tau }^{k}}$, $k=0,1,\dots ,n-1$, will be understood as right-continuous extensions at $t=a$ and left-continuous extensions at $t=\tau$ and $t=b$. In this way the Green’s function of problem (6) is built (cf. Remark 6).

Remark 3 In order to state one of the main results of [20] we introduce the set of all functions u continuous on the intervals $\left[a,{t}_{1}\right],\left({t}_{1},{t}_{2}\right],\dots ,\left({t}_{p},b\right]$, with ${t}_{i}$ from (5), having their derivatives ${u}^{\prime },\dots ,{u}^{\left(n-1\right)}$ continuously extendable onto these intervals. This set is denoted by ${\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$. For $u\in {\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$ we define

Equipped with the standard addition, scalar multiplication, and with the norm

$\parallel u\parallel =\sum _{k=0}^{n-1}{\parallel {u}^{\left(k\right)}\parallel }_{\mathrm{\infty }},\phantom{\rule{1em}{0ex}}u\in {\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right),$

${\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$ forms a Banach space.

Now we are ready to state the operator representation theorem for the problem with impulses at fixed times $a<{t}_{1}<\cdots <{t}_{p} which has the form

(13)
${u}^{\left(j\right)}\left({t}_{i}+\right)-{u}^{\left(j\right)}\left({t}_{i}\right)={J}_{ij}\left(u\left({t}_{i}\right),{u}^{\prime }\left({t}_{i}\right),\dots ,{u}^{\left(n-1\right)}\left({t}_{i}\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,p,j=0,\dots ,n-1,$
(14)
${\ell }_{k}\left(u,{u}^{\prime },\dots ,{u}^{\left(n-1\right)}\right)={c}_{k},\phantom{\rule{1em}{0ex}}k=1,\dots ,n.$
(15)

Theorem 4 [[20], Theorem 17]

Let (4), (9) hold, and let W, w, $\ell \left(W\right)$ and ${g}_{j}$, $j=1,\dots ,n$ be defined in (7), (8), and (12). Then $u\in {\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$ is a fixed point of an operator $\mathcal{H}:{\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)\to {\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$ defined by

$\begin{array}{l}\left(\mathcal{H}u\right)\left(t\right)={\int }_{a}^{b}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}h\left(s,u\left(s\right),\dots ,{u}^{\left(n-1\right)}\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\left(\mathcal{H}u\right)\left(t\right)=}+\sum _{j=1}^{n}\sum _{i=1}^{p}{g}_{j}\left(t,{t}_{i}\right){J}_{i,j-1}\left(u\left({t}_{i}\right),\dots ,{u}^{\left(n-1\right)}\left({t}_{i}\right)\right)\\ \phantom{\left(\mathcal{H}u\right)\left(t\right)=}+w\left(t\right){\left[\ell \left(W\right)\right]}^{-1}{\left({c}_{1},\dots ,{c}_{n}\right)}^{T},\end{array}\right\}$
(16)

$t\in \left[a,b\right]$, if and only if u is a solution of problem (13)-(15). Moreover, the operator is completely continuous.

Remark 5 Let us note that the row vector

$w\left(t\right){\left[\ell \left(W\right)\right]}^{-1}$

does not depend on the choice of a fundamental system of solutions ${\stackrel{˜}{u}}_{1},\dots ,{\stackrel{˜}{u}}_{n}$, but only on the data of problem (6).

Remark 6 Let us put

${J}_{ij}=0,\phantom{\rule{1em}{0ex}}i=1,\dots ,p,j=0,\dots ,n-1,\phantom{\rule{2em}{0ex}}{c}_{k}=0,\phantom{\rule{1em}{0ex}}k=1,\dots ,n$

and

Then the operator in Theorem 4 can be written as

$\left({\mathcal{H}}_{0}u\right)\left(t\right)={\int }_{a}^{b}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}{h}_{0}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

Theorem 4 implies that u is a fixed point of ${\mathcal{H}}_{0}$ if and only if u is a solution of the problem

$\sum _{j=0}^{n}{a}_{j}\left(t\right){u}^{\left(j\right)}\left(t\right)={h}_{0}\left(t\right),\phantom{\rule{2em}{0ex}}{\ell }_{j}\left(u,{u}^{\prime },\dots ,{u}^{\left(n-1\right)}\right)=0,\phantom{\rule{1em}{0ex}}j=1,\dots ,n.$
(17)

Therefore a (unique) solution of problem (17) has the form

$u\left(t\right)={\int }_{a}^{b}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}{h}_{0}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,$

and consequently $\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}$ is the Green’s function of (6).

Remark 7 Under the assumption (9) we are allowed using (11) to define the functions

$\begin{array}{l}{g}_{j}^{\left[1\right]}\left(t,\tau \right)=\sum _{k=1}^{n}{\stackrel{˜}{u}}_{k}\left(t\right){H}_{kj}\left(\tau \right),\\ {g}_{j}^{\left[2\right]}\left(t,\tau \right)=\sum _{k=1}^{n}{\stackrel{˜}{u}}_{k}\left(t\right)\left({H}_{kj}\left(\tau \right)+{\omega }_{kj}\left(\tau \right)\right)\end{array}\right\}$
(18)

for $t,\tau \in \left[a,b\right]$, $j=1,\dots ,n$. Obviously, due to (12),

(19)

for $j=1,\dots ,n$. Let us stress that ${g}_{j}^{\left[\nu \right]}$, as well as ${g}_{j}$, do not depend on the choice of fundamental system ${\stackrel{˜}{u}}_{1},\dots ,{\stackrel{˜}{u}}_{n}$, but only on the data of problem (6). The functions ${g}_{j}^{\left[\nu \right]}$ possess crucial properties for our approach. From their definition it follows that for each $\tau \in \left[a,b\right]$

$\frac{{\partial }^{k}{g}_{j}^{\left[\nu \right]}}{\partial {t}^{k}}\left(\cdot ,\tau \right)\in \mathbb{AC}\left(\left[a,b\right];\mathbb{R}\right)$
(20)

for $\nu =1,2$, $j=1,\dots ,n$, $k=0,\dots ,n-1$. Moreover, for each $\nu =1,2$, $j=1,\dots ,n$, $k=0,\dots ,n-1$, there exists a constant ${C}_{\nu jk}>0$ such that

$|\frac{{\partial }^{k}{g}_{j}^{\left[\nu \right]}}{\partial {t}^{k}}\left(t,\tau \right)|\le {C}_{\nu jk}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}|\frac{{\partial }^{k}{g}_{j}}{\partial {t}^{k}}\left(t,\tau \right)|\le \underset{\nu =1,2}{max}{C}_{\nu jk}\phantom{\rule{1em}{0ex}}t,\tau \in \left[a,b\right].$
(21)

This follows from the definition of ${g}_{j}^{\left[\nu \right]}$ ($\nu =1,2$), from the fact $w\in {\mathbb{C}}^{n-1}\left(\left[a,b\right];{\mathbb{R}}^{1×n}\right)$ and from the boundedness of the matrices ${W}^{-1}$ and H (cf. (7), (10) and (11)).

## 3 Transversality conditions

The most results for differential equations with state-dependent impulses concern initial value problems. Theorems about the existence, uniqueness or extension of solutions of initial value problems, and about intersections of such solutions with barriers ${\gamma }_{i}$ can be found for example in [[35], Chapter 5].

A different approach has to be used when boundary value problems with state-dependent impulses are discussed and boundary conditions are imposed on a solution anywhere in the interval $\left[a,b\right]$ including unknown points of impulses. This is the case of problem (1)-(3).

Our approach is based on the existence of a fixed point of an operator in some set $\overline{\mathrm{\Omega }}={\overline{\mathcal{B}}}^{p+1}$ (cf. Lemma 12), where $\overline{\mathcal{B}}\subset {\mathbb{C}}^{n-1}\left(\left[a,b\right];\mathbb{R}\right)$ is a ball defined in (28). In order to get a fixed point, we need to prove for functions of $\overline{\mathcal{B}}$ assertions about their transversality through barriers. Such assertions are contained in Lemmas 9 and 10 and it is important that they are valid for all functions in $\overline{\mathcal{B}}$ and not only for solutions of problem (1), (2).

Remark 8 Having the lemmas about the transversality, we will prove in Section 4 the existence of a solution u of problem (1)-(3), which has the following property:

(22)

Consider real numbers ${K}_{j}$, $j=0,1,\dots ,n-1$, and denote

${\mathcal{A}}_{n}=\left\{\left({x}_{0},{x}_{1},\dots ,{x}_{n-1}\right)\in {\mathbb{R}}^{n}:|{x}_{0}|\le {K}_{0},\dots ,|{x}_{n-1}|\le {K}_{n-1}\right\}.$
(23)

Now, we are ready to formulate the following transversality conditions:

$a<\underset{{\mathcal{A}}_{n-1}}{min}{\gamma }_{1}\le \underset{{\mathcal{A}}_{n-1}}{max}{\gamma }_{i-1}<\underset{{\mathcal{A}}_{n-1}}{min}{\gamma }_{i}\le \underset{{\mathcal{A}}_{n-1}}{max}{\gamma }_{p}
(24)
(25)
(26)
$\begin{array}{l}{\gamma }_{i}\left({x}_{0}+{J}_{i0}\left({x}_{0},\dots ,{x}_{n-1}\right),\dots ,{x}_{n-2}+{J}_{i,n-2}\left({x}_{0},\dots ,{x}_{n-1}\right)\right)\\ \phantom{\rule{1em}{0ex}}\le {\gamma }_{i}\left({x}_{0},\dots ,{x}_{n-2}\right),\phantom{\rule{1em}{0ex}}\left({x}_{0},\dots ,{x}_{n-1}\right)\in {\mathcal{A}}_{n},i=1,\dots ,p.\end{array}\right\}$
(27)

Let us define the set

(28)

Our current goal is to find a continuous functional ${\mathcal{P}}_{i}$ for $i=1,\dots ,p$, which maps each function u from $\overline{\mathcal{B}}$ to some time instant ${\tau }_{i}$ of (2).

Lemma 9 Let ${K}_{j}$, $j=0,\dots ,n-1$, ${L}_{ik}$, $i=1,\dots ,p$, $k=0,\dots ,n-2$, be real numbers satisfying (26), and let ${\mathcal{A}}_{n}$ and be given by (23) and (28), respectively. Finally, assume that ${\gamma }_{i}$, $i=1,\dots ,p$, satisfy (24), (25), and choose $u\in \overline{\mathcal{B}}$. Then the function

$\sigma \left(t\right)={\gamma }_{i}\left(u\left(t\right),{u}^{\prime }\left(t\right),\dots ,{u}^{\left(n-2\right)}\left(t\right)\right)-t,\phantom{\rule{1em}{0ex}}t\in \left[a,b\right],$
(29)

is continuous and decreasing on $\left[a,b\right]$ and it has a unique root in the interval $\left(a,b\right)$, i.e. there exists a unique solution of the equation

$t={\gamma }_{i}\left(u\left(t\right),\dots ,{u}^{\left(n-2\right)}\left(t\right)\right).$
(30)

Proof Let $u\in \overline{\mathcal{B}}$, $i\in \left\{1,\dots ,p\right\}$. By (24),

$\begin{array}{c}\sigma \left(a\right)={\gamma }_{i}\left(u\left(a\right),{u}^{\prime }\left(a\right),\dots ,{u}^{\left(n-2\right)}\left(a\right)\right)-a>0,\hfill \\ \sigma \left(b\right)={\gamma }_{i}\left(u\left(b\right),{u}^{\prime }\left(b\right),\dots ,{u}^{\left(n-2\right)}\left(b\right)\right)-b<0\hfill \end{array}$

is valid. This together with the fact that σ is continuous shows that σ has at least one root in $\left(a,b\right)$. Now, we will prove that σ is decreasing, by a contradiction. Let ${s}_{1},{s}_{2}\in \left(a,b\right)$, ${s}_{1}<{s}_{2}$ be such that

$\sigma \left({s}_{1}\right)=\sigma \left({s}_{2}\right),$

i.e.

${\gamma }_{i}\left(u\left({s}_{1}\right),\dots ,{u}^{\left(n-2\right)}\left({s}_{1}\right)\right)-{\gamma }_{i}\left(u\left({s}_{2}\right),\dots ,{u}^{\left(n-2\right)}\left({s}_{2}\right)\right)={s}_{1}-{s}_{2}.$

From (25), (26), (28), and the Mean Value Theorem we obtain

$\begin{array}{rl}0& <|{s}_{1}-{s}_{2}|=|{\gamma }_{i}\left(u\left({s}_{1}\right),\dots ,{u}^{\left(n-2\right)}\left({s}_{1}\right)\right)-{\gamma }_{i}\left(u\left({s}_{2}\right),\dots ,{u}^{\left(n-2\right)}\left({s}_{2}\right)\right)|\\ \le \sum _{j=0}^{n-2}{L}_{ij}|{u}^{\left(j\right)}\left({s}_{1}\right)-{u}^{\left(j\right)}\left({s}_{2}\right)|\le \sum _{j=0}^{n-2}{L}_{ij}{K}_{j+1}|{s}_{1}-{s}_{2}|<|{s}_{1}-{s}_{2}|,\end{array}$

According to Lemma 9, we can define a functional ${\mathcal{P}}_{i}:\overline{\mathcal{B}}\to \left(a,b\right)$ by

${\mathcal{P}}_{i}u={\tau }_{i},\phantom{\rule{1em}{0ex}}u\in \overline{\mathcal{B}},$
(31)

where ${\tau }_{i}$ is a solution of (30), i.e. a unique root of the function σ from Lemma 9, for $i=1,\dots ,p$. □

Lemma 10 Let the assumptions of Lemma 9 be satisfied. The functionals ${\mathcal{P}}_{i}$, $i=1,\dots ,p$, are continuous.

Proof Let ${u}_{m},u\in \overline{\mathcal{B}}$, for $m\in \mathbb{N}$ such that

(32)

Let us choose $i\in \left\{1,\dots ,p\right\}$ and prove that ${\mathcal{P}}_{i}{u}_{m}\to {\mathcal{P}}_{i}u$ as $m\to \mathrm{\infty }$. We denote

$\tau ={\mathcal{P}}_{i}u,\phantom{\rule{2em}{0ex}}{\tau }_{m}={\mathcal{P}}_{i}{u}_{m},\phantom{\rule{1em}{0ex}}m\in \mathbb{N}.$

From Lemma 9 it follows that $\tau ,{\tau }_{m}\in \left(a,b\right)$ are the unique roots of the functions

$\sigma \left(t\right)={\gamma }_{i}\left(u\left(t\right),\dots ,{u}^{\left(n-2\right)}\left(t\right)\right)-t,\phantom{\rule{2em}{0ex}}{\sigma }_{m}\left(t\right)={\gamma }_{i}\left({u}_{m}\left(t\right),\dots ,{u}_{m}^{\left(n-2\right)}\left(t\right)\right)-t,\phantom{\rule{1em}{0ex}}t\in \left[a,b\right],$

and these functions are strictly decreasing. Let $ϵ\in \mathbb{R}$, $ϵ>0$ be such that $\tau -ϵ,\tau +ϵ\in \left(a,b\right)$. Then $\sigma \left(\tau -ϵ\right)>0$ and $\sigma \left(\tau +ϵ\right)<0$. According to (32) we see that ${\sigma }_{m}\to \sigma$ uniformly on $\left[a,b\right]$, in particular ${\sigma }_{m}\left(\tau -ϵ\right)\to \sigma \left(\tau -ϵ\right)$ and ${\sigma }_{m}\left(\tau +ϵ\right)\to \sigma \left(\tau +ϵ\right)$ as $m\to \mathrm{\infty }$. These facts imply that

From the continuity of ${\sigma }_{m}$ and the Intermediate Value Theorem it follows that

which completes the proof. □

Our next step is to define an appropriate operator representation of the BVP with state-dependent impulses. The first idea would be a direct exploitation of the operator from Theorem 4, putting ${\mathcal{P}}_{i}u$ in place of ${t}_{i}$. This is not possible for many reasons. First, each ${\mathcal{P}}_{i}$ acts on the space of functions having continuous derivatives - but we need functions having p discontinuities. Even if we would overcome this difficulty we arrive at a problem of choosing an appropriate Banach space on which would be acting. According to Remark 8, we search a solution u of problem (1)-(3), which has its jumps (together with $u,{u}^{\prime },\dots ,{u}^{\left(n-1\right)}$) at the points ${\tau }_{i}={\mathcal{P}}_{i}u\in \left(a,b\right)$, $i=1,\dots ,p$ (see (31)). In general, these points are different for different solutions. Consequently, such solutions have to be searched in the Banach space ${\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{n}\right)$. But then there is a difficulty with the continuity of such operator. In fact the operator from (16) having ${\mathcal{P}}_{i}u$ in place of ${t}_{i}$ is not continuous in the space ${\mathbb{G}}_{\mathrm{L}}\left(\left[a,b\right];{\mathbb{R}}^{n}\right)$ (cf. Remark 6.2 and Example 6.3 in [36]).

Therefore, we choose the way here, which we have developed in our joint papers [810]. The main idea of our approach lies in representing the solution u of problem (1)-(3) by an ordered $\left(p+1\right)$-tuple $\left({u}_{1},\dots ,{u}_{p+1}\right)\in {\left[{\mathbb{C}}^{n-1}\left(\left[a,b\right];\mathbb{R}\right)\right]}^{p+1}$ as follows:

$u\left(t\right)=\left\{\begin{array}{cc}{u}_{1}\left(t\right),\hfill & t\in \left[a,{\mathcal{P}}_{1}{u}_{1}\right],\hfill \\ {u}_{2}\left(t\right),\hfill & t\in \left({\mathcal{P}}_{1}{u}_{1},{\mathcal{P}}_{2}{u}_{2}\right],\hfill \\ \dots \hfill & \dots \hfill \\ {u}_{p+1}\left(t\right),\hfill & t\in \left({\mathcal{P}}_{p}{u}_{p},b\right].\hfill \end{array}$
(33)

Consequently, we work with the space

$X={\left[{\mathbb{C}}^{n-1}\left(\left[a,b\right];\mathbb{R}\right)\right]}^{p+1}$

equipped with the norm

It is well known that X is a Banach space.

## 4 Main results

Let us turn our attention to problem (1)-(3) with state-dependent impulses under the assumptions (4) and (9). In our approach we will make use of the tools introduced in the previous sections.

(34)

Consider ${c}_{1},\dots ,{c}_{n}$ from (3), w from (7) and $\ell \left(W\right)$ from (8), and denote

$M={\int }_{a}^{b}m\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t,\phantom{\rule{2em}{0ex}}{c}_{0}={\left({c}_{1},\dots ,{c}_{n}\right)}^{T},\phantom{\rule{2em}{0ex}}{D}_{r}=\underset{t\in \left[a,b\right]}{max}{w}^{\left(r\right)}\left(t\right){\left[\ell \left(W\right)\right]}^{-1}{c}_{0},$
(35)

and

${K}_{r}=M\underset{\nu =1,2}{max}\left\{{C}_{\nu nr}\right\}+\sum _{j=1}^{n}\sum _{k=1}^{p}\underset{\nu =1,2}{max}\left\{{C}_{\nu jr}\right\}{A}_{k,j-1}+{D}_{r},$
(36)

for $r=0,\dots ,n-1$, where ${C}_{\nu jr}$ are constants from (21).

Remark 11 Let us note that the constants ${D}_{r}$ from (35) do not depend on the choice of the fundamental system of solutions ${\stackrel{˜}{u}}_{1},\dots ,{\stackrel{˜}{u}}_{n}$, but only on the coefficients ${a}_{i}$ of the differential equation (1) and on the operators ${\ell }_{j}$ from (3) (and, of course, on the constants ${c}_{j}$).

Now, we are ready to construct a convenient operator for a representation of problem (1)-(3). Let us choose its domain as the closure of the set

$\mathrm{\Omega }={\mathcal{B}}^{p+1}\subset X,$

where is defined in (28) with ${K}_{j}$ from (36).

Now, we have to modify the operator from Theorem 4 using ${g}_{j}^{\left[1\right]}$ and ${g}_{j}^{\left[2\right]}$ instead of the Green’s functions ${g}_{j}$, that is, we define an operator $\mathcal{F}:\overline{\mathrm{\Omega }}\to X$ by $\mathcal{F}\left({u}_{1},\dots ,{u}_{p+1}\right)=\left({x}_{1},\dots ,{x}_{p+1}\right)$ with

$\begin{array}{l}{x}_{i}\left(t\right)=\sum _{k=1}^{p+1}{\int }_{{\tau }_{k-1}}^{{\tau }_{k}}{g}_{n}\left(t,s\right)\frac{h\left(s,{u}_{k}\left(s\right),\dots ,{u}_{k}^{\left(n-1\right)}\left(s\right)\right)}{{a}_{n}\left(s\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{{x}_{i}\left(t\right)=}+\sum _{j=1}^{n}\left(\sum _{i\le k\le p}{g}_{j}^{\left[1\right]}\left(t,{\tau }_{k}\right){J}_{k,j-1}\left({u}_{k}\left({\tau }_{k}\right),\dots ,{u}_{k}^{\left(n-1\right)}\left({\tau }_{k}\right)\right)\\ \phantom{{x}_{i}\left(t\right)=}+\sum _{1\le k
(37)

for $i=1,\dots ,p+1$, $t\in \left[a,b\right]$, where

and W, w, ${g}_{j}$, ${g}_{j}^{\left[1\right]}$, ${g}_{j}^{\left[2\right]}$, $j=1,\dots ,n$, and ${c}_{0}$ are from (7), (12), (18), and (35), respectively.

Let us compare (16) for the operator with (37) for the operator . The first term in (16) expresses a solution of homogeneous boundary value problem without impulses. This term is decomposed in (37) on subintervals which depend on the choice of $\left(p+1\right)$-tuple $\left({u}_{1},\dots ,{u}_{p+1}\right)$. The second term in (16) caused (according to the discontinuity of functions ${g}_{j}$) needed impulses of solutions of the fixed-time impulsive problem (13)-(15). We significantly modify this term in (37) in such a way that, instead of discontinuous functions ${g}_{j}$ which have jumps at the points ${\tau }_{k}={P}_{k}{u}_{k}$, we use smooth functions ${g}_{j}^{\left[1\right]}$, ${g}_{j}^{\left[2\right]}$ defined in (18). Due to this modification the operator maps one tuple of smooth functions ${u}_{1},\dots ,{u}_{p+1}$ onto another tuple of smooth functions ${x}_{1},\dots ,{x}_{p+1}$, and we will be able to prove the compactness of on $\overline{\mathrm{\Omega }}$.

In the next lemma we arrive at a justification of our definition.

Lemma 12 Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. If $\left({u}_{1},\dots ,{u}_{p+1}\right)$ is a fixed point of the operator , then the function u defined by (33) is a solution of problem (1)-(3) satisfying (22).

Proof Let be defined by (28) and $\mathrm{\Omega }={\mathcal{B}}^{p+1}$. Further, let $\left({u}_{1},\dots ,{u}_{p+1}\right)\in \overline{\mathrm{\Omega }}$ be such that $\mathcal{F}\left({u}_{1},\dots ,{u}_{p+1}\right)=\left({u}_{1},\dots ,{u}_{p+1}\right)$. For each $i\in \left\{1,\dots ,p+1\right\}$, we have ${u}_{i}\in \overline{\mathcal{B}}$, and hence by Lemma 9 and (31), there exists a unique solution ${\tau }_{i}={P}_{i}{u}_{i}$ of the equation $t={\gamma }_{i}\left({u}_{i}\left(t\right),\dots ,{u}_{i}^{\left(n-2\right)}\left(t\right)\right)$. Due to (24), the inequalities $a<{\tau }_{1}<\cdots <{\tau }_{p} are valid and u can be defined by (33). We will prove that u is a fixed point of the operator from Theorem 4, taking the space ${\mathbb{PC}}^{n-1}\left(\left[a,b\right]\right)$ from Remark 3 with

${t}_{i}={\tau }_{i},\phantom{\rule{1em}{0ex}}i=1,\dots ,p.$

Denote

$\begin{array}{c}{\tau }_{0}=a,\phantom{\rule{2em}{0ex}}{\tau }_{p+1}=b,\phantom{\rule{2em}{0ex}}{\mathcal{I}}_{1}=\left[{\tau }_{0},{\tau }_{1}\right],\phantom{\rule{2em}{0ex}}{\mathcal{I}}_{2}=\left({\tau }_{1},{\tau }_{2}\right],\hfill \\ {\mathcal{I}}_{3}=\left({\tau }_{2},{\tau }_{3}\right],\phantom{\rule{2em}{0ex}}\dots ,\phantom{\rule{2em}{0ex}}{\mathcal{I}}_{p+1}=\left({\tau }_{p},{\tau }_{p+1}\right],\hfill \end{array}$

and choose $i\in \left\{1,\dots ,p+1\right\}$, $t\in {\mathcal{I}}_{i}$. Then, according to (33), we have

$\begin{array}{rl}u\left(t\right)=& {u}_{i}\left(t\right)=\sum _{k=1}^{p+1}{\int }_{{\mathcal{I}}_{k}}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}h\left(s,{u}_{k}\left(s\right),\dots ,{u}_{k}^{\left(n-1\right)}\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +\sum _{j=1}^{n}\left(\sum _{i\le k\le p}{g}_{j}^{\left[1\right]}\left(t,{\tau }_{k}\right){J}_{k,j-1}\left({u}_{k}\left({\tau }_{k}\right),\dots ,{u}_{k}^{\left(n-1\right)}\left({\tau }_{k}\right)\right)\\ +\sum _{1\le k

Of course we have

$\sum _{k=1}^{p+1}{\int }_{{\mathcal{I}}_{k}}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}h\left(s,u\left(s\right),\dots ,{u}^{\left(n-1\right)}\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int }_{a}^{b}\frac{{g}_{n}\left(t,s\right)}{{a}_{n}\left(s\right)}h\left(s,u\left(s\right),\dots ,{u}^{\left(n-1\right)}\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

Let $k\in \mathbb{N}$ be such that $i\le k\le p$. Then $t\le {\tau }_{i}\le {\tau }_{k}$ and therefore (19) gives

Let $k\in \mathbb{N}$ be such that $1\le k (such k exists only if $i>1$). Then $t>{\tau }_{i-1}\ge {\tau }_{k}$ and therefore we get by (19)

These facts imply that

$\begin{array}{r}\sum _{i\le k\le p}{g}_{j}^{\left[1\right]}\left(t,{\tau }_{k}\right){J}_{k,j-1}\left(u\left({\tau }_{k}\right),\dots ,{u}^{\left(n-1\right)}\left({\tau }_{k}-\right)\right)\\ \phantom{\rule{2em}{0ex}}+\sum _{1\le k

for $j=1,\dots ,n$. Consequently, by virtue of (16) and Theorem 4, u is a solution of problem (13)-(15). Clearly u fulfils equation (1) a.e. on $\left[a,b\right]$ and satisfies the boundary conditions (3). In addition, since u fulfils the impulse conditions (14) with ${t}_{i}={\tau }_{i}$, where ${\tau }_{i}={\gamma }_{i}\left({u}_{i}\left({\tau }_{i}\right),\dots ,{u}_{i}^{\left(n-2\right)}\left({\tau }_{i}\right)\right)={\gamma }_{i}\left(u\left({\tau }_{i}\right),\dots ,{u}^{\left(n-2\right)}\left({\tau }_{i}-\right)\right)$, $i=1,\dots ,p$, we see that u also fulfils the state-dependent impulse conditions (2). According to Remark 8, it remains to prove that ${\tau }_{1},\dots ,{\tau }_{p}$ are the only instants at which the function u crosses the barriers ${\gamma }_{1},\dots ,{\gamma }_{p}$, respectively. To this aim, due to (24) and (33), it suffices to prove that

(38)

Choose an arbitrary $i\in \left\{1,\dots ,p\right\}$ and consider σ from (29). Since u fulfils (2), we have

$\sigma \left({\tau }_{i}-\right)=0.$

Let us denote

$\psi \left(t\right)={\gamma }_{i}\left({u}_{i+1}\left(t\right),{u}_{i+1}^{\prime }\left(t\right),\dots ,{u}_{i+1}^{\left(n-2\right)}\left(t\right)\right)-t,\phantom{\rule{1em}{0ex}}t\in \left[a,b\right].$

From Lemma 9 it follows that ψ is decreasing. So, by virtue of (38), it suffices to prove that

$\psi \left({\tau }_{i}\right)\le 0.$
(39)

Using (33), (2), and (27), we have

$\begin{array}{rl}\psi \left({\tau }_{i}\right)=& {\gamma }_{i}\left({u}_{i+1}\left({\tau }_{i}\right),\dots ,{u}_{i+1}^{\left(n-2\right)}\left({\tau }_{i}\right)\right)-{\tau }_{i}={\gamma }_{i}\left(u\left({\tau }_{i}+\right),\dots ,{u}^{\left(n-2\right)}\left({\tau }_{i}+\right)\right)-{\tau }_{i}\\ =& {\gamma }_{i}\left(u\left({\tau }_{i}-\right)+{J}_{i0}\left(u\left({\tau }_{i}-\right),\dots ,{u}^{\left(n-1\right)}\left({\tau }_{i}-\right)\right),\dots ,{u}^{\left(n-2\right)}\left({\tau }_{i}-\right)\\ +{J}_{i,n-2}\left(u\left({\tau }_{i}-\right),\dots ,{u}^{\left(n-1\right)}\left({\tau }_{i}-\right)\right)\right)-{\tau }_{i}\\ \le & {\gamma }_{i}\left(u\left({\tau }_{i}-\right),\dots ,{u}^{\left(n-2\right)}\left({\tau }_{i}-\right)\right)-{\tau }_{i}=0,\end{array}$

which yields (39). This completes the proof. □

Lemma 13 Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. Then the operator from (37) has a fixed point in $\overline{\mathrm{\Omega }}$.

Proof The last term $\omega \left(t\right){\left[\ell \left(W\right)\right]}^{-1}{c}_{0}$ in (37) is the same as in (16) for the compact operator . Therefore it suffices to prove the compactness of the operator on $\overline{\mathrm{\Omega }}$ for ${c}_{0}=0$. To do it we can use the same arguments as in the proof of Lemma 6 in [9], where the second-order state-dependent impulsive problem is investigated. In particular, the compactness of on $\overline{\mathrm{\Omega }}$ is a consequence of the following properties of functions and functionals contained in (37):

• the first term in (37) can be written in the form

$\begin{array}{r}\sum _{k=1}^{p+1}{\int }_{{\tau }_{k-1}}^{{\tau }_{k}}{g}_{n}\left(t,s\right)\frac{h\left(s,{u}_{k}\left(s\right),\dots ,{u}_{k}^{\left(n-1\right)}\left(s\right)\right)}{{a}_{n}\left(s\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}={\int }_{a}^{b}{g}_{n}\left(t,s\right)\sum _{k=1}^{p+1}\frac{h\left(s,{u}_{k}\left(s\right),\dots ,{u}_{k}^{\left(n-1\right)}\left(s\right)\right)}{{a}_{n}\left(s\right)}{\chi }_{\left({\tau }_{k-1},{\tau }_{k}\right)}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\end{array}$

where ${\tau }_{k}={\mathcal{P}}_{k}{u}_{k}$ for $k=1,\dots ,p$, ${\tau }_{0}=a$, ${\tau }_{p+1}=b$,

• ${\mathcal{P}}_{k}$ are continuous on $\overline{\mathcal{B}}$ (due to Lemma 10),

• $\frac{h\left(t,x\right)}{{a}_{n}\left(t\right)}\in Car\left(\left[a,b\right]×{\mathbb{R}}^{n};\mathbb{R}\right)$,

• ${g}_{j}^{\left[1\right]}$, ${g}_{j}^{\left[2\right]}$ satisfy (20), ${g}_{n}$ satisfies (19),

• ${J}_{kj}$ are continuous on ${\mathbb{R}}^{n}$.

For the application of the Schauder Fixed Point Theorem it remains to prove that

$\mathcal{F}\left(\overline{\mathrm{\Omega }}\right)\subset \overline{\mathrm{\Omega }}.$
(40)

Let $\left({x}_{1},\dots ,{x}_{p+1}\right)=\mathcal{F}\left({u}_{1},\dots ,{u}_{p+1}\right)$ for some $\left({u}_{1},\dots ,{u}_{p+1}\right)\in \overline{\mathrm{\Omega }}$. Then, by (21), (34), (35), and (37), we have

$|{x}_{i}^{\left(r\right)}\left(t\right)|\le M\underset{\nu =1,2}{max}\left\{{C}_{\nu nr}\right\}+\sum _{j=1}^{n}\sum _{k=1}^{p}\underset{\nu =1,2}{max}\left\{{C}_{\nu jr}\right\}{A}_{k,j-1}+{D}_{r}$

for $i=1,\dots ,p+1$, $r=0,\dots ,n-1$, $t\in \left[a,b\right]$. From (36) we get

${\parallel {x}_{i}^{\left(r\right)}\parallel }_{\mathrm{\infty }}\le {K}_{r},\phantom{\rule{1em}{0ex}}i=1,\dots ,p+1,r=0,\dots ,n-1,$

and so $\mathcal{F}\left({u}_{1},\dots ,{u}_{p+1}\right)\in \overline{\mathrm{\Omega }}$. We have proved (40), and consequently there exists at least one fixed point of in $\overline{\mathrm{\Omega }}$. □

Theorem 14 Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. Then there exists at least one solution to problem (1)-(3) satisfying (22).

Proof The assertion follows directly from Lemma 12 and Lemma 13. □

Remark 15 The existence result from Theorem 14 can be extended to unbounded functions h and ${J}_{ij}$ by means of the method of a priori estimates. This can be found for the special case $n=2$ in [10].

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## Acknowledgements

The authors were supported by the grant IGA_PrF_2014028. The authors sincerely thank the anonymous referees for their valuable comments and suggestions.

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Correspondence to Irena Rachůnková.

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Rachůnková, I., Tomeček, J. Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem. Bound Value Probl 2014, 118 (2014). https://doi.org/10.1186/1687-2770-2014-118

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• DOI: https://doi.org/10.1186/1687-2770-2014-118