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 ${\sum }_{j=0}^n{a}_j(t)u^{(j)}(t)=h(t,u(t),\dots ,u^{(n-1)}(t))$, a.e. $t\in [a,b]\subset \mathbb{R}$, ${\ell }_k(u,u^{\prime },\dots ,u^{(n-1)})=c_k$, $k=1,\dots ,n$, with the state-dependent impulses $u^{(j)}(t+)-u^{(j)}(t-)=J_{ij}(u(t-),u^{\prime }(t-),\dots ,u^{(n-1)}(t-))$, where the impulse points t are determined as solutions of the equations $t={\gamma }_i(u(t-),u^{\prime }(t-),\dots ,u^{(n-2)}(t-))$, $i=1,\dots ,p$, $j=0,\dots ,n-1$. 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 $[a,b]$ and $h/a_n$ satisfies the Carathéodory conditions on $[a,b]\times {\mathbb{R}}^n$. 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 $[a,b]$ 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 $[a,b]\subset \mathbb{R}$. Studies of real-life problems with state-dependent impulses can be found e.g. in [1–6]. Here we consider the equation

$$\sum _{j=0}^n{a}_j(t)u^{(j)}(t)=h(t,u(t),\dots ,u^{(n-1)}(t)),\text{a.e. }t\in [a,b],$$

(1)

subject to the impulse conditions

$$\begin{array}{l}{u}^{(j)}(t+)-u^{(j)}(t-)=J_{ij}(u(t-),u^{\prime }(t-),\dots ,u^{(n-1)}(t-)),\\ \text{where }t={\gamma }_i(u(t-),u^{\prime }(t-),\dots ,u^{(n-2)}(t-))\\ \text{for }i=1,\dots ,p,j=0,\dots ,n-1,\end{array}\}$$

(2)

and the linear boundary conditions

$${\ell }_k(u,u^{\prime },\dots ,u^{(n-1)})=c_k,k=1,\dots ,n.$$

(3)

In what follows we use this notation. Let $k,m,n\in \mathbb{N}$. By ${\mathbb{R}}^{m\times n}$ we denote the set of all matrices of the type $m\times n$ with real valued coefficients. Let $A^T$ denote the transpose of $A\in {\mathbb{R}}^{m\times n}$. Let ${\mathbb{R}}^n={\mathbb{R}}^{n\times 1}$ be the set of all n-dimensional column vectors $c={(c_1,\dots ,c_n)}^T$, where $c_i\in \mathbb{R}$, $i=1,\dots ,n$, and $\mathbb{R}={\mathbb{R}}^{1\times 1}$. By $\mathbb{C}({\mathbb{R}}^n;{\mathbb{R}}^m)$ we denote the set of all mappings $x:{\mathbb{R}}^n\to {\mathbb{R}}^m$ with continuous components. By ${\mathbb{L}}^{\mathrm{\infty }}([a,b];{\mathbb{R}}^{m\times n})$, ${\mathbb{L}}^1([a,b];{\mathbb{R}}^{m\times n})$, ${\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^{m\times n})$, $\mathbb{AC}([a,b];{\mathbb{R}}^{m\times n})$, $\mathbb{BV}([a,b];{\mathbb{R}}^{m\times n})$, ${\mathbb{C}}^k([a,b];{\mathbb{R}}^{m\times n})$, we denote the sets of all mappings $x:[a,b]\to {\mathbb{R}}^{m\times n}$ 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 $[a,b]$. By $Car([a,b]\times {\mathbb{R}}^n;\mathbb{R})$ we denote the set of all functions $f:[a,b]\times {\mathbb{R}}^n\to \mathbb{R}$ satisfying the Carathéodory conditions on the set $[a,b]\times {\mathbb{R}}^n$. Finally, by ${\chi }_M$ we denote the characteristic function of the set $M\subset \mathbb{R}$.

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

$$u(t)=u(t-)=\underset{\tau \to t-}{lim}u(\tau ),u(s+)=\underset{\tau \to s+}{lim}u(\tau ).$$

${\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n)$ 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

$${\parallel u\parallel }_{\mathrm{\infty }}=\underset{i\in \{1,\dots ,n\}}{max}\left(\underset{t\in [a,b]}{sup}|u_i(t)|\right)\text{for }u={(u_1,\dots ,u_n)}^T\in {\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n).$$

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

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

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

• for each compact set $K\subset {\mathbb{R}}^n$ there exists a function $m_K\in {\mathbb{L}}^1([a,b];\mathbb{R})$ such that $|f(t,x)|\le m_K(t)$ for a.e. $t\in [a,b]$ 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 [8–10] 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 [11–19]. To get the existence results for problem (1)-(3), we exploit the paper [20] with fixed-time impulsive problems.

Here we assume that

$$\begin{array}{l}n\ge 2,\frac{a_j}{a_n}\in {\mathbb{L}}^1([a,b];\mathbb{R}),j=0,\dots ,n-1,\frac{h(t,x)}{a_n(t)}\in Car([a,b]\times {\mathbb{R}}^n;\mathbb{R}),\\ c_j\in \mathbb{R},J_{ij}\in \mathbb{C}({\mathbb{R}}^n;\mathbb{R}),{\gamma }_i\in \mathbb{C}({\mathbb{R}}^{n-1};\mathbb{R}),i=1,\dots ,p,j=0,\dots ,n-1,\\ {\ell }_k:{\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n)\to \mathbb{R}\text{ is a linear bounded functional},\mathit{\text{ i.e.}}\\ {\ell }_k(z)=K_{k}z(a)+{\int }_a^b{V}_k(t)\mathrm{d}[z(t)],z\in {\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^{n\times 1}),\\ \text{where }{K}_k\in {\mathbb{R}}^{1\times n},V_k\in \mathbb{BV}([a,b];{\mathbb{R}}^{1\times n}),k=1,\dots ,n,n,p\in \mathbb{N}.\end{array}\}$$

(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 ${\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^{n\times 1})$ 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 $u\in {\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n)$ is said to be a solution of problem (1)-(3) if u satisfies (1) for a.e. $t\in [a,b]$ 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 $a<t_1<\cdots <t_p<b$ such that

$${\gamma }_i(x_0,x_1,\dots ,x_{n-2})=t_i\text{for }i=1,\dots ,p,x_0,x_1,\dots ,x_{n-2}\in \mathbb{R}.$$

(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 [23–31] 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

$$\begin{array}{l}\sum _{j=0}^n{a}_j(t)u^{(j)}(t)=0,\text{a.e. }t\in [a,b],\\ {\ell }_k(u,u^{\prime },\dots ,u^{(n-1)})=0,k=1,\dots ,n,\end{array}\}$$

(6)

has only the trivial solution. Let $\{{\tilde{u}}_1,\dots ,{\tilde{u}}_n\}$ be a fundamental system of solutions of the differential equation from (6), W be their Wronski matrix and w its first row, i.e.

$$W(t)=\left(\begin{array}{ccc}{\tilde{u}}_1(t)& \cdots & {\tilde{u}}_n(t)\\ {\tilde{u}}_1^{\prime }(t)& \cdots & {\tilde{u}}_n^{\prime }(t)\\ {\tilde{u}}_1^{(n-1)}(t)& \cdots & {\tilde{u}}_n^{(n-1)}(t)\end{array}\right),w(t)=({\tilde{u}}_1(t),\dots ,{\tilde{u}}_n(t)),t\in [a,b].$$

(7)

Denote

$$\ell (W)={\left({\ell }_i({\tilde{u}}_j,{\tilde{u}}_j^{\prime },\dots ,{\tilde{u}}_j^{(n-1)})\right)}_{i,j=1}^n.$$

(8)

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

$$det\ell (W)\ne 0.$$

(9)

Further assume (9), consider $V_j$, $j=1,\dots ,n$, from (4), and denote

$$V(t)=\left(\begin{array}{c}{V}_1(t)\\ V_2(t)\\ \cdots \\ V_n(t)\end{array}\right),A(t)=\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ 0& 0& 0& \cdots & 1\\ -\frac{a_0(t)}{a_n(t)}& -\frac{a_1(t)}{a_n(t)}& -\frac{a_2(t)}{a_n(t)}& \cdots & -\frac{a_{n-1}(t)}{a_n(t)}\end{array}\right),$$

$t\in [a,b]$ and

$$H(\tau )=-{[\ell (W)]}^{-1}({\int }_{\tau }^{b}V(s)A(s)W(s)\mathrm{d}s\cdot W^{-1}(\tau )+V(\tau )),\tau \in [a,b].$$

(10)

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

$$H(\tau )={(H_{ij}(\tau ))}_{i,j=1}^n,W^{-1}(\tau )={({\omega }_{ij}(\tau ))}_{i,j=1}^n,$$

(11)

we can define functions $g_j$, $j=1,\dots ,n$, as

$$g_j(t,\tau )=\sum _{k=1}^n{\tilde{u}}_k(t)(H_{kj}(\tau )+{\chi }_{(\tau ,b]}(t){\omega }_{kj}(\tau )),t,\tau \in [a,b].$$

(12)

For each fixed $\tau \in [a,b]$ the functions $\frac{{\partial }^k{g}_j(t,\tau )}{\partial {\tau }^k}$, $k=0,1,\dots ,n-1$, 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 $[a,t_1],(t_1,t_2],\dots ,(t_p,b]$, with $t_i$ from (5), having their derivatives $u^{\prime },\dots ,u^{(n-1)}$ continuously extendable onto these intervals. This set is denoted by ${\mathbb{PC}}^{n-1}([a,b])$. For $u\in {\mathbb{PC}}^{n-1}([a,b])$ we define

$$u^{(k)}(a)=u^{(k)}(a+),u^{(k)}(t_i)=u^{(k)}(t_i-)\text{for }k=1,\dots ,n-1,i=1,\dots ,p.$$

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

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

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

Now we are ready to state the operator representation theorem for the problem with impulses at fixed times $a<t_1<\cdots <t_p<b$ which has the form

$$\sum _{j=0}^n{a}_j(t)u^{(j)}(t)=h(t,u(t),\dots ,u^{(n-1)}(t)),\text{a.e. }t\in [a,b],$$

(13)

$$u^{(j)}(t_i+)-u^{(j)}(t_i)=J_{ij}(u(t_i),u^{\prime }(t_i),\dots ,u^{(n-1)}(t_i)),i=1,\dots ,p,j=0,\dots ,n-1,$$

(14)

$${\ell }_k(u,u^{\prime },\dots ,u^{(n-1)})=c_k,k=1,\dots ,n.$$

(15)

Theorem 4 [[20], Theorem 17]

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

$$\begin{array}{l}(\mathcal{H}u)(t)={\int }_a^b\frac{g_n(t,s)}{a_n(s)}h(s,u(s),\dots ,u^{(n-1)}(s))\mathrm{d}s\\ \phantom{(\mathcal{H}u)(t)=}+\sum _{j=1}^n\sum _{i=1}^p{g}_j(t,t_i)J_{i,j-1}(u(t_i),\dots ,u^{(n-1)}(t_i))\\ \phantom{(\mathcal{H}u)(t)=}+w(t){[\ell (W)]}^{-1}{(c_1,\dots ,c_n)}^T,\end{array}\}$$

(16)

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

Remark 5 Let us note that the row vector

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

does not depend on the choice of a fundamental system of solutions ${\tilde{u}}_1,\dots ,{\tilde{u}}_n$, but only on the data of problem (6).

Remark 6 Let us put

$$J_{ij}=0,i=1,\dots ,p,j=0,\dots ,n-1,c_k=0,k=1,\dots ,n$$

and

$$h(t,x)=h_0(t)\in {\mathbb{L}}^1([a,b];\mathbb{R})\text{for }x\in {\mathbb{R}}^n.$$

Then the operator ℋ in Theorem 4 can be written as

$$({\mathcal{H}}_{0}u)(t)={\int }_a^b\frac{g_n(t,s)}{a_n(s)}{h}_0(s)\mathrm{d}s.$$

Theorem 4 implies that u is a fixed point of ${\mathcal{H}}_0$ if and only if u is a solution of the problem

$$\sum _{j=0}^n{a}_j(t)u^{(j)}(t)=h_0(t),{\ell }_j(u,u^{\prime },\dots ,u^{(n-1)})=0,j=1,\dots ,n.$$

(17)

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

$$u(t)={\int }_a^b\frac{g_n(t,s)}{a_n(s)}{h}_0(s)\mathrm{d}s,$$

and consequently $\frac{g_n(t,s)}{a_n(s)}$ is the Green’s function of (6).

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

$$\begin{array}{l}{g}_j^{[1]}(t,\tau )=\sum _{k=1}^n{\tilde{u}}_k(t)H_{kj}(\tau ),\\ g_j^{[2]}(t,\tau )=\sum _{k=1}^n{\tilde{u}}_k(t)(H_{kj}(\tau )+{\omega }_{kj}(\tau ))\end{array}\}$$

(18)

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

$$g_j(t,\tau )=\{\begin{array}{cc}{g}_j^{[1]}(t,\tau )\hfill & \text{for }a\le t\le \tau \le b,\hfill \\ g_j^{[2]}(t,\tau )\hfill & \text{for }a\le \tau <t\le b,\hfill \end{array}$$

(19)

for $j=1,\dots ,n$. Let us stress that $g_j^{[\nu ]}$, as well as $g_j$, do not depend on the choice of fundamental system ${\tilde{u}}_1,\dots ,{\tilde{u}}_n$, but only on the data of problem (6). The functions $g_j^{[\nu ]}$ possess crucial properties for our approach. From their definition it follows that for each $\tau \in [a,b]$

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

(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

$$|\frac{{\partial }^k{g}_j^{[\nu ]}}{\partial t^k}(t,\tau )|\le C_{\nu jk}\text{and}|\frac{{\partial }^k{g}_j}{\partial t^k}(t,\tau )|\le \underset{\nu =1,2}{max}{C}_{\nu jk}t,\tau \in [a,b].$$

(21)

This follows from the definition of $g_j^{[\nu ]}$ ($\nu =1,2$), from the fact $w\in {\mathbb{C}}^{n-1}([a,b];{\mathbb{R}}^{1\times n})$ 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 $[a,b]$ 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 ℱ in some set $\overline{\mathrm{\Omega }}={\overline{\mathcal{B}}}^{p+1}$ (cf. Lemma 12), where $\overline{\mathcal{B}}\subset {\mathbb{C}}^{n-1}([a,b];\mathbb{R})$ 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:

$$\begin{array}{l}\text{for each }i\in \{1,\dots ,p\}\text{ there exists a unique }{\tau }_i\in (a,b)\text{ such that}\\ {\tau }_i={\gamma }_i(u({\tau }_i-),u^{\prime }({\tau }_i-),\dots ,u^{(n-2)}({\tau }_i-)),a<{\tau }_1<\cdots <{\tau }_p<b,\\ \text{and the restrictions }u{|}_{[a,{\tau }_1]},u{|({\tau }_1,{\tau }_2],\dots ,u|}_{({\tau }_p,b]}\text{ have absolutely}\\ \text{continuous derivatives of the }(n-1)\text{th order}.\end{array}\}$$

(22)

Consider real numbers $K_j$, $j=0,1,\dots ,n-1$, and denote

$${\mathcal{A}}_n=\{(x_0,x_1,\dots ,x_{n-1})\in {\mathbb{R}}^n:|x_0|\le K_0,\dots ,|x_{n-1}|\le K_{n-1}\}.$$

(23)

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

$$a<\underset{{\mathcal{A}}_{n-1}}{min}{\gamma }_1\le \underset{{\mathcal{A}}_{n-1}}{max}{\gamma }_{i-1}<\underset{{\mathcal{A}}_{n-1}}{min}{\gamma }_i\le \underset{{\mathcal{A}}_{n-1}}{max}{\gamma }_p<b,i=2,\dots ,p,$$

(24)

$$\begin{array}{l}\text{for each }i=1,\dots ,p,k=0,\dots ,n-2\text{ there exists }{L}_{ik}\in [0,\mathrm{\infty })\text{ such that}\\ \text{if }(x_0,x_1,\dots ,x_{n-2}),(y_0,y_1,\dots ,y_{n-2})\text{ belong to }{\mathcal{A}}_{n-1},\text{ then}\\ |{\gamma }_i(x_0,x_1,\dots ,x_{n-2})-{\gamma }_i(y_0,y_1,\dots ,y_{n-2})|\le {\sum }_{j=0}^{n-2}{L}_{ij}|x_j-y_j|,\\ i=1,\dots ,p,\end{array}\}$$

(25)

$$\sum _{j=0}^{n-2}{L}_{ij}{K}_{j+1}<1\text{for }i=1,\dots ,p,$$

(26)

$$\begin{array}{l}{\gamma }_i(x_0+J_{i0}(x_0,\dots ,x_{n-1}),\dots ,x_{n-2}+J_{i,n-2}(x_0,\dots ,x_{n-1}))\\ \le {\gamma }_i(x_0,\dots ,x_{n-2}),(x_0,\dots ,x_{n-1})\in {\mathcal{A}}_n,i=1,\dots ,p.\end{array}\}$$

(27)

Let us define the set

$$\mathcal{B}=\{u\in {\mathbb{C}}^{n-1}([a,b];\mathbb{R}):{\parallel u^{(j)}\parallel }_{\mathrm{\infty }}<K_j\text{ for }j=0,\dots ,n-1\}.$$

(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 ℬ 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

$$\sigma (t)={\gamma }_i(u(t),u^{\prime }(t),\dots ,u^{(n-2)}(t))-t,t\in [a,b],$$

(29)

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

$$t={\gamma }_i(u(t),\dots ,u^{(n-2)}(t)).$$

(30)

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

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

is valid. This together with the fact that σ is continuous shows that σ has at least one root in $(a,b)$. Now, we will prove that σ is decreasing, by a contradiction. Let $s_1,s_2\in (a,b)$, $s_1<s_2$ be such that

$$\sigma (s_1)=\sigma (s_2),$$

i.e.

$${\gamma }_i(u(s_1),\dots ,u^{(n-2)}(s_1))-{\gamma }_i(u(s_2),\dots ,u^{(n-2)}(s_2))=s_1-s_2.$$

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

$$\begin{array}{rl}0& <|s_1-s_2|=|{\gamma }_i(u(s_1),\dots ,u^{(n-2)}(s_1))-{\gamma }_i(u(s_2),\dots ,u^{(n-2)}(s_2))|\\ \le \sum _{j=0}^{n-2}{L}_{ij}|u^{(j)}(s_1)-u^{(j)}(s_2)|\le \sum _{j=0}^{n-2}{L}_{ij}{K}_{j+1}|s_1-s_2|<|s_1-s_2|,\end{array}$$

which is a contradiction.

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

$${\mathcal{P}}_{i}u={\tau }_i,u\in \overline{\mathcal{B}},$$

(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

$$u_m\to u\text{in }{\mathbb{C}}^{n-1}([a,b];\mathbb{R})\text{ as }m\to \mathrm{\infty }.$$

(32)

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

$$\tau ={\mathcal{P}}_{i}u,{\tau }_m={\mathcal{P}}_i{u}_m,m\in \mathbb{N}.$$

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

$$\sigma (t)={\gamma }_i(u(t),\dots ,u^{(n-2)}(t))-t,{\sigma }_m(t)={\gamma }_i(u_m(t),\dots ,u_m^{(n-2)}(t))-t,t\in [a,b],$$

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

$${\sigma }_m(\tau -ϵ)>0\text{and}{\sigma }_m(\tau +ϵ)<0\text{for a.e. }m\in \mathbb{N}.$$

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

$${\mathcal{P}}_i{u}_m={\tau }_m\in (\tau -ϵ,\tau +ϵ)=({\mathcal{P}}_{i}u-ϵ,{\mathcal{P}}_{i}u+ϵ)\text{for a.e. }m\in \mathbb{N},$$

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 ℋ 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 ℋ would be acting. According to Remark 8, we search a solution u of problem (1)-(3), which has its jumps (together with $u,u^{\prime },\dots ,u^{(n-1)}$) at the points ${\tau }_i={\mathcal{P}}_{i}u\in (a,b)$, $i=1,\dots ,p$ (see (31)). In general, these points are different for different solutions. Consequently, such solutions have to be searched in the Banach space ${\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n)$. But then there is a difficulty with the continuity of such operator. In fact the operator ℋ from (16) having ${\mathcal{P}}_{i}u$ in place of $t_i$ is not continuous in the space ${\mathbb{G}}_{\mathrm{L}}([a,b];{\mathbb{R}}^n)$ (cf. Remark 6.2 and Example 6.3 in [36]).

Therefore, we choose the way here, which we have developed in our joint papers [8–10]. The main idea of our approach lies in representing the solution u of problem (1)-(3) by an ordered $(p+1)$-tuple $(u_1,\dots ,u_{p+1})\in {[{\mathbb{C}}^{n-1}([a,b];\mathbb{R})]}^{p+1}$ as follows:

$$u(t)=\{\begin{array}{cc}{u}_1(t),\hfill & t\in [a,{\mathcal{P}}_1{u}_1],\hfill \\ u_2(t),\hfill & t\in ({\mathcal{P}}_1{u}_1,{\mathcal{P}}_2{u}_2],\hfill \\ \dots \hfill & \dots \hfill \\ u_{p+1}(t),\hfill & t\in ({\mathcal{P}}_p{u}_p,b].\hfill \end{array}$$

(33)

Consequently, we work with the space

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

equipped with the norm

$$\parallel (u_1,\dots ,u_{p+1})\parallel =\sum _{i=1}^{p+1}\sum _{j=0}^{n-1}{\parallel u_i^{(j)}\parallel }_{\mathrm{\infty }}\text{for }(u_1,\dots ,u_{p+1})\in X.$$

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.

In addition we assume

$$\begin{array}{l}\text{there exists }m\in {\mathbb{L}}^1([a,b];\mathbb{R}),A_{ij}\in \mathbb{R}\text{ such that}\\ |\frac{h(t,x)}{a_n(t)}|\le m(t)\text{ for a.e. }t\in [a,b]\text{ and all }x\in {\mathbb{R}}^n,\\ |J_{ij}(x)|\le A_{ij}\text{ for each }i=1,\dots ,p,j=0,\dots ,n-1.\end{array}\}$$

(34)

Consider $c_1,\dots ,c_n$ from (3), w from (7) and $\ell (W)$ from (8), and denote

$$M={\int }_a^{b}m(t)\mathrm{d}t,c_0={(c_1,\dots ,c_n)}^T,D_r=\underset{t\in [a,b]}{max}{w}^{(r)}(t){[\ell (W)]}^{-1}{c}_0,$$

(35)

and

$$K_r=M\underset{\nu =1,2}{max}\{C_{\nu nr}\}+\sum _{j=1}^n\sum _{k=1}^p\underset{\nu =1,2}{max}\{C_{\nu jr}\}{A}_{k,j-1}+D_r,$$

(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 ${\tilde{u}}_1,\dots ,{\tilde{u}}_n$, 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 ℬ is defined in (28) with $K_j$ from (36).

Now, we have to modify the operator ℋ from Theorem 4 using $g_j^{[1]}$ and $g_j^{[2]}$ instead of the Green’s functions $g_j$, that is, we define an operator $\mathcal{F}:\overline{\mathrm{\Omega }}\to X$ by $\mathcal{F}(u_1,\dots ,u_{p+1})=(x_1,\dots ,x_{p+1})$ with

$$\begin{array}{l}{x}_i(t)=\sum _{k=1}^{p+1}{\int }_{{\tau }_{k-1}}^{{\tau }_k}{g}_n(t,s)\frac{h(s,u_k(s),\dots ,u_k^{(n-1)}(s))}{a_n(s)}\mathrm{d}s\\ \phantom{x_i(t)=}+\sum _{j=1}^n(\sum _{i\le k\le p}{g}_j^{[1]}(t,{\tau }_k)J_{k,j-1}(u_k({\tau }_k),\dots ,u_k^{(n-1)}({\tau }_k))\\ \phantom{x_i(t)=}+\sum _{1\le k<i}{g}_j^{[2]}(t,{\tau }_k)J_{k,j-1}(u_k({\tau }_k),\dots ,u_k^{(n-1)}({\tau }_k)))+w(t){[\ell (W)]}^{-1}{c}_0\end{array}\}$$

(37)

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

$${\tau }_k={\mathcal{P}}_k{u}_k\text{for }k=1,\dots ,p,{\tau }_0=a,{\tau }_{p+1}=b,$$

and W, w, $g_j$, $g_j^{[1]}$, $g_j^{[2]}$, $j=1,\dots ,n$, and $c_0$ are from (7), (12), (18), and (35), respectively.

Let us compare (16) for the operator ℋ with (37) for the operator ℱ. 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 $(p+1)$-tuple $(u_1,\dots ,u_{p+1})$. 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 $g_j^{[1]}$, $g_j^{[2]}$ defined in (18). Due to this modification the operator ℱ 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 ℱ 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 $(u_1,\dots ,u_{p+1})$ is a fixed point of the operator ℱ, then the function u defined by (33) is a solution of problem (1)-(3) satisfying (22).

Proof Let ℬ be defined by (28) and $\mathrm{\Omega }={\mathcal{B}}^{p+1}$. Further, let $(u_1,\dots ,u_{p+1})\in \overline{\mathrm{\Omega }}$ be such that $\mathcal{F}(u_1,\dots ,u_{p+1})=(u_1,\dots ,u_{p+1})$. For each $i\in \{1,\dots ,p+1\}$, 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 $t={\gamma }_i(u_i(t),\dots ,u_i^{(n-2)}(t))$. Due to (24), the inequalities $a<{\tau }_1<\cdots <{\tau }_p<b$ are valid and u can be defined by (33). We will prove that u is a fixed point of the operator ℋ from Theorem 4, taking the space ${\mathbb{PC}}^{n-1}([a,b])$ from Remark 3 with

$$t_i={\tau }_i,i=1,\dots ,p.$$

Denote

$$\begin{array}{c}{\tau }_0=a,{\tau }_{p+1}=b,{\mathcal{I}}_1=[{\tau }_0,{\tau }_1],{\mathcal{I}}_2=({\tau }_1,{\tau }_2],\hfill \\ {\mathcal{I}}_3=({\tau }_2,{\tau }_3],\dots ,{\mathcal{I}}_{p+1}=({\tau }_p,{\tau }_{p+1}],\hfill \end{array}$$

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

$$\begin{array}{rl}u(t)=& u_i(t)=\sum _{k=1}^{p+1}{\int }_{{\mathcal{I}}_k}\frac{g_n(t,s)}{a_n(s)}h(s,u_k(s),\dots ,u_k^{(n-1)}(s))\mathrm{d}s\\ +\sum _{j=1}^n(\sum _{i\le k\le p}{g}_j^{[1]}(t,{\tau }_k)J_{k,j-1}(u_k({\tau }_k),\dots ,u_k^{(n-1)}({\tau }_k))\\ +\sum _{1\le k<i}{g}_j^{[2]}(t,{\tau }_k)J_{k,j-1}(u_k({\tau }_k),\dots ,u_k^{(n-1)}({\tau }_k)))+w(t){[\ell (W)]}^{-1}{c}_0\\ =& \sum _{k=1}^{p+1}{\int }_{{\mathcal{I}}_k}\frac{g_n(t,s)}{a_n(s)}h(s,u(s),\dots ,u^{(n-1)}(s))\mathrm{d}s\\ +\sum _{j=1}^n(\sum _{i\le k\le p}{g}_j^{[1]}(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-))\\ +\sum _{1\le k<i}{g}_j^{[2]}(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-)))+w(t){[\ell (W)]}^{-1}{c}_0.\end{array}$$

Of course we have

$$\sum _{k=1}^{p+1}{\int }_{{\mathcal{I}}_k}\frac{g_n(t,s)}{a_n(s)}h(s,u(s),\dots ,u^{(n-1)}(s))\mathrm{d}s={\int }_a^b\frac{g_n(t,s)}{a_n(s)}h(s,u(s),\dots ,u^{(n-1)}(s))\mathrm{d}s.$$

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

$$g_j^{[1]}(t,{\tau }_k)=g_j(t,{\tau }_k)\text{for }j=1,\dots ,n.$$

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

$$g_j^{[2]}(t,{\tau }_k)=g_j(t,{\tau }_k)\text{for }j=1,\dots ,n.$$

These facts imply that

$$\begin{array}{r}\sum _{i\le k\le p}{g}_j^{[1]}(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-))\\ +\sum _{1\le k<i}{g}_j^{[2]}(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-))\\ =\sum _{i\le k\le p}{g}_j(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-))\\ +\sum _{1\le k<i}{g}_j(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-))\\ =\sum _{k=1}^p{g}_j(t,{\tau }_k)J_{k,j-1}(u({\tau }_k),\dots ,u^{(n-1)}({\tau }_k-)),\end{array}$$

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 $[a,b]$ and satisfies the boundary conditions (3). In addition, since u fulfils the impulse conditions (14) with $t_i={\tau }_i$, where ${\tau }_i={\gamma }_i(u_i({\tau }_i),\dots ,u_i^{(n-2)}({\tau }_i))={\gamma }_i(u({\tau }_i),\dots ,u^{(n-2)}({\tau }_i-))$, $i=1,\dots ,p$, 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

$$t\ne {\gamma }_i(u_{i+1}(t),u_{i+1}^{\prime }(t),\dots ,u_{i+1}^{(n-2)}(t))\text{for }t\in ({\tau }_i,b],i=1,\dots ,p.$$

(38)

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

$$\sigma ({\tau }_i-)=0.$$

Let us denote

$$\psi (t)={\gamma }_i(u_{i+1}(t),u_{i+1}^{\prime }(t),\dots ,u_{i+1}^{(n-2)}(t))-t,t\in [a,b].$$

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

$$\psi ({\tau }_i)\le 0.$$

(39)

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

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

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

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

Proof The last term $\omega (t){[\ell (W)]}^{-1}{c}_0$ in (37) is the same as in (16) for the compact operator ℋ. Therefore it suffices to prove the compactness of the operator ℱ 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 ℱ 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

  $$\begin{array}{r}\sum _{k=1}^{p+1}{\int }_{{\tau }_{k-1}}^{{\tau }_k}{g}_n(t,s)\frac{h(s,u_k(s),\dots ,u_k^{(n-1)}(s))}{a_n(s)}\mathrm{d}s\\ ={\int }_a^b{g}_n(t,s)\sum _{k=1}^{p+1}\frac{h(s,u_k(s),\dots ,u_k^{(n-1)}(s))}{a_n(s)}{\chi }_{({\tau }_{k-1},{\tau }_k)}(s)\mathrm{d}s,\end{array}$$

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),

• $\frac{h(t,x)}{a_n(t)}\in Car([a,b]\times {\mathbb{R}}^n;\mathbb{R})$,

• $g_j^{[1]}$, $g_j^{[2]}$ 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

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

(40)

Let $(x_1,\dots ,x_{p+1})=\mathcal{F}(u_1,\dots ,u_{p+1})$ for some $(u_1,\dots ,u_{p+1})\in \overline{\mathrm{\Omega }}$. Then, by (21), (34), (35), and (37), we have

$$|x_i^{(r)}(t)|\le M\underset{\nu =1,2}{max}\{C_{\nu nr}\}+\sum _{j=1}^n\sum _{k=1}^p\underset{\nu =1,2}{max}\{C_{\nu jr}\}{A}_{k,j-1}+D_r$$

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

$${\parallel x_i^{(r)}\parallel }_{\mathrm{\infty }}\le K_r,i=1,\dots ,p+1,r=0,\dots ,n-1,$$

and so $\mathcal{F}(u_1,\dots ,u_{p+1})\in \overline{\mathrm{\Omega }}$. We have proved (40), and consequently there exists at least one fixed point of ℱ 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].