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

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 $M\subset \mathbb{R}$.

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

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.

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).

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].

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).

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

Theorem 4 [[20], Theorem 17]

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

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).

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

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)).

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).

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).

which is a contradiction.

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.

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]).

It is well known that X is a Banach space.

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.

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 $c_j$).

where ℬ is defined in (28) with $K_j$ from (36).

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).