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

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.

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

subject to the impulse conditions

and the linear boundary conditions

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

{\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

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

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 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 such that

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

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.

Denote

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

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

t\in [a,b] and

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

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

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

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

{\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 which has the form

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

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

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

and

Then the operator ℋ in Theorem 4 can be written as

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

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

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

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

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]

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

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).

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:

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

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

Let us define the set

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

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

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

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), be such that

i.e.

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

which is a contradiction.

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

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

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

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

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 . 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

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 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:

Consequently, we work with the space

equipped with the norm

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.

In addition we assume

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

and

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

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

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

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 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

Denote

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

Of course we have

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 (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