Open Access

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

Boundary Value Problems20142014:118

DOI: 10.1186/1687-2770-2014-118

Received: 20 November 2013

Accepted: 1 May 2014

Published: 14 May 2014

Abstract

The paper provides an existence principle for a general boundary value problem of the form j = 0 n a j ( t ) u ( j ) ( t ) = h ( t , u ( t ) , , u ( n 1 ) ( t ) ) , a.e. t [ a , b ] R , k ( u , u , , u ( n 1 ) ) = c k , k = 1 , , n , with the state-dependent impulses u ( j ) ( t + ) u ( j ) ( t ) = J i j ( u ( t ) , u ( t ) , , u ( n 1 ) ( t ) ) , where the impulse points t are determined as solutions of the equations t = γ i ( u ( t ) , u ( t ) , , u ( n 2 ) ( t ) ) , i = 1 , , p , j = 0 , , n 1 . Here, n , p N , c 1 , , c n R , the functions a j / a n , j = 0 , , n 1 , are Lebesgue integrable on [ a , b ] and h / a n satisfies the Carathéodory conditions on [ a , b ] × R n . The impulse functions J i j , i = 1 , , p , j = 0 , , n 1 , and the barrier functions γ i , i = 1 , , p , are continuous on R n and R n 1 , respectively. The functionals k , k = 1 , , 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 i j are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier γ i at the unique impulse point τ i are presented, and consequently the existence of a solution to the problem is proved.

MSC:34B37, 34B10, 34B15.

Keywords

nonlinear higher-order ODE state-dependent impulses general linear boundary conditions transversality conditions fixed point

1 Introduction

In this paper we are interested in the nonlinear ordinary differential equation of the n th-order ( n 2 ) with state-dependent impulses and general linear boundary conditions on the interval [ a , b ] R . Studies of real-life problems with state-dependent impulses can be found e.g. in [16]. Here we consider the equation
j = 0 n a j ( t ) u ( j ) ( t ) = h ( t , u ( t ) , , u ( n 1 ) ( t ) ) , a.e.  t [ a , b ] ,
(1)
subject to the impulse conditions
u ( j ) ( t + ) u ( j ) ( t ) = J i j ( u ( t ) , u ( t ) , , u ( n 1 ) ( t ) ) , where  t = γ i ( u ( t ) , u ( t ) , , u ( n 2 ) ( t ) ) for  i = 1 , , p , j = 0 , , n 1 , }
(2)
and the linear boundary conditions
k ( u , u , , u ( n 1 ) ) = c k , k = 1 , , n .
(3)

In what follows we use this notation. Let k , m , n N . By 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 R m × n . Let R n = R n × 1 be the set of all n-dimensional column vectors c = ( c 1 , , c n ) T , where c i R , i = 1 , , n , and R = R 1 × 1 . By C ( R n ; R m ) we denote the set of all mappings x : R n R m with continuous components. By L ( [ a , b ] ; R m × n ) , L 1 ( [ a , b ] ; R m × n ) , G L ( [ a , b ] ; R m × n ) , AC ( [ a , b ] ; R m × n ) , BV ( [ a , b ] ; R m × n ) , C k ( [ a , b ] ; R m × n ) , we denote the sets of all mappings x : [ a , b ] 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 [ a , b ] . By Car ( [ a , b ] × R n ; R ) we denote the set of all functions f : [ a , b ] × R n R satisfying the Carathéodory conditions on the set [ a , b ] × R n . Finally, by χ M we denote the characteristic function of the set M R .

Note that a mapping u : [ a , b ] R n is left-continuous regulated on [ a , b ] if for each t ( a , b ] and each s [ a , b ) there exist finite limits
u ( t ) = u ( t ) = lim τ t u ( τ ) , u ( s + ) = lim τ s + u ( τ ) .
G L ( [ a , b ] ; R n ) is a linear space, and equipped with the sup-norm it is a Banach space (see [[7], Theorem 3.6]). In particular, we set
u = max i { 1 , , n } ( sup t [ a , b ] | u i ( t ) | ) for  u = ( u 1 , , u n ) T G L ( [ a , b ] ; R n ) .

A function f : [ a , b ] × R n R satisfies the Carathéodory conditions on [ a , b ] × R n if

  • f ( , x ) : [ a , b ] R is measurable for all x R n ,

  • f ( t , ) : R n R is continuous for a.e. t [ a , b ] ,

  • for each compact set K R n there exists a function m K L 1 ( [ a , b ] ; R ) such that | f ( t , x ) | m K ( t ) for a.e. t [ a , b ] and each x 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
n 2 , a j a n L 1 ( [ a , b ] ; R ) , j = 0 , , n 1 , h ( t , x ) a n ( t ) Car ( [ a , b ] × R n ; R ) , c j R , J i j C ( R n ; R ) , γ i C ( R n 1 ; R ) , i = 1 , , p , j = 0 , , n 1 , k : G L ( [ a , b ] ; R n ) R  is a linear bounded functional ,  i.e. k ( z ) = K k z ( a ) + a b V k ( t ) d [ z ( t ) ] , z G L ( [ a , b ] ; R n × 1 ) , where  K k R 1 × n , V k BV ( [ a , b ] ; R 1 × n ) , k = 1 , , n , n , p N . }
(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 G L ( [ a , b ] ; R n × 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 G L ( [ a , b ] ; R n ) is said to be a solution of problem (1)-(3) if u satisfies (1) for a.e. t [ 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 γ i in (2) are constant functions, i.e. there exist t 1 , , t p R satisfying a < t 1 < < t p < b such that
γ i ( x 0 , x 1 , , x n 2 ) = t i for  i = 1 , , p , x 0 , x 1 , , x n 2 R .
(5)

In this case, each solution of the problem crosses i th barrier at same time instant τ i = t i for i = 1 , , 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
j = 0 n a j ( t ) u ( j ) ( t ) = 0 , a.e.  t [ a , b ] , k ( u , u , , u ( n 1 ) ) = 0 , k = 1 , , n , }
(6)
has only the trivial solution. Let { u ˜ 1 , , 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 ) = ( u ˜ 1 ( t ) u ˜ n ( t ) u ˜ 1 ( t ) u ˜ n ( t ) u ˜ 1 ( n 1 ) ( t ) u ˜ n ( n 1 ) ( t ) ) , w ( t ) = ( u ˜ 1 ( t ) , , u ˜ n ( t ) ) , t [ a , b ] .
(7)
Denote
( W ) = ( i ( u ˜ j , u ˜ j , , u ˜ j ( n 1 ) ) ) 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 ( W ) 0 .
(9)
Further assume (9), consider V j , j = 1 , , n , from (4), and denote
V ( t ) = ( V 1 ( t ) V 2 ( t ) V n ( t ) ) , A ( t ) = ( 0 1 0 0 0 0 1 0 0 0 0 1 a 0 ( t ) a n ( t ) a 1 ( t ) a n ( t ) a 2 ( t ) a n ( t ) a n 1 ( t ) a n ( t ) ) ,
t [ a , b ] and
H ( τ ) = [ ( W ) ] 1 ( τ b V ( s ) A ( s ) W ( s ) d s W 1 ( τ ) + V ( τ ) ) , τ [ a , b ] .
(10)
If we denote by H i j and ω i j elements of the matrices H and W 1 , respectively, that is,
H ( τ ) = ( H i j ( τ ) ) i , j = 1 n , W 1 ( τ ) = ( ω i j ( τ ) ) i , j = 1 n ,
(11)
we can define functions g j , j = 1 , , n , as
g j ( t , τ ) = k = 1 n u ˜ k ( t ) ( H k j ( τ ) + χ ( τ , b ] ( t ) ω k j ( τ ) ) , t , τ [ a , b ] .
(12)

For each fixed τ [ a , b ] the functions k g j ( t , τ ) τ k , k = 0 , 1 , , n 1 , will be understood as right-continuous extensions at t = a and left-continuous extensions at t = τ 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 ] , , ( t p , b ] , with t i from (5), having their derivatives u , , u ( n 1 ) continuously extendable onto these intervals. This set is denoted by PC n 1 ( [ a , b ] ) . For u PC n 1 ( [ a , b ] ) we define
u ( k ) ( a ) = u ( k ) ( a + ) , u ( k ) ( t i ) = u ( k ) ( t i ) for  k = 1 , , n 1 , i = 1 , , p .
Equipped with the standard addition, scalar multiplication, and with the norm
u = k = 0 n 1 u ( k ) , u PC n 1 ( [ a , b ] ) ,

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 < < t p < b which has the form
j = 0 n a j ( t ) u ( j ) ( t ) = h ( t , u ( t ) , , u ( n 1 ) ( t ) ) , a.e.  t [ a , b ] ,
(13)
u ( j ) ( t i + ) u ( j ) ( t i ) = J i j ( u ( t i ) , u ( t i ) , , u ( n 1 ) ( t i ) ) , i = 1 , , p , j = 0 , , n 1 ,
(14)
k ( u , u , , u ( n 1 ) ) = c k , k = 1 , , n .
(15)

Theorem 4 [[20], Theorem 17]

Let (4), (9) hold, and let W, w, ( W ) and g j , j = 1 , , n be defined in (7), (8), and (12). Then u PC n 1 ( [ a , b ] ) is a fixed point of an operator H : PC n 1 ( [ a , b ] ) PC n 1 ( [ a , b ] ) defined by
( H u ) ( t ) = a b g n ( t , s ) a n ( s ) h ( s , u ( s ) , , u ( n 1 ) ( s ) ) d s ( H u ) ( t ) = + j = 1 n i = 1 p g j ( t , t i ) J i , j 1 ( u ( t i ) , , u ( n 1 ) ( t i ) ) ( H u ) ( t ) = + w ( t ) [ ( W ) ] 1 ( c 1 , , c n ) T , }
(16)

t [ a , b ] , 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 ( t ) [ ( W ) ] 1

does not depend on the choice of a fundamental system of solutions u ˜ 1 , , u ˜ n , but only on the data of problem (6).

Remark 6 Let us put
J i j = 0 , i = 1 , , p , j = 0 , , n 1 , c k = 0 , k = 1 , , n
and
h ( t , x ) = h 0 ( t ) L 1 ( [ a , b ] ; R ) for  x R n .
Then the operator in Theorem 4 can be written as
( H 0 u ) ( t ) = a b g n ( t , s ) a n ( s ) h 0 ( s ) d s .
Theorem 4 implies that u is a fixed point of H 0 if and only if u is a solution of the problem
j = 0 n a j ( t ) u ( j ) ( t ) = h 0 ( t ) , j ( u , u , , u ( n 1 ) ) = 0 , j = 1 , , n .
(17)
Therefore a (unique) solution of problem (17) has the form
u ( t ) = a b g n ( t , s ) a n ( s ) h 0 ( s ) d s ,

and consequently 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
g j [ 1 ] ( t , τ ) = k = 1 n u ˜ k ( t ) H k j ( τ ) , g j [ 2 ] ( t , τ ) = k = 1 n u ˜ k ( t ) ( H k j ( τ ) + ω k j ( τ ) ) }
(18)
for t , τ [ a , b ] , j = 1 , , n . Obviously, due to (12),
g j ( t , τ ) = { g j [ 1 ] ( t , τ ) for  a t τ b , g j [ 2 ] ( t , τ ) for  a τ < t b ,
(19)
for j = 1 , , n . Let us stress that g j [ ν ] , as well as g j , do not depend on the choice of fundamental system u ˜ 1 , , u ˜ n , but only on the data of problem (6). The functions g j [ ν ] possess crucial properties for our approach. From their definition it follows that for each τ [ a , b ]
k g j [ ν ] t k ( , τ ) AC ( [ a , b ] ; R )
(20)
for ν = 1 , 2 , j = 1 , , n , k = 0 , , n 1 . Moreover, for each ν = 1 , 2 , j = 1 , , n , k = 0 , , n 1 , there exists a constant C ν j k > 0 such that
| k g j [ ν ] t k ( t , τ ) | C ν j k and | k g j t k ( t , τ ) | max ν = 1 , 2 C ν j k t , τ [ a , b ] .
(21)

This follows from the definition of g j [ ν ] ( ν = 1 , 2 ), from the fact w C n 1 ( [ a , b ] ; R 1 × 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 γ 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 in some set Ω ¯ = B ¯ p + 1 (cf. Lemma 12), where B ¯ C n 1 ( [ a , b ] ; R ) is a ball defined in (28). In order to get a fixed point, we need to prove for functions of 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 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:
for each  i { 1 , , p }  there exists a unique  τ i ( a , b )  such that τ i = γ i ( u ( τ i ) , u ( τ i ) , , u ( n 2 ) ( τ i ) ) , a < τ 1 < < τ p < b , and the restrictions  u | [ a , τ 1 ] , u | ( τ 1 , τ 2 ] , , u | ( τ p , b ]  have absolutely continuous derivatives of the  ( n 1 ) th order . }
(22)
Consider real numbers K j , j = 0 , 1 , , n 1 , and denote
A n = { ( x 0 , x 1 , , x n 1 ) R n : | x 0 | K 0 , , | x n 1 | K n 1 } .
(23)
Now, we are ready to formulate the following transversality conditions:
a < min A n 1 γ 1 max A n 1 γ i 1 < min A n 1 γ i max A n 1 γ p < b , i = 2 , , p ,
(24)
for each  i = 1 , , p , k = 0 , , n 2  there exists  L i k [ 0 , )  such that if  ( x 0 , x 1 , , x n 2 ) , ( y 0 , y 1 , , y n 2 )  belong to  A n 1 ,  then | γ i ( x 0 , x 1 , , x n 2 ) γ i ( y 0 , y 1 , , y n 2 ) | j = 0 n 2 L i j | x j y j | , i = 1 , , p , }
(25)
j = 0 n 2 L i j K j + 1 < 1 for  i = 1 , , p ,
(26)
γ i ( x 0 + J i 0 ( x 0 , , x n 1 ) , , x n 2 + J i , n 2 ( x 0 , , x n 1 ) ) γ i ( x 0 , , x n 2 ) , ( x 0 , , x n 1 ) A n , i = 1 , , p . }
(27)
Let us define the set
B = { u C n 1 ( [ a , b ] ; R ) : u ( j ) < K j  for  j = 0 , , n 1 } .
(28)

Our current goal is to find a continuous functional P i for i = 1 , , p , which maps each function u from B ¯ to some time instant τ i of (2).

Lemma 9 Let K j , j = 0 , , n 1 , L i k , i = 1 , , p , k = 0 , , n 2 , be real numbers satisfying (26), and let A n and be given by (23) and (28), respectively. Finally, assume that γ i , i = 1 , , p , satisfy (24), (25), and choose u B ¯ . Then the function
σ ( t ) = γ i ( u ( t ) , u ( t ) , , u ( n 2 ) ( t ) ) t , t [ 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 = γ i ( u ( t ) , , u ( n 2 ) ( t ) ) .
(30)
Proof Let u B ¯ , i { 1 , , p } . By (24),
σ ( a ) = γ i ( u ( a ) , u ( a ) , , u ( n 2 ) ( a ) ) a > 0 , σ ( b ) = γ i ( u ( b ) , u ( b ) , , u ( n 2 ) ( b ) ) b < 0
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 ( a , b ) , s 1 < s 2 be such that
σ ( s 1 ) = σ ( s 2 ) ,
i.e.
γ i ( u ( s 1 ) , , u ( n 2 ) ( s 1 ) ) γ i ( u ( s 2 ) , , u ( n 2 ) ( s 2 ) ) = s 1 s 2 .
From (25), (26), (28), and the Mean Value Theorem we obtain
0 < | s 1 s 2 | = | γ i ( u ( s 1 ) , , u ( n 2 ) ( s 1 ) ) γ i ( u ( s 2 ) , , u ( n 2 ) ( s 2 ) ) | j = 0 n 2 L i j | u ( j ) ( s 1 ) u ( j ) ( s 2 ) | j = 0 n 2 L i j K j + 1 | s 1 s 2 | < | s 1 s 2 | ,

which is a contradiction.

According to Lemma 9, we can define a functional P i : B ¯ ( a , b ) by
P i u = τ i , u B ¯ ,
(31)

where τ i is a solution of (30), i.e. a unique root of the function σ from Lemma 9, for i = 1 , , p . □

Lemma 10 Let the assumptions of Lemma 9 be satisfied. The functionals P i , i = 1 , , p , are continuous.

Proof Let u m , u B ¯ , for m N such that
u m u in  C n 1 ( [ a , b ] ; R )  as  m .
(32)
Let us choose i { 1 , , p } and prove that P i u m P i u as m . We denote
τ = P i u , τ m = P i u m , m N .
From Lemma 9 it follows that τ , τ m ( a , b ) are the unique roots of the functions
σ ( t ) = γ i ( u ( t ) , , u ( n 2 ) ( t ) ) t , σ m ( t ) = γ i ( u m ( t ) , , u m ( n 2 ) ( t ) ) t , t [ a , b ] ,
and these functions are strictly decreasing. Let ϵ R , ϵ > 0 be such that τ ϵ , τ + ϵ ( a , b ) . Then σ ( τ ϵ ) > 0 and σ ( τ + ϵ ) < 0 . According to (32) we see that σ m σ uniformly on [ a , b ] , in particular σ m ( τ ϵ ) σ ( τ ϵ ) and σ m ( τ + ϵ ) σ ( τ + ϵ ) as m . These facts imply that
σ m ( τ ϵ ) > 0 and σ m ( τ + ϵ ) < 0 for a.e.  m N .
From the continuity of σ m and the Intermediate Value Theorem it follows that
P i u m = τ m ( τ ϵ , τ + ϵ ) = ( P i u ϵ , P i u + ϵ ) for a.e.  m 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 from Theorem 4, putting P i u in place of t i . This is not possible for many reasons. First, each 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 , , u ( n 1 ) ) at the points τ i = P i u ( a , b ) , i = 1 , , p (see (31)). In general, these points are different for different solutions. Consequently, such solutions have to be searched in the Banach space G L ( [ a , b ] ; R n ) . But then there is a difficulty with the continuity of such operator. In fact the operator from (16) having P i u in place of t i is not continuous in the space G L ( [ a , b ] ; 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 [810]. The main idea of our approach lies in representing the solution u of problem (1)-(3) by an ordered ( p + 1 ) -tuple ( u 1 , , u p + 1 ) [ C n 1 ( [ a , b ] ; R ) ] p + 1 as follows:
u ( t ) = { u 1 ( t ) , t [ a , P 1 u 1 ] , u 2 ( t ) , t ( P 1 u 1 , P 2 u 2 ] , u p + 1 ( t ) , t ( P p u p , b ] .
(33)
Consequently, we work with the space
X = [ C n 1 ( [ a , b ] ; R ) ] p + 1
equipped with the norm
( u 1 , , u p + 1 ) = i = 1 p + 1 j = 0 n 1 u i ( j ) for  ( u 1 , , u p + 1 ) 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
there exists  m L 1 ( [ a , b ] ; R ) , A i j R  such that | h ( t , x ) a n ( t ) | m ( t )  for a.e.  t [ a , b ]  and all  x R n , | J i j ( x ) | A i j  for each  i = 1 , , p , j = 0 , , n 1 . }
(34)
Consider c 1 , , c n from (3), w from (7) and ( W ) from (8), and denote
M = a b m ( t ) d t , c 0 = ( c 1 , , c n ) T , D r = max t [ a , b ] w ( r ) ( t ) [ ( W ) ] 1 c 0 ,
(35)
and
K r = M max ν = 1 , 2 { C ν n r } + j = 1 n k = 1 p max ν = 1 , 2 { C ν j r } A k , j 1 + D r ,
(36)

for r = 0 , , n 1 , where C ν j r 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 u ˜ 1 , , u ˜ n , but only on the coefficients a i of the differential equation (1) and on the operators 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
Ω = B p + 1 X ,

where is defined in (28) with K j from (36).

Now, we have to modify the operator 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 F : Ω ¯ X by F ( u 1 , , u p + 1 ) = ( x 1 , , x p + 1 ) with
x i ( t ) = k = 1 p + 1 τ k 1 τ k g n ( t , s ) h ( s , u k ( s ) , , u k ( n 1 ) ( s ) ) a n ( s ) d s x i ( t ) = + j = 1 n ( i k p g j [ 1 ] ( t , τ k ) J k , j 1 ( u k ( τ k ) , , u k ( n 1 ) ( τ k ) ) x i ( t ) = + 1 k < i g j [ 2 ] ( t , τ k ) J k , j 1 ( u k ( τ k ) , , u k ( n 1 ) ( τ k ) ) ) + w ( t ) [ ( W ) ] 1 c 0 }
(37)
for i = 1 , , p + 1 , t [ a , b ] , where
τ k = P k u k for  k = 1 , , p , τ 0 = a , τ p + 1 = b ,

and W, w, g j , g j [ 1 ] , g j [ 2 ] , j = 1 , , 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 ( p + 1 ) -tuple ( u 1 , , 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 τ k = P k u k , we use smooth functions g j [ 1 ] , g j [ 2 ] defined in (18). Due to this modification the operator maps one tuple of smooth functions u 1 , , u p + 1 onto another tuple of smooth functions x 1 , , x p + 1 , and we will be able to prove the compactness of on Ω ¯ .

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 , , u p + 1 ) 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 Ω = B p + 1 . Further, let ( u 1 , , u p + 1 ) Ω ¯ be such that F ( u 1 , , u p + 1 ) = ( u 1 , , u p + 1 ) . For each i { 1 , , p + 1 } , we have u i B ¯ , and hence by Lemma 9 and (31), there exists a unique solution τ i = P i u i of the equation t = γ i ( u i ( t ) , , u i ( n 2 ) ( t ) ) . Due to (24), the inequalities a < τ 1 < < τ p < b 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 PC n 1 ( [ a , b ] ) from Remark 3 with
t i = τ i , i = 1 , , p .
Denote
τ 0 = a , τ p + 1 = b , I 1 = [ τ 0 , τ 1 ] , I 2 = ( τ 1 , τ 2 ] , I 3 = ( τ 2 , τ 3 ] , , I p + 1 = ( τ p , τ p + 1 ] ,
and choose i { 1 , , p + 1 } , t I i . Then, according to (33), we have
u ( t ) = u i ( t ) = k = 1 p + 1 I k g n ( t , s ) a n ( s ) h ( s , u k ( s ) , , u k ( n 1 ) ( s ) ) d s + j = 1 n ( i k p g j [ 1 ] ( t , τ k ) J k , j 1 ( u k ( τ k ) , , u k ( n 1 ) ( τ k ) ) + 1 k < i g j [ 2 ] ( t , τ k ) J k , j 1 ( u k ( τ k ) , , u k ( n 1 ) ( τ k ) ) ) + w ( t ) [ ( W ) ] 1 c 0 = k = 1 p + 1 I k g n ( t , s ) a n ( s ) h ( s , u ( s ) , , u ( n 1 ) ( s ) ) d s + j = 1 n ( i k p g j [ 1 ] ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) + 1 k < i g j [ 2 ] ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) ) + w ( t ) [ ( W ) ] 1 c 0 .
Of course we have
k = 1 p + 1 I k g n ( t , s ) a n ( s ) h ( s , u ( s ) , , u ( n 1 ) ( s ) ) d s = a b g n ( t , s ) a n ( s ) h ( s , u ( s ) , , u ( n 1 ) ( s ) ) d s .
Let k N be such that i k p . Then t τ i τ k and therefore (19) gives
g j [ 1 ] ( t , τ k ) = g j ( t , τ k ) for  j = 1 , , n .
Let k N be such that 1 k < i (such k exists only if i > 1 ). Then t > τ i 1 τ k and therefore we get by (19)
g j [ 2 ] ( t , τ k ) = g j ( t , τ k ) for  j = 1 , , n .
These facts imply that
i k p g j [ 1 ] ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) + 1 k < i g j [ 2 ] ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) = i k p g j ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) + 1 k < i g j ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) = k = 1 p g j ( t , τ k ) J k , j 1 ( u ( τ k ) , , u ( n 1 ) ( τ k ) ) ,
for j = 1 , , 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 = τ i , where τ i = γ i ( u i ( τ i ) , , u i ( n 2 ) ( τ i ) ) = γ i ( u ( τ i ) , , u ( n 2 ) ( τ i ) ) , i = 1 , , p , we see that u also fulfils the state-dependent impulse conditions (2). According to Remark 8, it remains to prove that τ 1 , , τ p are the only instants at which the function u crosses the barriers γ 1 , , γ p , respectively. To this aim, due to (24) and (33), it suffices to prove that
t γ i ( u i + 1 ( t ) , u i + 1 ( t ) , , u i + 1 ( n 2 ) ( t ) ) for  t ( τ i , b ] , i = 1 , , p .
(38)
Choose an arbitrary i { 1 , , p } and consider σ from (29). Since u fulfils (2), we have
σ ( τ i ) = 0 .
Let us denote
ψ ( t ) = γ i ( u i + 1 ( t ) , u i + 1 ( t ) , , u i + 1 ( n 2 ) ( t ) ) t , t [ a , b ] .
From Lemma 9 it follows that ψ is decreasing. So, by virtue of (38), it suffices to prove that
ψ ( τ i ) 0 .
(39)
Using (33), (2), and (27), we have
ψ ( τ i ) = γ i ( u i + 1 ( τ i ) , , u i + 1 ( n 2 ) ( τ i ) ) τ i = γ i ( u ( τ i + ) , , u ( n 2 ) ( τ i + ) ) τ i = γ i ( u ( τ i ) + J i 0 ( u ( τ i ) , , u ( n 1 ) ( τ i ) ) , , u ( n 2 ) ( τ i ) + J i , n 2 ( u ( τ i ) , , u ( n 1 ) ( τ i ) ) ) τ i γ i ( u ( τ i ) , , u ( n 2 ) ( τ i ) ) τ i = 0 ,

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 Ω ¯ .

Proof The last term ω ( t ) [ ( W ) ] 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 Ω ¯ 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 Ω ¯ 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
    k = 1 p + 1 τ k 1 τ k g n ( t , s ) h ( s , u k ( s ) , , u k ( n 1 ) ( s ) ) a n ( s ) d s = a b g n ( t , s ) k = 1 p + 1 h ( s , u k ( s ) , , u k ( n 1 ) ( s ) ) a n ( s ) χ ( τ k 1 , τ k ) ( s ) d s ,

where τ k = P k u k for k = 1 , , p , τ 0 = a , τ p + 1 = b ,

  • P k are continuous on B ¯ (due to Lemma 10),

  • h ( t , x ) a n ( t ) Car ( [ a , b ] × R n ; R ) ,

  • g j [ 1 ] , g j [ 2 ] satisfy (20), g n satisfies (19),

  • J k j are continuous on R n .

For the application of the Schauder Fixed Point Theorem it remains to prove that
F ( Ω ¯ ) Ω ¯ .
(40)
Let ( x 1 , , x p + 1 ) = F ( u 1 , , u p + 1 ) for some ( u 1 , , u p + 1 ) Ω ¯ . Then, by (21), (34), (35), and (37), we have
| x i ( r ) ( t ) | M max ν = 1 , 2 { C ν n r } + j = 1 n k = 1 p max ν = 1 , 2 { C ν j r } A k , j 1 + D r
for i = 1 , , p + 1 , r = 0 , , n 1 , t [ a , b ] . From (36) we get
x i ( r ) K r , i = 1 , , p + 1 , r = 0 , , n 1 ,

and so F ( u 1 , , u p + 1 ) Ω ¯ . We have proved (40), and consequently there exists at least one fixed point of in Ω ¯ . □

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 i j by means of the method of a priori estimates. This can be found for the special case n = 2 in [10].

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University

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© Rachůnková and Tomeček; licensee Springer. 2014

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