Fixed point problem associated with state-dependent impulsive boundary value problems
© Rachůnková and Tomeček; licensee Springer. 2014
Received: 27 March 2014
Accepted: 27 June 2014
Published: 24 September 2014
The paper investigates a fixed point problem in the space which is connected to boundary value problems with state-dependent impulses of the form , a.e. , , . Here, the impulse instants are determined as solutions of the equations , . We assume that , , the vector function f satisfies the Carathéodory conditions on , the impulse functions , , are continuous on , and the barrier functions , , are continuous on . The operator ℓ is an arbitrary linear and bounded operator on the space of left-continuous regulated on vector valued functions and is represented by the Kurzweil-Stieltjes integral. Provided the data functions f and are bounded, transversality conditions which guarantee that this fixed point problem is solvable are presented. As a result it is possible to realize the construction of a solution of the above impulsive problem.
MSC: 34B37, 34B10, 34B15.
In the literature most of impulsive boundary value problems deals with impulses at fixed times. This is the case that moments, where impulses act in state variables, are known (cf. Section 2). The theory of these impulsive problems is widely developed and presents direct analogies with methods and results for problems without impulses. Important texts in this area are –.
A different situation arises, when impulse moments satisfy a predetermined relation between state and time variables, see e.g.–. This case, which is represented by state-dependent impulses, is studied here, where we are interested in a system of n () nonlinear ordinary differential equations of the first order with state-dependent impulses and general linear boundary conditions on the interval . The main reason that boundary value problems with state-dependent impulses are developed significantly less than those with impulses at fixed moments is that new difficulties with an operator representation of the problem appear when examining state-dependent impulses (cf. Section 4). Therefore almost all existence results for boundary value problems with state-dependent impulses have been reached for periodic problems which can be transformed to fixed point problems of corresponding Poincaré maps in . Hence, the difficulties with the construction of a functional space and an operator have been cleared in the periodic case. See e.g.–. Other types of boundary value problems with state-dependent impulses have been studied very rarely, see , .
For nonzero impulse functions , , this solution is discontinuous on and, since discontinuity points , , are not fixed and depend on the solution through (2), such a solution has to be searched in the space ; see the notation below. Note that conditions which guarantee the solvability of problem (1)-(3) have not been known before. Some results for special cases of problem (1)-(3) can be found in our previous papers –.
By we denote the set of all mappings satisfying the Carathéodory conditions on the set . Finally, by we denote the characteristic function of the set .
A mapping satisfies the Carathéodory conditions on if
is measurable for all ,
is continuous for a.e. ,
for each compact set there exists a function such that for a.e. and each .
2 Problem with impulses at fixed times
and each solution of the problem crosses i th barrier at the same time instant for .
In order to get an operator representation of this problem (cf. Theorem 4) the Green’s matrix is constructed.
(, Definition 7)
is continuous on and on for each ,
for each ,
- (c)for any the mapping
(, Lemma 8)
G is bounded on ,
is absolutely continuous on and for each and its columns satisfy the differential equation from (9) a.e. on ,
for each ,
- (iv)for each and
(, Theorem 11)
Similar results can be found also in , Chapter 6].
3 Transversality conditions
In Section 4 we define an operator (cf. (26)) whose fixed point is used for the construction of a solution z of problem (1)-(3) (cf. (28)). In order to get a correct definition of we need to describe intersection point t of a function with the barriers , . These intersection points are roots of the functions , and their uniqueness is stated in Lemma 7.
where is a solution of (19), i.e. a unique root of the function from Lemma 7, for .
Since solutions are affected by impulses at the points , the functionals , , are used in the definition of the operator (cf. (26)), it is important to prove their properties which are presented in Lemma 8 and Corollary 9 and which are necessary for the compactness of (cf. Lemma 13).
which is the desired inequality. □
Let the assumptions of Lemma 7be satisfied. Then the functionals, , which are given by (20), are continuous onin the norm of.
4 Fixed point problem
where ℬ is defined in (18) with constants , , , satisfying the assumptions of Lemma 7.
where , , and are defined by (24). As we will show this will be enough for our needs (cf. Theorem 11).
This fact obstructs the compactness of the operator in X.
Then we are ready to prove the following theorem.
5 Existence results
where is defined in (23). Let us choose a fixed .
Therefore there exists a subsequence which is convergent in X. □
Consequently, by virtue of (18), for , that is, . □
This work was supported by the grant No. 14-06958S of the Grant Agency of the Czech Republic.
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