On the other hand, 2.

Using linearly independent solutions of 2. Then, from the impulsive condition 2. Let us consider any two solutions of 2.

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## Spectral properties of an impulsive Sturm–Liouville operator

Let F and G be associated with the Jost solution of the boundary value problem 2. Then we obtain. Furthermore, using the impulsive condition 2. Clearly, inserting these coefficients into 2. Now by 2. Note that, the Wronskian of the solutions of 2. Moreover, from 2. A necessary and sufficient condition to investigate the eigenvalues and spectral singularities of the impulsive Sturm—Liouville operator L is to investigate the zeros of the function M We introduce the sets of spectral singularities and eigenvalues of impulsive operator L as.

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Under condition 2. From 2. This completes the proof of 3. Similarly, 3. We give a lemma, which is necessary to discuss the properties of eigenvalues and spectral singularities of L. Assume 2. The set S 1 is bounded , and no more than a countable number of elements and its limit points can lie on a bounded subinterval of the real axis. The set S 2 is compact , and its linear Lebesgue measure is zero.

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- Spectral properties of an impulsive Sturm–Liouville operator.

Asymptotic equation 3. Thus the boundedness of the sets S 1 and S 2 follows from 3.

Using the uniqueness theorem of analytic functions [ 23 ], we obtain that S 2 is a closed set and its linear Lebesgue measure is zero. The set of eigenvalues of L is bounded and has at most a countable number of elements , and its limit points can lie only on a bounded subinterval of the real axis.

The set of spectral singularities of L is compact , and its linear Lebesgue measure is zero. Using 2. By the uniqueness theorem of analytic functions, we find that. From the continuity of all derivatives of M 22 up to the real axis, we obtain that. Since M 22 cannot be continued analytically from the real line to the lower half-plane under condition 3.

On the other hand, from 2. Moreover, it follows from 2. Then, using 2. From the continuity of the functions P and P x we get. Thus, from 3.

## A Multivalued Discrete System and Its Properties

Now we can write. Also, for A n , using the gamma function, we can write. Therefore it is obvious that. From 3.

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Assume 3. Then the operator L has a finite number of eigenvalues and spectral singularities , and each of them is of finite multiplicity. Then it follows from 3. We remark that, by the help of asymptotic equation 3. In this section, we determine the scattering function of the impulsive Sturm—Liouville operator L. Let us consider the solutions F and G of 2. Then it follows from 2. This gives that. The operator L has no spectral singularities. Then the scattering function of the operator L is defined by.

Since q is a real-valued potential function, it is evident from 2. To investigate the eigenvalues and spectral singularities of L 0 , we examine the zeros of M For this purpose, we see that. Otherwise, the problem has no eigenvalues and spectral singularities. A special order item has limited availability and the seller may source this title from another supplier. In this event, there may be a slight delay in shipping and possible variation in description.

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Sign up to receive offers and updates: Subscribe. All Rights Reserved. We mention only the book [7] where Runge—Kutta methods used in the paper are comprehensively studied. Many real problems are described with the help of impulsive differential equations, e. We refer the reader to [5,14,15,19] for the theory.

The basic theory of the impulsive differential equations can be found e. The book [5] is devoted mainly to existence of solutions of impulsive systems in a very general form.

The optimal control of such a systems is described in [22]. Among others notice [1], where the connection with the hybrid systems is studied. Impulsive systems can also be modeled via differential systems with measures see e. In [21], the pure convergence of the Euler-type discretization is proved by graph completion techniques. In this approach, trajectory dependent jumps have to be modeled via discrete measures which depend itself on the solution which would require a new solution concept. In the following, we will use explicit jump conditions in the form of [14,6,22,15] which will force a discontinuity of solutions.

The comparison of the following system formulation 1 with hybrid systems as e.