Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 479049, 9 pages

http://dx.doi.org/10.1155/2013/479049

Research Article

Second-Order Impulsive Differential Equations with Functional Initial Conditions on Unbounded Intervals

Luigi Muglia and Paolamaria Pietramala

Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy

Received 29 December 2012; Accepted 27 February 2013

Academic Editor: Feliz Minhós

Copyright © 2013 Luigi Muglia and Paolamaria Pietramala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present some results on the existence of solutions for second-order impulsive differential equations with deviating argument subject to functional initial conditions. Our results are based on Schaefer's fixed point theorem for completely continuous operators.

1. Introduction

Differential equations with impulses arise quite often in the study of different problems in particular are used as a model for evolutionary processes subject to a sudden rapid change of their state at certain moments. The theory of impulsive differential equations has become recently a quite active area of research. For an introduction to this theory we refer to the books [1–4], which also contain a variety of interesting examples and applications.

In this paper, we establish new results for the existence of solutions for the second-order impulsive differential equation with deviating argument subject to initial conditions:

where is in the space ( not necessarily the Euclidean norm), , is a real matrix, , and are the left and right limits of in .

Such type of problem arises from a few of scientific applications as, for example, the problem of impulsive maneuver of a spacecraft (see [5] in finite-dimentional setting and [6] in infinite-dimentional setting). More recently, impulsive second-order differential equations or inclusions on compact intervals subject to nonhomogeneous conditions have been studied by several authors (see for example [7–14] and the references therein). The study of differential problems on unbounded interval has been done, for example, in [14–18].

Differential equations with deviating argument are investigated, for example, in [6, 7, 9, 12, 14, 16, 17, 19, 20].

Various techniques are utilized in the above papers: Schauder's fixed point Theorem [6, 9, 16, 17, 19, 20], Leray-Schauder's nonlinear alternative [14, 18], Contractions Principle [8, 13], fixed point index theory [12], and Sadovskii's fixed point Theorem [7]. Moreover, in order to prove the compactness of involved operator, the Ascoli-Arzelà Theorem is often used (see [6, 9, 12, 14, 18]).

The methodology here is to write the problem as a perturbed integral equation, and we look for fixed points of an operator in a suitable functions space. For this purpose we want to utilize the Schaefer's fixed point theorem for completely continuous operators. In order to prove that is completely continuous, we make use of a variant of the compactness result in the Banach space of continuous bounded functions from a topological space into due to De Pascale, Lewicki and G. Marino. We have used similar techniques and tools in [16, 17, 20] to study first-order, impulsive, or nonimpulsive, differential or integrodifferential equations on unbounded intervals. In this paper we extend these ideas to second-order impulsive equations.

Moreover we prove that our method can be easily used also in the case of functional conditions that, to the best of the our knowledge, are not studied for this class of problems. In fact, we discuss in details the Problem in the case of the initial conditions

because it sheds light on the techniques used, but the same approach may be applied to impulsive equations subject to a more general functional initial condition that covers a large number of cases, namely

where is a bounded linear operator and is a completely continuous operator. This is done in the last section.

2. Notations and Preliminaries

Firstly, we recall definitions, notations, and useful facts regarding the cosine families (see [21–23] for the detailed study of cosine and sine families).

Definition 1. A one-parameter family of bounded linear mappings on into itself is called a strongly continuous cosine family if and only if

, for all ,, where is the identity map, , for all .

Moreover, the sine family is defined as

By definition one obtains that:

is continuous in , and , for all , , for all , commute, for all , , for all , , for all .

Definition 2. The infinitesimal generator of a strongly continuous cosine family is the operator defined by

and .

One obtains that, for and ,

and then

So, by one has

, for all .

Definition 3. A cosine family (resp., a sine family ) is uniformly bounded if there exists (resp., ) such that

where denotes the norm in the space of the matrices.

Example 4. In , let be the family of bounded linear operators

Let us observe that is a uniformly bounded cosine family and its infinitesimal generator is

We will utilize the Schaefer's fixed point theorem for completely continuous operators.

Theorem 5 (see [24]). Let be a normed linear space. Let be a completely continuous operator. Let

Then either in unbounded or has a fixed point.

We use the condition for compactness in the Banach space of continuous bounded functions from a topological space into , endowed with the norm , due to De Pascale, Lewicki and G. Marino.

Theorem 6 (see [25]). Let be a continuous operator. Suppose that, for any bounded set , is a bounded set and there exist bounded functions , , such that, for all and for all ,

Then is a compact operator.

3. The Integral Problem

From now on, we assume that the functions and the matrix have the following properties.

The matrix is the infinitesimal generator of a strongly continuous cosine family , uniformly bounded by . Suppose moreover that the corresponding sine family is uniformly bounded by a constant . is a continuous function such that there exist a continuous integrable function and a continuous nondecreasing function for which

is a continuous increasing function such that , for all . The function belongs to . and , are continuous functions such that there are for which

We will work in the Banach space

endowed with the supremum norm .

We define for :

For any let be the function defined on by

In the next propositions we show the properties of the operator useful for us.

Proposition 7. The operator maps into and . Moreover Tx has derivative in , , and .

Proof. The continuity of , , and guarantees that is continuous in , and there exists for . Moreover, since is integrable, for , one has

so . We need to show that, for , there exist and . Note that

(i). Indeed, using (u5), for

(ii)It results that . Indeed let . Then

 Analogously one can see that

 in such a way that . Since, by (u1), , a similar proof permits to verify that

By the above steps, it follows that .

Since there exists if , to conclude our proof we show that there exists , is finite, and . We also observe that, for and , by (u6),

Thus, following the same idea, when , we have

Moreover, for

Finally, for

while

and these permit to obtain that there exist and and that one has .

Proposition 8. The fixed points of are solutions of the problem .

Proof. In fact, and . Moreover, by the hypotheses on and on the sine and cosine families, one obtains that is derivable in , and it results in the following:

So one can conclude that

that is, satisfies the differential equation in problem . This combined with the results in Proposition 7 yields the thesis.

In order to prove that is a completely continuous operator, we make use of the following compactness criterion in .

Proposition 9 (see [16]). Let be a continuous operator. Suppose that, for any bounded set , is a bounded set, and there exist bounded functions , , such that, for all and, for all

Then is a compact operator.

Proof. The Banach space is isometric to the Banach space:

where . Of course, is closed in . So the thesis follows from Theorem 6.

Proposition 10. is a completely continuous operator.

Proof. First we prove that is a continuous operator. Let a sequence in such that in . We prove that . By the continuity of it results that, for fixed ,

Moreover, by the monotonicity of , we have

where . Therefore is dominated by an integrable function that does not depend on . Then, from the uniform boundedness of the sine family and by the dominated convergence theorem, we have

The uniform boundedness of the cosine family and the continuity of and imply that, as ,

Thus

Now, let

Thanks to Proposition 9, it is enough to show that is a bounded set and that it is possible to control the oscillations of each function in by means of a finite number of bounded functions. The boundedness of follows by the inequalities:

To control the oscillations of we should need to distinguish three cases: , , and .

Here we study the case only because the proofs of the other cases are similar.

Let and . Then

Now, by property , one obtains that

so we have

Using we obtain also that

To control the oscillations of the operators and , first we note that if , and we have nothing to prove because . The boundedness of and solves also the case since we have

Thus it remains to prove only the case . We have that

In a similar way we can show that

So, defining

where , and we obtain that for all

4. Main Result

Theorem 11. Assume that the hypotheses hold. Then the problem has at least one solution.

Proof. Our problem can be reduced, by Propositions 7 and 8, to find a fixed point for the operator . Proposition 10 assures that is a completely continuous operator.

To apply the Schaefer's fixed point theorem, it remains to prove that the set

is bounded. The proof is based on an idea in [19].

Let with . First of all, since , for we have

We consider now that . Thus we have

Consider the function defined by

Observe that is not necessarily continuous in but and exist, for all .

For , we have

Moreover, being increasing and for , one has

so, taking the supremum over in the inequality (52), we obtain that

Denoting by the right-hand side of the last inequality, we have that the function is continuous:

and for .

Moreover, since is nondecreasing, for ,

This implies that

and so, for any ,

Since is a continuous function for all , we have

This, together with condition (13), permits us to conclude that is bounded by a constant depending on the functions and only.

Summarizing, implies that .

5. Functional Initial Conditions

Let us consider the problem

The modularity of the operator used in the proof of our Theorem 11 permits to prove a result on the existence of solutions for Problem modifying few parts of the above proof.

For , we define the function as

We suppose that

is a bounded linear operator for which if are such that , then .  is a completely continuous operator such that for a certain constant . For , .

Theorem 12. Assume that the hypotheses hold. Then the problem has at least one solution.

Proof. Let us consider the operator defined on as

Following the proof of Theorem 11, it is necessary to study only the part of the operator arising from the functional initial condition. First, for , one has

and, by the continuity of , as

These, following Proposition 7, are enough to prove that and = .

Moreover, it is not difficult to verify that for , and this is enought to prove that so that one has

From the hypotheses on one has

so, it follows from that

Moreover, and so, using Proposition 8, the fixed points of are solutions of . It remains to prove that we can control the oscillations of by a finite number of bounded functions.

We need to control only the oscillations of . If (the other case are similar), we have

So, by defining we have

Hence is a completely continuous operator. Following the same proof of Theorem 11 one can see that the set

is bounded, so has fixed points that are solutions of . This complete the proof.

Remark 13. The reader is referred to [5, 9, 13, 19] for some examples and applications.

References

D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, New York, NY, USA, 1993.

M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006.

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.

A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14, World Scientific, River Edge, NJ, USA, 1995.

X. Liu and A. R. Willms, “Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft,” Mathematical Problems in Engineering, vol. 2, pp. 277–299, 1996.

R. Sakthivel, E. R. Anandhi, and N. I. Mahmudov, “Approximate controllability of second-order systems with state-dependent delay,” Numerical Functional Analysis and Optimization, vol. 29, no. 11-12, pp. 1347–1362, 2008.

G. Arthi and K. Balachandran, “Controllability of damped second-order impulsive neutral functional differential systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 3, pp. 799–813, 2012.

D. N. Chalishajar, “Controllability of second order impulsive neutral functional differential inclusions with infinite delay,” Journal of Optimization Theory and Applications, vol. 154, no. 2, pp. 672–684, 2012.

E. Hernández, M. Rabello, and H. R. Henríquez, “Existence of solutions for impulsive partial neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1135–1158, 2007.

G. Infante and P. Pietramala, “Nonlocal impulsive boundary value problems with solutions that change sign, CP1124, Mathematical Models in Engineering, Biology, and Medicine,” in Proceedings of the International Conference on Boundary Value Problems, A. Cabada, E. Liz, and J. J. Nieto, Eds., pp. 205–213, 2009.

G. Infante, P. Pietramala, and M. Zima, “Positive solutions for a class of nonlocal impulsive BVPs via fixed point index,” Topological Methods in Nonlinear Analysis, vol. 36, no. 2, pp. 263–284, 2010.

T. Jankowski, “Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 179–189, 2008.

R. Sakthivel, N. I. Mahmudov, and J. H. Kim, “On controllability of second order nonlinear impulsive differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 45–52, 2009.

S. Sivasankaran, M. M. Arjunan, and V. Vijayakumar, “Existence of global solutions for second order impulsive abstract partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6747–6757, 2011.

D. N. Chalishajar and F. S. Acharya, “Controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 48, no. 4, pp. 813–838, 2011.

G. Marino, P. Pietramala, and L. Muglia, “Impulsive neutral semilinear equations on unbounded intervals,” Nonlinear Functional Analysis and Applications, vol. 9, no. 4, pp. 527–543, 2004.

G. Marino, P. Pietramala, and L. Muglia, “Impulsive neutral integrodifferential equations on unbounded intervals,” Mediterranean Journal of Mathematics, vol. 1, no. 1, pp. 93–108, 2004.

S. K. Ntouyas and D. O'Regan, “Existence results on semi-infinite intervals for nonlocal evolutions equations and inclusions via semigroup theory,” Numerical Functional Analysis and Optimization, vol. 29, no. 3-4, pp. 419–444, 2008.

J. P. Dauer and K. Balachandran, “Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 93–105, 2000.

G. Marino, P. Pietramala, and H.-K. Xu, “Nonlinear neutral integrodifferential equations on unbounded intervals,” International Mathematical Forum, vol. 1, no. 17–20, pp. 933–946, 2006.

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.

J. A. Goldstein, Semigroups of Linear Operators and Applications, The Clarendon Press Oxford University Press, New York, NY, USA, 1985.

C. C. Travis and G. F. Webb, “Cosine families and abstract nonlinear second order differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 32, no. 1-2, pp. 75–96, 1978.

H. Schaefer, “Über die Methode der a priori-Schranken,” Mathematische Annalen, vol. 129, pp. 415–416, 1955.

E. De Pascale, G. Lewicki, and G. Marino, “Some conditions for compactness in and their application to boundary value problems,” Analysis, vol. 22, no. 1, pp. 21–32, 2002.