Table of Contents
International Journal of Engineering Mathematics
Volume 2016, Article ID 1478482, 5 pages
http://dx.doi.org/10.1155/2016/1478482
Research Article

On Stability of Vector Nonlinear Integrodifferential Equations

Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beersheba, Israel

Received 15 March 2016; Accepted 5 May 2016

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2016 Michael Gil’. 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

Let be a bounded domain in a real Euclidean space. We consider the equation , where and are matrix-valued functions and is a nonlinear mapping. Conditions for the exponential stability of the steady state are established. Our approach is based on a norm estimate for operator commutators.

1. Introduction and Statement of the Main Result

Throughout this paper, is the complex -dimensional Euclidean space with a scalar product and norm ; is the set of -matrices; is the unit operator in corresponding space; is a bounded domain with a smooth boundary in a real Euclidean space; is the Hilbert space of functions defined on with values in , the scalar product and the norm .

Our main object in this paper is the equation where and are matrix-valued functions defined on and , respectively, with values in , and satisfy conditions pointed out below, and is unknown.

Traditionally, (2) is called the Barbashin type integrodifferential equation or simply the Barbashin equation. It plays an essential role in numerous applications, in particular, in kinetic theory [1], transport theory [2], continuous mechanics [3], control theory [4], radiation theory [5, 6], and the dynamics of populations [7]. Regarding other applications, see [8]. The classical results on the Barbashin equation are represented in the well-known book [9]. The recent results about various aspects of the theory of the Barbashin equation can be found, for instance, in [1014] and the references given therein. In particular, in [11], the author investigates the solvability conditions for the Cauchy problem for a Barbashin equation in the space of bounded continuous functions and in the space of continuous vector-valued functions with the values in an ideal Banach space. The stability and boundedness of solutions to a linear scalar nonautonomous Barbashin equation have been investigated in [15].

The literature on the asymptotic properties of integrodifferential equations is rather rich (cf. [1622] and the references given therein), but the stability of nonlinear vector integrodifferential equations is almost not investigated. It is at an early stage of development.

A solution of (2) is a function having a measurable derivative bounded on each finite interval.

It is assumed that under consideration provides the existence and uniqueness of solutions (e.g., it is Lipschitz continuous). The zero solution of (2) is said to be exponentially stable, if there are constants , and , such that , provided . It is globally exponentially stable if .

Suppose that, for a positive , For example, for an integer , let . Here, with a matrix kernel satisfying Then, by the Schwarz inequality, Thus, Hence, for any , we have condition (3) with .

The following notations are introduced: for a linear operator , is the adjoint operator, is the operator norm, and is the spectrum. For -matrix , put where , are the eigenvalues of , counted with their multiplicities; is the Frobenius (Hilbert-Schmidt) norm of . The following relations are checked in [23, Section ]: , If is a normal matrix, , then . Furthermore, denote and assume that In addition, with the notation , put This integral is simply calculated. If is a normal matrix for all , then

Now, we are in a position to formulate our main result.

Theorem 1. Let conditions (3), (11), and hold. Then, the zero solution to (2) is exponentially stable. If, in addition, in (3), then the zero solution is globally exponentially stable.

This theorem is proved in the next 3 sections. It gives us “good” results when is “small,” that is, if matrices and “almost commute” and is “small.” If (2) is scalar, then , So, in the scalar case, condition (14) takes the form This condition is similar to the stability test derived in [24] for scalar integrodifferential equations.

2. Auxiliary Results

Let be a Hilbert space with a scalar product and the norm ; denotes the set of bounded linear operators in and is the commutator of .

Lemma 2. Let and . Then,

Proof. Put . Then, . On the other hand, So, , as claimed.

Let Then, the Lyapunov equation has a unique solution and it can be represented as (cf. [25]). Denote , where .

Lemma 3. Under condition (19), one has

Proof. Making use of (21), we can write But . So , where We have If , then . If , then . So .
In addition, by Lemma 2, This proves the lemma.

3. Equations in a Hilbert Space

In this section, for simplicity, we put . Put . Consider in the equation where and continuously maps into and satisfies The solution and stability are defined as in Section 1. The existence and uniqueness of solutions are assumed. Recall that is a solution of (20).

Lemma 4. Let conditions (19) and (29) with hold. Then, any solution of (28) satisfies the inequality

Proof. For brevity, we write . Multiplying (28) by and doing the scalar product, we get Since , due to (20) and Lemma 3, it can be written that Taking into account the fact that due to (29) we get From this inequality, we have . Hence, as claimed.

Lemma 5. Let conditions (29) and hold. Then, the zero solution to (28) is exponentially stable. If in (29), then the zero solution to (28) is globally exponentially stable.

Proof. If , then the required result is due to the previous lemma. If , then, taking due to the previous lemma, . Hence, we easily obtain the required result.

4. Proof of Theorem 1

Take Then, and So Due to [23, Example ], where . Hence, since . Consequently, . In addition, Now, the required result is due to Lemma 5.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

References

  1. C. Cercignani, Mathematical Methods in Kinetic Theory, Macmillian, New York, NY, USA, 1969. View at MathSciNet
  2. H. G. Kaper, C. G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear Transport Theory, vol. 5 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1982. View at MathSciNet
  3. V. M. Aleksandrov and E. V. Kovalenko, Problems in Continuous Mechanics with Mixed Boundary Conditions, Nauka, Moscow, Russia, 1986 (Russian).
  4. A. L. Khoteev, “An optimal control problem for integro-differential equations of Barbashin type,” in Problemy Optimizacii Upravlenija, pp. 74–87, Minsk, Russia, 1976 (Russian).
  5. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, Mass, USA, 1967. View at MathSciNet
  6. G. I. Marchuk, The Methods of Calculation for Nuclear Reactors, Atomizdat, Moscow, Russia, 1961 (Russian).
  7. H. R. Thieme, A Differential-Integral Equation Modelling the Dynamics of Populations with a Rank Structure, vol. 68 of Lecture Notes in Biomathematics, 1986.
  8. A. W. England, “Thermal microwave emission from a halfspace containing scatterers,” Radio Science, vol. 9, no. 4, pp. 447–454, 1974. View at Publisher · View at Google Scholar · View at Scopus
  9. J. M. Appell, A. S. Kalitvin, and P. P. Zabrejko, Partial Integral Operators and Integro-Differential Equations, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet
  10. A. S. Kalitvin, “On two problems for the Barbashin integro-differential equation,” Journal of Mathematical Sciences, vol. 126, no. 6, pp. 1600–1606, 2005. View at Google Scholar
  11. A. S. Kalitvin, “Some aspects of the theory of integro-differential Barbashin equations in function spaces,” Journal of Mathematical Sciences, vol. 188, no. 3, pp. 241–249, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. B. G. Pachpatte, “On a parabolic type Fredholm integrodifferential equation,” Numerical Functional Analysis and Optimization, vol. 30, no. 1-2, pp. 136–147, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. B. G. Pachpatte, “On a nonlinear Volterra integral equation in two variables,” Sarajevo Journal of Mathematics, vol. 6, no. 19, pp. 59–73, 2010. View at Google Scholar · View at MathSciNet
  14. B. G. Pachpatte, “On a parabolic integrodifferential equation of Barbashin type,” Commentationes Mathematicae Universitatis Carolinae, vol. 52, no. 3, pp. 391–401, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. M. Gil', “On stability of linear Barbashin type integrodifferential equations,” Mathematical Problems in Engineering, vol. 2015, Article ID 962565, 5 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. R. P. Agarwal, A. Domoshnitsky, and Y. Goltser, “Stability of partial functional integro-differential equations,” Journal of Dynamical and Control Systems, vol. 12, no. 1, pp. 1–31, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. N. M. Chuong, T. D. Ke, and N. N. Quan, “Stability for a class of fractional partial integro-differential equations,” Journal of Integral Equations and Applications, vol. 26, no. 2, pp. 145–170, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. A. Domoshnitsky and Y. M. Goltser, “Approach to study of bifurcations and stability of integro-differential equations,” Mathematical and Computer Modelling, vol. 36, no. 6, pp. 663–678, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. A. D. Drozdov and M. I. Gil, “Stability of linear integro-differential equations with periodic coefficients,” Quarterly of Applied Mathematics, vol. 54, no. 4, pp. 609–624, 1996. View at Google Scholar
  20. Ya. Goltser and A. Domoshnitsky, “Bifurcations and stability of integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 2, pp. 953–967, 2001. View at Publisher · View at Google Scholar
  21. J. Cao and Z. Huang, “Existence and exponential stability of weighted pseudo almost periodic classical solutions of integro-differential equations with analytic semigroups,” Differential Equations and Dynamical Systems, vol. 23, no. 3, pp. 241–256, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. N. T. Dung, “On exponential stability of linear Levin-Nohel integro-differential equations,” Journal of Mathematical Physics, vol. 56, Article ID 022702, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. M. I. Gil', Operator Functions and Localization of Spectra, vol. 1830 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  24. M. I. Gil', “Stability of Fredholm type integro-parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 244, no. 2, pp. 318–332, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, RI, USA, 1974.