About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 935089, 10 pages
http://dx.doi.org/10.1155/2013/935089
Research Article

Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Mathematics, 10000 Zagreb, Croatia

Received 24 January 2013; Accepted 18 March 2013

Academic Editor: Elena Braverman

Copyright © 2013 Siniša Miličić and Mervan Pašić. 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 study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

1. Introduction

Let and a given real or complex Banach space with the zero , and let denote the space of all linear bounded operators from into . We consider the linear nonautonomous differential equation in as where , , is an operator-valued function defined on interval with the value in and . If , then (1) becomes a linear nonautonomous differential system.

As a basic result, in Theorem 1, we state the existence and uniqueness of a solution of (1) which is proved by using a fixed-point theorem in Fréchet spaces for an -contraction mapping. Next, in Theorems 2 and 3, we give a necessary and sufficient condition for the nonintegrability of on provided blows up near and allows a precise asymptotic behaviour near . Based on these results, in the case when , we study the so-called nonrectifiable attractivity of the zero solution (see Definition 4) of the so-called linear integrable two-dimensional system (see Definition 5) as follows: with , , and some real constants.

Precisely, as a kind of singular behaviour near of all solutions of linear integrable system (2), in Theorem 6, we involve on the matrix elements and a necessary and sufficient condition for the infiniteness of the length of every solution curve of (in our best knowledge, it is the first paper dealing with this kind of problems). Theorem 6 is a consequence of the precise asymptotic formula for all solutions near of integrable differential system (2) presented in Lemmas 11 and 12. Of course, instead of interval and , we can also state our main results on and .

In applications, the last years have seen an increasing interest in the analysis of differential equations and systems on finite time intervals. This is because the Finite-Time Stability (FTS) and the Finite-Time Lyapunov Exponent (FTLE) were introduced, respectively, in the control of systems within a finite time as well as in the Lagrangian Coherent Structure (LCS) on finite-time intervals in fluid, ocean, and atmosphere dynamics [13] and in the biological application [4]. It includes the time-varying vector fields known only on a finite-time interval, but not on the whole half line .

In the theory of differential equations, the importance of studying the different kind of asymptotic behaviours of the nonautonomous linear differential systems comes from their application in the study of asymptotic and oscillatory behaviour of the second and higher order ordinary differential equations. For instance, in [5], authors study the asymptotic behaviour near of oscillatory solutions of the nonlinear second-order differential equation via the asymptotic formula for solutions of an auxiliary linear differential system having elements which are absolutely continuous functions on . In [6], authors derive a precise asymptotic behaviour near of solutions of the third-order nonlinear differential equation such that , , , and on , by using the asymptotic solution formula for the corresponding linear differential system based on Hartman and Wintner’s asymptotic integration (see [7, 8] and for its generalization [9, 10]). Similarly, in [11], authors prove an asymptotic solution formula for the second-order nonlinear differential equations depending on the asymptotic behaviour of fundamental solutions of the corresponding homogeneous equation . On certain types of asymptotic behaviour of linear and nonlinear differential and integro-differential systems, we refer reader to some recently published papers [1214] and references therein. The attractivity of solutions of scalar delay differential equations is widely studied and it is important in mathematical biology; see, for instance, [15] and references therein.

This paper is mainly based on a part of the P.h.D degree thesis [16] of the first author.

2. Statement of the Main Results

Preliminarily, we state the following auxiliary result.

Theorem 1. For each , there exists a unique solution of (1).

Next, we derive a necessary condition for a kind of singularities of (1) at in terms of singular behaviour of the operator norm of near .

Theorem 2. Let there exist a solution of (1) such that as . If , then one has:

Thus, if (3) is not satisfied, then the integrability of occurs for all solutions of (1), which is out of our interest. All results from this section will be proved in the next sections.

At the second, we impose on the operator-valued function the following structural hypotheses:

 () there are positive constants , and a positive real function such that  () the solution of (1) is given by where , , is an operator-valued function called the fundamental solution, such that and there are positive constants , and a positive real function such that Such hypotheses on (1) imply the following properties of every solution and its .

Theorem 3. Let the hypotheses - be fulfilled. Then, for every , the corresponding solution of (1) satisfies(i) as ;(ii) is an injective function on if and only if is not an eigenvector of with an eigenvalue for all , ;(iii) if and only if .

The meaning and importance of hypotheses - as well as the conclusions (i)–(iii) of Theorem 3 will be verified by Theorem 6, where and (1) is a large class of two-dimensional linear differential systems.

The second aim of the paper is to use previous theorem in the study of the rectifiable and nonrectifiable attractivity of zero solution of two-dimensional linear differential system as where and the prime denotes , and the matrix-valued function , (where denotes the space of real matrices), is continuous on . Therefore, we may apply previous theorems to system (8). As in the general case of , the zero solution of system (8) is said to be (global) attractive if for every solution of (8), we have as . On the asymptotic stability and attractivity for several kinds of linear systems in the case when , we refer reader to [710, 1214] and references therein.

We know that a plain curve is a Jordan curve in if there exists a continuous injection , , , such that . It is often said that is parametrized by or that is associated to . The length of , denoted by , is defined by where the supremum is taken over all finite partitions of the interval (see [17, 18]).

Definition 4. The zero solution of system (8) is rectifiable attractive (resp., nonrectifiable attractive) if it is attractive and the curve of every solution of (8) is a rectifiable (resp., nonrectifiable) Jordan curve in .

Following [17], we are interested in Jordan curves so that the parametrization of our solutions faithfully represents the length and rectifiability properties (omitting injectivity, we might find solutions that self-intersect on large sets or are just self-winding and, hence, artificially nonrectifiable).

The following well-known fact (see [19]) gives us a main support to relate the singularity in (1) with the nonrectifiability in system (8):

On the rectifiability of graph of solutions of scalar second-order linear differential equations, we refer reader to [20].

Here the particular attention is paid to the case of the so-called linear integrable systems defined as follows.

Definition 5. Let denote the linear space of all matrix with the elements in . We say that (8) is a linear integrable system if there exists an invertible matrix such that for every the matrix is a diagonal matrix for every .

Note that the matrices are themselves always real. As commented in Section 6, we proceed with presentation of our results and arguments in terms of the coefficients of such matrices.

Denoting the matrix elements of by when we have and , we speak of integrable systems (see for instance [21]). In such a case, both eigenvalues of the matrix admit eigenvectors that do not depend on variable . This motivates us to introduce Definition 5 which gives a more general notion of the integrable system than previous one. It is because (see Lemma 11 in Section 4) the integrable system (8) has the following form: where . For and in (8), we have and ; so, the classic integrable system is a special case.

In Section 5, we show that the set of all matrix-valued functions that satisfy Definition 5 make an algebra.

Furthermore, we show in Section 4 that if (8) is a linear integrable system, then the matrix-valued function satisfies the required hypotheses - in particular for where and are two eigenvalues of for each . We also show that the solution curve of every solution of the integrable system (8) is a Jordan curve if for every pair , , at least one of the statements holds true, where .

This implies the third main result of the paper.

Theorem 6. Let . One supposes (15) and The zero solution of an integrable system (8) with of form (12) is nonrectifiable (resp., rectifiable) attractive if and only if

Remark 7. Under the assumptions of Theorem 6, we conclude that if , then the zero solution of integrable system (8) is rectifiable attractive.

The proof of the previous theorem is given in Section 4 based on Theorem 3. As an important consequence of preceding Theorem 6, we are able to study the following two-parametric model-system, which is singular at : where .

Corollary 8. Let and . One has that:(i)if , the zero solution of (18) is rectifiable attractive provided and nonrectifiable attractive provided ;(ii)if , then the zero solution of (18) is rectifiable attractive for all and .

As we see, in the case of , the rectifiable attractivity of (18) depends on the order of growth of the singular term appearing in the antidiagonal coefficients of . The existence of solutions for the integrable systems (8) and (18) is guaranteed by their explicit forms given in Lemma 11.

3. Proofs of Theorems 1, 2, and 3

In this section, we study the existence and uniqueness of solutions to (1) as well as its attractivity of the zero solution.

In the case when the operator-valued function , is defined and continuous on the whole interval , then the existence of a classic solution is well known; see, for instance, [22, Chapter 3], [23, Chapter 2], and [24, Chapter 5.1]; when are unbounded linear operators, we refer reader to [24, Chapter 5.4] and some selected papers from [25].

However, our operator-valued function , , is mainly singular and not defined at . By this reason, we cannot use the existence results and methods obtained and used on . One way to understand the main difficulties because of the pressumed singular behaviour of near , we can make the reflexion from into by the transformation . Then, the existence of a solution of equation , , , can be related with the existence of a global solution of equation , , where , , , and . Since is an infinite interval, our approach to the existence and uniqueness of solution of (1) is motivated by the works [2629].

At first, it is clear that (1) can be rewritten in the form of corresponding integral operators equation: where is defined by The existence and uniqueness of solution of problem (19)-(20) will be explored by the following -contraction fixed point result in Fréchet spaces.

Definition 9. Let be a Fréchet space. A mapping is said to be an -contraction on if for each there exists such that

In applications, the inequality (21) need not be satisfied for the family of seminorms , rather only for another family of seminorms which is equivalent to . Hence, the key point of the following -contraction principle is that an operator has a fixed point provided (21) is satisfied in .

Lemma 10. Let be a Fréchet space and be an -contraction on with respect to a family of seminorms equivalent to . Then, has a fixed point on .

The previous lemma is a particular case of [29, Theorem 1.2].

Proof of Theorem 1. Let , , and It is known that is a family of seminorms on and is a Fréchet space.
Let be an operator defined in (20). Since may be singular at , it is easy to check that in general is not an -contraction with respect to the family of seminorms given in (22).
Next, let and be two sequences of real numbers determined by Let us remark that if satisfies (3), then obviously as . In respect to these two sequences and , we define Such sequence is a family of seminorms which are equivalent with respect to because
Next, according to (20) and (23), for , we calculate Multiplying previous inequality by , where is defined in (24), we obtain Taking the maximum in the previous inequality over the interval and using (24) and (25), we conclude that Now, applying Lemma 10, we obtain the existence of satisfying problem (19)-(20). Since and , from (19)-(20) follows . Hence, by differentiating (19), we conclude that is also a solution of equation (1). Thus, the existence of a solution of (1) is proved.
Now, we will show the uniqueness of solution of equation (1). Let and be two solutions of (1) such that . Integrating (1) with respect to and , we obtain and which yields . Puting this equality into (29), we conclude that On the other hand, if the solutions and are different, then there is such that , which implies that . In particular, there is a big enough such that , and, hence, that is, Putting (32) into (30), we obtain that . But it is not possible since . Hence, the assumption about the existence of such that is not possible. Therefore, we conclude that for all , and, thus, the uniqueness of solution of (1) is shown.

Proof of Theorem 2. It is enough to show that if the statement (3) does not hold, then for every and the corresponding solution of (1) such that . In fact, if we suppose contrary to (3), then there exists such that , . Also, because of as , there exists such that , . Hence, from (1), we obtain which shows that .

Proof of Theorem 3. The conclusion (i) immediately follows from the hypothesis because as .
Next, we give the proof of the conclusion (ii). Let , , be such that is an eigenvector of with an eigenvalue . Then, Applying to the pervious formula, we obtain the expression , which by definition of equals to . Hence, is not an injective function. Note that, since is postulated invertible for all , all the steps of this proof are reversible. Thus, the required equivalence holds.
Finally, the conclusion (iii) follows from the hypotheses - because Hence, the required equivalence in (iii) holds.

Note that conclusion (ii) does not depend on any of the hypotheses .

4. Proofs of Theorem 6 and Corollary 8

In this section, we study the rectifiable and nonrectifiable attractivity of the linear integrable systems (8).

At the first, we state the following lemma in which a specific form of the matrix of linear integrable systems is proposed.

Lemma 11. Suppose that system (8) is integrable. Then, there exist functions and constants such that The eigenvalues of are given by the formula

Proof. Let be two eigenvalues of the matrix-valued function . Since system (8) is supposed to be an integrable system, from Definition 5, there is a regular matrix such that If and , then we can change the columns in which causes and instead of . Also, if and or , then Hence, we may suppose that ; that is, for some complex numbers , , , with , we may set Now from (38) and (40), we obtain where the matrix elements of are real numbers for all .
We denote that and if for all , then let and if form a complex-conjugate pair for all , then let
Putting previous notations in (41), we get the desired conclusion (36). Note that all of these functions and constants are real, since is defined as a matrix-element of a real matrix, is defined real in both cases, and the rest of the constants are necessarily real by virtue of satisfying (36). Now, the conclusion (37) immediately follows from (36).

By Theorem 1, we know that there exists a unique solution of system (8). Moreover, in the next lemma, we show that if (8) is a linear integrable system, then the solution is explicitly expressed by the matrix elements of .

Lemma 12. Let be a solution of linear integrable system (8) and let be a matrix determined in Definition 5. Then, there exists a matrix-valued function , , such that and satisfies the following properties: (1) is the identity matrix;(2) is invertible for every ;(3) is a diagonal matrix for every ;(4)if is an eigenvalue of , then is an eigenvalue of ;(5) for every .

Proof. Let be the matrix that diagonalizes . Let be of form We define Now, is obviously the identity matrix, and thus . Since is invertible for all , so is . Next, is diagonal by construction. For an eigenvalue of , by construction, is a diagonal element of ; so, it is an eigenvalue of . Also, And finally, the function solves the integrable system (8) because and has the property .

According to [23, p. 188], the matrix-valued function is called the evolution operator.

In the next lemma, we give some necessary and sufficient conditions such that the solution's curve of every solution of integrable system (8) is a Jordan curve. Results of that kind are important because Definition 4 requires the Jordan property.

Lemma 13. Let (8) be a linear integrable system such that and , , where the real numbers , , , and are determined by Lemma 11. Then, the solution's curve of every solution is a Jordan curve in if and only if for every pair , , at least one of the statements in (15) holds true.

Proof. In order to prove this lemma, we use the equivalence stated in the conclusion (ii) of Theorem 3 (we may use Theorem 3 because of Lemma 12; see also the remark at the end of Theorem 3). According to that, we need to find pairs and such that is an eigenvalue of . Since the evolution operator can be diagonalized into the form , we compute The diagonalized form of is then given by the formula Now, the eigenvalues of are and . Since , we only need to solve the equation This equation is solved whenever where the order of integration has been reversed by multiplying by .
Since by this argument, if and only if is an eigenvalue of , we get the full equivalence.

Proof of Theorem 6. The key point of this proof is to show that the integrable system (8) satisfies the required hypotheses -, because all conclusions of Theorem 6 follow immediately from the conclusions (i), (ii), and (iii) of Theorem 3. Before we show that, we state the following two propositions which will be proved in Section 5.

Proposition 14. Let be defined by , where , , are real numbers such that . Then, there exist constants such that

Proposition 15. Let be a diagonal matrix of the form for all . Then, , and

Now we proceed with the proof of Theorem 6. By definition, , where is an invertible matrix and is a matrix-valued function given in (45). Since is invertible, there exist constants such that

Next, from (36), we get where is defined in Proposition 14. Hence, by (57) and Propositions 14 and 15 applied to , we get two constants such that where . Now from (56) and (58), for all and , we have which shows that the hypothesis is fulfilled with respect to given in (13).

Next, let the evolution operator and matrix-valued function be defined by (46). The required condition is satisfied because of the following arguments. First, from Lemma 12 and (37), we have Hence, by Proposition 15 applied to , we obtain Now from (56) and (61), for all and , we have which shows that satisfied the required inequalities in with respect to given in (13).

Finally, by Lemma 13, we obtain that the solution's curve of every solution is a Jordan curve. Thus, all assumptions of Theorem 3 are fulfilled, and, therefore, we may apply Theorem 3 here. Consequently, the proof of this theorem is complete.

Proof of Corollary 8. It is enough to show that all assumptions of Theorem 6: , (15), (16), and (17) are fulfilled in particular for where , , and . In fact, the following elementary calculation is shown: Moreover, for , so, condition (17) is fulfilled if and only if . Also, for , Therefore, we may apply Theorem 6 to system (18) which proves this corollary.

5. Appendix

In this section, we first state a result saying that the set of all matrix-valued functions which satisfy Definition 5 forms an algebra. In this direction, let be a set defined by

Theorem 16. The set is an algebra with respect to the classic matrix operations + and ; that is to say, if , then the following properties hold: (1) for all ; (2); (3)if for all , then ;(4)if is a real analytic function such that converges absolutely for all , then .

Proof. We denote the diagonal matrices and . (1), which is diagonal. (2), which is diagonal. (3), which is also diagonal. (4), which is also diagonal.

Next, we give the proofs of some technical results used in previous sections for proving the main results.

Proof of Proposition 14. Denote by the vector . It is enough to show that is a norm as a function , since all the norms in are equivalent [30, p. 38], which, by definition, is the claim of this lemma. Let By direct computation, we see that . Since is symmetric and positive definite, it has a unique symmetric positive definite square root , so that [31, p. 231]. Hence, , where ; so, the function is a norm.

Proof of Proposition 15. Let be such that . Then, ; so, for any , we have since . The remaining equality is true because

6. Final Remarks

In our approach to the model over , we have chosen to emphasize the a priori structure of the real matrices describing the integrable systems. Thus, we have stated and proved this theory in terms of such real functions, complexifying only when absolutely necessary.

We could have taken another approach, complexifying from the start and then easily diagonalizing all the relevant matrices, but this would not essentially simplify any argument except in notation, while we would run the risk of confusing the reader even further about which term or function is real and which one is complex.

References

  1. F. Amato, M. Ariola, M. Carbone, and C. Cosentino, “Finite-time control of linear systems: a survey,” in Current Trends in Nonlinear Systems and Control, Systems & Control: Foundations & Applications, pp. 195–213, Birkhäuser, Boston, Mass, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  2. G. Haller, “A variational theory of hyperbolic Lagrangian coherent structures,” Physica D, vol. 240, no. 7, pp. 574–598, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. Peacock and J. Dabiri, “Introduction to focus issue: lagrangian coherent structures,” Chaos, vol. 20, no. 1, Article ID 017501, 2010. View at Scopus
  4. K. Rateitschak and O. Wolkenhauer, “Thresholds in transient dynamics of signal transduction pathways,” Journal of Theoretical Biology, vol. 264, no. 2, pp. 334–346, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. I. Rachůnková and L. Rachůnek, “Asymptotic formula for oscillatory solutions of some singular nonlinear differential equation,” Abstract and Applied Analysis, vol. 2011, Article ID 981401, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. B. B. Singh and I. M. Chandarki, “On the asymptotic behaviours of solutions of third order non-linear autonomous differential equation governing the MHD flow,” Differential Equations & Applications, vol. 3, no. 3, pp. 385–397, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. P. Hartman, Ordinary Differential Equations, Birkhäauser, Boston, Mass, USA, 2nd edition, 1982. View at MathSciNet
  8. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, USA, 1965. View at MathSciNet
  9. S. Castillo and M. Pinto, “Asymptotic integration of ordinary different systems,” Journal of Mathematical Analysis and Applications, vol. 218, no. 1, pp. 1–12, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Bodine, “A dynamical systems result on asymptotic integration of linear differential systems,” Journal of Differential Equations, vol. 187, no. 1, pp. 1–22, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Medina and M. Pinto, “On the asymptotic behavior of solutions of certain second order nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 399–405, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  12. H. Matsunaga, “Stability switches in a system of linear differential equations with diagonal delay,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 145–152, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. K. Choi and N. J. Koo, “Asymptotic property for linear integro-differential systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 1862–1872, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. K. Choi, N. J. Koo, and S. Dontha, “Asymptotic property in variation for nonlinear differential systems,” Applied Mathematics Letters, vol. 18, no. 1, pp. 117–126, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Qian and Y. Sun, “Global attractivity of solutions of nonlinear delay differential equations with a forcing term,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 3, pp. 689–703, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Miličić, Geometric and fractal properties of solutions of linear differential systems [Ph.D. thesis], University of Zagreb, Zagreb, Croatia.
  17. K. Falconer, Fractal Geometry: Mathematical Fondations and Applications, John Wiley & Sons, New York, NY, USA, 1999. View at MathSciNet
  18. C. Tricot, Curves and Fractal Dimension, Springer, New York, NY, USA, 1995. View at MathSciNet
  19. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York, NY, USA, 1999. View at MathSciNet
  20. M. K. Kwong, M. Pašić, and J. S. W. Wong, “Rectifiable oscillations in second-order linear differential equations,” Journal of Differential Equations, vol. 245, no. 8, pp. 2333–2351, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Onitsuka and J. Sugie, “Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients,” Proceedings of the Royal Society of Edinburgh A, vol. 141, no. 5, pp. 1083–1101, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Ju. L. Daleckii and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, vol. 43 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1974. View at MathSciNet
  23. S. G. Kreĭn, Linear Differential Equations in Banach Space, vol. 29 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1971. View at MathSciNet
  24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at MathSciNet
  25. A. Favini and E. Obrecht, Differential Equations in Banach Spaces, vol. 1223 of Lectures notes in Mahtematics, Springer, New York, NY, USA, 1985.
  26. L. H. Hao and K. Schmitt, “Fixed point theorems of Krasnoselskii type in locally convex spaces and applications to integral equations,” Results in Mathematics, vol. 25, no. 3-4, pp. 290–314, 1994. View at MathSciNet
  27. C. Avramescu, “Some remarks on a fixed point theorem of Krasnoselskii,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2003, no. 5, pp. 1–15, 2003. View at Zentralblatt MATH · View at MathSciNet
  28. L. T. P. Ngoc and N. T. Long, “On a fixed point theorem of Krasnoselskii type and application to integral equations,” Fixed Point Theory and Applications, vol. 2006, Article ID 30847, 24 pages, 2006. View at MathSciNet
  29. L. T. P. Ngoc and N. T. Long, “Applying a fixed point theorem of Krasnosel'skii type to the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3769–3774, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. S. Lang, Real and Functional Analysis, Springer, New York, NY, USA, 3rd edition, 1993. View at MathSciNet
  31. S. Roman, Advanced Linear Algebra, Springer, New York, NY, USA, 2008. View at MathSciNet