Petr ZEMÁNEK
Homepage
Every mathematical discipline goes through three periods of development: the naive, the formal, and the
critical.
D. Hilbert
RESEARCH INTERESTS
My main research interests fall into the qualitative theory of differential, difference, and dynamic equations. More specifically, I am enthralled by the spectral theory of the Sturm—Liouville equations, linear Hamiltonian (differential or difference) systems, and (discrete or dynamic) symplectic systems, which includes, e.g., the study of square integrable or summable solution, self-adjoint extensions, eigenfunctions expansion etc. |
MY PROFILES AT ACADEMIC DATABASES
MathSciNet Reviews (by AMS);
Zentralblatt MATH
(by EMS, reviews);
Scopus
(by Elsevier).
Web of Science (Journal Citation Reports, ESI, InCites) or ORCID or ResearcherID (by Clarivate Analytics). Academia.edu; Google Scholar (by Google); Microsoft Academic (by Microsoft Corporation). |
MY MATH GENEALOGY
Mathematical genealogy of Petr Zemánek (by Mathematics Genealogy Project). |
LIST OF MY PUBLICATIONS
Move the cursor on the icon to view the abstract of the document and click on the icon (if possible) to get the full text of the document.Some manuscripts are also available via arXiv.
Journals and publishers
Research papers in refereed journals and proceedings
25. |
Linear operators associated with differential and difference systems: What is different?, Proceedings of ICDEA 2019, to appear. |
ABSTRACT ------------------------------------------ The existence of a densely defined operator associated with (time-reversed) discrete symplectic systems is discussed and the necessity of the development of the spectral theory for these systems by using linear relations instead of operators is shown. An explanation of this phenomenon is provided by using the time scale calculus. In addition, the density of the domain of the maximal linear relation associated with the system is also investigated. |
24. |
(with R. Šimon Hilscher) On square integrable solutions and principal and antiprincipal solutions for linear Hamiltonian systems, Ann. Mat. Pura Appl. (4) 197 (2018), no. 1, 283–306. MR3747532; SCO; WOS. |
ABSTRACT ------------------------------------------ New results in the Weyl—Titchmarsh theory for linear Hamiltonian differential systems are derived by using principal and antiprincipal solutions at infinity. In particular, a non-limit circle case criterion is established and a close connection between the Weyl solution and the minimal principal solution at infinity is shown in the limit point case. In addition, the square integrability of the columns of the minimal principal solution at infinity is investigated. All results are obtained without any controllability assumption. Several illustrative examples are also provided. |
23. |
Principal solution in Weyl—Titchmarsh theory for second order Sturm—Liouville equation on time scales, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 2, 18 pp. (electronic). MR3606980; SCO; WOS. |
ABSTRACT ------------------------------------------ A connection between the oscillation theory and the Weyl—Titchmarsh theory for the second order Sturm—Liouville equation on time scales is established by using the principal solution. In particular, it is shown that the Weyl solution coincides with the principal solution in the limit point case, and consequently the square integrability of the Weyl solution is obtained. Moreover, both limit point and oscillatory criteria are derived in the case of real-valued coefficients, while a generalization of the invariance of the limit circle case is proven for complex-valued coefficients. Several of these results are new even in the discrete time case. Finally, some illustrative examples are provided. |
22. |
(with S. L. Clark) Characterization of self-adjoint extensions for discrete symplectic systems, J. Math. Anal. Appl. 440 (2016), no. 1, 323–350. MR3479602; Zbl 06570297; SCO; WOS; arXiv. |
ABSTRACT ------------------------------------------ All self-adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized. Especially, for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given expression are discussed and the Krein—von Neumann extension is described explicitly. In addition, a limit point criterion for symplectic systems is established. The result partially generalizes even the classical limit point criterion for the second order Sturm—Liouville difference equations. |
21. |
Limit point criteria for second order Sturm—Liouville equations on time scales, in: "Differential and Difference Equations with Applications", Proceedings of the International Conference on Differential & Difference Equations and Applications (Amadora, 2015)", S. Pinelas, Z. Došlá, O. Došlý, and P. E. Kloeden (editors), Springer Proceedings in Mathematics & Statistics, Vol. 164, pp. 331–338, Springer, Berlin, 2016. MR3571739; Zbl 06674324; SCO; WOS. |
ABSTRACT ------------------------------------------ Necessary and sufficient conditions for the classification of the second order Sturm—Liouville equation on time scales being in the limit point case are established. They unify and extend some of the criteria known for the second order Sturm—Liouville differential and difference equations. |
20. |
(with R. Šimon Hilscher) Time scale symplectic systems with analytic dependence on spectral parameter, J. Difference Equ. Appl. 21 (2015), no. 3, 209–239. MR3316777; Zbl 06429991; SCO; WOS. |
ABSTRACT ------------------------------------------ This paper is devoted to the study of time scale symplectic systems with polynomial and analytic dependence on the complex spectral parameter λ. We derive fundamental properties of these systems (including the Lagrange identity) and discuss their connection with systems known in the literature, in particular with linear Hamiltonian systems. In analogy with the linear dependence on λ, we present a construction of the Weyl disks and determine the number of linearly independent square integrable solutions. These results extend the discrete time theory considered recently by the authors. To our knowledge, in the continuous time case this concept is new. We also establish the invariance of the limit circle case for a special quadratic dependence on λ and its extension to two (generally nonsymplectic) time scale systems, which yields new results also in the discrete case. The theory is illustrated by several examples. |
19. |
(with S. L. Clark) On discrete symplectic systems: Associated maximal and minimal linear relations and nonhomogeneous problems, J. Math. Anal. Appl. 421 (2015), no. 1, 779–805. MR3250508; Zbl 06334563; SCO; WOS; arXiv. |
ABSTRACT ------------------------------------------ In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental properties of the corresponding deficiency indices, including a relationship between the number of square summable solutions and the dimension of the defect subspace, are also derived. Moreover, a sufficient condition for the existence of a densely defined operator associated with the symplectic system is provided. |
18. |
(with R. Šimon Hilscher) Limit circle invariance for two differential systems on time scales, Math. Nachr. 288 (2015), no. 5-6, 696–709. MR3338923; Zbl 06434792; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ_{0}, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non-Hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors. |
17. |
(with R. Šimon Hilscher) Generalized Lagrange identity for discrete symplectic systems and applications in Weyl—Titchmarsh theory, in: "Theory and Applications of Difference Equations and Discrete Dynamical Systems", Proceedings of the 19th International Conference on Difference Equations and Applications (Muscat, 2013), Z. AlSharawi, J. Cushing, and S. Elaydi (editors), Springer Proceedings in Mathematics & Statistics, Vol. 102, pp. 187–202, Springer, Berlin, 2014. MR3280207; Zbl 06459801; SCO. |
ABSTRACT ------------------------------------------ In this paper we consider discrete symplectic systems with analytic dependence on the spectral parameter. We derive the Lagrange identity, which plays a fundamental role in the spectral theory of discrete symplectic and Hamiltonian systems. We compare it to several special cases well known in the literature. We also examine the applications of this identity in the theory of Weyl disks and square summable solutions for such systems. As an example we show that a symplectic system with the exponential coefficient matrix is in the limit point case. |
16. |
(with R. Šimon Hilscher) Limit point and limit circle classification for symplectic systems on time scales, Appl. Math. Comput. 233 (2014), 623–646. MR3215014; Zbl 06571339; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we study the limit point and limit circle classification for symplectic systems on time scales, which depend linearly on the spectral parameter. In a broader context, we develop a unified Weyl—Titchmarsh theory for continuous and discrete linear Hamiltonian and symplectic systems. Both separated and coupled boundary conditions are allowed. Our results include the study of the Weyl disks and circles and their limiting behavior, as well as a precise analysis of the number of linearly independent square integrable solutions. We also prove an analogue of the famous Weyl alternative. We connect and unify many known results in the Weyl—Titchmarsh theory for continuous, discrete, and special time scales systems and explain the differences between them. Some of our statements, in particular those connected with coupled endpoints or the Weyl alternative, are new even in the continuous time setting. |
15. |
(with R. Šimon Hilscher) Weyl disks and square summable solutions for discrete symplectic systems with jointly varying endpoints, Adv. Difference Equ. 2013 (2013), no. 232, 18 pp. (electronic). MR3101933; Zbl 06813505; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we develop the spectral theory for discrete symplectic systems with general jointly varying endpoints. This theory includes a characterization of the eigenvalues, construction of the M-lambda function and Weyl disks, their matrix radii and centers, statements about the number of square summable solutions, and limit point or limit circle analysis. These results are new even in some particular cases, such as for the periodic and antiperiodic endpoints, or for discrete symplectic systems with special linear dependence on the spectral parameter. The method utilizes a new transformation to separated endpoints, which is simpler and more transparent than the one in the known literature. |
14. |
(with R. Šimon Hilscher) Weyl—Titchmarsh theory for discrete symplectic systems with general linear dependence on spectral parameter, J. Difference Equ. Appl. 20 (2014), no. 1, 84–117. MR3173539; Zbl 06259235; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we develop the Weyl—Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. Our new theory includes characterizations of the Weyl disks and Weyl circles, their limiting behavior, properties of square summable solutions including the analysis of the exact number of linearly independent square summable solutions, and limit point/circle criteria. Some illustrative examples are also provided. |
13. |
Rofe—Beketov formula for symplectic systems, in: "Oscillation of Difference, Differential, and Dynamic Equations", M. Bohner, Z. Došlá, and S. Pinelas (editors), Adv. Difference Equ. 2012 (2012), no. 104, 9 pp. (electronic). MR3085714; Zbl 06634010; SCO; WOS. |
ABSTRACT ------------------------------------------ We establish the Rofe—Beketov formula for symplectic systems on time scales. This result generalizes the well-known d'Alembert formula (or the Reduction of Order Theorem) and the Rofe—Beketov formula published for the second order Sturm—Liouville equations on time scales. Moreover, this result is new even in the discrete time case. |
12. |
(with P. Hasil) Friedrichs extension of operators defined by even order Sturm—Liouville equations on time scales, Appl. Math. Comput. 218 (2012), no. 22, 10829–10842. MR2942368; Zbl 06242014; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we characterize the Friedrichs extension of operators associated with the 2n-th order Sturm—Liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions. A nontrivial example is also provided. |
11. |
A note on the equivalence between even order Sturm—Liouville equations and symplectic systems on time scales, Appl. Math. Lett. 26 (2013), no. 1, 134–139. MR2971414; Zbl 06107010; SCO; WOS. |
ABSTRACT ------------------------------------------ The 2n-th order Sturm—Liouville differential and difference equations can be written as the linear Hamiltonian differential systems and symplectic difference systems, respectively. In this paper, a similar result is given for the 2n-th order Sturm—Liouville equation on time scales with using time reversed symplectic dynamic systems. Moreover, we show that this transformation preserves the value of the corresponding quadratic functionals which enables a further generalization of the theory for continuous and discrete Sturm—Liouville equations. |
10. |
(with R. Šimon Hilscher) New results for time reversed symplectic dynamic systems and quadratic functionals, in: "Proceedings of The 9'th Colloquium on the Qualitative Theory of Differential Equations" (Szeged, 2011), L. Hatvani, T. Krisztin, and R. Vajda (editors), Electron. J. Qual. Theory Differ. Equ. (2012), no. 15, 11 pp. (electronic). MR3338534; Zbl 06439041. |
ABSTRACT ------------------------------------------ In this paper, we examine time scale symplectic (or Hamiltonian) systems and the associated quadratic functionals which contain a forward shift in the time variable. Such systems and functionals have a close connection to Jacobi systems for calculus of variations and optimal control problems on time scales. Our results, among which we consider the Reid roundabout theorem, generalize the corresponding classical theory for time reversed discrete symplectic systems, as well as they complete the recently developed theory of time scale symplectic systems. |
9. |
(with R. Šimon Hilscher) Overview of Weyl—Titchmarsh theory for second order Sturm—Liouville equations on time scales, Int. J. Difference Equ. 6 (2011), no. 1, 39–51. MR2900837. |
ABSTRACT ------------------------------------------ In this paper we present an overview of the basic Weyl—Titchmarsh theory for second order Sturm—Liouville equations on time scales. We construct m(λ)-function, the Weyl solution, and Weyl disk. We justify the terminology “disk” by its geometric properties, show explicitly the coordinates of the center of the disk, and calculate its radius. We show that the dichotomy regarding the square integrable solutions known in the continuous time and discrete theory works in the same way for general time scales. |
8. |
(with P. Hasil) Critical second order operators on time scales, in: "Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Dresden, 2010), W. Feng, Z. Feng, M. Grasselli, X. Lu, S. Siegmund, and J. Voigt (editors), AIMS Proceedings, Discrete Contin. Dyn. Syst. 2011 (2011), suppl., 653–659. MR2987447; Zbl 06409960; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we introduce the concept of critical operators for dynamic operators of second order. Next, we show that an arbitrarily small (in a certain sense) negative perturbation of a non-negative critical operator leads to an operator which is no longer non-negative. |
7. |
(with R. Šimon Hilscher) Weyl—Titchmarsh theory for time scale symplectic systems on half line, in: "Recent Progress in Differential and Difference Equations", Proceedings of the Conference on Differential and Difference Equations and Applications (Rajecké Teplice, 2010), J. Diblík, E. Braverman, Y. Rogovchenko, and M. Růžičková (editors), Abstr. Appl. Anal. 2011 (2011), Art. ID 738520, 41 pp. (electronic). MR2773642; Zbl 05888151; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we develop the Weyl—Titchmarsh theory for time scale symplectic systems. We introduce the M(λ)-function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and the second author in Appl. Math. Comput. It also unifies the results in many other papers on the Weyl—Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second order Sturm—Liouville equations on time scales. |
6. |
Krein—von Neumann and Friedrichs extensions for second order operators on time scales, in "Dynamic Equations on Time Scales and Applications", L. H. Erbe and A. C. Peterson (editors), Int. J. Dyn. Syst. Differ. Equ. 3 (2011), no. 1-2, 132–144. MR2797045; Zbl 05873604; SCO. |
ABSTRACT ------------------------------------------ We consider an operator defined by the second order Sturm—Liouville equation on an unbounded time scale. For such an operator we give characterizations of the domains of its Krein—von Neumann and Friedrichs extensions by using the recessive solution. This generalizes and unifies similar results obtained for operators connected with the second order Sturm—Liouville differential and difference equations. |
5. |
(with S. L. Clark) On a Weyl—Titchmarsh theory for discrete symplectic systems on a half line, Appl. Math. Comput. 217 (2010), no. 7, 2952–2976. MR2733742; Zbl 05828077; SCO; WOS. |
ABSTRACT ------------------------------------------ Recently, Bohner and Sun introduced basic elements of a Weyl—Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl—Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Green′s function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy λ-dependent boundary conditions at infinity. |
4. |
(with R. Šimon Hilscher) Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval, in: "Equadiff 12", Proceedings of the Conference on Differential Equations and their Applications (Brno, 2009), J. Diblík, O. Došlý, P. Drábek, and E. Feistauer (editors), Math. Bohem. 135 (2010), no. 2, 209–222. MR2723088; Zbl 05850417. |
ABSTRACT ------------------------------------------ In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm—Liouville differential equations. |
3. |
(with R. Šimon Hilscher) Trigonometric and hyperbolic systems on time scales, Dynam. Systems Appl. 18 (2009), no. 3-4, 483–506. MR2562285; Zbl 05635428; SCO; WOS. |
ABSTRACT ------------------------------------------ In this paper we discuss trigonometric and hyperbolic systems on time scales. These systems generalize and unify their corresponding continuous-time and discrete-time analogies, namely the systems known in the literature as trigonometric and hyperbolic linear Hamiltonian systems and discrete symplectic systems. We provide time scale matrix definitions of the usual trigonometric and hyperbolic functions and show that many identities known from the basic calculus extend to this general setting, including the time scale differentiation of these functions. |
2. |
(with R. Šimon Hilscher) Definiteness of quadratic functionals for Hamiltonian and symplectic systems: A Survey, Int. J. Difference Equ. 4 (2009), no. 1, 49–67. MR2553888. |
ABSTRACT ------------------------------------------ In this paper we provide a survey of characterizations of the nonnegativity and positivity of quadratic functionals arising in the theory of linear Hamiltonian and symplectic systems. We study these functionals on traditional continuous time domain (under and without controllability), on discrete domain, and on time scale domain which unifies and extends both previous types. For each case we distinguish functionals with zero, separated, and jointly varying endpoints. The presented conditions are formulated in terms of the properties of a special conjoined basis of the considered linear system. It is now easy to compare all the results ‒ between continuous, discrete, and time scale cases, between the zero, separated, and jointly varying endpoints, and between the nonnegativity and positivity. |
1. |
Discrete trigonometric and hyperbolic systems: An overview, in: "Ulmer Seminare über Funktionalanalysis und Differentialgleichungen", Vol. 14, pp. 345–359, University of Ulm, Ulm, 2009. arXiv. |
ABSTRACT ------------------------------------------ In this paper we present an overview of results for discrete trigonometric and hyperbolic systems. These systems are discrete analogues of trigonometric and hyperbolic linear Hamiltonian systems. We show results which can be viewed as discrete n-dimensional extensions of scalar continuous trigonometric and hyperbolic formulas. |
Textbooks
3. |
(with P. Hasil) Sbírka řešených příkladů z matematické analýzy II (in Czech, [Collection of Solved Problems in Mathematical Analysis II]) Masaryk University, Brno, 2016 [vid. 2016‑12‑10]. |
ABSTRAKT ------------------------------------------ Jedná se o druhý díl trilogie sbírek s řešenými příklady ze základních partií matematické analýzy. Tetokrát jsme se zaměřili na obyčejné diferenciální rovnice (153 příkladů), metrické prostory (18 příkladů) a diferenciální počet funkcí více proměnných (185 příkladů). Sbírka je koncipována jako doplněk pro studenty předmětu „M2100: Matematická analýza II“ vyučovaného na PřF MU. ABSTRACT ------------------------------------------ Solutions of 153 examples from the theory of ordinary differential equations, 18 examples from the theory of metric spaces, and 185 examples from the differential calculus of functions of several variables are presented. Although it is intended primarily for course M2100, it can be useful in any course concerning the topics listed above. |
3a. |
(with P. Hasil) Sbírka příkladů z matematické analýzy II (in Czech, [Collection of Problems in Mathematical Analysis II]) Masaryk University, Brno, 2014 [vid. 2014‑02‑12]. |
ABSTRAKT ------------------------------------------ ABSTRACT ------------------------------------------ |
2. |
(with O. Došlý) Integrální počet v R (in Czech, [Integral Calculus in R]), Masaryk University, Brno, 2011. ISBN 978‑80‑210‑5635‑0. |
ABSTRAKT ------------------------------------------ Integrální počet je druhou základní partií úvodního kurzu matematické analýzy. Toto skriptum je určeno posluchačům bakalářského studia odborné i učitelské matematiky, fyziky, matematické ekonomie a informatiky. Celý text je rozdělen do šesti kapitol. V úvodní kapitole jsou probrány základní metody určování primitivních funkcí. Druhá kapitola je věnována konstrukci, vlastnostem a výpočtu určitého (Riemannova) integrálu. Třetí kapitola pojednává o nevlastních integrálech, a to jak o integrálech přes neohraničený obor, tak i o integrálech z neohraničených funkcí. Ve čtvrté kapitole jsou studovány geometrické a některé základní fyzikální aplikace určitého integrálu. Pátá kapitola je zaměřena na některé alternativní konstrukce určitého integrálu (zejména na Newtonův, Lebegueův a Kurzweilův integrál). Text je uzavřen doplňkem o konstrukci míry, která úzce souvisí s teorií určitého integrálu. ABSTRACT ------------------------------------------ The integral calculus is the second fundamental part of the one-variable calculus. The book consists of six chapters. Basic methods of finding the antiderivatives (primitive functions) are given in the first chapter. The second chapter is devoted to the theory of the definite (Riemann) integral. Improper integrals (over an unbounded interval or for functions with a singular point) are studied in the third chapter. Geometric and some fundamental physical applications of the integral calculus can be found in the fourth chapter. In Chapter 5, some alternative constructions of the definite integral are presented (especially, the Newton, Lebesgue, and Kurzweil integrals). The book is concluded with an appendix about the theory of a measure (especially, of the Jordan and Lebesgue measures). |
1. |
(with P. Hasil) Sbírka řešených příkladů z matematické analýzy I (in Czech, [Collection of Solved Problems in Mathematical Analysis I]) [online], 3rd edition, Elportál, Masaryk University, Brno, 2012 [vid. 2012‑06‑06]. ISBN 978‑80‑210‑5882‑8. ERRATA. |
ABSTRAKT ------------------------------------------ V této publikaci je zpracováno řešení 313 příkladů z diferenciálního počtu funkcí jedné proměnné (výpočet limit posloupností a funkcí přímými úpravami i pomocí l'Hospitalova pravidla, výpočet derivace, vyšetřování průběhu funkce, aplikace diferenciálního počtu ve slovních úlohách, užití diferenciálu funkce, Taylorova věta a Taylorův polynom) a také 144 úloh z integrálního počtu funkcí jedné proměnné (základní integrační metody, integrace racionální lomené funkce, určitý integrál, nevlastní integrály, aplikace integrálního počtu v geometrii a ve fyzice). Tyto příklady pokrývají praktickou část předmětů „M1100: Matematická analýza I“ a „M1101: Matematická analýza I“. ABSTRACT ------------------------------------------ Solutions of 313 examples from the differential calculus of one variable and 144 examples from the integral calculus of one variable are presented. Although it is intended primarily for courses M1100 and M1101, it can be useful in any course concerning the differential and integral calculus of functions of one variable. |
Theses and dissertations
4. |
Discrete Symplectic Systems and Square Summable Solutions, habilitation thesis, Masaryk University, Brno, 2016. |
ABSTRACT ------------------------------------------ In this habilitation thesis we present our recent contributions to the ongoing development of the theory of square summable solutions of discrete symplectic systems. It is based on results, which were achieved by the author and his scientific collaborators (S. Clark and R.Šimon Hilscher) during his postdoctoral research in the period 2011‑2016. The major part of them was published in papers [A14,A15,A17,A19,A21] and in a more general setting also in papers [A16,A18,A20]. Systematic research in this area began in 2010, when M. Bohner and S. Sun and independently (and more extensively) S. Clark and the author in [A5] investigated square summable solutions of discrete symplectic systems with a special linear dependence on the spectral parameter, see the beginning of Chapter 2 for more details. In the present work we collect our results for discrete symplectic systems with general linear dependence on the spectral parameter as well as with a polynomial or analytic dependence. However, we emphasize that these results do not only improve the type of the dependence on the spectral parameter, but they significantly generalize and extend the results of Bohner and Sun and [A5]. We also lay foundations of the operator theoryfor discrete symplectic systems, which is intimately connected with the topic of square summable solutions. The thesis consists of seven chapters and an appendix. |
3. |
New Results in Theory of Symplectic Systems on Time Scales, doctoral dissertation, Masaryk University, Brno, 2011. ISBN 978‑80‑210‑5515‑5. |
ABSTRACT ------------------------------------------ In this dissertation we present new results in the theory of symplectic systems on time scales (also symplectic dynamic systems) obtained and published by the author (jointly with collaborators) during his doctoral study between the years 2007 and 2011. The dissertation is organized into five chapters. The study of symplectic systems is motivated in the introductory chapter, where an overview of the new results contained in the text is also given. In the second chapter, the reader will find fundamental parts of the time scale calculus indispensable for the understanding of the subsequent chapters. The main body of the text is represented by the following chapters. In Chapter 3, we define trigonometric and hyperbolic systems on time scales and study their properties. Solutions of these systems generalize the well known trigonometric functions sine, cosine, tangent, cotangent, and their hyperbolic analogies. They also satisfy formulas generalizing some of the known trigonometric and hyperbolic identities from the scalar continuous case (e.g., Pythagorean trigonometric identity, double angle, product-to-sum, and sum-to-product formulas). In the following Chapter 4, the Weyl—Titchmarsh theory for symplectic dynamic systems is established. We generalize results for linear Hamiltonian differential systems obtained particularly during the second half of the 20th century. The theory given in both of these chapters is new even for symplectic difference systems, which are a special case of the symplectic systems on time scales. In the final chapter, we pay our attention to the most special case of the symplectic systems on time scales, namely to the Sturm—Liouville dynamic equations of the second order. For operators associated with these equations we characterize the domains of their Krein—von Neumann and Friedrichs extensions and also introduce the concept of the critical, subcritical, and supercritical operators. Some results obtained in Chapter 5 are also new in this special case, therefore the most important results of the Weyl—Titchmarsh theory for the second order Sturm—Liouville dynamic equations are given in the last part of this chapter. For completeness, this dissertation is finished with a sketch of a further research in the presented theory, author′s current list of publications, and his curriculum vitae. |
2. |
Symplektické diferenční systémy (in Czech, [Symplectic Difference Systems]), MSc. thesis, Masaryk University, Brno, 2007. |
ABSTRACT ------------------------------------------ This master thesis gives a survey of all currently known results concerning symplectic difference systems. Especially, basic properties, oscilatory properties, connection with the calculus of variation, the theory of quadratic functionals, trigonometric and hyperebolic systems, and transformations theory are studied. In addition, a new Lyapunov-type inequality for symplectic systems is derived. |
1. |
Lineární diferenciální rovnice n-tého řádu s konstantními koeficienty a jejich aplikace (in Czech, [Linear Differential Equations of n-th Order with Constant Coefficients and Their Applications]), BSc. thesis, Masaryk University, Brno, 2005. |
ABSTRACT ------------------------------------------ Linear differential equations n-th order with constant coefficients and their applications are the theme of this bachelor thesis. The text is parted into four chapters. The first chapter contains basic definitions and some fundamental theorems of the general theory of ordinary differential equations. The second chapter is focused on the theory of homogeneous differential equations and their general solutions. In the third chapter, the theory of inhomogeneous differential equations is described and the variation of constants is shown. Next, there are given general forms of solutions of inhomogeneous differential equations with special right-hand sides. In the last section of this chapter, Euler differential equation is studied. In the fourth chapter, three physical examples (elastic oscillation, harmonious oscillator with applied outside power, and RLC circuit) are solved. |