· table of contents professor nikolay viktorovich azbelev. 263 professor lina fazulovna...

VOLUME 9, 2002 No. 3~4 Dedicated to

AL IAL s

L. Jl:l. RAKHMATULUNA and N. V. AZBELEV on tile occasion of tlleir seventieth and eiglltietll birthdays

THE RESEARCH INSTITUTE THE COLLEGE Olf:l JUDEA & SAMARIA

ARIEL, ISRAEL

The College of Judea and Samaria The Research Institute Executive Director: Dr. Y. Eshel

© All Rights Reserved 2002 Printed in Israel ISSN 0793-1786

TABLE OF CONTENTS

Professor Nikolay Viktorovich Azbelev. 263

Professor Lina Fazulovna Rakhmatul!ina. 269

L. JEierezansky and E. Braverman. Non-oscillation properties of a linear neutral differential equations. 275

A. Chentsov. To the question about the duality of different versions of the programmed iterations method,! 289

K. Chndinov. Partial stability criterion for autonomous systems of differential equations. 315

V. Derr. A generalization of Riemann-Stieltjes integral. 325

M. Dodkin. Asymptotic behavior for one class of forced delay differential equations. 343

A. Domoshnitsky. Wronskian of fundamental system of delay differential equations. 353

S. Gnsarenko. On solvability of a minimization problem for a quadratic functional with linear restrictions in Hilbert space. 377

D. Khusalnov, D. JEienditkis, and J.Diblik. Weak delay in systems with an aftereffect. 385

I. Kignradze and lEI. Puza. Conti-Opial type existence and uniqueness theorems for nonlinear singular boundary value problems. 405

D. Medvedev and V. Vlasov. On certain properties of exponential solutions of difference differential equations in Sobolev spaces. 423

A. Ponosov, A. Shindiapin, and J. Miguel. TheW-transform links delay and ordinary differential equations. 437

I. Rachunkova and M. Tvrdy. Impulsive periodic boundary value problem and topological degree. 4 71

A. Rumyantsev. The reliable computing experiment in the study of boundary value problems for functional differential equations. 499

I. Shragin. Baire-Caratheodory classification of the real-valued standard functions. 521

S. Stanek. Global properties of solutions of the functional differential equation x(t)x'(t) = kx(x(t)), 0 < lkl < 1. 527

V. Tsalyuk. Multivalued Stieltjes integral for discontinuous functions of bounded variation.

A. Rodkina and V. Nosov. On stability of discrete Kiefer-Wolfowitz procedures in nonlipshitz case.

551

577

263

Professor Nikolai Viktorovich Azbelev

(on the occasion of his 80th birthday).

Professor Nikolai Viktorovich Azbelev was born on April 15, 1922 in selo (small village) Bazlovo, Pskov Region, Russia, in a physician's family. His mother, Antonina Fyodorovna Khlebnikova, was scholar and collabora-tor of a famous botanist V.L.Komarov, later the President of the Academy of Sciences of the USSR. His father, Viktor Nikolaevich, graduated in 1905 the Military Medical Academy in St.Petersburg, further attended lectures in microbiology at the Robert Koch's Institute, Berlin, Germany, and was a physician in a field hospital during the World War I. Later he was a direc-tor of the Polar Institute of Bacteriology in Arkhangelsk, Russia. In 1941 N.V.Azbelev enrolled the Moscow State University (MSU). His studies at the Faculty of Mechanics and Mathematics of MSU were interrupted because of his military service in the Soviet Army during the World War II. In 1945 he entered the Moscow Aviation Institute, from which he graduated in 1949 with the degree in engineering, and started to work at the Design Bureau headed by Prof.A.A.Mikulin, a member of the Academy of Sciences. At this place N.V.Azbelev gained an experience in several areas of applied mathe-matics, and solved several important problems. For example he proposed an original computational method in the study of strength properties of a kind of ball bearing. In 1947 he was among the first to make use of the so-called method of electrical analogy as applied to turbine dynamics. He also designed an analog computer to find the frequencies of the shift vibra-tion in turbo-jet engines. In 1951-54 he was a post-graduate student at the Department of Higher Mathematics of the Moscow Machine and Instruments Institute under supervision of Prof. B. I. Segal. In 1954 Azbelev defended his Candidate of Sciences (the Soviet equivalent of Ph.D. degree) thesis "On the boundaries of feasibility of Chaplygin's theorem on differential inequali-ties" at Moscow State University (In his report, V.V.Nemytskyi, the official reader of the thesis, emphasized a very high mathematical level of the work). In the same year N. Azbelev left Moscow for Izhevsk, a town situated in the vicinity of the Ural Mountains, to head the Department of Higher Math-ematics at the Izhevsk Mechanical Institute (IMI). One of the first things N. Azbelev did upon arrival at IMI was to found the Izhevsk Mathemati-cal Seminar. It soon became the central meeting point for mathematicians and engineers in the Izhevsk area. Azbelev's warmth and sensitiveness were tremendously important for the creation of the mathematical community around IMI. The works of the participants of this seminar concerning the theory of integral, differential and difference inequalities allowed to solve a

264

number of problems on existence, uniqueness and asymptotic behavior of solutions to differential equations. Most of those results are based on the Azbelev's "fork principle" which is very useful in finding invariant sets of operators. Other works of the Izhevsk Seminar are devoted to search for effective conditions and criteria of the unique solvability for boundary value problems of ordinary differential equations and investigation of properties of the Green function to those problems. Since 1961 the major attention of N .Azbelev and his seminar was focused on the problems of the general theory of equations with discontinuous operators and later on differential equations with deviating argument. In Izhevsk N.Azbelev wrote his Doc-toral thesis "On the Chaplygin problem", which was defended in 1962 at the Kazan State University. In 1964 he was granted the title of professor. In 1966 Professor Azbelev was elected to be the Head of the Department of Mathematics of the Tambov Institute for Chemical Engineering. A large group of postgraduate students and colleagues of Prof. Azbelev followed him to Tambov. Soon after this the Tambov Seminar started its work. It dealt with equations with deviating argument. The activity of the Tambov Semi-nar implied the creation of an effective theory of differential equations with deviating argument. This theory became a basis of the contemporary Theory of Functional Differential Equations. In 1975 N.Azbelev accepted invitation of the Rector of the Perm Polytechnic Institute (PPI) Prof. M.N .Dedyukin, and moved to Perm. N.Azbelev founded at PPI the Department of Math-ematical Analysis. Azbelev's scientific expertise and leadership contributed immensely to the development of this department. It became one of the most famous mathematical centers and the core of the Perm Seminar on Functional Differential Equations. Since 1994 N .Azbelev is the head of the Scientific Research Center for Functional Differential Equations at the Perm State Technical University (former PPI). Members of the Perm Seminar were awarded 9 Doctor's degrees and more than 50 Candidate's degrees. Annual issues of the proceedings "Boundary Value Problems" and "Functional Dif-

. Ierential Equations" were published. Prof. Azbelev in collaboration with his colleagues V.P.Maksimov, L.F.Rakhmatullina, and P.M.Simonov wrote seven books, three of them in English. Now N.Azbelev works on a new book. The contemporary theory of FDEs was treated thoroughly in the monographs "Introduction to the Theory of Functional Differential Equations", "The-ory of Linear Abstract Functional Differential Equations and Applications", and "Methods of the Contemporary Theory of Linear Functional Differential Equations". New concepts and approaches worked out by N.Azbelev jointly with L.Rakhmatullina became the basis of this theory. Now this theory covers

265

many classes of equations containing the ordinary derivatives of the solution function. Let us point out some contributions of N.Azbelev to the theory of FDEs: results on the reducibility of equation (the property of the equation to allow a kind of regularization); results on boundary value problems con-cerning the solvability, representation of solutions, estimates and properties of solutions; results on the stability of solutions, in particular, the famous W-method and its extensive development. Of special importance are the con-tributions of N.Azbelev to creation and development of the theory of abstract functional differential equations (AFDEs). Such class of equations in Banach spaces is an essential further generalization of the equations with ordinary derivatives. Some n-th order FDEs, systems with impulses, equations with quasi-derivatives, and some classes of singular equations may be considered in the context of the Theory of AFDE. It is worth noticing also that this theory has become a very useful tool for solving some variational problems, especially in the cases where the problem of minimization of a functional is unsolvable within the framework of the classical calculus of variations, as well as for the study of boundary value problems with arbitrary finite number of boundary conditions in the form of equalities and inequalities N.Azbelev's influence is not limited to the original and fundamental contributions he has made to the theory of integral and functional differential equations. An invariable feature of N .Azbelev's activity is his ability to unite around him-self colleagues and students. He significantly contributed to the education " of young mathematicians, supervised over 60 Candidates and 10 Doctors of Sciences. He was a founder of mathematical schools for gifted children in the 60's at Izhevsk, Russia. For more than 25 years N.Azbelev was a member of editorial board of "Differentsial'nye Uravneniya" (Belarus). He is a member of editorial boards of "Nonlinear Dynamic and Systems Theory", "Memoirs on Differential Equations and Mathematical Physics", "Functional Differen-tial Equations", "Izvestiya RANS. Differentsial'nye Uravneniya" and some others. Professor Azbelev is a great master of exposition of scientific results at seminars and conferences. His works are characterized by nonstandard thinking, geometric intuition, aspiration to find simple solution of problems, and dislike to any sort of pseudo-mathematical speculations. Despite be-ing 80 years old, Professor Azbelev is full of new ideas and creative plans, energetic and industrious. Congratulating Nikolai Viktorovich on the occa-sion of his 80th birthday, we wish him good health and long years of further successful activity.

On behalf of the Editorial Board of"FDE" M.Drakhlin and V.Maksimov.

266

Selected Bibliography of N.V.Azbelev

(the list of N.Azbelev's publications dated by 1951-96 is provided in the articles dedicated to his 60th and 75th birthdays, see "Differential'nye Urav-neniya", 18, No. 4, 1982 and 33, No.4, 1997)

1. On an estimate of the spectral radius of the linear operator in the space of continuous functions (with L.F.Rakhmatullina). (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1996, No.11,23-28; translation in Russian Math. (Iz. VUZ) 40(1996), No. 11, 21-26.

2. The contemporaneous state of the theory of functional differential equations. Mathematica. Statistica. Informatica I Universidade Eduardo Mondlane. Maputo, (1996),No. 4, 1-11.

3. Stability and asymptotic behavior of solutions of equations with de-lay. Mathematica. Statistica. Informatica I Universidade Eduardo Mondlane. Maputo, (1996), No. 4, 15-31.

4. Some examples of regularization of singular equations (with A.I. Shindiapin). Mathematica. Statistica. Informatica I Universidade

· Eduardo Mondlane. Maputo, (1996), No. 4, 32-37. 5. On the question of formalization of mathematical models. (Russian)

Vestnik Perm State Techn. Univ., Functional Differential Equations, (1997), No. 4, 7-14.

6. Conditions of monotonicity for the Green operators of singular bound-ary value problems (with M.J.Alves and E.I.Bravyi). (Russian) Vest-nik Perm State Techn. Univ., Functional Differential Equations, (1997), No. 4, 15-22.

7. Singular problems in the theory of functional differential equations: problems and repspectives (with E.I.Bravyi, S.A.Gusarenko and P.M.Simonov). (Russian) Vestnik Perm State Techn. Univ., Functional Differential Equations, (1997), No. 4, 22-35.

8. Stability of equations with delayed argument (with P.M.Simonov). (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1997, No. 6, 3-16; translation in Russian Math. (Iz. VUZ}41 (1997), No. 6, 1-14.

9. Stability of solutions of the equations with aftereffect (with L.F.Rakhmatullina). Functional Different. Equat., 5 (1998), No. 1-2, 39-55.

267

10. On singular boundary value problems for the linear functional differ-ential equation of the second order (with M.J.Alves and E.I.Bravyi). (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1999, No. 2, 3-11; translation in Russian Math. {Iz. VUZ} 43(1999), No. 2, 1-9.

11. Impact of certain traditions on development of the theory of differ-ential equations. Comput. and Appl. Math., 37(1999), No. 4/5, 1-8.

12. Stability and asymptotic behavior of solutions of equations with af-tereffect. Volterra equations and applications {Arlington, TX, Jgg6}, 27-38, Stability and Control: Theory Methods and Appl., 10, Gor-don and Breach, London, 2000.

13. Stability of equations with delayed argument (with P.M.Simonov). II. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., :woo, No. 4, 3-13; translation in Russian Math. {Iz. VUZ} 44(2000), No. 4, 1-11.

14. Methods of the contemporary theory of linear functional differential equations (with V.P. Maksimov and L.F.Rakhmatullina). (Russian) Regular and Chaotic Dynamics, Moscow-Izhevsk, 2000.

15. To the 25th anniversary of the Perm Seminar on functional differ-ential equations. (Russian) Differentsial'nye Uravneniya, 37(2001), No. 8, 1136-1139; translation in Different. Equat. 37(2001), No. 8, 1194-1198.

16. On the question of effective sufficient conditions for solvability of variational problems (with E.I.Bravyi and S.A.Gusarenko). (Russian) Dokl. Russian Acad. Nauk, 381(2001), No. 2, 1-4.

17. Stability of solutions of equations with ordinary derivatives (with P .. M.Simonov). (Russian) Perm State Univ. Publishing, Perm, 2001.

18. Elements of the contemporary theory of functional differential equa-tions. Methods and applications (with V.P. Maksimov and L.F. Rakhmatullina). (Russian) Institute of Computer-Assisted Investi-gations, Moscow, 2002.

19. How it happened (a historical essay of the origin and the develop-ment of the theory of FDEs). (Russian) Vestnik Perm State Techn. Univ., Functional Differential Equations, (2002), 13-40.

268

20. On the question of minimization of a qudratic functional perturba-tion (with E.I.Bravyi and S.A.Gusarenko). (Russian) Vestnik Perm State Techn. Univ., FUnctional Differential Equations, (2002), 47-51.

21. Functional differential equations and the theory of stability of equa-tions with aftereffect (with P.M. Simonov). (Russian) Vestnik Perm State Techn. Univ., Functional Differential Equations, (2002), 52-69.

22. Stability of differential equations with aftereffect (with P.M. Si-monov). Taylor and Francis, London and New-York, 2002.

Professor Lina Fazylovna Ra.khmatullina (on the occasion of her 70th birthday).

269

Professor Lina Fazylovna Ra.khmatullina was born on January 3, 1932 in village Baltasi, Tatarstan, Russia, where her parents were on the staff of the District Administration. In 1949 Lina Fazylovna became a stu-dent of the Physics and Mathematics Faculty of the Kazan State University (KSU). She graduated in 1954. Her graduation essay was published in a KSU scientific journal. In 1955-60 L.Rakhmatullina worked as an assistant professor at the U dmurt State Pedagogical Institute (USPI), Izhevsk, Rus-sia. In 1963 she defended her Candidate's Thesis on problems in the Stabil-ity Theory at the KSU. She continued her work at the USPI, and later at the Izhevsk Mechanical Institute as a Lecturer of the Department of Higher Mathematics. There she took a very active part in organizing a new pro-gram awarding simultaneously engineering and mathematical degrees. In 1966 L.Rakhmatullina, with her husband, N.Azbelev, moved to Tambov. She worked as a Lecturer at the Tambov Institute for Chemical Engineering. In Tambov, linear equations with deviating argument became the primary interest of L.Rakhmatullina. She introduced the modern definition of the no-tion of the solution to such equations. Now this definition underlies the new concept of differential equations with deviating argument. There she was an organizer of the Tambov Seminar on Functional Differential Equations. The aforementioned new concept allowed to work out the general theory of linear functional differential equations. Nowdays this theory covers many classes of equations containing the derivative of the solution function. The basis of the new theory, which was actively developed in succeeding years, were the results obtained by Lina Fazylovna and her post-graduate students M.P.Berdnikova and V.P.Maksimov. Her talent of a scientist and teacher was recognized among her colleagues: in all discussions on difficult questions of Mathematical Analysis, L.Rakhmatullina eventually became the main au-thority as well as a patient expert being ready to help in preparing reports, lectures and manuscripts. In 1975, following invitation of the rector of the Perm Polytechnic Institute (PPI) Prof. M. N.Dedyukin, L.F.Rakhmatullina and N.V.Azbelev moved with a large group of members of the Tambov Semi-nar to Perm and founded a new Department of Mathematical Analysis. Here Lina Fazylovna worked as a Lecturer, later a Professor, till 1994, when the Department was reorganized and there was founded the Scientific Research Center on Functional Differential Equations, where Lina Fazylovna works now. L.F.Rakhmatullina defended her Doctoral Thesis in 1982 at the In-stitute of Mathematics of Academy of Sciences of the Ukraine. The thesis

270

was concerned with linear functional differential equations, has aroused con-siderable interest. Lina Fazylovna continued her research and pedagogical activity. Under her supervision the Doctoral theses by V.P.Maksimov and L.M.Berezanskyi as well as 12 Candidate's Theses were prepared. In 1984 she was granted the title of professor. In 1991 the book "Introduction to the Theory of Functional Differential Equations" by L.F.Rakhmatullina coau-thored by N.V.Azbelev and V.P.Maksimov was published. In this book the results of the Tambov and Perm Seminars were presented in a systematic form. Of special importance are the contributions of Professor Rakhmatul-lina to the creation and the development of the Theory of Abstract Func-tional Differential Equations. The main ideas of this theory are surveyed in the book "Theory of Linear Abstract Functional Differential Equations and Applications" written with N.Azbelev. Lina Fazylovna is a coauthor of the books "Methods of the Contemporary Theory of Linear Functional Differen-tial Equations" and "Elements of the Contemporary Theory of Functional Differential Equations. Methods and Applications." Now Prof. Rakhmatul-lina works on a new book. The works of L.F.Rakhmatullina are characterized by original setting of the problems and a high rigor of proofs. The features of her pedagogical activity are the particularity and benevolence with respect to colleagues and students. Lina Fazylovna is a devoted traveller. Every year, Lina Fazylovna together with her husband travels by motorcycle or by car across Russia or FSU republics. Lina Fazylovna is a ballet and opera lover. Congratulating Lina Fazylovna on the occasion of her 70th birthday, we wish her good health, long years of further successful activity, new gifted students.

On behalf of the Editorial Boarder ofFDE M.Drakhlin and V.Maksimov.

Selected Bibliography of L.F .Rakhmatullina

Monographs

1. Introduction to the theory of functional differential equations (with N.V.Azbelev and V.P.Maksimov). (Russian) Nauka, Moscow, 1991.

2. Introduction to the theory of linear functional differential equations (with N.V.Azbelev and V.P.Maksimov). World Federation Publish-ers Company, Inc., Atlanta, 1995.

3. Theory of linear abstract functional differential equations and ap-plications (with N.V.Azbelev). Mem. Differential Equations Math. Phys. 8(1996), 1-102.

271

4. Methods of the contemporary theory of linear functional differential equations (with N.V.Azbelev and V.P.Maksimov). (Russian) Regu-lar and Chaotic Dynamics, Moscow-Izhevsk, 2000.

5. Elements of the contemporary theory of functional differential equa-tions. Methods and applications (with N.V.Azbelev and V.P.Maksimov). (Russian) Institute for Computer-Assisted Investi-gations, Moscow, 2002.

Journal Articles

1. On a class of nonlinear singular integral equations (Russian) U ch. Zapiski of the Kazan University, 1955, v. 115, No. 7, 25-29.

2. On an extension of the solution of the Chaplygin theorem outside the feasibility boundary of the differential inequality theorem (with N.V.Azbelev and Z.B.Tsalyuk. (Russian) Nauchn. Dokl. Vyssh. Skholy. Phys. and Math., 1958, No 2, 3-5.

3. On an application of solvability conditions of the Chaplygin theorem to the questions of boundedness and stability of solutions to differ-ential equations (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1959, No. 2, 198-201.

4. On the problem of stability of the solution of the nonlinear heat-conduction equation. (Russian) Prikl. Math.Mekh., v. 25, issue 3, 1961, 591-592; translation in PMM, J. Appl. Math. Mech. 25(1961), 880-883.

5. Theorems of solutions existence for equations with deviating argu-ment (with N.V.Azbelev and M.P.Berdnikova). (Russian) Trudy Tambov Inst. Chem. Engineering, 1969, No. 3, 65-68.

6. Existence, uniqueness and convergence of successive approximations for nonlinear integral equations with deviating argument (with N.V.Azbelev and A.I.Chigirev). (Russian) Differentsial'nye Urav-neniya, 1970, v. 6, No. 2, 223-229; translation in Different. Equat. 6(1972), 173-178.

7. On linear equations with deviating argument (with N.V.Azbelev). (Russian) Differentsial'nye Uravneniya, 1970, v. 6, No. 4, 616-628; translation in Different. Equat.6(1973), 473-482.

8. W -method in the theory of differential equations with deviating ar-gument (with N.V.Azbelev and A.G.Terent'ev). (Russian) Trudy Tambov Inst. Chem. Engineering, 1970, No. 4, 60-63.

9. Integral equations with deviating argument (with N.V.Azbelev and M.P.Berdnikova). (Russian) Dokl. Akad. Nauk SSSR, 1970, v. 192,

272

No. 3,479-482; translation in Sov. Math., Dokl.11(1970), 643-647. 10. To the theory of linear equations with functional argument. (Rus-

sian) Differentsial'nye Uravneniya, 1972, v. 8, No. 3, 523-528. 11. The Cauchy problem for differential equations with retarded ar-

gument (with N.Azbelev).(Russian) Differentsial'nye Uravneniya, 1972, v. 8, No. 9, 1542-1552; translation in Different. Equat. 8 (1972), 1190-1198.

12. On the question of regularization to linear boundary value problems (with A.G.Terent'ev). (Russian) Differentsial'nye Uravneniya, 1973, v. 9, No. 5, 868-873; translation in Different. Equat.9(1975), 660-664.

13. On a representation of solutions to linear functional differential equa-tions (with V.P.Maksimov).(Russian) Differentsial'nye Uravneniya, 1973, v. 9, No. 6, 1026-1036; translation in Different.Equat.9(1975), 781-790.

14. Linear functional differential equation solved with respect the deriva-tive (with V.P.Maksimov). (Russian) Differentsial'nye Uravneniya, 1973, v. 9, No. 12, 2231-2240.

15. Conditions of continuous dependence on parameters for the solu-tion of the linearfunctional differential equation (with N.V.Azbelev). (Russian) Trudy Moscow Inst. Chem. Engineering, 1974, No. 53,

. 70-72. 16. On canonical forms of linear functional differential equations. (Rus-

sian) Differentsial'nye Uravneniya, 1975, v. 11, No. 12, 2143-2153; translation in Different. Equat.11(1976), 1589-1597.

17. Linear functional differential equations (with N.V.Azbelev). (Rus-sian) In: Differential equations with deviating argument, N aukova Dumka, Kiev, 1977, 11-19.

18. On the linear functional differential equation of the evolutionary type (with N.V.Azbelev and L.M.Berezanskii). (Russian) Differ-entsial'nye Uravneniya, 1977, v. 13, No. 11, 1915-1925; translation in Different. Equat.l3(1977), 1331-1339.

19. The adjoint equation for the general boundary value problem (with V.P.Maksimov). (Russian) Differentsial'nye Uravneniya, 1977, v. 13, No. 11, 1966-1973; translation in Different. Equat.l3(1977), 1368-1373.

20. Functional differential equations (with N.V.Azbelev). (Russian) Dif-ferentsial'nye Uravneniya, 1978, v. 14, No. 5, 771-797; translation in Different. Equat.l4(1978), 547-565.

:---~ .. .. ,

273

21. The Green operator and regularization of linear boundary value problems. (Russian) Differentsial'nye Uravneniya, 1979, v. 15, No. 3, 425-435; translation in Different. Equat.15(1979), 297-304.

22. Functional differential equations (with N.V.Azbelev and V.P.Maksimov). (Russian) Uspekhi Mat. Nauk, 1981, v. 36, No. 4, 205.

23. On the Green function of the general linear boundary value problem (with N.V.Azbelev and V.P.Maksimov). (Russian) Uspekhi Mat.Nauk, 1982, v.37, No. 4, 119-120.

24. On reducibility offunctional differential equations (with N.V.Azbelev and V.P.Maksimov). (Russian) Uspekhi Mat. Nauk, 1983, v.38, No. 5, 128-129.

25. On definition of the notion of the solution to the equation with devi-ating argument. (Russian) Functional Differential Equations (Rus-sian), Perm Polytech. Inst., Perm, 1985, 13-19.

26. On the question of functional differential inequalities and monotone operators (with N.V.Azbelev). (Russian) Functional Differential Equations {Russian), Perm Polytech. Inst., Perm, 1986, 3-9.

27. Abstract functional differential equation (with N.V.Azbelev). (Rus-sian) Functional Differential Equations {Russian), Perm Polytech. Inst., Perm, 1987, 3-11.

28. On regularization of linea boundary value problems. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1987, No. 7, 37-43; translation in Sov. Math.:H(1987), No. 7, 46-55.

29. Representation of solutions to linear functional differential equations which are not everywhere solvable. (Russian) Functional Differential Equations (Russian), Perm Polytech. Inst., Perm, 1988, 10-13.

30. Theory of the abstract linear functional differential equation. (Rus-sian) In: Questions of the qualitative theory of differential equations, Nauka, Novosibirsk, 1988, 114-117.

31. The generalized Green operator of the overdeterminated boundary value problem for the linear functional differential equation. (Rus-sian) Izv. Vyssh. Uchebn. Zaved. Mat., 1993, No. 5, 95-100; translation in Russian Math. (Iz. VUZ}37(1993), No. 5, 85-89.

32. On extension of the Vallee-Poussin theorem to equations with after-effect (with N.V.Azbelev) In: Boundary Value Problems for Func-tional Differential Equations, World Scientific Publishing Co. Pte., Ltd., Singapore, 1995, 23-36.

274

33. The Vallee-Poussin theorem on differential inequality for equations with aftereffect (with N.V.Azbelev). (Russian) Trudy Mat. Inst. Steklov, 1995, v. 211, 32-39; translation in Proc. Steklov Inst. Math.211(1995), 28-34.

34. Continuous dependence on parameters of the solution to the linear boundary value problem (with A.V.Anokhin).I (Russian) Izv. Vyssh. Uchebn. Zaved. Mat.,1996, No. 11, 29-38; translation in Russian Math. (Iz. VUZ)40(1996), No. 11, 27-36.

35. On an estimate of the spectral radius of the linear operator in the space of continuous functions (with N.V.Azbelev). (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 1996, No.11, 23-28; translation in Russian Math. (lz. VUZ) 40{1996), No. 11, 21-26.

36. The upper estimate of the spectral radius of the isotonic operator in the space of continuous functions. Mem. Different. Equat. Math. Phys., 1997, v. 11, 163-167.

37. Stability of solutions of the equations with aftereffect (with N.V.Azbelev). Functional Differential Equations, 1998, v. 5, No. 1-2, 39-55.

38. The upper estimate of the spectral radius of the isotonic operator. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 2000, No. 1, 56-65; translation in Russian Math. {lz. VUZ}44(2000), No. 1, 54-63.

39. Continuous dependence on parameters of the solution to the linear boundary value problem. (Russian) lzv. Vyssh. Uchebn. Zaved. Mat., 2000, No. 4, 41-49; translation in Russian Math. {lz. VUZ}44(2000), No. 4, 39-47.

40. On the problem of convergence of solutions of linear boundary value problem. Functional Differential Equations, 2000, v. 7, No. 3-4, 325-334.

FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 9 2002, NO 3-4 PP. 275-288

NON-OSCILLATION PROPERTIES OF A LINEAR NEUTRAL DIFFERENTIAL EQUATIONS *

L. BEREZANSKY t AND E. BRAVERMAN l

Abstract. For a neutral differential equation

±{t)- a(t)±(g(t)) + b(t)x(h(t)) = 0,

0 :S a(t) < 1, g(t) :S t, h(t) :S t,

a connection between oscillation properties of the differential equation and differential inequalities is established.

Explicit non-oscillation conditions, comparison theorems and criterion for existence of a positive solution are presented.

Key Words. Non-oscillation, neutral equation, comparison theorems

AMS(MOS) subject classification. 34Kll, 34K40

1. Introduction. This paper deals with nonoscillation properties of a scalar neutral differential equation.

Linear neutral type equation can be written in any one of the following two forms:

m

(1) (x(t)- a(t)x(g(t)))' + L bk(t)x(hk(t)) = 0 k=l

' Partially supported by Israel Ministry of Absorption t Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105,

Israel I Department of Mathematics and Statistics, University of Calgary, 2500 University

Drive N.W., Calgary, Alberta T2N 1N4, Canada 07'>

276 L. BEREZANSKY AND E. BRAVERMAN

or m

(2) x(t)- a(t)x(g(t)) + L: bk(t)x(hk(t)) = o, k=l

where g(t) ~ t, hk(t) ~ t. Equations (1) and (2) are similar; however there is difference between

them. For example, unlike (2), solution x of (1) is an arbitrary continuous function, such that x(t) - a(t)x(g(t)) is differentiable. Thus (1) in general cannot be rewritten in form (2), and vice versa.

Concerning the connection of (1) with (2), we mention here the paper [9] where the oscillation of (1) was studied by applying an adjoint equation which has form (2). For the autonomous case in the "neutral part" when a(t) = a, g(t) = t- a (1) and (2) are the same equation. In this case the results of this paper coincide with the known ones.

It is to be emphasized that equation ( 1) is much better studied than equation (2). Extensive literature on (1) is concerned with existence and uniqueness theorems and especially stability and oscillation theories (see monographs [10], [11], [13], [14] and references therein).

Equation (2) is a natural representative of neutral type equations. There exist applied problems which can be written in form (2) [15]. Monograph [2] contains solvability and uniqueness results, the solution representation for (2) and elements of stability theory. Recent monograph [16] is devoted to the stability of equation (2). We also mention here papers [1] where a new method based on Bohl-Perron theorem was applied to stability investigation of (2).

Though there exists a developed stability theory for (2) surprisingly there are only few publications on its oscillation. We mention here paper [12] where comparison results for (2) were obtained and two papers [5, 6] where the positiveness for the fundamental function of equation (2) is studied. The purpose of [3] and the present paper is to fill up this gap and to investigate the oscillation of (2).

In [3] we considered Eq.(2) with nonnegative coefficient b(t) ;::: 0 and obtain explicit nonoscillation and oscillation conditions. In this paper we generalize some results of [3] to the case b(t) is not necessarily nonnegative. Besides we consider here a new problem for a neutral equation which was previously investigated for delay equations only [13]. This problem is to find conditions on initial functions providing that the solution of the initial value problem is positive.

We also obtain some new comparison results and criteria for oscillation by using so-called "slowly oscillatory solutions" (see [7], [8] for this notion

NON-OSCILLATION OF NEUTRAL EQUATIONS 277

and its applications in oscillation theory). For simplicity we consider an equation with a single delay.

The paper is organized as follows. Section 2 contains relevant defini-tions and notations and auxiliary lemmas. Section 3 includes the main result of the paper which is the equivalence of the nonoscillation of (2), the exis-tence of a positive solution for the corresponding differential inequality and the existence of a nonnegative solution of some explicitly constructed by (2) nonlinear integral inequality. Section 4 contains a comparison theorem and nonoscillation results for equation (2). These results are obtained by applying nonoscillation criteria and comparison with a differential equation contain-ing an infinite number of delays. In section 5 we compare two nonoscillatory solutions of Eq.(2) and obtain a sufficient condition on equation parameters and initial functions which provide the solution of the initial value problem being positive. Such conditions are well-known for delay differential equa-tions. For neutral equations we do not know such results. Section 6 contains a new sufficient oscillation condition which employs the existence of a slowly oscillatory solution. Similar result for equation (1) was obtainedin [9].

2. Preliminaries. We consider a scalar delay differential equation

(3) ±(t)- a(t)x(g(t)) + b(t)x(h(t)) = 0, t ~ t0 ,

under the following conditions (a1) a(t), b(t), g(t), h(t) are Lebesgue measurable locally essentially bounded functions; (a2) a(t) ~ O,limHoosupa(t) < 1; (a3) g(t) :::; t, mes E = 0 => mes g-1(E) = 0, limHoo g(t) = oo, where mes E is the Lebesgue measure of the set E; (a4) h(t) :::; t, limt-+oo h(t) = oo, for every t > t0 there exists a nature number n(t) such that after n number of superpositions we have h(h( ... (h(t) ... ) < to.

Together with (3) we consider for each t1 ~ t0 an initial value problem

( 4) x(t)- a(t)x(g(t)) + b(t)x(h(t)) = f(t), t ~ tt.

(5) x(t) = rp(t), x(t) = 1/;(t), t < t1, x(ti) = xo.

We also assume that the following hypothesis holds: ( a5) f : [it, oo) ---+ R is a Lebesgue measurable locally essentially bounded function, rp, 1/J : ( -oo, t1) ---+ R are Borel measurable bounded functions.


DEFINITION 1. An absolutely continuous on each interval (t1, c] function x :. R-+ R is called a solution of problem {4), {5), if it satisfies equation (4) for almost all t E [t1, oo) and equalities {5) fort::; t1.

Denote by L00 [t0 , c] the space of all Lebesgue measurable essentially bounded in the interval (t0 , c] functions with the usual sup-norm. Define in this space a linear operator

(6) (Sy)(t) = { a(t)y(g(t)), g(t) ~to, 0, g(t) < to.

LEMMA 1. [ 2] Suppose a, g are Lebesgue measurable locally essentially bounded functions, limt-too sup la(t)l < 1 and condition {a3) holds. Then for every c > t0 operator S acts in the space L 00 [t0 , c], its operator norm IISII < 1, and for the inverse operator we have a representation

(I- s)-1 =I+ S + 82 + ... ,

where I is the identical operator. Operator (I- S)-1 is positive if a(t) ~ 0. Consider now a differential equation with the infinite number of delays

00

(7) x(t) + :E bk(t)x(hk(t)) = 0, t ~to, k=O

where

(8) b0 (t) = b(t), bk+l(t) = (Sbk)(t), ho(t) = h(t), hk+l(t) = hk(g(t)).

By induction it is easy to see that

sup lbk(t)l ::; sup ia(tW sup lb(t)l. tE[to,c] tE[to,c] tE[to,c]

Then 00

B(t) = L lbk(t)i k=O

is an essentially locally bounded function, where the series converges uni-formly on any bounded interval (t0 , c]. Besides, (a4) implies for every t >to there exists a nature number n(t) such that hk(t)


is called a fundamental function of equation {3). We assume X(t, s) = 0, 0::; t < s. Note that the fundamental functions of Eq.(3) and Eq.(7) coincide. In-

deed, a condition x(t) = 0, x(t) = 0, t < s implies Eq.(9) can be rewritten in the form

(I- B)x(t) + b(t)x(h(t)) = o, t;:::: s,

which is equivalent to Eq. (7). THEOREM 1. Let {a1)-{a5) hold. Then there exists one and only one

solution of problem {4), {5) that can be presented in the form

(10) x(t) = X(t, ti)x0 + {t X(t, s)[(I- Bt1 f](s)ds lt,

+ rt X(t, s)[(I- B)-1 F](s)ds, lt, where F(t) = a(t)'lj;(g(t))- b(t)rp(h(t)) and '1/J(g(t)) = O,g(t) 2:: t1, rp(h(t)) = 0, h(t) 2:: tr.

Proof Let us rewrite problem (4)-(5) in the form

(I- B)x(t) + b(t)x(h(t)) = f(t) + F(t), t 2:: tr,

x(t) = x(t) = 0, t < tr, x(to) = Xo,

which is equivalent to

00

(11) i:(t) + L bk(t)x(hk(t)) =[(I- Bt1 f](t) +[(I- Bt1 F](t), t 2:: t1, k=O

(12) x(t) = i:(t) = 0, t < 0, x(ti) = xo.

For solution of (11)-(12) we have [4] representation (10). Hence (10) is also representation of problem (4)-(5). 0

In the following we will need some estimations of the fundamental func-tion.

LEMMA 2. For the fundamental function of Eq.{3) the following inequal-ity holds

(13) jX(t, s)j ::0 exp {[[(I- Bt1 lbl](r)dr}, s ::0 t.


Proof X(t, s) is also the fundamental function of Eq.(7). Hence it is the solution of the following problem

00

x(t) + L: bk(t)x(hk(t)) = o, x(t) = x(t) = 0, t < s, x(s) = 1. k=O

Then for x(t) = X(t, s) we have

rt oo :51+}, t; lbk(r)l t~~)x(~)ldr.

Denote y(t) = maxto::;~91x(~)l. We have

y(t):::; 1 + t[(I- s)-1lbl](r)y(r)dr. Hence by the Gronwall-Bellman inequality

y(t):::; exp{t[(J- s)-1 lbiJ(r)dr}, s:::; t.

The last inequality implies (13). 0 COROLLARY 1. Suppose c > t0 is an arbitrary number. lft0 :5 s :5 t:::; c

then

(14) IX(t, s)l :5 exp {l)(I- s)-1lbl](r)dr},

Proof Inequality (14) is evident. To prove (15) we have for x(t) = X(t, s)

00

l:i:(t)l :5 L: lbk(t)iix(hk(t))i k=O


0

3. Non-oscillation criteria. DEFINITION 3. We will say that equa-tion {3} has a nonoscillatory solution if there exists a solution of (4)-{5), which is eventually positive or eventually negative. Otherwise all solutions of ( 3) are oscillatory.

The following theorem establishes a non-oscillation criteria. THEOREM 2. Suppose {a1)-(a4) hold, b(t) 2:: 0. Then the following

hypotheses are equivalent: 1) Differential inequality

(16) y(t)- a(t)y(g(t)) + b(t)y(h(t)) ::; 0, t 2:: t0 ,

has an eventually positive solution. 2) For some t1 2:: t0 and for t 2:: t 1 an integral inequality

(17) u(t) :::: a(t)u(g(t)) exp { rt u(s)ds} + b(t) exp { rt u(s)ds}' }g(t) Jh(t)

has a nonnegative locally integrable solution, where the first {the second) of two terms in the right hand side is added only if g(t) 2:: t1 or h(t) 2:: t1, respectively. 3) Equation {3} has a nonoscillatory solution. 4) For some t2 2:: to, X(t, s) > 0, t 2:: s 2:: t2. 5) Eq.{7) has a nonoscillatory solution.

Proof Equivalence of implications 1), 2), and 3) was obtained in Theo-rem 1 [3], the equivalence of 4) and 5) in Theorem 2 [4].

4) =} 3). Denote x(t) = X(t, t1). Then x(t) is a nonoscillatory solution of (3).

3) =;. 4). There exists nonnegative solution ofu(t) ofineq.(17) fort 2:: t 1 . Let x(t) = X(t, t 1). We want to show, that x(t) > 0. This function is a sol uti on of the following problem

(18) x(t)- a(t)x(g(t)) + b(t)x(h(t)) = 0, t:::: tl,

x(t) = 0, x(t) = 0, t < tl, x(tl) = 1,


Let

(19) x(t) = { y(t) + exp{- J/, u(s)ds}, t?: t 1 0, t < tl·

After substituting (19) into (18) we have

(20) iJ(t) - a(t)iJ(g(t)) + b(t)y(h(t)) = f(t), t ?: t11

y(t) = 0, iJ(t) = 0 t < t11 y(tt) = 0,

where

f(t) = exp {- £: u(s)ds}

x [u(t) -a(t)u(g(t))exp{i;t) u(s)ds} -b(t)exp{L:t) u(s)ds}].

Ineq.(17) implies f(t) ?: 0, t?: t1. In the proof of Th.1 [3] it was shown that if in (20) f(t) ?: 0 then y(t) ?: 0. Eq.(19) implies, that x(t) = X(t, t1) > 0. Similarly we can prove, that X(t, s) > 0, t > s?: t1• 0

Remark. Implication 5) => 3) is right without the assumption b(t) ?: 0. Indeed, if x is a positive solution of (7) for t ?: t1 ?: to that this function with additional initial value :i:(t) = 0, t ::::; t 1 is also positive solution of Eq.(3) for t ?: tl·

4. Comparison Theorem and Non-oscillation Conditions. Com-pare Eq.(3) with the following equation

(21) :i:(t)- at(t):i:(gt(t)) + bt(t)x(ht(t)) = 0, t?: to,

where for parameters of (21) hypotheses (al)-(a4) hold. In [3] we compared nonoscillation and oscillation properties of equations

(3) and (21) under condition that b(t)?: 0, b1(t)?: 0. In this section b1(t) is not assumed to be nonnegative.

THEOREM 3. Suppose b(t) ?: 0, a1(t) ::::; a(t), b1(t) ::::; b(t), g(t) ::::; g1(t), h(t) ::::; ht(t) and Eq.(9) has a nonoscillatory solution. Then Eq.(21) also has a nonoscillatory solution.

Proof Inequality b1(t)::::; b(t) implies lb1(t)!::::; b(t). Th.2 [3] implies that the equation

(22)


has a nonoscillatory solution. Consider corresponding to Eq.(22) equation with infinite number of delays:

00

{23) x(t) + :E lbl,k(t)ix(hl,k(t)) = 0, t::::: to, k=O

where

{24)

h1,o(t) = h1(t), hl,k+l(t) = h1,k(g(t)),

and operatorS is denoted by (6). Theorem 2 implies Eq.(23) has a nonoscil-latory solution. Corollary 2.2 in [4] yields that the equation

00

x(t) + :E b1,k(t)x(h1,k(t)) = o, t::::: to, k=O

has a nonoscillatory solution. The remark to Theorem 2 implies now that Eq. {21) has a nonoscillatory solution. 0

COROLLARY 2. Suppose {a1}-(a4) hold, there exists b1(t) f= 0 almost everywhere, such that b1(t) 2:: 0, b1 (t) 2:: b(t), and at least one of the following conditions holds: 1}

(1 a(t)b1(g(t))) { 1/.h(t) }

0 < >. < lim inf - - b ( ) exp \ b1 ( s )ds , t-->oo e 1 t " g(t)

where >. = limt-+oo sup J;(t) b1 (s)ds; 2)

0 < >. < lim inf (~- a(t):lg(t)) exp {~ [Y(t) b1(s)ds}), t-+oo e 1 t " Jh(t)

where >. = limt-+oo sup f~(t) b1 ( s )ds; 3}

0 < >. < lim inf ~ (1- a(t):~;(t)) exp {~ r bl(s)ds})' t-+oo e 1 t "19(t)

where >. = limt->oo sup f~(t) b1 (s)ds. 4) h(t)- g(t) is eventually positive or eventually negative, and

0 < lim sup {t b1(s)ds < lim inf (~- a(t):J((l(t))). Hoo jh(t) !-too e 1 t


5) h(t) - g(t) is eventually positive or eventually negative, and

0 I. 1t b1(s) d 1. . f ( 1 a(t)b1(g(t)) ) < 1m sup s < 1m m - . t->oo h(t) 1- a(s) t->oo e(1- a(t)) b1(t)[1- a(g(t))]

6}

lim supa(t) =a< 1, roo bl(s)ds < oo, t~oo ito

there exist..\> 0 andt1 2:: to such that ..\a< 1 and bt(g(t)) ~ ..\bt(t), t 2:: t1 • Then equation ( 3} has a nonoscillatory solution. Proof follows from Theorem 3 of the present paper, Theorems 3,4 and

5 in [3] and their corollaries. Remark. As bt(t) in Corollary 2 one can apply b1(t) = lb(t)l. This

function was used for Eq.(l) in Th.3.2.3 [10].

5. Comparison Theorem and Existence of a Positive Solution. In this section we will obtain explicit conditions on initial data which implies positiveness of the solution of initial value problem (4)-(5).

First we will compare solutions of two neutral equations. To this end consider together with ( 4)-(5) the following problem

(25) y(t)- a1(t)y(g(t)) + b1(t)y(h(t)) = h(t), t 2:: t1,

(26) y(t) = IPt (t), iJ(t) = '1/!1 (t), t < tt. y(tt) = yo, iJ(tt) = Yl·

Suppose for parameters of (25)-(26) conditions (al)-(a5) hold. Denote by Y(t, s) the fundamental function of (25).

THEOREM 4. Suppose

a(t) 2:: a1(t) 2:: 0, b(t) 2:: b1(t) 2:: 0, f(t) 2:: ft(t),

cp(t) 2:: ¥?1 (t), '1/!1 (t) 2:: 'if;(t), Yo 2:: Xo.

Suppose in addition the solution x of (4}-(5) is positive. Then y(t) 2:: x(t) > 0, where y is the solution of {25}-{26}.

Proof Let us rewrite Eq.(4) in the form

x(t)- a1(t)x(g(t)) + b1(t)x(h(t))

= [a(t)- a1(t)]x(g(t))- [b(t)- b1(t)]x(h(t)) + f(t).


Then the solution representation in Theorem 1 (see Eq.(lO)) has the following form

x(t) = Y(t, to)xo + 1t Y(t, s)f(s)ds t,

+ 1t Y(t, s) [(a(s)- a1(s))x(g(s))- (b(s)- b1 (s))x(h(s))] ds h

+ ft Y(t, s) [a(s)1j!(g(s))- b(s)(f>(h(s)] ds. ito For solution y of (25)-(26) Th.l yields

y(t) = Y(t, tt)Yo + 1t Y(t, s)!J(s)ds t,

+ 1t Y(t, s) [at(s)1ft(g(s))- bt(s)(f'(h(s)] ds. t,

By the proof of Th.2 [3] we have x(t) :'::: 0, t ~ t1. Hence x(t) ~ y(t). D Let us proceed to the positiveness conditions for the solution of problem

(4)-(5). THEOREM 5. Suppose {a1)-{a5) hold, a(t) ~ 0, b(t) ~ 0 and there exists

for t ~ t1 a nonnegative solution of the following inequality

(27) u(t) ~ a(t)u(g(t)) exp { rt u(s)ds} + b(t) exp { {t u(s)ds}, }g(t) lmax{t,,h(t))

where u(g(t)) = 0, g(t) ~ t 1 . If

(28) x(to) > 0, (f'(t) ::; x(tt), 1/!(t) ~ 0, f(t) ~ 0,

then for the solution of {4)-{5) we have x(t) > 0, t ~ t1. Proof. Let u be a nonnegative solution of (17). Denote

v(t) = { xoexp{- J/, u(s)ds}, t ~ tt. xo, t:S::tt.

We have

v(t)- a(t)v(g(t)) + b(t)v(h(t)) =


-xou(t) exp { -{ u(s)ds} + x0a(t)u(g(t)) exp {- f(t) u(s)ds}

{ rh(t) }

+x0b(t)exp -it, u(s)ds

= -x0 exp { -{ u(s)ds} [u(t)- a(t)u(g(t)) exp { i:t) u(s)ds}

-b(t) exp { r u(s)ds}] :::; 0. lmax{tt,h(t))

Hence v(t) is a solution of the following problem

v(t)- a(t)v(g(t)) + b(t)v(h(t)) = g(t),

v(t) = xo, v(t) :- 0, t :::; t1; g(t) :::; 0.

The theorem conditions and Theorem 4 imply x(t) :?: v(t) > 0, where x is the solution of (4)-(5). 0

6. Slowly Oscillatory Solutions. The following oscillation criterion is known for ordinary linear differential equations of the second order: if the equation has an oscillatory solution then all its solutions are oscillatory. As is well known, for delay differential equations this is not true.

Y. Domshlak revised [7] the above mentioned result for differential equa-tion with monotone delays. He demonstrated that if an associated equation has a slowly oscillatory solution then every solution of the given equation is oscillatory. In [8],[9] several new explicit sufficient conditions of oscillation are obtained by an explicit construction of such slowly oscillating solutions.

We present here a similar oscillation criterion for a neutral equation. Unlike Domshlak's result the existence of a slowly oscillatory solution is as-sumed for equation (3) and not for the associated one. Moreover the delays are not necessarily monotone.

DEFINITION 4. A solution x of equation {3) is said to be slowly oscilla-tory if for every t0 :?: to there exist t1 > t0 , h > t0 such that

(29) g(t) :?: t1, h(t) :?: t1 for t:?: t2;


Remark. In the case of constant delays g(t) = t- r, h(t) = t = a, solution x is slowly oscillatory if there exists a sequence of intervals [t1n, t2n) such that x( t2n) = 0, the solution is positive and its derivative is nonpositive on these intervals and the lengths of the intervals are greater than max { r, a}.

THEOREM 6. Suppose a(t) ~ 0, b(t) ~ 0. If there exists a slowly os-cillatory solution of the equation {3) then all solutions of this equation are oscillatory.

Proof. Denote by x a slowly oscillatory solution of (3). Suppose this equation has a non-oscillating solution. Then by Theorem 1 for a certain t0 ~ t0 X(t, s) > 0 if t ~ s > t 0 .

There exist t1 > t0 , t2 > t0 such that Cond.(29) holds. Due to (10) for t ~ t2 solution x can be presented as follows

(30) x(t) = -1t X(t, s)[a(t)x(g(s))- b(t)x(h(s))Jds, t2

where x(g(s)) = 0 if g(s) > t2 and x(h(s)) = 0 if h(s) > t2 • The inequalities g(t) ~ t 1 , h(t) ~ t 1 fort~ t 2 yield that the expression under the integral in (30) can differ from zero only ift1 < g(t),h(s) < t2, therefore (29) yields that in (30) x(h(s)) > 0, x(g(t)) :


[7] Y. Domshlak, Sturmian Comparison method in inve3tigation of the behavior of so-lutions for Differential-Operator Equations, Ehn, Baku, USSR, 1986 (Russian).

[8] Y. Domshlak, Properties of delay differential equations with oscillating coefficients, Functional Differential Equations, Israel Seminar, 2 (1994), 59-68.

[9] Y. Domshlak, G. Kvinikadze and I.P. Stavroulakis, Sturmian comparison method: the version for first order neutral differential equations, submitted.

[10] L. N. Erbe, Q. Kong and B. G. Zhang, Oscillation theory for functional differential equations, Marcel Dekker, New York, Basel, 1995.

[11] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, ~luwer Academic Publishers, Dordrecht, Boston, London, 1992.

[12] I. Gyori, Oscillation and comparison results in neutral differential equations and their applications to the delay logistic equation, Computers and Mathematics with Applications, 18 (1989), 1-20.

[13] I. Gyori and G. Ladas, Oscillation theory of delay differential equations, Clarendon Press, Oxford, 1991.

[14] J.K. Hale and S.M. Verduyn Lunel, Introduction to functional differential equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[15] V. Kohnanovskii and A.D. Myshkis, Introduction to the theory and applications of functional differential equations, Kluwer Academic Publishers Group, Dordrecht, 1999.

[16] V.G. Kurbatov, Functional differential operators and equations, Kluwer Academic ,, Publishers, Dordrecht, 1999.

,,

FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 9 2002, NO 3-4 PP. 289-314

TO THE QUESTION ABOUT THE DUALITY OF DIFFERENT VERSIONS OF THE PROGRAMMED

ITERATIONS METHOD,l*

A. G. CHENTSovt

Abstract. Different vaxiants of the programmed iterations method used previously in the differential games theory axe compaxed. Some natural duality of the "direct" version realized in a space of set-valued mappings and the "indirect" version realized in a space of sets is established. This duality includes "programmed" operators, the iterated procedures generated by these operators and the corresponding conditions of nondegeneracy. The present work is devoted to investigation of game control problems for dynamical systems defined axiomatically. Such an approach has a natural analogy with the general methods of the theory of functional differential equations, which axe developed by N.V. Azbelev and L.F. Rakhmatullina and their pupils. The author lectured on his own results in their seminax in Perm', discussed scientific directions of the FDE theory, being intensively developed. It seems that game problems can concern settings studied in the framework of the FDE theory and promote new directions and new actual applications as well.

Key Words. The programmed iterations method, fixed point, nonanticipating mul-tiselector.

1. ][ntroduction. The method of programmed iterations (MPI) was used (see [1]-[7]) in the theory of differential games (DG) for construct-ing the value function (the Bellman function) and the stable bridge in the sense of N.N.Krasovskii (see [8],[9]). The determination of the correspond-ing control procedures was realized in terms of the value function or the stable bridge. Therefore, we can consider the versions of MPI in [1]-[7] as an indirect method of constructing control procedures. Later, in [10]-[14]

' Supported by the Russian Foundation for Basic Research, projects 00-01-00348, 01-01-96450 and International Science and Technology center, project no. 1293.

t Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sci., 16 Kovalevskaya St., 620219 Ekaterinburg Russia

289

290 A.G.CHENTSOV

the corresponding direct variant of MPI was constructed. Namely, for the problems of DG the procedures of [10]-[14] realize set-valued quasistrategies solving the corresponding problem. Here, we give a comparison of the direct and indirect versions of MPI. As a result, we obtain a duality of the above-mentioned versions of MPI. These duality envelops the operators generating the corresponding iterated procedures, the sequences of iterations, the ob-tained limits of these sequences and (finally) the conditions guaranteeng the natural nondegeneracy of the realized solutions. The given duality permits to investigate the direct iterated procedure in terms of the corresponding indirect procedure and conversely.

2. General designations and definitions. We use quantors and pro-positional connectives; in the following the symbol D. means the equality by definition. Denote by P(H) (by P'(H)) the family of all (all nonempty) subsets of a set H. Denote by BA the set of all mappings from a set A into a set B. If A and B are sets, f E BA and C E P(A), then

(! I C) ~ (f(x))xEO E B 0.

If A and Bare sets and z E Ax B, then we denote by pr1(z) and pr2 (z) elements of the sets A and B, respectively, for which z = (pr 1 ( z), pr 2 ( z)).

Each set whose elements are sets is called a family. We admit the axiom of choice. We denote by lR the real line; N D. {1; 2; ... }and No~ {0} UN= {0; 1; 2; ... }. We postulate that elements of No are not sets. If U is a set, then Iu E uu is by definition the identity mapping in U : for u E U the equality

b. Iu(u) = u holds. As usual, we define the sequence of powers of an operator acting in a set. Namely, if U is a set and a E uu, then

(2.1)

is defined by the natural rule:

If S is the family of all subsets of some set and a E S8 , then ~E S8 is defined by the rule: forM E S the set ~(M) is the intersection of all sets ak(M), k E NQ. Of course, here we use (2.1). For arbitrary sets A and B suppose M(A, B) D. P(B)A; as in [14], for

f3 : M(A, B) ----t M(A, B)

PROGRAMMED ITERATIONS METHOD ... 291

we define {300 : M(A, B) -+ M(A, B) by the rule: for C E M{A, B) and a E A the set {300 (C)(a) is the intersection of all sets f3k(C)(a), kENo. We use the traditional notion of the monotone convergence of sets: if E is a set, (A;)iEN E M{N, E) and A E P{E), then, as usual, by definition

((Ai)iEN .j. A){=} ((A= nAi)&(\fj EN: Ai+l C AJ)). iEN

If A and Bare sets, (ai)iEN E M(A, B)N and a E M(A, B), then (as in [14]) we suppose by definition that

(2.2) ((ai)iEN .If a) {=} (\fx E A: (ai(x))iEN .j. a(x)).

So, in (2.2) the pointwise monotone convergence of a sequence of set-valued mappings is defined. Following [14], we introduce the natural pointwise order: if A and B are sets, u E M(A, B) and v E M(A, B), then by definition

(u!;;;; v) {=} (\fx E A: u(x) c v(x)).

So, in M(A, B) the pointwise order relative to inclusion is introduced. If S is a nonempty family, then

sfo) D. {a E S 8 I \IE E s: a(E) c E}.

If H is a set, then (as in [15]) we suppose that

(2.3) (cr- .j.)[H]:: {1i E P'(P(H)) I \f(Ai)iEN E 1iN liA E P(H): ((Ai)iEN .j. A) ===} (A E tl)}.

In (2.3) we introduce the notion of "sequential closed" families of sets. If H is a set and 1i E (cr- .j.)[H], then (see [15])

292 A.G.CHENTSOV

In (2.4) we have the natural generalization of statements of [3,4, 7], connected with the indirect version of MPI.

If A and B are sets, then we suppose

(2.5) M[A; B] D. {a E M(A, B)M(A,B) I '11-l E M(A, B): a(1i)!; 1-l}.

From (2.2) and (2.5) we obtain the obvious property; if A and B are sets, (J E M[A; B] and C E M(A, B), then

(2.6) (fJk(C))wJ -1). (J""(C).

Of course, we can consider (2.6) as an iterated realization of the set-valued mapping fJ""(C). The corresponding analog takes place for iterated proce-dures in the spaces of sets: if S is the family of all subsets of some set, a E Sfo) and E E S, then

(2.7)

In the following, we consider some connection of the procedures (2.6) and (2.7) for very natural (in control theory) situation: we investigate two iter-ated procedures connected with the abstract problem of control under action of indeterminate factors.

In the following, we use some topological constructions. If ('ll', r) is a topological space (TS) and t E 'll', then by N 7 ( t) we denote the filter of all neighborhoods [18, ch .1] oft in ('ll', r). We use the Moore-Smith convergence (see [18],[19]). To designate a net in a set H, we use triplets of the type

(D,-::5_,h),

where (D, -::5.), D f 0, is a directed set and hE Hv. This designation corre-sponds to [20, p.33]. If H is a set and (D, -::5_, h) is a net in H, then

(H- ass)[D; -::5_; h] ~ {S E P(H) I 3d ED '18 ED: (d -::5. 8) =} (h(8) E S)}

is the filter (of H) assotiated with the net (D, -::5_, h). In this terms, the Moore-Smith convergence is defined very simple: if ('ll', r) is a TS, (D, -::5_, f) is a net in 'll' and t E 'll', then by definition

(2.8) ((D, -::5_, h) ~ t)


(I sot)[A; «; B; :5] is regarded in the sense of [20, p.34]: (I sot)[A; «; B; :5] is the set of all ( «, :5)-isotone mappings of the set BA having the cofinal (with respect to (B, :5)) image of the set A. We use "isotone-cofinal" map-pings of the set (Isot)[A; «; B; :5] for the corresponding representation of compactness (see [18, ch .5],[19]); in addition, we use the representation of [20, p.35].

3. The generalized quasidynamic system. Fix to E lR and {)0 E ]t0 , oo[; let 10 ~ [t0 , iJ0]. Consider 10 as the basic time interval. Fix a nonempty set X and a set IC E P'(X10 ). In addition, consider X as the phase space of a "system"; IC defines the set in which trajectories of this "system" can be realized. Fix a topology T of the set C; so,

(3.1) (IC, T)

is the basic TS. Denote by IF and R the families of all closed and all compact [19] subsets of C in TS (3.1) respectively. Moreover, denote by T 0 T the natural topology of the set IC x C corresponding to the product of the two samples ofTS (3.1); see [18, ch .3]. By :F0 we denote the family of all subsets of IC x IC closed in (IC x C, T 0 1). Let

(3.2) b. D = 10 x C.

We call elements of D (3.2) functional positions. Recall that for z E D the elements pr1 (z) E Io and pr2 (z) E IC have the property z = (pr 1 (z),pr2 (z)). Fix a nonempty set 1' and a set n E P'(1'10 ). Elements of n are considered as indeterminate factors. Introduce the following "system":

(3.3) S: DxO-tP'(IC);

in addition, for zED and wEn the set S(z,w) is considered as the bundle of the possible trajectories of (3.3) corresponding to the functional position z of the set (3.2) and the concrete realization w of indeterminate factors. It is possible to give many concrete problems for which the above-mentioned interpretation is natural. But now we consider several general conditions in terms of (3.1)-(3.3).

CONDITION 3.1. Vz ED Vw E Q: S(z,w) E IF. CONDITION 3.2. Vt E Io Vw En: {(y,h) E IC X c I hE S((t,y),w)} E

E :F®. CONDITION 3.3. Vt E Io VuE C Vw E Q 3H E Nr(u) 3K E R Vv E H:

S((t, v), w) c K.

294 A.G.CHENTSOV

CONDITION 3.4. 'r/z ED 'r/w E 0 'r/h E S(z,w) 'r/t E [pr1(z),19o(: hE. E S((t,h),w).

CONDITION 3.5. 'r/t E Io 'r/u E C 'r/v E C: ((u I [to, t]) = (v I [to, t])) ==*

==* ('r/w E 0: S((t,u),w) = S((t,v),w)). Since an arbitrary section of a set from :F® is closed in TS (3.1); we have

the statement: Condition 3.1 follows from Condition 3.2. PROPOSITION 3.1. If Conditions 3.1 and 3.3 are fulfilled, then 'r/z E

D 'r/w E 0: S(z,w) E Jl The proof is obvious: we use the property that each closed set in a

compact TS is compact. Condition 3.5 has the sense of a heredity of the system (3.3); moreover, if, in addition, Condition 3.4 is fulfilled, then in fact we have the semigroup property. In the following, we will consider other conditions. But, the correctness of the corresponding conditions is postulated as required.

4. The indirect iterated procedure. We consider an abstract analog of the construction of (3,4]. The basic element of this construction is the op-erator of programmed absorption. We introduce the corresponding abstract analog of this operator. But, first we introduce some new designations. For HE P(D) and t E Io we suppose

(4.1) /::,.

H(t} = {g E c I (t,g) E H}. In (4.1) we introduce the "usual" section of a subset of the corresponding product of two sets. Suppose

(4.2) F ~{FE P(D) I 'r/t E 10 : F(t} ElF}. Of course, F (4.2) is the family of all subsets of D closed in the natural topology of the product [18, ch .3] of the TS (10 , P(/0 )) and (3.1). As a corollary, from (4.2) we have

(4.3) F E (a- .j..)[D].

In the following, we fix a set M E lF and consider M as some goal set for the "system" (3.3). Namely, we strive to realize (in the "system" (3.3)) only trajectories of the set M. So, the realization of trajectories of M is our basic goal. To this end, we use some auxiliary programmed problems of control with functional constraints. Note some analogy with (3],(4]. In this connection, we suppose 'r/H E P(D) 'r/z ED 'r/w En:

(4.4) IT(w I z,H) ~::,.{hE S(z,w)nM I 'r/t E [pr1(z),19o(: (t,h) E H}.


Of course, the following property takes place: under Condition 3.1

(4.5) VF E F Vz ED \iw En: TI(w I z,F) E JF.

The proof of (4.5) follows from (4.1) and (4.2). Now we introduce the oper-ator

(4.6) A: P(D) -----+ P(D)

by the following rule: if HE P(D), then

(4.7) 6

A( H) = {z E H I Vw E n : ll(w I z, H) # 0}.

Note that in fact the relations (4.6) and (4.7) define an operator of the programmed absorption. In addition,

(4.8)

In the following, we use the combination of (2.7) and (4.8). But, now we consider auxiliary operators. Namely, if w E n, then the operator

(4.9) Aw: P(D) -----+ P(D)

is defined by the following relation

(4.10) 6

Aw(H) = {z E H I ll(w I z, H) # 0}.

We use (4.9) and (4.10) by analogy with [7] and [16, ch .V]. Of course, we have

\iw E S1: AwE P(D)fo\Dl.

In terms of ( Aw )wEn the following representation of A is realized: A is the "pointwise intersection" of all operators (4.9). Namely, VUE P(D):

(4.11) A(U) = nAw(U). wEn

The representation similar to (4.11) was introduced in [7]. PROPOSITION 4.1. If Condition 3.4 is fulfilled, then \iw E S1: Aw =

= AwoAw. The proof follows from definitions. From ( 4.11) in general case we have

the property

(4.12) VH E P(D): (H = A(H)) -

296 A.G.CHENTSOV

PROPOSITION 4.2. If Conditions 3.2 and 3.3 are fulfilled, then Vw E E !1 VF E F: Aw(F) E F.

Proof. Let Conditions 3.2 and 3.3 be fulfilled. Fix wE !1, FE F, t. E E Io and

(4.13) g E cl(Aw(F)(t.), 7).

Using (4.13) and the well known Birkhoff theorem (see [18],[19]), we choose some net (V, j, cp) in Aw(F)(t.) for which

(4.14) (V, :::;, cp) ~g.

Then, for d E V we have z;j !:. (t., cp(d)) E Aw(F). Since F(t.) E lF, from (4.14) we have z. ~ (t., g) E F. By (4.4), (4.10) and the choice of (D, j, cp) we have the mapping

(4.15) d >--t II(w I z:t, F) : V--+ P'(M).

Using the axiom of choice and (4.15), we choose a mapping

(4.16) p E rrn(w I z:i,F). dE1J

Then, by (4.16) p E Mv and Vd E V: p(d) E II(w I z:i, F). Using Condition 3.3, we choose

(HE N7 (g)) & (K E .it)

for which the following inclusion holds:

(4.17) Us((t.,u),w) c K. uEH

From (4.14) we have the obvious property: HE (C- ass)[V; j; cp]. Choose d E V for which

Vd E V: (d j d)=* (cp(d) E H).

Introduce the nonempty set V 0 "" { d E V I d j d} cofinal with respect to (V, j); V 0 E 'P'(V) is equipped with the direction~ for which by definition Vd1 E Vo Vrh. E Vo :


Of course, !;;; is the trace of ::S on 'Do : !;;;=::S n('Do x 'Do). Suppose (/5 ~ (cp I 'Do). So, ('Do,!;;;, q;) is a net in H. Consider p ~ (p I 'Do). Recall that by ( 4.4) and (4.16) we have 'Vd E 'D:

p(d) E S((t., cp(d)), w).

As a corollary, by ( 4.17) for d E 'Do we have p( d) E K. So, we obtain the net ('D0 , !;;;, p) in K. Using the compactness of K, we choose h E K, a directed set(~.~),~ f= (/),and a mapping l E (Jsot)[~;~;'D0 ;!;;;] for which

(4.18) (~, ~. p 0 l) ..I..., h.

In addition, for o E ~ we have

( 4.19) (p o l)(o) E IT(w I zi(o)• F).

By (4.4) and (4.19) we have 'Vo E ~ 'Vt E [t., '190 [: (t, (pol)(o)) E F. Therefore, 'Vt E [t., '19o[ 'io E ~ :

(pol)(o) E F(t).

But, by (4.2) we have 'it E [t., '190 [: F(t) E JF. Therefore, by (4.18) we obtain

( 4.20) 'Vt E [t., '19o[: (t, h) E F.

Recall that M E lF and (~,~.pol) is a net in M. Therefore, by (4.18) we have hEM. Finally, we note that by (4.4) and (4.19) foro E ~the inclusion

(pol)(o) E S(zi(o),w)

holds. As a corollary, we have Vo E ~ :

( 4.21) (pol)(o) E S((t.,(q;ol)(o)),w).

In connection with Condition 3.2 and ( 4.21), we introduce the set

(4.22) 2 ~ {(y,s) E C x CIs E S((t.,y),w)} E :F0 .

We recall that (/5 E Hvo and, in particular, (/5o l E C~. Moreover, pol E C~. From (4.21) and (4.22) we have Vo E ~:

( 4.23) ((q; o z)(o), (15 o t)(o)) E 2.

298 A.G.CHENTSOV

We note that from (4.14) by the choice of l the convergence

(4.24)

takes place. From (4.18),(4.23) and (4.24) we obtain

(4.25) ((~,~,1jjol) ~ g)&((~,~,pol) ~h)& &(V8 E ~: ((


Proof. Fix (F;);EN E FN and FE P(D) with the property

(4.26)

From (2.3),(4.3) and (4.26) we have F E F. Moreover, by (4.4),(4.6) and (4.26)

(4.27) A(F) c nA(F;). iEN

Finally, fork EN the inclusion A(Fk+l) C A(Fk) takes place. To prove the inclusion opposite to ( 4.27), we choose an arbitrary point z. of the intersec-tion of all sets A(F;), i E N. Then z. E F. Fix w E n. Then by (4.4) and (4.7) we have the sequence

i ~ IT(w I z.,F;): N--+ P'(S(z.,w) n M).

In particular, we have the sequence in P'(C). Using the axiom of choice, we choose some element

(4.28) (h;)iEN E 11 IT(w I z., F;). iEN

Of course, from (4.28) we obtain the sequence

( 4.29) (h;);EN: N--+ S(z.,w) n M.

We recall that in our case Condition 3.1 is fulfilled too. We use Proposition 3.1. Then S ( z., w) E .1{, In this proof we use the symbol ::::: only for designation of the natural order of N. Then, (N, ::;) is a directed set and by compactness of S(z.,w) we can choose some hE S(z.,w), a directed set ('D, ::S), 'D # 0, and l E (Isot)['D; ::S; N; ::;] with the property

(4.30)

From (4.29) and (4.30) we have the inclusion hE S(z.,w) n M. Lett. ~ pr1(z.) and t* E [t.,190 [. Then, by (4.28) we have

( 4.31) (hk)kEN E 11 Fk(t*}. kEN

Fix n EN and consider Fn(t*} E F (see (4.2)). From (4.26) fork EN with the property n ::::: k we have the inclusion

300 A.G.CHENTSOV

Then, by (4.31) we obtain that for k E N such that n ::; k the inclusion hk E Fn(t*) takes place. But, by the choice of l we have the property: for some o E V the inequality n::; l(o) holds. Moreover, ford E V we have the implication

(o ~d)==* (l(o) ::; l(d)).

Therefore, for d E V with the property o ~ d we have n ::; l(d) and, as a corollary, h!(d) E. Fn(t*). From (4.30) we obtain h E Fn(t*) and, as a result, (t*, h) E Fn. Since the choice of n was arbitrary, by (4.26) we have (t*, h) E F. We obtain 'it E [t., -Do[: (t, h) E F. By (4.4) we have h E II(w I z,, F). So, IT(w I z., F) # 0. But, the choice of w was arbitrary. Therefore, z. E F has the property

'iv E n : II(v I z., F) # 0. By (4.7) we obtain z. E A(F). So, we have established the inclusion inverse with respect to (4.27), which completes the proof. 0

PROPOSITION 4.6. If Conditions 3.2 and 3.3 are fulfilled, then A E :> 2to(F).

Proof. Let Conditions 3.2 and 3.3 are correct. We use (4.3) and (4.8). By (4.4) for 81 E P(D) and 82 E P(D) we have the implication

(4.32) (81 c 82) ==} ('iz ED 'iw En: IT(w I z, 81) c IT(w I z, 82)). From (4.7) and (4.32) we have the monotonocity of A: if 81 E P(D), 82 E P(D) and 81 C 82 , then A(81) c A(82). By Proposition 4.4 for F E F we have A(F) E F. Note (see Section 3) that in our case Condition 3.1 is fulfilled. Then Conditions 3.1 and 3.3 are fulfilled. Therefore, A has the property of the sequential continuity in the sense of Proposition 4.5. Then, by the definition of Section 2 we have the required statement. 0

00

THEOREM 4.1. If Conditions 3.2 and 3.3 are fulfilled, then A(D) E F and the given set is the extremal fixed point of A :

00 00

(A(A(D)) = A(D))&('/8 E P(D) : (8 = A(8)) ==* 00 00

==* (8 C A(D)))&((Ak (D))kEN {. A(D)).

The proof follows from (2.4). We note only the obvious interpretation 00

connected with iterated procedures. Namely, by Theorem 4.1 the set A(D) is the limit of the following iterated process


We note that ( 4.33) is similar to the procedure of [3],[4]. We consider some properties connected with the property of heredity. Suppose that

b. (4.34) Ph(D) ={HE P(D) I Vt E Io VuE IC Vv E IC: (((t, u) E H)&

&((u I [to, t]) = (v I [to, t]))) ==> ((t, v) E H)}.

From (4.34) we obtain the obvious property

( 4.35) V1l E P'(Ph(D)): n HE Ph(D). HE1l

We note several obvious statements connected with the family ( 4.34) of all hereditary sets.

PROPOSITION 4. 7. If Condition 3. 5 is fulfilled, then 'ift E I0 'ifu E IC Vv E IC:

((u I [to,t]) =(vI [to,t])) ==> (VH E P(D) Vw En: II(w I (t,u),H) = = II(w I (t, v), H)).

The corresponding proof follows from (4.4). Now, from (4.10) we have PROPOSITION 4.8. If Condition 3.5 is fulfilled, then VH E Ph(D) Vw E

!1 : Aw(H) E Ph(D). PROPOSITION 4.9. If Condition 3.5 is fulfilled, then VH E Ph(D)

A(H) E Ph(D). The proof follows from ( 4.35) and Proposition 4.8. COROLLARY 4.2. Under Condition 3.5 we have Vk E No : Ak(D) E

Ph(D). In this proof we use the obvious induction. Using ( 4.35), we obtain the

00

useful property: if Condition 3.5 is fulfilled, then A(D) E Ph(D). So, under Condition 3.5 we have the A-invariance of the space of hereditary sets. In the following, we consider some weakening of the given requirement. But, in addition, we will obtain some weakened variant of heredity too.

5. The iterated procedure in the space of set-valued mappings and a duality. We first recall some concrete variant of the construction of [10]-[14]. We consider the given construction as the direct version of the MPI procedure. Later we consider the natural connection of the two variants of MPI. As a result, we will obtain some duality.

In the following we fix To E [to, '19o[ and go E C; the pair

(5.1) b. zo =(To, go) ED

302 A.G.CHENTSOV

plays the role of the fixed initial functional position. Therefore, we consider the sets S(z0 ,w), wEn, as the basic reactions of our system (3.3). Since the realization of trajectories of M is our goal, we introduce the following goal set~valued mapping

(5.2) !:;

Co= II(· I zo, D)= (II(w I zo, D))wen

corresponding to the initial functional position (5.1). From (3.2),(4.4) and (5.2) we have the property: Co is the mapping

(5.3) w o---+ S(z0 ,w) n M: n ----t P(M).

We consider (5.3) as the goal mapping. So, (5.2), (5.3) is some set-valued pseudostrategy. Now, we introduce the special iterated construction which realizes constructing a nonanticipating multiselector of Co (5.3). Of course, for wE 0 we have Co(w) = S(zo, w) n M. Now, introduce '


of course, (5.8) is a nonempty set (the mapping cg E M(fl, C), for which cg(w) = f/J, is a concrete element ofN (5.8)). The set (5.8) has the t;;::-greatest element

(5.9) (na)[Co] ::_ ( U C(w))wErl EN. CEN

So, for (na)[Co] (5.9) we have the representation

w I-+ UC(w): n----+ P(C). CEN

In addition, for£ E N we have £ I; (na)[C0]. Of course, r (5.7) generates the sequence (rk)kENo of operators acting in M(fl, IC). Moreover, we have

(5.10) roo : M(fl, C) ----+ M(fl, C).

Of course, in our problem it is sufficient to use r (5.5) and roo (5.10) for constructing the basic iterated sequence with the initial element C0 :

(5.11) (r0 (C0 ) = C0)&(\fk EN: rk(C0) = r(rk-l(C0))).

We can consider (5.11) as a version of MPI; in addition, we have the natural limit passage

(5.12)

In (5.12) we realize the approach similar to [1]-[4]. But, we call (5.11) and (5.12) the direct version of MPI, keeping in mind the question of constructing set-valued quasistrategies. Of course, r has the sense of some "programmed" operator. But, we note that in (5.6) the "system" (3.3) is not used. We operate only with germs of mappings on ! 0 . Later, under some additional conditions we consider the connection of A and r.

Now, we consider some family of subsets of D; in addition, we introduce a weakened variant of the heredity property. Namely, suppose

c,. Po(D) ={HE P(D) I Vt E [ro,-6'o[lfwl E fl\fu E Co(wl): ((t,u) E H)*

=} (\fw2 E fit(wl) \fv E Co(w2) n Zt(u): (t, v) E H)}. (5.13) From (4.34) and (5.13) we obtain the following inclusion

(5.14)

304 A.G.CHENTSOV

From (5.14) we have a useful property of P 0(D) (5.13): hereditary subsets of D are elements of (5.13). Moreover, we note that

(5.15) '


differential equations. It is known {8} that sometimes in the given systems the necessity of the employment of the control procedures with an informa-tional memore arises. The given procedures are required for realization of conditions of the alternative solvability of DG with a functional goal set (see {8}).

So, in the following we postulate Conditions 5.1-5.4. THEOREM 5.1. If HE Po(D), then r(II(· I Zo, H))= II(· I Zo, A( H)). Proof. Fix H E P 0 (D). Suppose that a ~ II(· I zo, H) and b ~:; II(· I

zo, A( H)). Fix w E 0 and compare r(a)(w) and b(w). Let

306 A.G.CHENTSOV

we have II(w* I z*,H) #0. Of course, here we use (4.7).Choose arbitrarily v E II(w* I z*, H). By {4.4) we have v E S(z*, w*) n M; moreover,

(5.21) 'r/t E [t*,Do[: (t,v) E H.

By the choice of w* we have the property

w* = (wDw*)t• E !1.

Then we use Condition 5.4, supposing w1 = w, w2 = w•, h = u and t = t*. By this condition and (5.20)

S(z*,w*) n M c S(zo,w*) n Zt· (u).

By properties of v we obtain

(5.22) v E S(zo,w*) n Zt·(u).

But, v E M. Therefore, from (5.2) and (5.22) we have v E C0 (w*). Using (5.22), we obtain

(5.23)

Recall that HE P 0 (D). We use (5.13). Let~ E [r0 , t*[. Since~ E [r0 , D0[, we set in (5.13) t = ~ and w1 = w. By the choice of u we obtain the implication

((~, u) E H) =i- ('rlw E f2e(w) 'r/v E Co(w) n Ze(u) : (~, v) E H).

But, (~, u) E A( H) and, in particular, (~, u) E H. Therefore, for each w E ne(w) and v E Co(w) n Ze(u) we have (~, v) E H. But, w* E ne(w) and v E C0(w*) n Ze(u) (see (5.23)). Here we use the inclusion [t0 , ~] c [t0 , t*]. Then (~, v) E H. Since the choice of~ was arbitrary, we have 'r/t E [r0 , t*[: (t, v) E H. From (5.21) and (5.23) we obtain that v E C0 (w*) n M has the property: fortE [r0 , Do[ the inclusion (t, v) E H holds. Then, by (4.4),(5.2) and (5.23)

v E a(w*) n Zt· (u).

Since the choice of t* and w* was arbitrary, we have

(5.24)

(of course, here we use the obvious equality Q.?0 (w) = {w }; see (5.4)). Since u E a(w), from (5.6) and (5.24) we have u E r(a)(w). The inclusion b(w) c


r(a)(w) is established; therefore, r(a)(w) = b(w). But, the choice of w was arbitrary. Then we have r(a) =b. o

COROLLARY 5.1. Vk E No: rk(Co) =II(· I Zo, _Ak(D)). Proof. Recall that DE Po(D) and (5.17) holds. Of course,

r 0 (Co) =Co= II(·[ zo,D) =II(·[ zo,A0 (D))

(see (5.2)). Let n E No and the equality rn(Co) =II(· I Zo,An(D)) holds. Then An(D) E 1'0 (D) and by Theorem 5.1

rn+l(Co) = r(rn(Co)) == r(II(·I zo,An(D))) =II(· I Zo,A(An(D))) = =II(· [ zo, _An+l (D)). 0

00

COROLLARY 5.2. roo(Co) =II(· I Zo, A(D)). 00

Proof. Fix wEn and compare the sets II(w I Zo, A(D)) and r""(Co)(w). Recall that by Corollary 5.1 we have

00 k Vk E No : II(w I Zo, A(D)) c r (Co)(w).

Then, by the corresponding definition of Section 2, we obtain the inclusion

(5.25) 00

II(w I Zo, A(D)) c roo(Co)(w).

Choose arbitrarily 'P E r 00 (C0)(w). Then Vk E No : 'P E rk(C0 )(w). Using Corollary 5.1, we obtain

(5.26) Vk E No:

308 A.G.CHENTSOV

6. The conditions of nondegeneracy. In this section, we consider the question of constructing the set-valued mapping (5.9). Moreover, we consider the procedure of constructing this mapping by the procedure (5.11), (5.12). The corresponding conditions are given in (10]-(14]. These conditions mean that Co is required to be the compact-valued mapping. It is known that in general case

(na)[C0 j c r""(Co);

see (10]-(14]. In the following, we postulate CONDITION 6.1. Vw E Q: S(zo,w) E j{. PROPOSITION 6.1. Vw E 0: Co(w) E j{. Proof. Fix w E n and consider the set S(z0 ,w) E .it. In addition,

Co(w) = S(zo, w) nM is the set closed in the compact TS (S(z0 ,w), 8), where 8 is the topology induced in S(z0 ,w) from the TS (3.1). Then C0 (w) is the set compact in (S(zo,w), 8): the topology 80 of Co(w) induced from (S(z0 ,w), 8) is the topology of a compact space. But, by the transitivity of the operator of the passage to a subspace of a TS we have that (C0 (w), 80 ) is a compact subspace of the TS (3.1). Then C0 (w) E .it. 0

In the following, we consider a special case of the TS (3.1). Namely, we suppose that (3.1) is a subspace of some Tychonoff product of TS. So, lett be a Hausdorf topology of X. We consider the Hausdorf space

(6.1) (X,t).

Denote by 0 10 ( t) the natural topology of X 1• corresponding to the Tychonoff product [18],(19] of samples of the TS ( 6.1 ). Of course, ! 0 is used as an index set. In fact, we have the Tychonoff power of the TS (6.1). We note that (see (18],(19]) the convergence of nets in the TS

(6.2) (X10 ,010 (t))

is realized as the pointwise convergence; in this connection, see (20, p.35]. In the following we postulate

CONDITION 6.2. The topology 7 of the set C is induced from the TS {6.2): the TS (3.1) is a subspace of the TS {6.2}.

REMARK 6.1. So, we further consider (3.1) only as a subspace of the TS {6.2). Of course, in this case we follow the stipulation about the validity of Condition 6.1. So, {3.1} is the set C in the topology of pointwise convergence with the compact-valued mapping S(z0 , ·).

We note that (6.2) is a Hausdorff TS and by Proposition 6.1 we have Co E Jt0 . So, we have the case of a compact-valued in the TS (3.1) that is


defined by Condition 6.2. Therefore, from results of [11-14] we have the very important

PROPOSITION 6.2. r 00 (Co) = (na)[Co] E .ft0 • 00

CoROLLARY 6.1. (na)[Co] =II(· I zo, A(D)). The proof follows from Corollary 5.2 and Proposition 6.2. Consider the very important question of nondegeneracy. Namely, we

investigate the property: (na)[Co](w) f 0 for wE 0. Let '

310 A.G.CHENTSOV

00

Proof. By Theorem 4.1, z0 E A(A (D)). As a corollary, by (4.7) and Corollary 6.1

00

Vw En: (na)(Co](w) = IT(w I Zo, A (D)) -:1 0. 0 THEOREM 6.1. Under Conditions 3.2 and 3.3 we have

00

(zo EA. (D))-


The representation (7.2) is more interesting. Recall that in our construction the verification of the property II(w I z., H) f. 0 is required. In (7.2) we have some concrete variant of this condition. Of course, in typical cases of the basic problem the verification of the property

S(z.,w)nMn( n H(t))f.-.0 tE[t.,~o[

is most essential (see (4.7),(4.10)), since the analogous verification in the case t. = iJo is obvious.

We recall that by ( 4.33)

(7.3) A1 (D) = A(D) ={zED I Vw En: S(z,w) n M f. 0}.

Of course, we use the following obvious (see (4.4)) property: II(w I z, D) = S(z,w) n M for zED and wE 0. We have in (7.3) the set of programmed absorption. From (4.1) and (7.3) we obtain

(7.4) A1 (D)(iJo) = A(D)(iJo) = {g E C I (iJo, g) E A(D)}.

But, forgE C (from (7.3)) we have the equivalence

(7.5) ((iJ0 ,g) E A(D)) {==} (Vw E 0: S((iJo,g),w) n M f. 0).

If S((iJo,g),w) = {g} for allw E 0 and g E C, then

((iJo,g) E A(D)) {==}(gEM),

and from (7.4) and (7.5) we have the equality

(7.6) A(D)(iJo) = M.

Of course, the general setting of Section 3 assumes very distinctive variants of the set-valued mapping (3.3). In our more general cases, we have the above-mentioned representation of a duality. But, for typical (for DG) settings, the considered particular case is very natural. So, in the following, the next condition is assumed to be hold:

(7.7) 'r/g E CVw E 0: S((iJo,g),w) = {g}

(the coincidence with a singleton takes place). Then (7.6) is fulfilled. PROPOSITION 7.1. Vk EN: Ak(D)(iJ0) = M.

312 A.G.CHENTSOV

Proof. For k = 1 the required equality follows from (7.6). Let n EN and the equality An(D)('00 } = M holds. Then

(7.8) An+! (D)= A(An(D)) = {z E An(D) I Vw E 0:

II(w I z,An(D)) =10}. By the choice of n, we have

(7.9)

Choose m E M. Then m E IC and J1o ~ (D0 , m) E An(D). Let w E 0. Since pr1 (JJo) = 1Jo and pr2 (JJo) = m, by (4.4) we have

II(wiJJo, An(D)) = {hE S(JJo, w) n MIVt E [pr1 (JJo), 1J0{: (t, h) E An(D)} = = S(f.t,w) n M = S((1J0 ,m),w) n M = {m} n M = {m} =10.

Since the choice of w was arbitrary, from (7.8) we have J1o E An+l (D) and, as a corollary, mE An+1 (D)(1J0}. So,

(7.10)

From (7.9) and (7.10) we have the equality

An+l(D)(1Jo}=M. D

00

COROLLARY 7.1. A (D)(1Jo} = M. The proof follows from Theorem 4.1 and Proposition 7.1. From Propo-

sition 7.1 and Corollary 7.1 we obtain (under the condition (7.7)) that the procedure ( 4.33) to the analogous one characterizing the transformation of the sequence of the mappings

(7.11)

The passage (in (7.11)) from the case k = n- 1 to the case k = n is realized by the employment of (7.2). Namely, A0 (D)(t} = IC for each t E (t0 ,1J0 (. If n EN and t. E (to, Do[, then by (4.1),(4.7) and (7.2),

An(D)(t.} = {g E IC I (t.,g) E An(D)} = {g E An-l(D)(t.} I Vw E 0: (7.12) II(w I (t.,g),An-l(D)) =10} = {g E An-l(D)(t,} I Vw E 0:

S((t., g),w) n M n ( n An-I (D)(t}) =10}. tE[t.,Uo(

So, we realize the transformation

(7.13) (An-! (D)(t} )tE[to,Uo[ -+ (An (D)(t} )tE[to,Uo[i


of course, the choice of n E N is arbitrary. So, in fact, (7.13) realizes some "new" iterated process in the space of all mappings from [to, i9o[ into P(IC). On the basis of (7.12), we can introduce the corresponding analog of A. Such modification is obvious. Note only that the mapping

00

t >------? A(D)(t) : [to, i9o[-----+ P(IC)

is the corresponding limit of the above-mentioned "new" iterated process, since Vt. E [to, i9o[:

(7.14) k 00

(A (D)(t.) )kEN.). A(D)(t.).

Proposition 7.1 and Corollary 7.1 supplement the given iterated construction for some unessential transformation of ( 4.33). So, we can use transformations similar to (7.13) in the above-mentioned duality constructions. We note that (7.12) is similar (in some sense) to the iterated principle of [4]. Moreover, we note that by ( 4.4), for the realization of r 00 (C0 ) with the employment

00

of the indirect iterated procedure, only the sets A(D)(t), t E [t0 , 190 [, are used (see (4.1) and (7.14)). In this connection, we recall Corollary 5.2. The analogous conclusion is fulfilled with respect to rk(C0), kENo, where only the set-valued mappings defined by (7.11) are used. Therefore, the iterated procedure connected with (7.12) and (7.13) is sufficient for the realization of our duality (Proposition 7.1 and Corollary 7.1 play an unessential role).

REFERENCES

[1] A.G. Chentsov, On the structure of a game problem of convergence, Soviet Math. Dokl., 16 (1975), 1404- 1408.

[2] A. G. Chentsov, On a game problem of converging at a given instant of time, Math. USSR Sbornik, 28 (1976), 353-376.

[3] A.G. Chentsov, On a game problem of guidance, Soviet Math. Dokl., l'i (1976), 73-77.

[4] A.G. Chentsov, On a game problem of guidance with information memory, Soviet Math. Dokl., 17 (1976), 411-414.

[5] S.V. Chistyakov, To the solution of game problem of pursuit. Priklad. Mat. i Mech., 41 (1977) (Russian).

[6] A.G. Chentsov, On a game problem of converging at a given instant of time, Izv. Acad. Nauk USSR, Matematika, 42 (1978) (Russian).

[7] A. G. Chentsov, The programmed iterations method for a differential pursuit-evasion game, Dep. in VINITI, 1933-79, Sverdlovsk, 1979 (Russian).

[8] N.N. Krasovskii and A.I. Subbotin, Game-theoretical control problems, Springer-Verlag, Berlin, 1988.

[9] N .N. Krasovskii, Dynamic system control. Problem of the minimum of guaranteed result, Nauka, Moscow, 1985 (Russian).

314 A.G.CHENTSOV

[10] A.G. Chentsov, The programmed iterations method in the class of finitely additive controls-measures, Differents. Uravneniya, no 11 (1997), 1528-1536. (Russian)

[11] A.G. Chentsov, The iterative realization of nonanticipating set-valued mappings. Dokl. Akad. Nauk, 357 (1997), 595-598 (Russian).

[12] A.G.Chentsov, To the question about a parallel version of the abstract analog of the programmed iterations method. Dokl. Akad. Nauk, 362 (1998), 602-605 (Rus-sian).

[13] A.G. Chentsov, On the question about iterative realization of nonanticipating set-valued mappings. Izvestija VUZov. Matematika, 3 (2000), 6Q-76 (Russian).

[14] A. G. Chentsov, Nonanticipating selectors of set-valued mappings and iterated pro-cedures. Functional Differential Equations, 6, no. 3-4 (1999), 249-274.

[15] A.I. Subbotin, A.G. Chentsov, An Iterative Procedure for

· table of contents professor nikolay viktorovich azbelev. 263 professor lina fazulovna...

Documents