Considerații generale privind integrarea asimptotică a ecuațiilor diferențiale cu dihitomii

Integrarea asimptotica a ecuatiilor diferentiale ordinare este in zilele noastre o parte fundamentala a matematicii aplicate. Lucrari de matematica precum [1], [5], [8] ori mecanica cereasca [4], [9] sunt relevante in ceea ce priveste utilitatea si eficienta metodelor de integrare asimptotica a ecuatiilor si sistemelor de ecuatii diferentiale ordinare care modeleaza probleme importante ale stiintelor naturii.

Problema dichotomiilor in integrarea asimptotica a fost investigata de numerosi autori iar metodele de lucru apeleaza la discipline diverse. Citam articolele [10] - [14]. Rezultate generale de integrare asimptotica, ingloband cazul ecuatiilor diferentiale cu dichotomii, pot fi citite in [6], [7], [15] (teoria Hartman-Wintner), [16] (teoria Massera-Schaffer), [5] (teoria Levinson-Weyl si dezvoltari ale sale, cu accent pe problematica defectologiei), [8] (metode Kiguradze-Kvinikadze privind ecuatiile Emden-Fowler, proprietati S, etc). Lucrarea de fata se refera la teoriile dezvoltate in [6], [15] precum si la o anumita continuare a lor in [1], [2]. Ea este impartita in doua capitole, dedicate abordarii lui Aurel Wintner [15] a integrarii asimptotice in cazul hiperbolic (Capitolul 1), respectiv teoriei lui Philip Hartman [6] (Capitolul 2). Fie f=f (t) o functie continua cu valori reale definita pe semiaxa reala nenegativa astfel incat f (a^z) sa existe (ca limita finita). Suntem interesati de integrarea asimptotica a ecuatiei diferentiale liniare in cazul hiperbolic, f (a^z) < 0; facand, eventual, o schimbare de variabila, putem considera ca f (a^z) = -1. Astfel, ecuatia (1) poate fi scrisa sub forma (2) x (1+I?) x = 0, unde (3) I?

(t) a+' 0 (t a+' +a^z). (4) p (t) are semn constant nenul si unde (6) q (t) = 0, atunci exista o limita finita, y (a^z), pentru fiecare solutie y=y (t) a ecuatiei diferetiale liniare (7) (p (t) y) + q (t) y = 0, marimea y (a^z) fiind nenula pentru cel putin o solutie y = y (t) a ecuatiei (7). si Mai precis, C1 = y (T) p (T), C2 = y (T). Invers, se observa din (4) ca orice T si oricare pereche de constante C1, C2 introduse in integrala (9) determina o solutie de y=y (t) a lui (7) care satisface (8) si (9). Pentru o solutie y (t) a lui (7), fie Mt =Mt (T) si m=m (T) definite ca si respectiv convergenta integralei din (11) rezultand din (5), (6) daca se tine seama de continuitatea lui q (t). Fie t >T. Atunci conform (8), (10) si (11), Deci, conform (10), avem (12) Mt < KMt + m, unde K =K (T) este integrala Din (5) rezulta ca integrala (13) este convergenta si valoarea sa tinde catre 0 daca Ta+' +a^z. Fie T asa de mare incat K ...

R.P. AGARWAL D. - "OREGAN, INFINITE INTERVAL PROBLEMS FOR DIFFERENTIAL, DIFFERENCE AND INTEGRAL EQUATIONS", KLUWER, DORDRECHT, 2001

R.P. AGARWAL, O.G. MUSTAFA, Y.V. ROGOVCHENKO - "EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A BOUNDARY VALUE PROBLEM ON AN INFINITE INTERVAL" - MATHEMATICAL AND COMPUTER MODELLING, 2005

C. AVRAMESCU, O.G. MUSTAFA, S.P. ROGOVCHENKO, Y.V. ROGOVCHENKO - "CONDITIONAL STABILITY FOR A CLASS OF SECOND-ORDER DIFFERENTIAL EQUATIONS" - APPLIED MATHEMATICS LETTERS, 2005

A. COOK - "THE MOTION OF THE MOON" - ADAM HILGER PRESS, BRISTOL, 1988

M.S.P. EASTHAM - "THE ASYMPTOTIC SOLUTION OF LINEAR DIFFERENTIAL SYSTEMS" - CLARENDON PRESS, OXFORD, 1989

P. HARTMAN - "UNRESTRICTED SOLUTION FIELDS OF ALMOST-SEPARABLE DIFFERENTIAL EQUATIONS" - TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 63, 1948

P. HARTMAN, A. WINTNER, ASYMPTOTIC INTEGRATION OF LINEAR DIFFERENTIAL EQUATIONS" - AMERICAN JOURNAL OF MATHEMATICS 77, 1955

I.T. KIGURADZE, T.A. CHANTURIA - "ASYMPTOTIC PROPERTIES OF SOLUTIONS OF NONAUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS", KLUWER, DORDRECHT, 1993

C. MARCHAL - "THE THREE-BODY PROBLEM" - STUDIES IN ASTRONAUTICS, VOL. 4, ELSEVIER, OXFORD, 1990

R. NAULIN, J. URBINA - "ASYMPTOTIC INTEGRATION OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS OF ORDER N" - ACTA MATHEMATICA HUNGARICA 80, 1988

R. NAULIN - "WEAK DICHOTOMIES AND ASYMPTOTIC INTEGRATION OF NONLINEAR DIFFERENTIAL SYSTEMS", NONLINEAR STUDIES 5, 1998

K.J. PALMER - "EXPONENTIAL DICHOTOMY, INTEGRAL SEPARATION AND DIAGONALIZA-BILITY OF LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS" - JOURNAL OF DIFFERENTIAL EQUATIONS 43, 1982

K.J. PALMER - "EXPONENTIAL DICHOTOMIES AND TRANSVERSAL HOMOCLINIC POINTS" - JOURNAL OF DIFFERENTIAL EQUATIONS 55, 1984

R.J. SACKER, G.R. SELL - "A SPECTRAL THEORY FOR LINEAR DIFFERENTIAL SYSTEMS" - JOURNAL OF DIFFERENTIAL EQUATIONS 27, 1978

A. WINTNER - "ASYMPTOTIC INTEGRATIONS OF THE ADIABATIC OSCILLATOR IN ITS HYPERBOLIC RANGE" - DUKE MATHEMATICAL JOURNAL 15, 1948

J.L. MASSERA, J.J. SCHAFFER - "LINEAR DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS" - ANNALS OF MATHEMATICS 67, 1958