TEZA DE ABILITARE

PROBLEME LA LIMITA PENTRU SISTEME DE ECUATII

DIFERENTIALE, CU DIFERENTE SI FRACTIONARE

REZUMAT

Domeniul fundamental: MATEMATICA SI STIINTE ALE

NATURII

Domeniul de abilitare: MATEMATICA

Autor: Prof.univ.dr. RODICA TUDORACHE (LUCA)

Teza elaborata ın vederea obtinerii atestatului de abilitare ın scopul conducerii lucrarilor de

doctorat ın domeniul MATEMATICA.

BUCURESTI, 15 decembrie 2016

ACADEMIA ROMÂNĂ SCOSAAR

Prezentarea domeniului de cercetare al autoarei

Studiul sistemelor de ecuatii cu derivate partiale cu ajutorul operatorilor monotoni si a

semigrupurilor de contractii neliniare a capatat o dezvoltare foarte mare ın ultimele decenii.

Printre primele lucrari ın acest domeniu amintim lucrarea Acad. V. Barbu [17], ın care s-a

demonstrat cu ajutorul operatorilor monotoni existenta solutiilor pentru un sistem hiperbo-

lic cu conditii la limita neliniare. Contributii ın aceeasi directie au adus prin lucrarile lor

Barbu, Benilan, Brezis, Crandall, Haraux, Morosanu, Pazy, Petrovanu, etc. (vezi de exemplu

lucrarile [16], [17], [18], [22], [26], [32], [47], [102], [167], [168], [169], [171], [172], [173], [170],

[175]).

In acelasi timp, ın ultimele decenii problemele la limita nelocale pentru ecuatii diferentiale

ordinare, ecuatii cu diferente si ecuatii diferentiale fractionare au fost dezvoltate foarte mult.

Studiul acestui tip de probleme este motivat nu numai de interesul teoretic al problemelor,

ci si de faptul ca numeroase fenomene din inginerie, fizica si stiintele vietii pot fi modelate

ın acest fel. De exemplu, problemele cu control feedback, cum ar fi starile stationare ale

unui termostat, unde regulatorul la unul dintre capete permite cresterea sau scaderea tem-

peraturii, ın functie de temperatura ınregistrata ıntr-un alt punct, pot fi interpretate ca o

ecuatie diferentiala ordinara de ordinul al doilea cu o conditie la limita cu trei puncte. Un alt

exemplu este reprezentat de vibratiile unui ”guy wire” (un cablu ıntins utilizat pentru stabil-

itatea unor structuri, ca de exemplu catargele corabiilor, stalpi de radio, turbine de vant sau

corturi) cu sectiune transversala uniforma si compus din N parti de densitati diferite, care

poate fi privita ca o problema la limita cu mai multe puncte (vezi lucrarea [174]). Studiul

problemelor la limita cu mai multe puncte pentru ecuatii diferentiale de ordinul al doilea

a fost initiat de Il’in si Moiseev (vezi [97], [98]). De atunci, astfel de probleme la limita

cu mai multe puncte (cazuri continue sau discontinue) au fost studiate (si sunt studiate si

ın continuare) folosind diverse metode, cum ar fi: teoreme de punct fix pe multimi conuri,

teoreme de continuitate de tip Leray-Schauder, alternative neliniare de tip Leray-Schauder,

teoria indexului punctului fix si teoria gradului de coincidenta.

Probleme la limita cu solutii pozitive descriu multe fenomene din stiintele aplicate cum

ar fi difuzia neliniara generata de surse neliniare, arderea gazelor, si probleme de concentratii

din chimie si biologie (vezi lucrarile [24], [28], [40], [44], [45], [108], [113]). Diverse probleme

care apar ın conductia caldurii, curgerea fluidelor subterane, ın termo-elasticitate si ın fizica

plasmei pot fi reduse la probleme diferentiale neliniare cu conditii la limita integrale (vezi

2

de exemplu lucrarile [29], [30], [102], [181]). In ultimele decenii, multi autori au studiat

probleme scalare cu conditii la limita integrale ([6], [23], [103], [106], [110], [162], [185],

[188]). Mentionam de asemenea lucrarile [33], [41], [46], [99], [100], [101], [109], [116], [183],

[187], [189], [190], unde autorii studiaza existenta solutiilor pozitive pentru cateva sisteme

de ecuatii diferentiale cu conditii la limita integrale. Conditii la limita cuplate apar ın

studiul ecuatiilor de reactie-difuzie si probleme de tip Sturm-Liouville, si au aplicatii ın

multe domenii ale stiintei si ingineriei cum ar fi conductia termica si biologie matematica

(vezi de exemplu lucrarile [10], [11], [12], [35], [36], [119], [122], [176]).

Modelarea matematica a numeroase probleme din informatica, teoria calculatoarelor,

economie, mecanica, sisteme de control, retele neuronale biologice si altele conduc la ecuatii

cu diferente neliniare (vezi lucrarile [114], [118]). In ultimele decenii, multi autori au studiat

asemenea probleme folosind diverse metode, cum ar fi teoreme de punct fix, teoria punc-

tului critic, metoda supra- si sub-solutiilor, teoria indexului punctului fix si teoria gradului

topologic.

Ecuatiile diferentiale fractionare descriu multe fenomene ın diverse domenii ale ingineriei

si ın alte domenii stiintifice cum ar fi fizica, biofizica, chimie, biologie (de exemplu fenomene

de flux sanguin), economie, teoria controlului, procesarea imaginilor si semnalelor, aero-

dinamica, vısco-elasticitate, electro-magnetism, etc. (vezi lucrarile [19], [34], [115], [178],

[180], [182]). Pentru cateva cercetari recente ın acest domeniu mentionam lucrarile [2], [4],

[5], [7], [8], [14], [15], [20], [37], [43], [107], [121], [177], [191], [192] si bibliografiile lor. De

exemplu, ın [177], autorii au dezvoltat un model pentru infectia primara cu HIV care este

un virus ce ataca celulele albe din sange limfocite CD4+T. Acest model poate fi descris

ca un sistem cu trei ecuatii fractionare de ordine diferite (α, β, γ > 0) ın variabilele T

(concentratia celulelor neinfectate CD4+T), I (celulele infectate CD4+T) si V (particulele

de virus HIV libere din sange). Ecuatiile diferentiale fractionare pot fi de asemenea privite

ca un instrument mai bun pentru descrierea proprietatilor ereditare ale diverselor materiale

si procese decat cele care corespund ecuatiilor diferentiale de ordin ıntreg. Modelele de ordin

fractionar s-au dovedit a fi mai precise si realiste decat modelele de ordin ıntreg, si cu acest

avantaj ın aplicatiile acestor modele, este important de stabilit teoretic conditiile pentru

existenta solutiilor pozitive, deoarece rezultatele teoretice pot ajuta oamenii sa ınteleaga mai

bine comportarea dinamica ın procesele practice, astfel ıncat studiul modelelor fractionare

abstracte este cu siguranta oportuna si relevanta ın zilele noastre.

Realizarile stiintifice ale autoarei

In aceasta teza de abilitare sunt prezentate rezultatele mele stiintifice, precum si planul

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de dezvoltare profesionala. Activitatea mea de cercetare ın domeniul ecuatiilor diferentiale

a ınceput ın iunie 1990 o data cu programul de pregatire a doctoratului sub ındrumarea

prof.univ.dr. Gheorghe Morosanu de la Universitatea ”Al.I.Cuza” din Iasi. Pe data de 28

octombrie 1995 am sustinut teza de doctorat cu titlul ”Probleme la limita pentru sisteme cu

derivate partiale de tip hiperbolic si aplicatii”. In teza de doctorat am studiat cateva clase

de sisteme hiperbolice de tipul sistemului telegrafistilor, precum si o generalizare de ordin

superior, cu conditii la limita care pot contine extra-functii si conditii initiale. Am obtinut

rezultate de existenta, unicitate, proprietati de regularitate si comportarea asimptotica a

solutiilor tari si slabe, precum si existenta solutiilor periodice si aproape periodice, unde

variabila spatiala x apartine lui (0, 1), iar coeficientii sunt independenti sau dependenti de

variabila t (timp), neliniaritatile din sisteme sunt functii univoce sau multivoce, iar sursele

(functiile din conditiile la limita) sunt constante sau dependente de timp. Dupa anul 1995 am

continuat sa public rezultatele obtinute ın teza de doctorat ımpreuna cu cateva generalizari

ale acestora, folosind diverse ipoteze pentru coeficientii si neliniaritatile sistemelor.

Dintre rezultatele stiintifice obtinute ıncepand cu anul 2001, voi prezenta ın aceasta teza

de abilitare pe urmatoarele, grupate ın noua capitole.

In Capitolul 1 studiem existenta, unicitatea si proprietati de regularitate ale solutiilor

unui sistem hiperbolic neliniar cu coeficienti dependenti de timp cu conditii la limita si

conditii initiale, care are aplicatii ın electrotehnica. Capitolul 2 se ocupa cu existenta, unic-

itatea si comportarea asimptotica a solutiilor tari si slabe, precum si existenta solutiilor

periodice pentru o problema la limita neliniara pentru un sistem hiperbolic de tipul sistemu-

lui telegrafistilor pe semi-axa pozitiva a variabilei spatiale, unde conditia la limita contine o

extra-functie si un vector (vectorul surselor) constant sau dependent de t. Aceasta problema

are aplicatii ın teoria circuitelor integrate. Capitolul 3 se concentreaza pe existenta, unic-

itatea si comportarea asimptotica ale solutiilor tari si slabe, precum si existenta solutiilor

periodice pentru un sistem de ecuatii cu derivate partiale de ordinul al doilea ın raport

cu variabila spatiala pe semi-axa pozitiva a variabilei spatiale, cu o conditie la limita care

contine o extra-functie si un vector constant sau dependent de t, si cu conditii initiale. In

Capitolele 2 si 3 sunt facute de asemenea cateva observatii ın cazul ın care variabila spatiala

apartine lui R. Capitolul 4 este focalizat pe existenta, unicitatea si comportarea asimptotica

ale solutiilor pentru cateva sisteme de ecuatii diferentiale ordinare cu un numar finit sau

infinit de ecuatii, ın spatii Hilbert, cu conditii extreme si conditii initiale. Aceste probleme

sunt versiuni semi-discretizate ın raport cu variabila spatiala (sisteme cu diferente de ordinul

ıntai) ale unor probleme la limita pentru sisteme hiperbolice. Pentru demonstratiile rezul-

tatelor noastre din Capitolele 1-4, folosim cateva teoreme din teoria operatorilor monotoni si

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a ecuatiilor neliniare de evolutie ın spatii Hilbert, iar pentru existenta solutiilor periodice ale

problemelor din Capitolele 2 si 3, aplicam o teorema de punct fix a lui Browder si Petryshyn.

Problemele din aceste prime patru capitole au fost scrise ca probleme Cauchy ın spatii de

functii convenabil alese, cu ajutorul unor operatori maximal monotoni, astfel ıncat teoria

ecuatiilor neliniare de evolutie ın spatii Hilbert a fost folosita concomitent cu exploatarea car-

acterului particular al acestor probleme. Sunt de asemenea studiate si problemele stationare

asociate acestor probleme. In Capitolul 4 prezentam si cateva generalizari ale teoremei lui

Minty pentru caracterizarea unui operator maximal monoton ın spatii Hilbert sau spatii

Banach.

Capitolul 5 este consacrat studierii existentei, multiplicitatii si non-existentei solutiilor

pozitive pentru cateva clase de sisteme de ecuatii diferentiale ordinare neliniare de ordinul

al doilea cu parametri sau fara parametri, cu conditii la limita integrale Riemann-Stieltjes,

si pentru care neliniaritatile sunt functii nesingulare sau singulare. Capitolul 6 este dedicat

existentei, multiplicitatii si non-existentei solutiilor pozitive pentru clase de sisteme de ecuatii

diferentiale neliniare de ordin superior cu parametri sau fara parametri, cu conditii la limita

cu mai multe puncte, pentru care neliniaritatile sunt functii nesingulare sau singulare. Este

de asemenea studiat un sistem de ecuatii diferentiale de ordin superior cu neliniaritati care ısi

schimba semnul si are conditii la limita integrale Riemann-Stieltjes. In Capitolul 7 studiem

existenta, multiplicitatea si non-existenta solutiilor pozitive pentru clase de sisteme de ecuatii

neliniare cu diferente de ordinul al doilea, cu parametri sau fara parametri, si cu conditii la

limita cuplate sau necuplate cu mai multe puncte.

Capitolul 8 se ocupa cu existenta, multiplicitatea si non-existenta solutiilor pozitive

pentru clase de sisteme de ecuatii diferentiale neliniare fractionare Riemann-Liouville cu

parametri sau fara parametri, cu conditii la limita integrale Riemann-Stieltjes necuplate,

pentru care neliniaritatile sunt functii nesingulare sau singulare. Este de asemenea studiat

un sistem de ecuatii fractionare cu neliniaritati care ısi schimba semnul. Capitolul 9 este ded-

icat studiului existentei, multiplicitatii si non-existentei solutiilor pozitive pentru sisteme de

ecuatii diferentiale neliniare fractionare Riemann-Liouville cu parametri sau fara parametri,

cu conditii la limita integrale Riemann-Stieltjes cuplate, si pentru care neliniaritatile sunt

functii nesingulare sau singulare. Este de asemenea studiat un sistem de ecuatii fractionare

cu neliniaritati nesingulare sau singulare care ısi schimba semnul, si cu conditii la limita in-

tegrale. In Capitolele 5-9 sunt prezentate diverse exemple care arata validitatea rezultatelor

principale.

In centrul rezultatelor fiecaruia dintre ultimele cinci capitole sunt aplicatii ale teoremei

de punct fix a lui Guo-Krasnosel’skii pentru operatori ne-expansivi si ne-contractivi ıntr-un

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con. Ca o caracteristica unica a aplicatiilor acestei teoreme de punct fix este reprezentarea

noua a functiilor Green, care ın cele din urma ofera aproape o lista de verificare ın deter-

minarea conditiilor pentru existenta solutiilor pozitive ın raport cu neliniaritatile date. In

demonstratiile multor rezultate principale folosim de asemenea teorema de punct fix a lui

Schauder, alternativa neliniara de tip Leray-Schauder si cateva teoreme din teoria indexului

punctului fix.

Rezultatele prezentate ın aceasta teza de abilitare au fost publicate ın lucrarile: [129]

(Capitolul 1), [136] si [142] (Capitolul 2), [133], [148] si [156] (Capitolul 3), [137], [138], [139],

[140], [141], [143], [144] si [146] (Capitolul 4), [60], [71], [80], [50], [51], [58] si [85] (Capitolul

5), [53], [69], [63], [70], [48], [161] si [160] (Capitolul 6), [57], [68], [81], [54], [55], [72], [74],

[79] si [88] (Capitolul 7), [64], [82], [65], [86] si [159] (Capitolul 8), [66], [83], [84], [73], [87],

[76], [77] si [78] (Capitolul 9).

Rezultatele stiintifice prezentate ın Capitolele 2 si 4, precum si o parte din Capitolul 3

au fost raportate la granturile: CNCSIS GR 77/11.06.2008, Cod 125, Tema 40; CNCSIS GR

80/23.05.2007, Cod 125, Tema 54; CNCSIS GR 217/18.09.2006, Cod 125, Tema 9, ”Evolutii

neliniare ın spatii Banach si aplicatii”; PN-II-RU-MC-2008-2, Aprilie 2008, Cod CNCSIS

87. Rezultatele stiintifice prezentate ın Capitolele 5-9 au fost raportate la grantul PN-II-

ID-PCE-2011-3-0557 ”Probleme la limita pentru sisteme de ecuatii diferentiale neliniare cu

aplicatii ın electronica si mecanica” (2011-2016). La cele cinci granturi de mai sus am fost

directorul proiectelor.

Din anul 2001 am scris trei monografii:

1) Johnny Henderson, Rodica Luca, ”Boundary Value Problems for Systems of Differ-

ential, Difference and Fractional Equations. Positive Solutions”, ELSEVIER, Amsterdam,

2016 (2015 in Elsevier), (322 pag.), ISBN: 978-0-12-803652-5, raportata la grantul PN-II-ID-

PCE-2011-3-0557;

2) Rodica Luca-Tudorache, ”Probleme neliniare de evolutie ın spatii Hilbert”, Perfor-

mantica, Iasi, 2007 (148 pag.), ISBN: 978-973-730-359-2, raportata la grantul CNCSIS GR

80/23.05.2007, Cod 125, Tema 54;

3) Rodica Luca-Tudorache, ”Probleme la limita pentru sisteme neliniare hiperbolice si

aplicatii”, Casa de Editura Venus, Iasi, 2003 (236 pag.), ISBN: 973-8174-94-5.

Relevanta activitatii stiintifice si recunoasterea activitatii nationale si internationale ın

domeniul ecuatiilor diferentiale, ecuatiilor cu diferente si ecuatiilor fractionare sunt subliniate

de publicatiile mele, multe dintre ele fiind ın colaborare cu Prof. Johnny Henderson de la

Universitatea Baylor, Waco, Texas, SUA, precum si de citarile acestora ın reviste indexate

ISI sau din alte baze de date internationale.

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Proiecte de cercetare la care autoarea a fost director de proiect

Din anul 2001 am fost director de proiect la urmatoarele granturi cıstigate prin competitie

nationala:

1) PN-II-ID-PCE-2011-3-0557 ”Probleme la limita pentru sisteme de ecuatii diferentiale

neliniare cu aplicatii ın electronica si mecanica” (2011-2016), la care echipa de cercetare a

grantului a raportat 73 de lucrari (45 ın reviste ISI, 20 ın reviste BDI, 4 ın volume de

conferinte internationale - din care 3 ın volume ISI, si 4 lucrari ın evaluare la reviste ISI) si o

monografie ın editura ELSEVIER (vezi https:// sites.google.com/site/rodicalucatudorache/

home/ pn-ii-grant);

2) CNCSIS GR 77/11.06.2008, Cod 125, Tema 40; CNCSIS GR 80/23.05.2007, Cod

125, Tema 54; CNCSIS GR 217/18.09.2006, Cod 125, Tema 9, ”Evolutii neliniare ın spatii

Banach si aplicatii”, la care echipa de cercetare a raportat 50 de lucrari (din care 12 lucrari

ın reviste ISI) si 9 carti (din care 5 monografii), (vezi http://www.cncsis.ro, la Programe de

finantare - Granturi (1995-2008));

3) PN-II-RU-MC-2008-2, Aprilie 2008, Cod CNCSIS 87 - un grant de mobilitate pen-

tru ”Fifth World Congress of Nonlinear Analysts”, Orlando, Florida, SUA, 2-9 iulie 2008

(mobilitatea 1-10 iulie 2008), la care am raportat lucrarea ISI [146];

4) Grantul pentru ”6-th International Congress on Industrial and Applied Mathematics”

ICIAM 2007, 16-20 iulie 2007, Zurich, Elvetia, de la Swiss Federal Institute of Technology

Zurich, castigat prin competitie internationala.

De asemenea am fost membra ın echipele de cercetare ale altor 11 granturi sau proiecte

nationale.

Participarea mea ın mai multe granturi nationale ın calitate de director sau membra

a echipei a dezvoltat abilitatile si competentele mele privind managementul unor astfel de

proiecte.

Realizarile profesionale si didactice ale autoarei

In anul 2001 am devenit Conferentiar universitar, iar ın anul 2008 am fost promovata

Profesor Universitar ın cadrul Departamentul de Matematica al Universitatii Tehnice ”Ghe-

orghe Asachi” din Iasi.

In cadrul activitatii didactice, am scris urmatoarele cursuri si culegeri de probleme

pentru studentii din anii I si II - licenta si anul I - master:

1) Nicoleta Breaz, Lucia Cabulea, Ariana Pitea, Ioan Rasa, Rodica Tudorache, Gheorghita

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Zbaganu, ”Probabilitati si statistica”, 2013, (Cap. 3 si 4, 62 pag.), Editura StudIS, Vatra-

Dornei, ISBN: 978-606-624-309-4, (304 pag., format academic).

2) Rodica Luca-Tudorache, ”Probleme de analiza matematica. Calcul integral”, Casa

de Editura Venus, Iasi, 2007 (381 pag., format academic), ISBN: 973-756-039-6, 978-973-

756-039-1 (editura CNCSIS).

3) Rodica Luca-Tudorache, ”Probleme de analiza matematica. Calcul diferential”, Per-

formantica, Iasi, 2006 (346 pag., format academic), ISBN: 973-730-206-0, ISBN: 978-973-

730-206-9 (editura CNCSIS).

4) Rodica Luca-Tudorache, ”Probleme de teoria probabilitatilor”, Editura Tehnopress,

Iasi, 2006 (100 pag., format academic), ISBN-10 973-702-354-4, ISBN-13 978-973-354-4 (ed-

itura CNCSIS).

5) Rodica Luca-Tudorache, ”Analiza matematica. Calcul diferential”, Editura Tehno-

press, Iasi, 2005 (321 pag., format academic), ISBN: 973-702-151-7 (editura CNCSIS).

Pe langa activitatile mele didactice, adica cursuri si seminarii de Analiza matematica, Al-

gebra liniara, geometrie si ecuatii diferentiale (pentru studentii din anul I licenta) si Matem-

atici speciale (pentru studentii din anul II licenta si anul I master), am organizat ın perioada

2006-2011 si ın anul 2016 un Cerc de Matematica unde am lucrat cu cei mai buni studenti

din anul I probleme deosebite de Analiza matematica.

Planul de dezvoltare profesional

In viitorul apropiat as dori sa ma asociez la o Scoala Doctorala de Matematica, sau sa

colaborez cu profesori conducatori de doctorat ın cadrul Scolilor Doctorale de la Universitatea

Tehnica ”Gheorghe Asachi” din Iasi.

In plus, voi continua cercetarile din ultimii ani prin studierea unor noi probleme la

limita nelocale pentru sisteme de ecuatii diferentiale ordinare, sisteme de ecuatii cu diferente

si sisteme de ecuatii diferentiale fractionare.

Dinamica proceselor evolutive este adesea supusa unor schimbari bruste cum ar fi

socurile sau dezastrele naturale. Deseori aceste perturbari pe termen scurt sunt consid-

erate ca actionand instantaneu sau sub forma de impulsuri. Ecuatiile diferentiale impulsive

supuse efectelor de impuls au fost dezvoltate ın modelarea problemelor impulsive ın fizica,

dinamica populatiei, biologie, tehnologie chimica, control optimal, biotehnologie, farmaco-

cinetica, robotica industriala (a se vedea [21], [42], [124]). Probleme la limita cu argumente

deviate constitue o alta clasa importanta de probleme. Prin urmare este necesar sa se extinda

aceste cercetari.

Voi studia existenta, multiplicitatea si non-existenta solutiilor pozitive pentru sisteme

8

de ecuatii diferentiale ordinare neliniare de ordinul al doilea sau de ordin superior cu diverse

conditii la limita nelocale incluzand conditii la limita Riemann-Stieltjes si conditii impulsive.

Voi considera diverse clase de sisteme, si anume: sisteme cu parametri sau fara parametri,

sisteme cu neliniaritati nesingulare sau singulare, sisteme cu neliniaritati care ısi schimba

semnul, sisteme cu termeni integrali sau cu argumente deviate. Intervalul pentru variabila

t poate fi finit sau infinit, iar conditiile la limita pot fi necuplate sau cuplate ın functiile

necunoscute ale sistemelor. De asemenea voi investiga unele cazuri discrete ale acestor

probleme, adica sisteme finite sau infinite de ecuatii cu diferente cu conditii la limita cu mai

multe puncte. O alta clasa de sisteme pe care o voi studia este clasa sistemelor de ecuatii

diferentiale fractionare Riemann-Liouville cu conditii la limita integrale Riemann-Stieltjes

care pot contine derivate fractionare.

9

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