Teoreme de compresie-extensie de tip Krasnoselskii şi aplicaţii (Rezumatul tezei de doctorat)

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1 Teoreme de compresie-extensie de tip Krasnoselskii şi aplicaţii (Rezumatul tezei de doctorat) Sorin Monel Budişan Coordonator ştiinţi c: Prof. dr. Radu Precup

2 Cuprins Introducere 1 1 Generaliz¼ari ale teoremei de punct x în conuri a lui Krasnoselskii Teoreme de punct x într-o coroan¼a conic¼a obişnuit¼a Teoreme de punct x într-o coroan¼a conic¼a de nit¼a prin una sau dou¼a funcţionale Teoreme de punct x de tip Krasnoselskii în raport cu trei funcţionale O teorem¼a de tip Krasnoselskii pentru ecuaţia de coincidenţ¼a Aplicaţii Aplicaţii ale teoremei de punct x în conuri a lui Krasnoselskii la ecuaţii diferenţiale de întârziere Aplicaţii la ecuaţii cu p-laplacian Aplicaţii ale versiunii vectoriale a teoremei de punct x a lui Krasnoselskii Existenţa soluţiilor periodice pozitive ale sistemelor diferenţiale Aplicaţii ale generaliz¼arilor teoremei de punct x a lui Krasnoselskii 11 Cuvinte cheie: con, coroan¼a conic¼a, punct x, ecuaţie diferenţial¼a, sistem diferenţial, problem¼a cu valori pe frontier¼a, p-laplacian, teoreme de compresieextensie. 1

3 Introducere Teoremele de compresie-extensie de tip Krasnoselskii reprezint¼a un instrument principal în studiul problemelor neliniare pentru ecuaţii şi siteme de ecuaţii integrale, diferenţiale sau cu derivate parţiale. Ele sunt folosite nu doar pentru a asigura existenţa soluţiilor, ci şi pentru a localiza soluţiile într-o coroan¼a conic¼a sau în alte domenii de acest tip. Aceasta permite, în particular, s¼a obţinem soluţii multiple ale problemelor neliniare când neliniarit¼aţile sunt oscilante. În ultimele trei decenii, a fost produs¼a o bogat¼a literatur¼a pe acest subiect, atât în teorie cât şi în aplicaţii. Din punct de vedere teoretic, sunt cunoscute dou¼a mari tendinţe de abordare a subiectului: prima se desf¼aşoar¼a în cadrul teoriei punctului x şi foloseşte în principal teorema de punct x a lui Schauder şi generaliz¼arile sale, în timp ce a doua foloseşte teoria gradului topologic. Rezultatul originar datorat lui Krasnoselskii a fost obţinut folosind prima abordare, iar aceast¼a tez¼a urmeaz¼a acelaşi drum. Deci, toate rezultatele noastre teoretice sunt bazate pe teoria puntului x. Prezent¼am în continuare teoremele de compresie-extensie ale lui Krasnoselskii, în forma dat¼a de acesta. Teorema 1 (M. A. Krasnoselskii [36]) Fie X un spaţiu Banach real şi e K X un con. Fie N un operator pozitiv şi complet continuu cu N =. Dac¼a exist¼a numerele reale r; R > 0 astfel ca 8 < : Nx x pentru orice x 2 K cu j x j r; x 6= şi pentru orice " > 0 Nx (1 + ")x pentru orice x 2 K cu j x j R, atunci operatorul N are cel puţin un punct x nenul în K: Teorema 2 (M. A. Krasnoselskii [36])Fie X un spaţiu Banach real şi e K X un con. Fie N un operator pozitiv şi complet continuu cu N =. Dac¼a exist¼a numerele reale r; R > 0 astfel ca pentru orice " > 0 8 < : Nx (1 + ")x pentru orice x 2 K cu j x j r; x 6= şi Nx x pentru orice x 2 K cu j x j R; atunci operatorul N are cel puţin un punct x nenul în K: (1) (2) 1

4 În literatur¼a, pentru r < R, condiţiile de compresie din Teorema 1 sunt în majoritatea cazurilor înlocuite de condiţiile: k Nu kk u k; dac¼a k u k= r; (3) k Nu kk u k; dac¼a k u k= R: Asem¼an¼ator, condiţiile de extensie din Teorema 2 sunt înlocuite de: k Nu kk u k; dac¼a k u k= r; k Nu kk u k; dac¼a k u k= R: În aceast¼a tez¼a obţinem mai multe generaliz¼ari ale teoremelor de punct x în conuri ale lui Krasnoselskii, unde punctul x e localizat într-o coroan¼a conic¼a generalizat¼a având una din formele: unde ' pe K, K r;r := fu 2 K : r '(u) (u) Rg, (5) K r;r := fu 2 K : r (u); '(u) Rg, (6) (4) K r;r := fx 2 K : r (x) Rg (7) Condiţiile de compresie (1) şi (3), precum şi condiţiile de extensie (2) şi (4) vor generalizate folosind funcţionalele ', şi, ca mai jos: '(u) ' (Nu) dac¼a '(u) = r; (condiţii de compresie) (u) (Nu) dac¼a (u) = R; '(u) ' (Nu) dac¼a '(u) = r; (condiţii de extensie). (u) (Nu) dac¼a (u) = R; pentru formele (5), (6) ale coroanei conice, şi '(u) ' (Nu) dac¼a (u) = r (condiţii de compresie), (u) (Nu) dac¼a (u) = R; '(u) ' (Nu) dac¼a (u) = r (u) (Nu) dac¼a (u) = R; (condiţii de extensie), pentru forma (7) a coroanei conice. Comparând cu literatura existent¼a, rezultatele noastre completeaz¼a, extind şi generalizeaz¼a în mai multe direcţii rezultatele obţinute de R. Legget, L. Williams [38], R. Avery, J. Henderson, D. O Regan [4], [5], R. Precup [58], [59] şi alţii. Astfel de extensii folosind folosind funcţionalele '; şi au fost motivate de aplicaţii concrete unde acest tip de funcţionale apar într-un mod natural. De exemplu, putem considera funcţionalele(see Section 3.5): '(u) : = min t2i (u) : = max 8 < (u) : = : t2[0;1] u(t); (min t2i u(t))2 max u((t), t2[0;1] dac¼a u nu e identic zero, 0; dac¼a u 0: 2

5 Subliniem faptul c¼a funcţionala e folosit¼a în aceast¼a tez¼a pentru prima dat¼a. O parte consistent¼a a tezei este dedicat¼a aplicaţiilor teoremelor de compresieextensie la mai multe clase de probleme: probleme bilocale pentru ecuaţii diferenţiale de ordinul doi, ecuaţii funcţional-diferenţiale, sisteme de ecuaţii diferenţiale şi ecuaţii cu p-laplacian. Rezultatele noastre privind aplicaţiile completeaz¼a şi extind unele rezultate date de L.H. Erbe, H. Wang [14], D. O Regan, R. Precup [54], R. Avery, J. Henderson, D. O Regan [4] şi alţii. Aceast¼a tez¼a se bazeaz¼a pe lucr¼arile noastre S. Budisan [8], [9], [11], [12] şi S. Budisan, R. Precup [10]. Teza este structurat¼a pe trei capitole, o parte introductiv¼a şi o list¼a bibliogra c¼a. 3

6 Capitolul 1 Generaliz¼ari ale teoremei de punct x în conuri a lui Krasnoselskii Acest capitol conţine principalele contribuţii teoretice ale acestei teze. Acestea conţin condiţiile de compresie-extensie exprimate în termenii unor diferite funcţionale care înlocuiesc total sau doar parţial norma spaţiului. Acest capitol se bazeaz¼a pe lucr¼arile S. Budisan [11], [12]. 1.1 Teoreme de punct x într-o coroan¼a conic¼a obişnuit¼a. Prezentarea din aceast¼a secţiune este bazat¼a pe lucrarea S. Budisan [11]. În aceast¼a secţiune obţinem mai multe generaliz¼ari ale teoremelor de punct x ale lui Krasnoselskii în conul K; unde punctul x este localizat în coroana conic¼a K r;r := fu 2 K : r j u j Rg: Condiţiile de compresie-extensie sunt exprimate cu dou¼a funcţionale, anume: '(u) ' (Nu) dac¼a j u j= r (condiţii de compresie) (u) (Nu) dac¼a j u j= R; respectiv '(u) ' (Nu) dac¼a j u j= r (u) (Nu) dac¼a j u j= R: (condiţii de extensie) De-a lungul acestei secţiuni consider¼am c¼a (X; j : j) este un spaţiu liniar normat, K X este un con pozitiv, "" este relaţia de ordine indus¼a de K, 4

7 "" este relaţia de ordine strict¼a indus¼a de K şi R + = [0; 1):Dintre rezultatele din aceast¼a secţiune, prezent¼am aici Teorema 2.1 şi Observaţia2.2. Teorema 3 (S. Budisan [11]) Fie r; R 2 R + astfel încât 0 < r < R. Presupunem c¼a N : K r;r! K este un operator complet continuu şi e ' : K! R +, : K! R: De asemenea, presupunem c¼a sunt satisf¼acute urm¼atoarele condiţii: 8 '(0) = 0 şi exist¼a h 2 K f0g astfel ca >< '(h) > 0; pentru orice 2 (0; 1]; (i1) '(h) > 0; pentru orice 2 (0; 1]; >: '(x + y) ' (x) + ' (y) pentru orice x; y 2 K; (i2) (x) > (x) pentru orice > 1 şi pentru orice x 2 K cu j x j= R; '(u) ' (Nu) dac¼a j u j= r (i3) (u) (Nu) dac¼a j u j= R: Atunci N are un punct x în K r;r. Observatia 4 (S. Budisan [11]) (1) Dac¼a X := C[0; 1]; > 0; I [0; 1], I 6= [0; 1]; k x k:= max x(t) şi K := fx 2 C[0; 1] : x 0 pe [0; 1]; x(t) k x k t2[0;1] pentru orice t 2 Ig, o funcţional¼a ce satisface (i1) este '(x) := min t2i x(t): Într-adev¼ar, '(0) = 0; exist¼a h 2 K 2 (0; 1] şi f0g astfel ca '(h) > 0; pentru orice '(x + y) = min[x(t) + y(t)] minx(t) + miny(t) = '(x) + '(y): t2i t2i t2i (2) Norma este un exemplu de funcţional¼a ce satisface (i2). Alte rezultate obţinute în aceast¼a secţiune sunt: Teorema 2.3, Observaţia 2.4, Teorema Teoreme de punct x într-o coroan¼a conic¼a de nit¼a prin una sau dou¼a funcţionale Prezentarea din aceast¼a secţiune e bazat¼a pe lucrarea S. Budisan [12]. În aceast¼a secţiune obţinem mai multe generaliz¼ari ale teoremelor de punct x ale lui Krasnoselskii în conul K, unde punctul x este localizat într-o coroan¼a conic¼a generalizat¼a de una din formele: K r;r := fu 2 K : r '(u) Rg; sau K r;r := fu 2 K : r '(u) (u) Rg, unde ' pe K; K r;r := fu 2 K : r (u); '(u) Rg, unde ' pe K. 5

8 Atât condiţiile de compresie-extensie, cât şi condiţiile pe frontier¼a sunt exprimate folosind aceleaşi funcţionale ' and, anume: '(u) ' (Nu) dac¼a '(u) = r (condiţii de compresie) (u) (Nu) dac¼a (u) = R; or '(u) ' (Nu) dac¼a '(u) = r (u) (Nu) dac¼a (u) = R; (condiţii de extensie). De-a lungul acestei secţiuni consider¼am c¼a (X; j : j) este un spaţiu liniar normat, K X este un con pozitiv, "" este relaţia de ordine indus¼a de K, "" este relaţia de ordine strict¼a indus¼a de K şi R + = [0; 1):Dintre rezultatele din aceast¼a secţiune, prezent¼am aici Teorema 2.8, Teorema 2.10 şi Observaţia 2.9. Teorema 5 (S. Budisan [12]) Fie K r;r = fx 2 K : r ' (x) Rg o mulţime nevid¼a, unde r; R 2 R +, r < R şi e '; : K! R + funcţionale continue. Presupunem c¼a N : K r;r! K este un operator continuu cu N(K r;r ) relativ compact, ' (0) = 0 şi h 2 K n ' 1 (0) astfel încât ' (h) > 0 pentru orice > 0. De asemenea, presupunem c¼a urm¼atoarele condiţii sunt satisf¼acute: (i) ' pe K, (ii) ' (x) = ' (x) pentru orice x 2 K şi 2 (0; 1), (iii) '(x + y) '(x) + '(y), pentru orice x; y 2 K; (iv) (x) = (x) pentru orice x 2 K şi > 0, (v) N(' 1 (r)) este m¼arginit¼a, (vi) Atunci N are un punct x u 2 K r;r. ' (x) ' (Nx) dac¼a ' (x) = r; (x) (Nx) dac¼a (x) R: Observatia 6 (S. Budisan [12]) Dac¼a X := C([0; 1]; R + ); I [0; 1]; I 6= [0; 1]; > 0; k x k:= max x(t); K := fx 2 X : x(t) k x k pentru orice t 2 Ig atunci t2[0;1] '(x) := minx(t) şi (x) := max x(t) sunt funcţionale ce satisfac ipotezele Teoremei t2i t2[0;1] 5. Teorema 7 (S. Budisan [12]) Fie '; : K! R + funcţionale continue, ' (0) = 0 şi h 2 K n ' 1 (0) astfel ca '(h) > 0 pentru orice > 0. Presupunem c¼a ' pe K şi e K r;r := fx 2 K : r ' (x) (x) Rg o mulţime nevid¼a, unde r; R 2 R +, r < R. De asemenea, presupunem c¼a N : K r;r! K este un operator continuu cu N(K r;r ) relativ compact¼a : Presupunem c¼a urm¼atoarele condiţii sunt satisf¼acute: (i) ' (x) = ' (x) pentru orice x 2 K şi 2 (1; 1), (ii) '(x + y) '(x) + '(y), pentru orice x; y 2 K; 6

9 (iii) (x) = (x) pentru orice x 2 K şi 0, (iv) R'(x) r (x) pentru orice x 2 K; (v) N(' 1 (r)) este m¼arginit¼a, (vi) Atunci N are un punct x u 2 K r;r. ' (x) ' (Nx) dac¼a ' (x) = r; (x) (Nx) dac¼a (x) = R: Alte rezultate obţinute în aceast¼a secţiune sunt: Teorema 2.6, Observaţia 2.7, Teorema 2.11, Observaţia Teoreme de punct x de tip Krasnoselskii în raport cu trei funcţionale Prezentarea din aceast¼a secţiune este bazat¼a pe lucrarea S. Budisan [12]. În aceast¼a secţiune obţinem mai multe generaliz¼ari ale teoremelor de punct x ale lui Krasnoselskii în conul K, unde punctul x este localizat într-o coroan¼a conic¼a generalizat¼a de forma K r;r := fx 2 K : r (x) Rg; prin intermediul unei funcţionale, iar condiţiile de compresie-extensie sunt date în termenii a dou¼a funcţionale ' şi, în timp ce condiţiile pe frontier¼a sunt date folosind aceleaşi funcţionale ca în de niţia coroanei conice, anume: '(u) ' (Nu) dac¼a (u) = r (condiţii de compresie), (u) (Nu) dac¼a (u) = R; respectiv '(u) ' (Nu) dac¼a (u) = r (u) (Nu) dac¼a (u) = R; (condiţii de extensie). De-a lungul acestei secţiuni consider¼am c¼a (X; j : j) este un spaţiu liniar normat, K X este un con pozitiv, "" este relaţia de ordine indus¼a de K, "" este relaţia de ordine strict¼a indus¼a de K şi R + = [0; 1):Dintre rezultatele din aceast¼a secţiune prezent¼am aici Teorema Teorema 8 (S. Budisan [12]) Fie '; ; : K! R + funcţionale continue, (0) = 0 şi h 2 K n 1 (0) astfel ca '(h) > 0 pentru orice > 0. Fie K r;r := fx 2 K : r (x) Rg o mulţime nevid¼a, unde r; R 2 R +, r < R. Presupunem c¼a N : K r;r! K este un operator continuu cu N(K r;r ) relativ compact¼a : Presupunem c¼a urm¼atoarele condiţii sunt satisf¼acute: (i) (x) = (x) pentru orice x 2 K şi 0, (ii) '(x + y) '(x) + '(y), pentru orice x; y 2 K; 7

10 (iii) (x) (x) pentru orice x 2 K cu (x) = R pentru orice 2 (1; 1); (iv) (x) > 0 pentru orice x 2 K cu (x) = R, (v) N( 1 (r)) este m¼arginit¼a, (vi) Atunci N are un punct x u 2 K r;r. ' (x) ' (Nx) dac¼a (x) = r; (x) (Nx) dac¼a (x) = R: Un rezultat similar din aceast¼a secţiune este Teorema O teorem¼a de tip Krasnoselskii pentru ecuaţia de coincidenţ¼a În aceast¼a secţiune obţinem o teorem¼a abstract¼a de tip Krasnoselskii pentru urm¼atoarea ecuaţie de coincidenţ¼a: Lx = Nx Rezultatul din aceast¼a secţiune este Teorema

11 Capitolul 2 Aplicaţii Acest capitol conţine aplicaţiile principale ale tezei. Acestea se refer¼a la existenţa şi localizarea soluţiilor mai multor clase de probleme, cum ar ecuaţii şi sisteme diferenţiale sau ecuaţii cu derivate parţiale cu p-laplacian. Capitolul se bazeaz¼a pe lucr¼arile S. Budisan [8], [9], [12] şi S. Budisan, R. Precup [10]. 2.1 Aplicaţii ale teoremei de punct x în conuri a lui Krasnoselskii la ecuaţii diferenţiale de întârziere Rezultateledin aceast¼a secţiune au fost stabilite în lucrarea S. Budisan [8]. Rezultatele din aceast¼a secţiune se refer¼a la urm¼atoarea problem¼a bilocal¼a: 8 >< >: u 00 (t) + a (t) f (u (g (t))) = 0; 0 < t < 1 u (0) u 0 (0) = 0; u (1) + u 0 (1) = 0; u (t) = k; t < 0: Acestea sunt: Lema 3.1, Teorema 3.2, Observaţia 3.3 şi Observaţia Aplicaţii la ecuaţii cu p-laplacian Conţinutul acestei secţiuni se bazeaz¼a pe lucrarea [9], iar dintre rezultatele existente prezent¼am mai jos unele idei şi rezultate. În aceast¼a secţiune obţinem mai multe aplicaţii ale teoremei de punct x în conuri a lui Krasnoselskii la urm¼atoarea problem¼a cu valori pe frontier¼a pentru un sistem de ecuaţii cu p- Laplacian(p 2): 8 >< >: div(j ru js p 2 ru) = f(u) pentru j x j< T; u > 0 pentru 0 <j x j< T; u = 0 pentru x = 0; ru = 0 pentru j x j= T: 9

12 Rezultatele din aceast¼a secţiune sunt: Teorema 3.6, Observaţia 3.7, Observaţia 3.8, Teorema 3.9, Teorema 3.10, Observaţia 3.11, 2.3 Aplicaţii ale versiunii vectoriale a teoremei de punct x a lui Krasnoselskii Aceast¼a secţiune se bazeaz¼a pe lucrarea S. Budisan, R. Precup [10]. Prezent¼am o aplicaţie a teoremei de punct x în conuri a lui Krasnoselskii la sistemul funcţional-diferenţial de ordinul doi: u 00 1(t) + a 1 (t)f 1 (u 1 (g(t)); u 2 (g (t))) = 0; u 00 2(t) + a 2 (t)f 2 (u 1 (g(t)); u 2 (g (t))) = 0 (0 < t < 1) cu condiţiile pe frontier¼a 8 < i u i (0) i u 0 i (0) = 0; : i u i (1) + i u 0 i (1) = 0; u i (t) = k i pentru t < 0 (i = 1; 2): Rezultatele din aceast¼a secţiune sunt: Observaţia 3.12, Lema 3.13, Teorema 3.14, Observaţia Existenţa soluţiilor periodice pozitive ale sistemelor diferenţiale În aceast¼a secţiune complet¼am rezultatele din lucrarea R. Precup [57] prin furnizarea de condiţii su ciente ce asigur¼a ipotezele Teoremei 3.1 din lucrarea R. Precup [57]. Autorul lucr¼arii R. Precup [57] obţine condiţii su ciente în trei cazuri, iar obiectivul nostru este s¼a g¼asim condiţii su ciente în alte dou¼a cazuri. Aceste condiţii garanteaz¼a existenţa şi localizarea soluţiilor periodice pozitive ale sistemului diferenţial neliniar: 0 u 1(t) = u 0 2(t) = a 1 (t)u 1 (t) + " 1 f 1 (t; u 1 (t); u 2 (t)) a 2 (t)u 2 (t) + " 2 f 2 (t; u 1 (t); u 2 (t)); unde pentru i 2 f1; 2g : a i 2 C(R; R), Z! 0 a i (t)dt 6= 0, " i = sign Z! 0 a i (t)dt, f i 2 C(R R 2 +; R + ), şi a i, f i (:; u 1 ; u 2 ) sunt funcţii!-periodice pentru un anumit! > 0. Rezultatele din aceast¼a secţiune sunt: Teorema 3.17, Observaţia 3.18, Teorema 3.19, Observaţia

13 2.5 Aplicaţii ale generaliz¼arilor teoremei de punct x a lui Krasnoselskii În aceast¼a secţiune obţinem aplicaţii ale teoremelor abstracte din Capitolul 2. Prezentarea din aceast¼a secţiune se bazeaz¼a pe lucrarea S. Budisan [12]. Studiem în continuare existenţa cel puţin unei soluţii a urm¼atoarei probleme bilocale: u 00 (t) + f(t; u(t)) = 0; 0 t 1; (3.1) u(0) = u(1) = 0; (3.2) unde f : [0; 1] R +! R + este continu¼a. C¼aut¼am soluţii u 2 C 2 [0; 1] ale (3.1)-(3.2), care sunt nenegative şi concave pe [0; 1]. Vom aplica teoremele din precedentul capitol la un operator complet continuu al c¼arui nucleu este funcţia lui Green, funcţia G(t; s), a ecuaţiei u 00 = 0; supus¼a condiţiilor pe frontier¼a (3.2). Dintre rezultatele din aceast¼a secţiune menţion¼am um¼atorul rezultat de multiplicitate a soluţiilor(corolarul 3.29): Corolarul 9 (S. Budisan [12]) Presupunem c¼a exist¼a numerele pozitive r 1 ; r 2 ; R 1 şi R 2 astfel ca 0 < 16r 1 < R 1 11r2 1024, 16r 2 < R 2, iar f satisface urm¼atoarele condiţii: (i) f(s; x) 6r 1 pentru orice s 2 [0; 1]; pentru orice x 2 [0; 16r 1 ], şi (ii) f(s; x) R 1 pentru orice s 2 I, pentru orice x 2 [R 1 ; 16R 1 ], (iii) f(s; x) 6r 2 pentru orice s 2 [0; 1]; pentru orice x 2 [0; 16r 2 ], (iv) f(s; x) R 2 pentru orice s 2 I, pentru orice x 2 [R 2 ; 16R 2 ], Atunci (3.1)-(3.2) are cel puţin dou¼a soluţii u 1 şi u 2 cu r 1 min u 1 4R 1 ; t2[ 1 4 ; 3 4 ] r 1 max t2[0;1] u 1 16R 1 ; r 2 min u 2 4R 2 ; t2[ 1 4 ; 3 4 ] r 2 max t2[0;1] u 2 16R 2 : Alte rezultate obţinute în aceast¼a secţiune sunt: Teorema 3.21, Observaţia 3.22, Teorema 3.23, Observaţia 3.24, Teorema 3.25, Observaţia 3.26, Teorema 3.27, Obsevaţia 3.28, Observaţia 3.30, Teorema 3.31, Observaţia

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