Positive solutions of singular (k,n-k) conjugate boundary value problem

Transcription

1 Joural of Applied Mathematic & Bioiformatic vol5 o 25-2 ISSN: prit olie Sciepre Ltd 25 Poitive olutio of igular - cojugate boudar value problem Ligbi Kog ad Tao Lu 2 Abtract Poitive olutio of igular oliear cojugate boudar value problem i tudied b emploig a priori etimate the coe theorem ad the fied poit ide Keword: Poitive olutio; Multiplicit; Fied poit ide Itroductio I thi paper we are cocered with poitive olutio for igular - cojugate boudar value problem f < < 2 Ligbi Kog School of Mathematic ad Statitic Northeat Petroleum Uiverit Daqig 68 Chia lbidq@26com Tao Lu School of Mathematic ad Statitic Northeat Petroleum Uiverit Daqig 68 Chia @qqcom Article Ifo: Received : October 2 Revied : November 2 Publihed olie : Jauar 5 25

2 Poitive olutio of igular - cojugate boudar value problem i j i j 2 where i a poitive umber ad > i a parameter Our aumptio throughout are: H h i a oegative meaurable fuctio defied i ad do ot vaih ideticall o a ubiterval i ad < d < d < + ; H 2 f :[ + [ + i a odecreaig cotiuou fuctio ad f > for > ; f H f > lim + ; + B a poitive olutio of the problem 2 we mea that atifie i ii C [ C ] C f > > i ad 2 hold; i locall abolutel cotiuou i ad h f ae i If for a particular the cojugate boudar value problem 2 ha a poitive olutio the i called a eigevalue ad a correpodig eigefuctio of 2 We let Λ be the et of eigevalue of the problem 2 ie Λ{ >; 2 ha a poitive olutio } The cojugate boudar value problem 2 ha bee tudied

3 Ligbi Kog ad Tao Lu 5 eteivel for detailee for itace[] ad referece therei For pecial cae of the eitece reult of poitive olutio of 2 ha bee etablihed i [ ] A for twi poitive olutio everal tudie to the problem 2 ca be foud i [5 7] For the cae of > eigevalue iterval characterizatio of theproblem The equatio i 2 ha bee dicued i[8 ] I thi paper we ol ue fied poit ide ad fied poit theorem i coe which allow u to etablih ot ol eitece of poitive olutio but alo multiplicit of poitive olutio of the problem 2 uder weaer coditio I fact we ma allow that h poee trog igularit at for eample h α+ β+ with < α β < atifie H Our reult geeralized ad eted ome ow theorem ad improve the wor of ome author of the above referece[8 ] 2 Prelimiar Note To obtai poitive olutio for the problem 2 we tate ome propertie of Gree fuctio for 2 A how i[] the problem 2 i equivalet to the itegral equatio G f d 2

4 6 Poitive olutio of igular - cojugate boudar value problem where + +!!!! dt t t dt t t G 22 Moreover the followig reult have bee recetl offered b Kog ad Wag[] Lemma For a ] [ we have g G g β α 2 g g G 2 where α } mi{ β!! g Let mi ] [ α α ma [] β β ad β α γ Defie the coe i Baach pace ] [ C give b } C[]; { : P } mi C[]; { : K γ We defie the operator P T:P b : d f h G T 25 Lemma 2 Suppoe that H H hold The P T:P i a completel

5 Ligbi Kog ad Tao Lu 7 cotiou mappig adtp K Moveover for K we have T C [ C ] C 26 T f ae 27 T i j T i j 28 Proof We ol provetp K The proof of the remaider of Lemma 2 ca be foud i[] For P b emploig 2 ad 25 we have mi T miα [ ] thi implietp K [ ] α ma β [] γ T g f d G f d From the Lemma we ow that TK K ad fied poit of T i K i a olutio of 2 ad vice vera Mai Reult The mai reult of thi paper are a follow Theorem Aume that H H hold The there eit a umber with < < + uch that

6 8 Poitive olutio of igular - cojugate boudar value problem i 2 ha two poitive olutio for ; ii 2 ha a poitive olutio for ; iii 2 ha o poitive olutio for + The followig theorem will be ued i our proof Theorem 2 [] Let E be a Baach pace ad K E a coe i E For > defie K { u K; u } Aume that Φ: K K i a compact map uch that Φu u for u K { u K; u } a If u b If u Φ u for u K the i Φ K K ; Φ u for u K the i Φ K K Lemma Suppoe that H H hold If i ufficietl large the Λ Proof if Λ the the problem 2 ha a poitive olutio ad Lemma 2 mea that K i a fied poit of T It follow from f lim + that there eit a M > uch that f for all M + If M ice mi γ M γ [ ] we get ma [] mi [ ] [ ] G f miα G f g d d d

7 Ligbi Kog ad Tao Lu 9 αγ g d Thu If αγ g d M < the γ which cotradict with ufficietl large M γ > miα [ ] g f d α f g d ie M αγf g d which i alo cotradictio Lemma Suppoe that H H hold The there eit a > uch that ] Λ Proof Let K { K; < } chooe 2βf g d For K { K; } ad ] uig H H2 ad 2 we have T ma [] G f d βf g d < ie T < for K Thu Theorem 2 implie i T K K Hece T ha a fied poit i K ad it atifie G f d f α g d thi how that i a poitive olutio of problem 2

8 2 Poitive olutio of igular - cojugate boudar value problem Lemma 5 Suppoe that H H hold Let up Λ the Λ Proof Without lo of geeralit let { } be a mootoe icreaig equece ad lim where Λ We claim that the correpodig poitive olutio equece { } i uiforml bouded I fact from f lim + there eit a M > uch that f N for all M + > where N i choe o that N αγ g d If M the γ mi [ ] γ M hece we obtai mi [ ] G f d N α g d N αγ g d > which i a cotradictio Thu there eit a umber L with < L < + uch that L for all I additio uig H ad 2 we have G f d f L!! d + d

9 Ligbi Kog ad Tao Lu 2 f L!! 2 d : Q thi how that { } { } i equicotiou Acoli-Arzela theorem claim that ha a uiforml coverget ubequece deoted agai b { } ad { } coverge to uiforml o [] Iertig ito 2 ad lettig uig the Lebegue domiated covergece theorem we obtai G f d Therefore i a poitive olutio of 2 ad Λ Lemma 6 Suppoe that H H holdthe the problem 2 ha two poitive olutio for Proof A how i[9] it follow from H 2 ad uiform cotiou of f that there eit a δ > uch that T G f d < + δ Let Ω { C[]; δ < < + δ} the Ω i a bouded ope ubet i C[] ad K Ω i a bouded ope ubet i K ad K Ω It i clear that Coider the homotop it i obviou that K Ω { K; + δ} H t t T + t

10 22 Poitive olutio of igular - cojugate boudar value problem H : [] K Ω K i completel cotiou For t [] K Ω form we have H t t T + t < t + t < + δ Hece H t K Ω Thu we get H t for t [] K Ω B the homotop ivariace ad ormalit of the fied poit ide we obtai i T K Ω K i K Ω K 2 thi how thatt ha a fied poit i K Ω ad i a poitive olutio of 2 f From lim + there eit a R > uch that f η for all + where η i choe o that ηγα g d Let R > R R ma{ + δ + } the for K R ice mi γ γr R γ we have thu Theorem 2 implie [] T ma G f d mi α [ ] g η d ηαγ g d > [ ] i T K R K Coequetl the additivit of the fied poit ide ad 2 together

11 Ligbi Kog ad Tao Lu 2 implie i T K R \ K Ω K 2 therefore T ha aother fied poit i K R \ K Ω ad i alo a poitive olutio of 2 2 Up to ow the proof of theorem i complete ACKNOWLEDGEMENTS Thi wor i upported b ciece ad techolog reearch project of Heilogjiag Provicial Departmet of Educatio 2576 Referece [] RPAgarwal DO Rega ad V Lahmiatham Sigular p -p focal ad p higher order boudar value problem Noliear Aali [2] RPAgarwal ad DO Rega Poitive olutio for p -p cojugate boudar value problem J Diff Equatio [] PW Eloe J Hedero Sigular - cojugate boudar value problem J Diff Equatio [] Ligbi Kog Juu Wag The Gree fuctio for - cojugate boudar value problem ad it applicatio J Math Aal Appl

12 2 Poitive olutio of igular - cojugate boudar value problem [5] RP Agarwal ad DO Rega Multiplicit reult for igular cojugate focal ad p problem J Diff Equatio [6] RP Agarwal DO Rega ad VLahmiatham Twi oegative olutio for higher order boudar value problem Noliear Aali [7] RP Agarwal ad DO Rega Twi olutio to igular boudar value problem Proc Amer Math Soc [8] PJY Wog ad RP Agarwal Eigevalue iterval ad twi eigefuctio of higher order boudar value problem J Autralia Math Soc Ser B [9] PJY Wog ad RP Agarwal O eigevalue iterval ad twi eigefuctio of higher order boudar value problem J Comput Appl Math [] PJY Wog ad RP Agarwal O eigevalue iterval ad twi poitive olutio of p boudar value problem Fuctioal Differetial Equatio [] RP Agarwal M Boher ad PJY Wog Poitive olutio ad eigevalue of cojugate boudar value problem Proc Ediburgh Math Soc [2] DR Duiger ad HY Wag Multiplicit of poitive radial olutio for a elliptic tem o a aulu Noliear Aali [] LHErbe HU Souchua ad Wag Haia Multiple poitive olutio of ome boudar value problem J Math Aal Appl