Applying p-balanced Energy Technique to Solve Liouville-Type Problems in Calculus

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1 Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 ly -lced Eey Techue o Solve Louvlle-Tye Poles lculus L Wu, Ye L, J Lu Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95 sc We e eesed solv Louvlle-ye oles o exloe coscy oees o s o deel os o Re olds Geoec sucues o olds, he exsece o coscy oees o s o deel os, d eey owh o s o deel os e ewed I hs cle, we cocee o dscovey o soluos o Louvlle-ye oles whee olds e Euclde sces e l Re olds d s ecoe el-vlued ucos Louvlle-ye esuls o vsh oees o ucos e oed The ol wok ou esech ds s o exed he -eey o uco o e L sce o e o-l sce y ly -lced echue whee = = lculo sklls such s Hölde's Ieuly d Tess o Sees hve ee used o evlue ls d eos o uco eey lculo des d couol echues o solv Louvlle-ye oles show hs cle, whch e ulzed Euclde sces, c e uveslzed s successul loh, whch woks o oh s d deel os o Re olds Ths ovve loh hs -ech c o esech wok o solv Louvlle-ye oles he eel ses volved wh e eey The -lced echue hs loh ovdes clue o success o he od o -eey exeso o e o e eywods Deel Fos, Hölde Ieuly, Louvlle-ye oles, -lced owh, -hoc s, -eey owh, ess o sees I INTRODUTION HE sudy o Louvlle-ye oles Deel TGeoey s o dscove coscy oees o s o deel os ewee he do d he e o Re olds Exsece o coscy oees s deeed y eoec sucues o olds d eey owh o s o deel os Louvlle-ye oles hve ee suded wo decos Oe o he esech decos s o sudy ll kds o olds eued wh vous ec sucues olds c e clssed ccod o he cuvue vlues Fl olds such s Euclde Sces e olds wh cuvue vlues eul o zeo No-l olds e cuved olds wh o-zeo cuvue vlues vy o osve o eve hecs hve ee eesed ves ll ossle old ec sucues o ssue exsece o L Wu s wh he Dee o hecs, oouh o h ouy ollee -The y Uvesy o New Yok, New Yok, NY 7, US coesod uho, hoe: ; x: ; e-l: lwu@cccuyedu Ye L s wh he Dee o hecs, el ch Uvesy, ou Ples, I 8859 e-l: ly@cchedu J Lu s wh he Dee o hecs d Sscs, Uvesy o Wes Flod, Pescol, FL 35, US e-l: lu@uwedu coscy oees o s o deel os The ohe esech deco s o sudy eey owh o s o deel os os esech wok ws o sudy -eey o s o deel os y Louvlle-ye esuls hve ee oed y hecs he coex o e -eey o s o deel os L sce Louvlle-ye esuls e suzed s ollows: I 976, Schoe d Yu solved Louvlle-ye oles o hoc s d oed he coscy oey o olds wh o-eve Rcc cuvue vlues [] I 98, Geee d Wu oed Louvlle-ye esuls o vsh oey o hoc -os wh e -eey L sce o o-evely cuved olds o > [] I 99, w dscoveed Louvlle-ye esuls o -hoc s o -olc olds o olds wh o-osve cuvue vlues whe [3] I 995, heu d Leu oved Louvlle heoes o -hoc s o wh e -eey L sce whee o he e sce o -Hdd olds [] I, Zh oved Louvlle-ye Theoes o closed d -co-closed deel -o > wh e -eey L sce > o osvely cuved olds [5] I 8, Pol e l exloed Louvlle-ye esuls o -hoc s o cuved olds, ovded he do olds suo Soolev Pocé Ieuly [6] I hs cle, we ocus o solv Louvlle-ye oles whee olds e Euclde sces d s e el-vlued ucos Ou esech ol s o ek coss o e -eey o cooud eey eules o o Louvlle-ye esuls I cul, we ly he -lced eey echue o ucos o eelze -eey o e o e oe ecsely, we ly Hölde's Ieuly d Tess o Sees o evlue ls d eos o uco eey he couol ehod Louvlle-ye esul o vsh oees o ucos s oed The ol wok ou esech ds s Louvlle-ye esuls s o exed he -eey o uco o e L sce o e o-l sce y ly -lced echue whee = = ouol ehods d eey eso echues led o ucos Euclde sces, whch e eseed hs cle, c e eelzed s successul loh led o s o deel os o Re olds Ths loh c ly o ole solv Louvlle-ye oles he eel ses wh eey och o e The -lced eey echue hs ovve loh hs -ech c o esech wok o -eey eelzo o e Ieol Scholly d Scec Resech & Iovo scholwseo/37-689/95

2 Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95 o e II PRELIINRY I hs seco, we ve deos o -hoc s, deel os, -eey d -lced eey o s o deel os esecvely We lso ecll oche- Wezeock Foul o olds, Hölde's Ieuly, uchy-schwz Ieuly, d Tess o Sees lculus Le e -desol colee o-coc Re old wh volue elee d x ; o e eodesc ll o dus ceeed o x o Le k k T V e he sce o sooh k-os o wh vlues he veco udle :V Le d: k k e he exeo deel oeo d d : k k e he do deel oeo o d ve y d e e whee { e } s locl ohool e x, d X s he eo oduc y X ve y X Y,, Y X, Y,, Y o y k k k X Tx, d Yl Tx, l k I cul,, d s lso deed y d ce dv The Hode Llc s deed o he V-vlued deel os y k k dd d d : V V The o o s deoed y, oe dels c e oud [7] Deo deel o s sd o e hoc dd d d, closed d, co-closed d Deo deel o s sd o e -seudo-co- closed > d Deo 3 uco o deel o hs e -eey o > L -sce Ohewse, hs e -eey o o-l -sce The coce o -lced owh o cosss o 5 cses: -e owh, -ld owh, -ouse owh, -odee owh, d -sll owh uco o deel o s sd o e wh -lced owh ovded hs oe o -e, -ld, -ouse, -odee, d -sll owh whee > Ohewse, uco o deel o s sd o e wh -lced owh [8] Deo uco o deel o hs -e owh sses l x ;, d hs -e owh ohewse o > d > uco o deel o hs -ld owh hee exss x, d scly ces seuece o o o y, such h o evey l, we hve l x ; \ x ;, { } d hs -sevee owh ohewse o > d > uco o deel o hs -ouse owh hee exss x such h o evey, we hve x ; ds d, d hs -cue owh ohewse o > d > uco o deel o hs -odee owh hee exss x d F F, such h lsu, F x; whee F { :[,, d F o soe }, F Noce h ucos o deel os F e o ecessly oooe d hs -odee owh ohewse o > d > uco o deel o hs -sll owh hee exss x such h o evey, we hve d, x ; d hs -le owh ohewse o > d > The ove deos o -e, -ld, -ouse, -odee, d -sll d he coue-s -e, -sevee, -cue, -odee, d -le owh deed o, d wll e seced he coex whch he deo s used Hee, s ovous o us o oseve h uco o deel o hs he vsh -e owh hs e -eey, h s: l x ; Ieol Scholly d Scec Resech & Iovo scholwseo/37-689/95

3 Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95 Deo 5 The -eey > ucol o u s ve y E u du whee du deoes he deel o u Deo 6 u s sd o e -hoc s ccl o o -eey ucol E u Euvlely, u s -hoc s soluo o dv u u u s sd o e hoc e -hoc o s soluo o u dv u Le oche-wezeock Foul Fo y deel o o, he ollow dey holds:, R, whee R, deoes he Rcc cuvue o he deco o Le Hölde's Ieuly Le,, wh Fo y osve ues,,, d,,, R, we hve, we hve Fo y osve ucos d I cul, o ==, we hve uchy-schwz Ieuly s secl cse o Hölde's Ieuly: Nex, le us ecll L o he -h Te o ovee Sees d es o elesco sees lculus Theoe I he sees covees o zeo, h s: I covees, he he seuece covees, he l Theoe elesco sees covees d oly he seuece s covee o e ue L Fuheoe, he su o he covee elesco sees wll och o he vlue o L, h s: covees l L d he su S l s l l L III RESULT ND PROOF I hs seco, we s ve see o he Louvlle Theoe o ucos e h, we ese he deled oo o how o o he vsh oey o uco s Louvlle-ye esuls o ou esech ds Resul o Louvlle Theoe o Fucos Theoe 3 ssue h wo ucos d ssy he ollow wo codos whee : o ; l h s, hs -e owh o = The, we o he vsh oey o he uco, h s Poo Sce, hee us exs ces seuece { } such h Le codo o d, we c ewe he s s ollows: Sce, v uchy-schwz Ieuly, we hve: Fo slcy, we dee he ollow oos: : : We oce h he seuece { } s ouded sce l l o he secod codo ssuo h hs -e owh o = The ecoes Ieol Scholly d Scec Resech & Iovo scholwseo/37-689/95

4 y su o oh sdes, we hve: Ths les: 3 whee we use he c d he ouded seuece whee s e osve ue We su u o e: 6 N N o N whee we ly d 3 Hee, we ove h he sees s covee y Theoe, we cl h l, h s Fuheoe, we vey We o Louvlle-ye esul o he vsh oey o uco he coex o -e owh s oe cse o -lced echue whee = = IV ONLUSIONS The ol wok ou esech ds s o exloe d vey ew eey echoloy d ovve loh s successul wy o exed -eey o e L sce o e o-l sce ouol ehods d loh ulzed o ucos o Louvlle-ye soluos l olds, whch e eseed hs cle, c e successully led o s o deel os o Louvlle-ye soluos o cuved olds V RESERH IPT oh -lced eey echue d hs ovve loh hve -ech esech c o solv Louvlle-ye oles he eel ses wh e eey s lcos o hs successul loh d he -lced eey echue, y Louvlle-ye esuls o s d deel os och e -eey o-l sce hve ee cheved Wu s esech wok [9]-[3] Red esech cs o deel os, we hve exloed he echue o -lced eey hs ovve loh o ovecoe dcules o -eey exeso u o e [9]-[] S wh he deo o -lced eey owh, we e eesed hoc o o old wh o-eve Rcc cuvue [9], closed d -seudo-co-closed deel -o o cuved old wh he suo o Soolev-Pocé Ieuly x o cuvue ss [], closed d co-closed deel k-o o colee o-coc old [], closed d -seudo-co-closed deel -o o old wh o-eve Rcc cuvue [] I suy, o y deel o o d oe e o [9], [], [], we cosde wo o-eve ucos d e h, we ue ou h oche-wezeock Foul led o deel os o olds hs lyed sc ole uee he s ssuo o Theoe 3 o e ssed oe ecsely, we cl h oche-wezeock Foul woks s he oudo o eslsh, whch es he ocess o hs ovve loh lsed s elow: Sce, hee us exs ces seuece } { such h o Le d, we c ewe he s codo o es o d I ddo, hs ee oded s \ \ Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 99 Ieol Scholly d Scec Resech & Iovo 6 8 scholwseo/37-689/95 Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95

5 Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95 whee, e he osve coss d he es uco o x;, s oolly syec Lschz couous uco wh he ollow oees: o x; o x; c o x; \ x; d e o o osve cos, whch s deede wh choces o { } Fo slcy, we e-dee he ollow oos: : : We oce h he seuece { } s ouded sce l l o he secod codo ssuo h he deel o hs -e owh o = + The ecoes 3, whee 3 s osve cos s he se oo show Theoe 3, we ove h he sees s covee y Theoe, we cl h l, e I he ohe wods, we ove sed o he c o, we vey h o d o he esul o cos Fuheoe, secl old sucues deeed y cuvue oees o Soolev Pocé Ieuly hve uled ou he exsece o deel os eul o o-zeo coss Theeoe, he exsece o deel os wh he zeo cos oey, s Louvlle-ye esuls o vsh oees o deel os, hs ee cheved he coex o -e owh s oe cse o -lced eey echue whee = The deled oos c e oud [9], [], [] I [], we cosde d d d o deel o The se loh hs ee ed o he ollow euly: d d d \ d lo wh he evsed oos lsed s elow: d d \ o osve cos Red esech cs o s, we hve led -lced eey echue d hs ovve loh o -hoc s o ovecoe dcules o -eey eelzo led o y [] S wh deo o -lced eey owh [], we ocus o he deel o -hoc u e du o old suoed y Soolev Pocé Ieuly wh he xed cuvue ss Jus s he se ue hs ovve loh y se du, we o du, whch dces u s cos Louvlle-ye esuls o coscy oey o -hoc hve ee oed he coex o -e owh s oe cse o -lced echue whee = VI FUTURE RESERH PLNS hecs hve ee sudy Louvlle-ye oles o ovecoe he chllee o e -eey o y yes y eecve esech oches d vlule esuls hve ee oud I hs cle, we oly dscove he -e owh s oe cse o he -lced eey echue o exed -eey o e o e cully, he e ou cses o -ld, -ouse, -odee, d -sll owh [5] he -lced eey echue c e couously exloed s eecve oches o ovecoe dcules o -eey eelzo led o e -eey The loh he coex o -lced eey echue o cses o -ld, -ouse, -odee, d -sll owh wll e ou ollow-u esech wok Fuheoe, seek successul lohs wh eecve eey eso echues o solve Louvlle-ye oles he eel ses ll ossle e eey suos wll e ou esech ees he uue NOWLEDGENT Ths esech oec ws suoed y Resech Schols Po The y Uvesy o New Yok UNY oouh o h ouy ollee Resech wok o he s uho hs oec ws suoed y PS-UNY wd, oly uded y The Poessol S Ieol Scholly d Scec Resech & Iovo 6 8 scholwseo/37-689/95

6 Wold cdey o Scece, Eee d Techoloy Ieol Joul o hecl d ouol Sceces Vol:, No:6, 8 oess d The y Uvesy o New Yok UNY Ieol Scece Idex, hecl d ouol Sceces Vol:, No:6, 8 wseo/pulco/95 REFERENES [] R Schoe, d S T Yu, Hoc s d ooloy o sle hyesuces d olds wh o-eve Rcc cuvue, oe h Helv, vol 5, o, 333-3, 976 [] R Geee, d H Wu, Hoc os o o-coc Re d hle olds, The ch hecl Joul, vol 8, o, 63-8, 98 [3] S w, P-Hoc s d covex ucos, Geoee Dedc, vol 7, o 3, 6-65, 99 [] L-F heu, d P-F Leu, ek o covex ucos d -hoc s, Geoee Dedc, vol 56, o 3, 69-7, 995 [5] X Zh, oe o -hoc -o o colee olds, d h ull, vol, o 3, , [6] S Pol, Rol, d G Se, oscy o -hoc s o e -eey o o-osvely cuved olds, hesche Zesch, vol 58, o, 3-36, 8 [7] H Wu, Eleey oduco o Re olds hese veso, Pek Uvesy Pess, e, h, 989 [8] S W We, J L, d L Wu, Geelzos o he uozo heoe d oche s ehod -hoc eoey, ou h l, o, vol, 6-68, 8 [9] S W We, d L Wu, Vsh heoes o -lced hoc os, Glol Joul o Pue d led hecs, vol, o, , 5 [] L Wu, Solv Louvlle-ye oles o olds wh Poce-Soolev Ieuly y ode -eey o e o e, Joul o hecs Resech, vol 9, o, -, 7 [] L Wu, d Y L, Euvlece ewee hoc o d closed co-closed o oh L d o-l sces, Euoe Joul o hecl Sceces, vol 3, o, -3, 7 [] L Wu, d Y L, Geelz Louvlle-ye oles o deel -os o L sces o o-l sces, Ieol Joul o hecl lyss, vol, o 8, , 6 [3] L Wu, J Lu, d Y L, Dscove Louvlle-ye oles o -eey z s closed hl-ellsods y lculus vo ehod, Ieol Joul o hecl, ouol, Physcl, Eleccl d oue Eee, vol, o, 96-5, 6 L Wu eceved he PhD hecs Uvesy o Oklho 6 She sed he ech cee lo wh esech wok s sss Poesso Uvesy o Toledo du 6 houh 9 She s cuely ssoce Poesso oouh o h ouy ollee, whch s oe o he y Uvesy o New Yok UNY schools She s eesed ue h esech d h educo esech He esech eess ue h e Deel Geoey d Pl Deel Euos PDE She hs de y eseos o he ulshed esech wok doesc d eol h coeeces She hs lso ee eces such s PS-UNY wds sed o he schol chevees Ye L eceved he PhD hecs The Uvesy o Oklho She hs ee wok el ch Uvesy e he duo He esech eess e Pl Deel Euos, Deel Geoey, d Geoec lyss J Lu eceved he PhD hecs Eoy Uvesy 6 e duo, she ws hed s he sss oesso y he dee o hecs d Sscs Uvesy o Wes Flod She hs ee ooed o he ssoce oesso sce D Lu s es o esech eess e he couol hecs, Pl Deel Euos d che le Ieol Scholly d Scec Resech & Iovo 6 8 scholwseo/37-689/95

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