Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of Nebrsk-Lincoln Lincoln,NE 68588-0323 lerbe@@mh.unl.edu peerso@@mh.unl.edu Absrc We re concerned wih he self-djoin equion Lx() [r()x ()] + q()x(σ()) 0. We will sudy cerin Green s funcions ssocied wih his equion. Comprison heorems for iniil vlue problems nd boundry vlue problems will be given. Key words: mesure chins,ime scle, Green s funcion AMS Subjec Clssificion: 39A0. Inroducion We re concerned wih he self-djoin equion Lx() [r()x ()] + q()x(σ()) 0. To undersnd his so-clled differenil equion on mesure chin(ime scle) we need some preliminry definiions. Definiion Le T be closed subse of he rel numbers R. We ssume hroughou h T hs he opology h i inheris from he sndrd opology on he rel numbers R. For < supt, define he forwrd jump operor by σ() : inf{τ > : τ T } T nd for > inf T define he bckwrdjump operor by ρ() : sup{τ < : τ T } T for ll T. If σ() >, we sy is righ scered, while if ρ() < we sy is lef scered. If σ() we sy is righ dense, while if ρ() we sy is lef dense. Throughou his pper we mke he blnke ssumpion h b re poins in T.
Definiion Define he inervl in T Oher ypes of inervls re defined similrly. [, b] : { T such h b}. We re concerned wih he clculus on mesure chins which is unified pproch o coninuous nd discree clculus. An excellen inroducion o his subjec is given by S. Hilger [7]. See lso he monogrph by Kymkcln, Lksmiknhm, nd Sivsudrm [8]. Agrwl nd Bohner [] refer o i s clculus on ime scles. Oher ppers in his re include Agrwl nd Bohner [3], Agrwl, Bohner, nd Wong [4], nd Hilger nd Erbe [6]. In forhcoming pper he uhors will pply he echniques of his pper o prove he exisence of posiive soluions o generl wo-poin boundry vlue problems. To do his we will use fixed poin heorems for operors defined on pproprie cones in Bnch spce. Definiion Assume x : T R nd fix T (if supt ssume is no lef-scered), hen we define x () o be he number (provided i exiss) wih he propery h given ny ɛ > 0, here is neighborhood U of such h [x(σ()) x(s)] x ()[σ() s] ɛ σ() s, for ll s U. We cll x () he del derivive of x(). I cn be shown h if x : T R is coninuous T nd is righ scered, hen If is righ dense, hen x () x(σ()) x(). σ() x x(σ()) x(s) () lim. s σ() s Definiion We sy x : T R is righ dense coninuous on T provided i is coninuous ll righ dense poins nd poins h re lef dense nd righ scered we jus ssume he lef hnd limi exiss (nd is finie). We denoe his by x C rd (T ). Definiion Define he se D o be he se of ll funcions x : T R, such h x () is coninuous on T nd [r()x ()] is righ dense coninuous on T. We sy h funcion x is soluion of Lx0 on T provided x D nd Lx()0 for ll T. Exmple If T Z, he se of inegers, hen x () x() : x( + ) x(). Furhermore he equion Lx0 reduces o he self-djoin difference equion [r() x()] + q()x(σ() 0. See he books [2] nd [9] nd he references here for resuls concerning his self-djoin difference equion. 2
Exmple 2 If T R, hen he equion Lx0 reduces o he self-djoin differenil equion Lx() [r()x ()] + q()x() 0, which hs been sudied exensively over he yers. Exmple 3 Consider he differenil equion on he mesure chin x () + π 2 x(σ()) 0, T [0, ] n2{n}. Le x() be he soluion of he bove equion sisfying he iniil condiions x(0) 0, x (0). One cn show h sinπ, 0 π x() -, 2 (2 π 2 )x(-)-x(-2), 3,4,... Noe h our soluion pieces ogeher soluion of differenil equion nd soluion of difference equion. Exmple 4 Consider he differenil equion on he mesure chin x () + π 2 x(σ()) 0, T n0[2n, 2n + ]. Le x() be he soluion of he bove equion sisfying he iniil condiions x(0) 0, x (0). One cn show h his soluion on [0,] [2,3] is given by { x() sinπ, 0 π -cosπ+ π2 sinπ, 2 3. π In he nex exmple our differenil equion on mesure chin leds o discree problem wih vrible sep size. Exmple 5 Solve he IVP on he mesure chin where x () + x(σ()) 0, x(0), x (0) b, T k0t k T k n{k + n n }. 3
For T 0, his soluion is deermined by he IVP x( n + 2n(n + )3 ) + n + 2 n(n + ) 2 (n + 2) x( n n + ) + x(0), x (0) b. n n + 2 x(n n ) 0. For T, using Mple we ge h his soluion is deermined by he IVP x( + n + 2n(n + )3 ) + n + 2 n(n + ) 2 (n + 2) x( + n n + ) + n n + 2 x( + n n ) 0. x() lim n x( n ).897 +.584b n x () lim n x ( n ).637 +.80b. n In priculr, noe h on ech se T k he soluion x() of he given IVP solves he sme difference equion, where for k he iniil condiions on T k re deermined by he vlues of he soluion on T k. Definiion If F () f(), hen we define n inegrl by f(τ) τ F () F (). In his pper we will use elemenry properies of his inegrl h eiher re in he references [,3-8] or re esy o verify. Definiion We sy x(,s) is he Cuchy funcion for Lx0 provided for ech fixed s T, x(,s) is he soluion of he IVP Lx(,s)0, x(σ(s), s) 0, x (σ(s), s) r(σ(s)). I is esy o verify he following exmple. Exmple 6 The Cuchy funcion for Lx [r()x ()] 0 is given by x(, s) σ(s) r(τ) τ. We will use he following resul in he nex secion, whose proof is srigh forwrd consequence of he definiion of he del derivive. Lemm 7 Le, b T nd ssume f (, s) is coninuous on [, σ(b)] [,b], hen { { b f(, τ) τ} f(, τ) τ} b f (, τ) τ + f(σ(), ), f (, τ) τ f(σ(), ). 4
2 Min Resuls We will use he firs formul in Lemm 7 o prove he following vriion of consns formul. Theorem 8 (Vriion of consns formul) Assume h() is coninuous on [,b] nd x(,s) is he Cuchy funcion for Lx()0, hen i follows h x() : x(, s)h(s) s is he soluion of he IVP Lx() h(), x() 0, x () 0. Proof: Le x(,s) be he Cuchy funcion for Lx0 nd se x() x(, s)h(s) s. Noe h x()0. Using he firs formul in Lemm 7, we ge h x () x (, s)h(s) s + x(σ(), )h() since x(σ(), )0. Hence, x ()0. Also r()x () x (, s)h(s) s, Agin using he firs formul in Lemm 7, we ge h [r()x (, s)] I follows h Lx() r()x (, s)h(s) s. [r()x (, s)] h(s) s + r(σ())x (σ(), )h() [r()x (, s)] h(s) s + h(). Lx(, s)h(s) s + h() h(). Definiion Le, b T. We wn o consider Lx()0 on he inervl [, σ 2 (b)]. We sy nonrivil soluion of Lx0 hs generlized zero iff x()0. We sy nonrivil soluion x() hs generlized zero 0 (, σ 2 (b)], provided eiher x( 0 )0 or x(ρ( 0 ))x( 0 ) < 0. Finlly we sy Lx0 is disconjuge on [, σ 2 (b)] provided here is no nonrivil soluion of Lx0 wih wo(or more) generlized zeros in [, σ 2 (b)]. 5
Theorem 9 (Comprison heorem for IVP s) Assume Lx 0 is disconjuge on [, σ 2 (b)]. If u, v D re funcions sisfying Lu() Lv(), [, b], i follows h u() v(), u() v(), u () v (), [, σ 2 (b)]. Proof: Le u(), v() be s in he semen of his heorem nd se w() : u() v(), for [, σ 2 (b)]. Then h() : Lw() Lu() Lv() 0, for [, b]. I follows h w() solves he IVP Lw() h(), w() w () 0. Hence by he vriion of consns formul w() x(, s)h(s) s. Since Lx0 is disconjuge on [,σ 2 (b)], x(,s) 0 for σ(s). Noe h w() If is lef scered, hen w() ρ() ρ() x(, s)h(s) s + ρ() x(, s)h(s) s. x(, s)h(s) s + x(, ρ())h(ρ()) ρ() x(, s)h(s) s. Since x(,s) 0 for σ(s) we ge h w() 0 which implies he desired resul. If is lef dense i is esy o see h w() 0. Now consider he generl wo-poin BVP Lx 0, 6
αx() βx () 0, γx(σ(b)) + δx (σ(b)) 0, where we ssume hroughou h α, β, γ, nd δ re conns such h Theorem 0 Assume h he BVP α 2 + β 2 > 0, γ 2 + δ 2 > 0. Lx 0, αx() βx () 0, γx(σ(b)) + δx (σ(b)) 0, hs only he rivil soluion. For ech fixed s [,b], le u(,s) be he unique soluion of he BVP Lu(,s)0, αu(, s) βu (, s) 0, γu(σ(b), s) + δu (σ(b), s) γx(σ(b), s) δx (σ(b), s)), where x(,s) is he Cuchy funcion for Lx0. Then { u(,s), s G(, s) u(,s)+x(,s), σ(s) is he Green s funcion for he BVP Lx 0, αx() βx () 0, γx(σ(b)) + δx (σ(b)) 0. Proof: I is esy o see, for ech fixed s [,b], h u(,s) is uniquely deermined by he BVP given in he semen of he heorem. Le u(,s), x(,s), nd G(,s) be s in he semen of his heorem nd ssume h() is given coninuous funcion on [,b]. Then define for [, σ 2 (b)]. Firs noe h x() : x() G(, s)h(s) s + Using he definiion of G(,s) we ge h x() G(, s)h(s) s. G(, s)h(s) s. [u(, s) + x(, s)]h(s) s + G(, s)h(s) s. u(, s)h(s) s. 7
Hence, x () [u (, s)+x (, s)]h(s) s+[u(σ(), )+x(σ(), )]h()+ Simplifying we ge h u (, s)h(s) s u(σ(), )h() I follows h nd Therefore x () [u (, s) + x (, s)]h(s) s + x () x (σ(b)) αx() βx () G (, s)h(s) s G (σ(b), s)h(s) s. u (, s)h(s) s. [αg(, s) βg (, s)]h(s) s, [αu(, s) βu (, s)]h(s) s, 0. γx(σ(b)) + δx (σ(b)) [γg(σ(b), s) + δg (σ(b), s)]h(s) s, {γ[x(σ(b), s) + u(σ(b), s)] + δ [x (σ(b), s) + u (σ(b), s)}h(s) s, 0. Hence we hve shown h x() sisfies he boundry condiions. From bove we ge h r()x () Therefore, [r()x ()] [r()u (, s) + r()x (, s)]h(s) s + {[r()u (, s)] + [r()x (, s)] }h(s) s + r(σ())u (σ(), )h() + r(σ())x (σ(), )h() + [r()u (, s)] h(s) s r(σ())u (σ(), )h() [r()u (, s)] h(s) s + 8 r()u(, s)h(s) s. [r()x (, s)] h(s) s + h().
From erlier in he proof we ge h I follows h x() u(, s)h(s) s + x(, s)h(s) s. q()x(σ()) If is righ dense, hen we ge h q()x(σ()) If is righ scered, hen σ() q()x(σ(), s)h(s) s q()u(σ(), s)h(s) s + q()u(σ(), s)h(s) s + σ() q()x(σ(), s)h(s) + q()x(σ(), s)h(s) s. q()x(σ(), s)h(s) s. σ() q()x(σ(), s)h(s) Using he fc h x(σ(), ) 0, we lso ge h he second erm is zero. Hence in eiher cse we ge h Finlly, we obin for [, b]. q()x(σ()) q()u(σ(), s)h(s) s + Lx() [r()x ()] + q()x(σ()) Lu(, s)h(s) s + q()x(σ(), s)h(s) s. Lx(, s)h(s) s + h() h(), Corollry (Green s funcion for he conjuge problem) Assume Lx 0 is disconjuge on [, σ 2 (b)]. Le x(,s) be he Cuchy funcion for Lx 0 nd for ech fixed s T le u(,s) be he unique soluion of he BVP Lx 0, u(, s) 0, u(σ 2 (b), s) x(σ 2 (b), s). Then { u(,s), s G(, s) : u(,s)+x(,s), σ(s) is he Green s funcion for he BVP Lx() 0, x() 0, x(σ 2 (b)) 0. 9
Proof: Noe h, since Lx0 is discojuge on [,σ 2 (b)], i follows h he BVP Lx0, x()0, x(σ 2 (b))0 hs only he rivil soluion. If σ(b) is righ scered hen his Corollry follows from Theorem 9 wih α 0, β 0, δ [σ 2 (b) σ(b)]γ, γ 0. On he oher hnd if σ(b) is righ dense hen his resul follows from Theorem 9 wih α 0, β 0, γ 0, δ 0. Corollry 2 The Green s funcion for he BVP Lx [r()x ()] 0, is given by G(, s) R R σ(s) x() 0, x(σ 2 (b)) 0, r(τ) τ R σ 2 (b) σ(s) R σ 2 (b) r(τ) τ r(τ) τ, s r(τ) τ R σ 2 (b) R σ 2 (b) r(τ) τ r(τ) τ, σ(s) Proof: I is esy o see h Lx [r()x ()] 0 is disconjuge on [, σ 2 (b)]. By Corollry, he u(,s) in he semen of Corollry for ech fixed s [,b] solves he BVP Lx 0, Since re soluions of Lx0, u(, s) 0, u(σ 2 (b), s) 0. x () nd x 2 () r(τ) τ u(, s) α(s) + β(s) r(τ) τ. Using he boundry condiions u(, s) 0 nd u(σ 2 (b)) 0 i cn be shown h u(, s) τ σ 2 (b) r(τ) σ(s) σ 2 (b) 0 τ r(τ) τ. r(τ)
Hence G(,s) hs he desired form for s. By Corollry for σ(s), G(, s) x(, s) + u(, s), where x(, s) is he Cuchy funcion for Lx [r()x ()]. From Exmple 6 Therefore G(, s) Geing common denominor, σ(s) x(, s) σ(s) r(τ) τ r(τ) τ. τ σ 2 (b) r(τ) σ(s) σ 2 (b) τ r(τ) τ. r(τ) G(, s) σ(s) τ σ 2 (b) r(τ) τ r(τ) σ 2 (b) τ σ 2 (b) r(τ) σ(s) τ r(τ) τ. r(τ) σ(s) Hence for σ(s), τ[ r(τ) which is he desired resul. τ + σ 2 (b) r(τ) σ 2 (b) τ[ σ 2 (b) r(τ) r(τ) τ] τ] + r(τ) σ(s) σ 2 (b) σ 2 (b) τ[ r(τ) σ 2 (b) G(, s) τ σ 2 (b) r(τ) σ(s) τ r(τ) τ. r(τ) τ σ 2 (b) r(τ) τ r(τ) τ. r(τ) r(τ) τ σ(s) τ] r(τ) τ. r(τ) σ(s) τ σ 2 (b) r(τ) σ 2 (b) τ r(τ) τ, r(τ) For he specil cse of Corollry 2 where TZ, he Green s funcion is given in Exmple.26 in [2]. See Secion 7.5 in [2] for he vecor cse. Theorem 3 (Comprison heorem for conjuge BVP s) Assume Lx 0 is disconjuge on [, σ 2 (b)]. If u(), v() re funcions in D sisfying Lu() Lv(), [, b], i follows h u() v(), u(σ 2 (b)) v(σ 2 (b)), u() v(), [, σ 2 (b)].
Proof: Le u() nd v() be s in he semen of he heorem nd se w() u() v(), for [, σ 2 (b)]. Se h() Lw(), [, b], hen h() Lu() Lv() 0, [, b]. I follows h w() solves he BVP Lw() h(), [, σ 2 (b)], where I follows h where φ() solves he BVP From he disconjugcy nd I follows h which gives he desired resul. w() A, w(σ 2 (b)) B, A : u() v() 0, B : u(σ 2 (b)) v(σ 2 (b)) 0. w() φ() + Lφ() 0 G(, s)h(s) s, φ() A, φ(σ 2 (b)) B. φ() 0, [, σ 2 (b)], G(, s) 0, [, σ 2 (b)], w() 0, s [, b]. The following resul shows h nonrivil soluion of Lx0 cn no hve double zero. Theorem 4 If x() is nonrivil soluion of Lx0 such h x() 0 on [,σ 2 (b)], hen x()) > 0 on (,σ(b)). Proof: Assume h x() is nonrivil soluion of Lx0 such h x() 0 on [,σ 2 (b)]. We will ssume here is τ in (,σ 2 (b)) such h x(τ)0 nd show h his leds o conrdicion. Firs we show h x (τ) > 0. 2
If τ is righ scered, hen x (τ) x(σ(τ)) x(τ) σ(τ) τ x(σ(τ)) σ(τ) τ > 0, since x() is nonrivil soluion. On he oher hnd if τ is righ dense, hen x (τ) lim s τ x(σ(τ)) x(s) σ(τ) s x(s) lim s τ τ s x(s) lim s τ + τ s 0. Since x(τ) 0 nd x() is nonrivil soluion we hve h x (τ) > 0. Hence in ll cses we hve h his ls inequliy holds. To ge conrdicion we consider he wo possibiliies h τ is lef scered or τ is lef dense. Firs ssume τ is lef scered. Using x() is soluion we ge fer n inegrion h Hence This implies h Bu {r()x ()} τ ρ(τ) τ ρ(τ) q(ρ(τ))x(σ(ρ(τ))) q(ρ(τ))x(τ) 0. q()x(σ()). r(τ)x (τ) r(ρ(τ))x (ρ(τ)). x (ρ(τ)) x (ρ(τ)) > 0. x(τ) x(ρ(τ)) τ ρ(τ) x(ρ(τ)) τ ρ(τ) 0, which is conrdicion. Finlly consider he cse where we ssume τ is lef dense. In his cse using he fc h x () is lef coninuous τ we ge h here is δ > 0 such h x () > x (τ) 2 3
when [τ δ, τ]. Le τ 0 (τ δ, τ) nd inegre from τ 0 o τ o ge h x(τ) x(τ 0 ) x (τ) 2 τ τ 0 s 0. Since x(τ)0 we ge h which is conrdicion. x(τ 0 ) 0, Corollry 5 If, in Theorem 8, we eiher ssume in ddiion h Lu() < Lv() on subse of [,b] of posiive mesure or if we ssume in ddiion h one of he inequliies u() v(), u(σ 2 (b)) v(σ 2 (b)) is sric, hen we ge h for (, σ 2 (b)). Proof: u() > v() Nex we consider he focl BVP Lx 0, x() 0, x (σ(b)) 0. The following resul follows from Theorem 0 wih β γ 0. Corollry 6 (Green s funcion for focl BVP) Assume he BVP Lx 0, x() 0, x (σ(b)) 0 hs only he rivil soluion. For ech fixed s [,b] le u(,s) be he soluion of he BVP Lu(, s) 0, u(, s) 0, u (σ(b), s) x (σ(b), s), where x(,s) is he Cuchy funcion of Lx0. Then { u(,s), s G(, s) : u(,s)+x(,s), σ(s) is he Green s funcion for he focl BVP Lx() 0, x() 0, x (σ(b)) 0. Corollry 7 The Green s funcion for he focl BVP Lx [r()x ()] 0, x() 0, 4
is given by x (σ(b)) 0, G(, s) { σ(s) τ, r(τ) τ, r(τ) s σ(s) Proof: I is esy o see h he focl BVP given in he semen of his heorem hs only he rivil soluion. Hence we cn pply Corollry o find he focl Green s funcion G(,s). For s, G(,s)u(,s), where for ech fixed s, u(,s) solves he BVP Solving his BVP we ge h [r()x ()] 0, u(, s) 0, u(σ(b)) x(σ(b), s). u(, s) r(τ) τ, which is he desired expression for G(,s) for s. For σ(s), G(, s) u(, s) + x(, s), Simplifying we ge he desired conclusion for σ(s). r(τ) τ + σ(s) r(τ) τ. σ(s) G(, s) r(τ) τ, Definiion We sy Lx0 is disfocl on [,σ 2 (b)] provided here is no nonrivil soluion x() such h x() hs generlized zero in [,σ 2 (b)] followed by generlized zero of x () in [,σ 2 (b)]. The proof of he following resul is similr o he proof of some erlier resuls in his pper nd will be omied. Corollry 8 (Comprison heorem for focl BVP s) Assume u,v D, Lx0 is disfocl on [, σ 2 (b)], Lu() Lv(), [, b], nd hen u() v() for [, σ 2 (b)]. u() v(), u (σ(b)) v (σ(b)) 5
References [] R. Agrwl nd M. Bohner, Bsic clculus on ime scles nd some of is pplicions, preprin. [2] C. Ahlbrnd nd A. Peerson, Discree Hmilonin Sysems: Difference Equions, Coninued Frcions, nd Ricci Equions, Kluwer Acdemic Publishers, Boson, 996. [3] R. Agrwl nd M. Bohner, Qudric funcionls for second order mrix equions on ime scles, preprin. [4] R. Agrwl, M. Bohner, nd P. Wong, Surm-Liouville eigenvlue problems on ime scles, preprin. [5] B. Aulbch nd S. Hilger, Liner dynmic processes wih inhomogeneous ime scle, Nonliner Dynmics nd Qunum Dynmicl Sysems, Akdemie Verlg, Berlin, 990. [6] L. Erbe nd S. Hilger, Surmin Theory on Mesure Chins, Differenil Equions nd Dynmicl Sysems, (993), 223-246. [7] S. Hilger, Anlysis on mesure chins- unified pproch o coninuous nd discree clculus, Resuls in Mhemics, 8 (990), 8-56. [8] B. Kymkcln, V. Lksmiknhm, nd S. Sivsundrm, Dynmicl Sysems on Mesure Chins, Kluwer Acdemic Publishers, Boson, 996. [9] W. Kelley nd A. Peerson, Difference Equions: An Inroducion wih Applicions, Acdemic Press, 99. 6