Green s Functions and Comparison Theorems for Differential Equations on Measure Chains
|
|
- Claude Holland
- 5 years ago
- Views:
Transcription
1 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 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.
2 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
3 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
4 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 ) b n x () lim n x ( n ) b. 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
5 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
6 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
7 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
8 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().
9 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
10 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)
11 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)].
12 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
13 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
14 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
15 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
16 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), [7] S. Hilger, Anlysis on mesure chins- unified pproch o coninuous nd discree clculus, Resuls in Mhemics, 8 (990), [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
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More informationA LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly
More informationAN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS
AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS Mrin Bohner Deprmen of Mhemics nd Sisics, Universiy of Missouri-Roll 115 Roll Building, Roll, MO 65409-0020, USA E-mil: ohner@umr.edu Romn Hilscher
More informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More informationApplication on Inner Product Space with. Fixed Point Theorem in Probabilistic
Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: 1792-6602 prin, 1792-6939 online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More information3. Renewal Limit Theorems
Virul Lborories > 14. Renewl Processes > 1 2 3 3. Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process
More informationAsymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales
Asympoic relionship beween rjecories of nominl nd uncerin nonliner sysems on ime scles Fim Zohr Tousser 1,2, Michel Defoor 1, Boudekhil Chfi 2 nd Mohmed Djemï 1 Absrc This pper sudies he relionship beween
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationGENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS
- TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA
More informationSome Inequalities variations on a common theme Lecture I, UL 2007
Some Inequliies vriions on common heme Lecure I, UL 2007 Finbrr Hollnd, Deprmen of Mhemics, Universiy College Cork, fhollnd@uccie; July 2, 2007 Three Problems Problem Assume i, b i, c i, i =, 2, 3 re rel
More informationHermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals
Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for
More informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationOn Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives
In J Nonliner Anl Appl 9 8 No, 69-8 ISSN: 8-68 elecronic hp://dxdoiorg/75/ijn8745 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo -frcionl derivives Ghulm Frid, Anum Jved Deprmen of Mhemics, COMSATS Universiy
More informationPositive and negative solutions of a boundary value problem for a
Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 2455-3689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 73-83 Poiive nd negive oluion of oundry vlue prolem for frcionl, -difference
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationThe solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.
[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries
More information1. Introduction. 1 b b
Journl of Mhemicl Inequliies Volume, Number 3 (007), 45 436 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies
More information1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
More informationYan Sun * 1 Introduction
Sun Boundry Vlue Problems 22, 22:86 hp://www.boundryvlueproblems.com/conen/22//86 R E S E A R C H Open Access Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order
More informationIntegral Transform. Definitions. Function Space. Linear Mapping. Integral Transform
Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX
Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationHenstock Kurzweil delta and nabla integrals
Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of Nebrsk-Lincoln Lincoln, NE 68588-0323 peterso@mth.unl.edu Mthemtics, SPS, The University
More informationFractional operators with exponential kernels and a Lyapunov type inequality
Abdeljwd Advnces in Difference Equions (2017) 2017:313 DOI 10.1186/s13662-017-1285-0 RESEARCH Open Access Frcionl operors wih exponenil kernels nd Lypunov ype inequliy Thbe Abdeljwd* * Correspondence: bdeljwd@psu.edu.s
More informationNew Inequalities in Fractional Integrals
ISSN 1749-3889 (prin), 1749-3897 (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp.493-497 New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics,
More informationMAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017
MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g
More informationHUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA
Communicions on Sochsic Anlysis Vol 6, No 4 2012 603-614 Serils Publicions wwwserilspublicionscom THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Absrc
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationFractional Calculus. Connor Wiegand. 6 th June 2017
Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher
More informationON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS
Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped
More informationRESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π
RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... () nd b τ F( ) ω d, τ,,... (3)
More informationNon-oscillation of perturbed half-linear differential equations with sums of periodic coefficients
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 DOI 10.1186/s13662-015-0533-4 R E S E A R C H Open Access Non-oscillion of perurbed hlf-liner differenil equions wih sums of periodic coefficiens
More informationLAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS
Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp. 257-26 257 Open forum LAPLAE TRANSFORM OVEROMING PRINIPLE DRAWBAKS IN APPLIATION OF THE VARIATIONAL
More informationASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH REGULARLY VARYING COEFFICIENTS
Elecronic Journl of Differenil Equions, Vol. 06 06), No. 9, pp. 3. ISSN: 07-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR
More informationFURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)
Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's
More informationProbability, Estimators, and Stationarity
Chper Probbiliy, Esimors, nd Sionriy Consider signl genered by dynmicl process, R, R. Considering s funcion of ime, we re opering in he ime domin. A fundmenl wy o chrcerize he dynmics using he ime domin
More informationAn integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.
Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl
More informationJournal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle
J. Mh. Anl. Appl. 353 009) 43 48 Conens liss vilble ScienceDirec Journl of Mhemicl Anlysis nd Applicions www.elsevier.com/loce/jm Two normliy crieri nd he converse of he Bloch principle K.S. Chrk, J. Rieppo
More informationChapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More informationIX.2 THE FOURIER TRANSFORM
Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 7 IX. THE FOURIER TRANSFORM IX.. The Fourier Trnsform Definiion 7 IX.. Properies 73 IX..3 Emples 74 IX..4 Soluion of ODE 76 IX..5
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
More informationAnalytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function
Anlyic soluion of liner frcionl differenil equion wih Jumrie derivive in erm of Mig-Leffler funcion Um Ghosh (), Srijn Sengup (2), Susmi Srkr (2b), Shnnu Ds (3) (): Deprmen of Mhemics, Nbdwip Vidysgr College,
More informationApproximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces
Approximion nd numericl mehods for Volerr nd Fredholm inegrl equions for funcions wih vlues in L-spces Vir Bbenko Deprmen of Mhemics, The Universiy of Uh, Sl Lke Ciy, UT, 842, USA Absrc We consider Volerr
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.
ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Asrc. In his noe, we oin new some ineuliies
More informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
More informationSome New Dynamic Inequalities for First Order Linear Dynamic Equations on Time Scales
Applied Memicl Science, Vol. 1, 2007, no. 2, 69-76 Some New Dynmic Inequliie for Fir Order Liner Dynmic Equion on Time Scle B. İ. Yşr, A. Tun, M. T. Djerdi nd S. Küükçü Deprmen of Memic, Fculy of Science
More informationMultiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales
Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, -3; http://www.mth.u-szeged.hu/ejqtde/ Multiple Positive Solutions for the System of Higher Order Two-Point Boundry Vlue Problems
More informationPositive solutions for system of 2n-th order Sturm Liouville boundary value problems on time scales
Proc. Indin Acd. Sci. Mth. Sci. Vol. 12 No. 1 Februry 201 pp. 67 79. c Indin Acdemy of Sciences Positive solutions for system of 2n-th order Sturm Liouville boundry vlue problems on time scles K R PRASAD
More informationCALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION
Avilble online hp://scik.org Eng. Mh. Le. 15, 15:4 ISSN: 49-9337 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION PANDEY, C. P. 1, RAKESH MOHAN AND BHAIRAW NATH TRIPATHI 3 1 Deprmen o Mhemics, Ajy
More informationResearch Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals
Hindwi Pulishing orporion Inernionl Journl of Anlysis, Aricle ID 35394, 8 pges hp://d.doi.org/0.55/04/35394 Reserch Aricle New Generl Inegrl Inequliies for Lipschizin Funcions vi Hdmrd Frcionl Inegrls
More informationProcedia Computer Science
Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationWeighted Hardy-Type Inequalities on Time Scales with Applications
Medierr J Mh DOI 0007/s00009-04-054-y c Sringer Bsel 204 Weighed Hrdy-Tye Ineuliies on Time Scles wih Alicions S H Sker, R R Mhmoud nd A Peerson Absrc In his er, we will rove some new dynmic Hrdy-ye ineuliies
More informationFRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES
FRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES M JIBRIL SHAHAB SAHIR Accepted Mnuscript Version This is the unedited version of the rticle s it ppered upon cceptnce by the journl. A finl edited
More informationOn the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations
Journl of Mhemics nd Sisics 5 ():136-14, 9 ISS 1549-3644 9 Science Publicions On he Pseudo-Specrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy
More information..,..,.,
57.95. «..» 7, 9,,. 3 DOI:.459/mmph7..,..,., E-mil: yshr_ze@mil.ru -,,. -, -.. -. - - ( ). -., -. ( - ). - - -., - -., - -, -., -. -., - - -, -., -. : ; ; - ;., -,., - -, []., -, [].,, - [3, 4]. -. 3 (
More informationgraph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationDynamic Systems and Applications 12 (2003) A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE
Dynamic Sysems and Applicaions 2 (2003) 20-25 A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE KIRSTEN R. MESSER Universiy of Nebraska, Deparmen of Mahemaics and Saisics, Lincoln NE, 68588, USA. E-mail:
More informationOSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES
Dynamic Sysems and Applicaions 6 (2007) 345-360 OSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES S. H. SAKER Deparmen of Mahemaics and Saisics, Universiy of Calgary,
More informationHonours Introductory Maths Course 2011 Integration, Differential and Difference Equations
Honours Inroducory Mhs Course 0 Inegrion, Differenil nd Difference Equions Reding: Ching Chper 4 Noe: These noes do no fully cover he meril in Ching, u re men o supplemen your reding in Ching. Thus fr
More informationExistence of positive solutions for second order m-point boundary value problems
ANNALES POLONICI MATHEMATICI LXXIX.3 (22 Exisence of posiive soluions for second order m-poin boundary value problems by Ruyun Ma (Lanzhou Absrac. Le α, β, γ, δ and ϱ := γβ + αγ + αδ >. Le ψ( = β + α,
More informationEIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES
Elecronic Journal of Differenial Equaions, Vol. 27 (27, No. 37, pp. 3. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON
More informationExistence of positive solution for a third-order three-point BVP with sign-changing Green s function
Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion
More informationFRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
FRACTIONAL ORNSTEIN-ULENBECK PROCESSES Prick Cheridio Deprmen of Mhemics, ET Zürich C-89 Zürich, Swizerlnd dio@mh.ehz.ch ideyuki Kwguchi Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn hide@999.jukuin.keio.c.jp
More informationarxiv: v1 [math.pr] 24 Sep 2015
RENEWAL STRUCTURE OF THE BROWNIAN TAUT STRING EMMANUEL SCHERTZER rxiv:59.7343v [mh.pr] 24 Sep 25 Absrc. In recen pper [LS5], M. Lifshis nd E. Seerqvis inroduced he u sring of Brownin moion w, defined s
More informationA General Dynamic Inequality of Opial Type
Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn
More informationAverage & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
More informationNumerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
Numericl Approximions o Frcionl Problems of he Clculus of Vriions nd Opiml Conrol Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres To cie his version: Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres.
More informationCopyright by Tianran Geng 2017
Copyrigh by Tinrn Geng 207 The Disserion Commiee for Tinrn Geng cerifies h his is he pproved version of he following disserion: Essys on forwrd porfolio heory nd finncil ime series modeling Commiee: Thlei
More informationJournal of Computational and Applied Mathematics. On positive solutions for fourth-order boundary value problem with impulse
Journl of Computtionl nd Applied Mthemtics 225 (2009) 356 36 Contents lists vilble t ScienceDirect Journl of Computtionl nd Applied Mthemtics journl homepge: www.elsevier.com/locte/cm On positive solutions
More informationMATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
More informationFRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR
Romnin Repors in Physics, Vol. 64, Supplemen, P. 7 77, Dediced o Professor Ion-Ioviz Popescu s 8 h Anniversry FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR D. BALEANU,,3, J. H. ASAD
More informationCommunications inmathematicalanalysis
Communicions inmhemicanysis Voume 11, Number 2,. 23 35 (211) ISSN 1938-9787 www.mh-res-ub.org/cm GRONWALL-LIKE INEQUALITIES ON TIME SCALES WITH APPLICATIONS ELVAN AKIN-BOHNER Dermen of Mhemics nd Sisics
More informationHILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS
HILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS L. ERBE, A. PETERSON AND S. H. SAKER Absrac. In his paper, we exend he oscillaion crieria ha have been esablished by Hille [15] and Nehari
More informationFirst-Order Recurrence Relations on Isolated Time Scales
Firs-Order Recurrence Relaions on Isolaed Time Scales Evan Merrell Truman Sae Universiy d2476@ruman.edu Rachel Ruger Linfield College rruger@linfield.edu Jannae Severs Creighon Universiy jsevers@creighon.edu.
More informationNecessary and Sufficient Conditions for Asynchronous Exponential Growth in Age Structured Cell Populations with Quiescence
JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 25, 49953 997 ARTICLE NO. AY975654 Necessry nd Sufficien Condiions for Asynchronous Exponenil Growh in Age Srucured Cell Populions wih Quiescence O. Arino
More informationEXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2 ISSN 223-727 EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS Yuji Liu By applying monoone ieraive meho,
More informationExistence of Solutions to First-Order Dynamic Boundary Value Problems
Interntionl Journl of Difference Equtions. ISSN 0973-6069 Volume Number (2006), pp. 7 c Reserch Indi Publictions http://www.ripubliction.com/ijde.htm Existence of Solutions to First-Order Dynmic Boundry
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationA 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
More information1. Find a basis for the row space of each of the following matrices. Your basis should consist of rows of the original matrix.
Mh 7 Exm - Prcice Prolem Solions. Find sis for he row spce of ech of he following mrices. Yor sis shold consis of rows of he originl mrix. 4 () 7 7 8 () Since we wn sis for he row spce consising of rows
More informationSolutions for Nonlinear Partial Differential Equations By Tan-Cot Method
IOSR Journl of Mhemics (IOSR-JM) e-issn: 78-578. Volume 5, Issue 3 (Jn. - Feb. 13), PP 6-11 Soluions for Nonliner Pril Differenil Equions By Tn-Co Mehod Mhmood Jwd Abdul Rsool Abu Al-Sheer Al -Rfidin Universiy
More information