Systems of nonlinear ODEs with a time singularity in the right-hand side

Transcription

1 Syem of nonlinear ODE wih a ime ingulariy in he righ-hand ide Jana Burkoová a,, Irena Rachůnková a, Svaolav Saněk a, Ewa B. Weinmüller b, Sefan Wurm b a Deparmen of Mahemaical Analyi and Applicaion of Mahemaic, Faculy of Science, Palacký Univeriy Olomouc, 7. liopadu 2, Olomouc, Czech Republic b Deparmen for Analyi and Scienific Compuing, Vienna Univeriy of Technology, Wiedner Haupraße 8, A-4 Wien, Auria Abrac We udy boundary value problem for yem of nonlinear ordinary differenial equaion wih a ime ingulariy, x = M x + f, x,, ], bx, x =, where M : [, ] R n n and f : [, ] R n R n are coninuou marix-valued and vecor-valued funcion, repecively. Moreover, b : R n R n R n i a coninuou nonlinear mapping which i pecified according o a pecrum of he marix M. For he cae ha M ha eigenvalue wih nonzero real par, we prove new reul abou exience of a lea one coninuou oluion on he cloed inerval [, ] including he ingular poin, =. We alo formulae ufficien condiion for uniquene. The heory i illuraed by a numerical imulaion baed on he collocaion mehod. Keyword: BVP, ODE, ime ingulariy, global exience, uniquene, fixed poin heorem 2 MSC: 34A2, 34A34, 34B6. Moivaion In he preen work, we focu on he olvabiliy of he ingular nonlinear boundary value problem BVP, x = M x + f, x,, ], bx, x = and our main aim i o generalize reul from [8, 9], where he exience analyi of linear BVP wih conan, M, and variable, M, coefficien marix were provided, repecively. Imporan in hi conex i o noe ha he exience and uniquene reul for he analyical oluion alo anwer he queion of he well-poedene of he relaed BVP. Thi queion ha o be reolved before urning o he analyi of any numerical algorihm applied o approximae he analyical oluion. I i epecially difficul o anwer for BVP wih ingulariie, becaue in hi cae only boundary condiion of a cerain rucure guaranee ha he analyical problem can be uccefully approximaed by a numerical mehod. In hi ene, he analyical paper [9] can be een a a neceary prerequiie for [] where he convergence of polynomial collocaion for he variable coefficien cae, M, wa udied. Thu, he preen aricle in which he boundary condiion, ufficien for he well-poedene of he nonlinear BVP, are preciely pecified, i he preparaion for he repecive convergence analyi. A he end of he paper, we horly dicu he experimenally oberved, quie inriguing, behavior of he collocaion cheme applied o olve ome nonlinear e example of he above ype. Our nex goal i o analyze and explain hi convergence behavior. Correponding auhor addree: jana.burkoova@upol.cz Jana Burkoová, irena.rachunkova@upol.cz Irena Rachůnková, vaolav.anek@upol.cz Svaolav Saněk, ewa.weinmueller@uwien.ac.a Ewa B. Weinmüller, efan.wurm@uwien.ac.a Sefan Wurm Preprin ubmied o Applied Numerical Mahemaic January 23, 28

2 2. Inroducion A already menioned, he aim for he preen work i o analyze he olvabiliy of BVP for yem of nonlinear ordinary differenial equaion ODE wih a ime ingulariy a he origin, x = M x + where f C[, ] R n ; R n, M C[, ]; R n n, b CR n R n ; R n, and n N. f, x,, ], bx, x =, 2 Definiion. We ay ha x : [, ] R n i a oluion of yem on [, ] if x C[, ]; R n C, ]; R n and hold for, ]. We ay ha x : [, ] R n i a oluion of BVP, 2, in cae ha x i a oluion of yem and if i aifie condiion 2. The boundary condiion in 2 are pecified according o a pecrum of he marix M. Throughou he paper, we aume ha eigenvalue of M have nonzero real par and dicu he exience of oluion o problem, 2 which are coninuou on [, ], including he ingular poin =. The nonlinear boundary value problem of he form, 2 wih ingulariie udied here arie in he modelling of now avalanche run-up and run-ou. The leading-edge model decribing he dynamic of dry-flowing avalanche i conidered in [24, 3, 3]. In hi model, five force are combined o give he oal force governing he avalanche dynamic; he driving force, momenum flux, dynamic Coulomb reiive force, urbulen reiive force and paive now preure force. The analyi preened in our paper can be een a he coninuaion of he work iniiaed in [, 3] and coninued in [24, 34], where fir and econd order linear BVP wih ingulariie were udied. Linear model in hee paper are ypically of he form x = M x + f,, ], B x + B x = c, 3 where f C[, ]; R n, B, B R n n, c R n. The cae of nonmooh inhomogeneiy in 3 wrien a x = M x + f,, ], B x + B x = c, 4 wa inveigaed in full deail in [9], where he aenion wa focued on he exience, uniquene and moohne of oluion. I urned ou ha i i neceary o precribe boundary condiion of a cerain rucure o guaranee he well-poedne of he problem. The form of uch condiion depend on pecral properie of he marix M. The convergence analyi of he collocaion mehod applied o olve 4 can be found in [], while he analyical reul and convergence of collocaion applied o 4 wih conan coefficien marix M are provided in [8]. The fir analyical framework for he nonlinear ingular BVP of he ype x = M x + f, x,, ], B x + B x = c, 5 wih f C[, ] R n ; R n, can be found in [, 3]. Boh paper do no provide he exience and uniquene proof. Only, a local argumen i given, baed on he aumpion ha an appropriaely mooh oluion x exi. The main reul from [, 3] ae ha for he ufficienly mooh problem daa, he linearized BVP a x i uniquely olvable and able. Thoe properie are crucial in he conex of any dicreizaion approach aiming a he approximae oluion of he BVP 5. Since [, 3] are mainly focued on he numerical approximaion of he problem, hi local analyi i a preciely ailored prerequiie for he convergence heory of he involved numerical mehod. Moreover, i accoun for he fac ha in applicaion of he form 5 muliple oluion may arie. Following paper, [2, 4, 2, 22, 23, 34, 35], provide furher analyi of variou numerical mehod applied o olve ingular BVP of he fir and econd order. Here, main focu i on collocaion mehod, error eimaion, grid adapaion and ofware developmen. 2

3 A differen analyi of he nonlinear yem of ype in he form u = g, u,, T], wa conidered in [33], where he local exience and uniquene of a oluion in C m [, T ]; R n, T T, m, wihou precribing boundary condiion wa hown. Thi reul wa obained under he aumpion ha g C m [, T] R n ; R n and ha here exi ω uch ha g, ω =. Moreover, he following aumpion are made on he Jacobian: A, u = g, u/ u C m [, T] R n ; R n n and he eigenvalue λ k of A, ω have o aify he rericion m > Reλ k. For he cae m =, equivalen o conidering g, A and u only coninuou wihou coninuou higher derivaive, he above condiion reric he analyi o negaive real par of he eigenvalue of A, ω, > Reλ k. We now hall dicu he novely and imporance of he preen paper. From he above dicuion, i i clear ha here i an exenive lieraure on ingular BVP. However, global exience and uniquene reul are rare. In hi paper we provide he miing global analyi for problem, 2. We dicu he global exience and uniquene of coninuou oluion of he nonlinear ingular yem. In hi eing, we provide a complee udy of he problem where M ha arbirary eigenvalue wih nonzero real par and general Jordan canonical form. We poin ou ha our heory preciely decribe iniial, erminal or boundary condiion, repecively, which are neceary and ufficien for he relaed IVP, TVP, and BVP o have a oluion in C[, ]. The global reul in Theorem 5, 2 and 26 are, o our knowledge, he fir uch exience reul in he lieraure. They even provide a new inigh for he problem 5 where a ime ingulariy occur only in he linear erm becaue previou paper dealing wih 5 only give local argumen baed on an aumpion ha an appropriaely mooh oluion exi. The comparion of he preen paper wih [33] i only poible in he pecial cae when we look a Lemma 4 in our paper and Theorem 3 in [33]. Boh reul deal wih he local exience and uniquene of a oluion o for M wih eigenvalue whoe real par are negaive. However, in [33] more moohne on he problem daa i required o how an analogou aemen. The now available global heory i an imporan preparaion for he numerical analyi and he following compuaional reamen, ince i provide an eenial rucural informaion on he underlying BVP, namely i well-poedne. Thi propery i indipenable for any analyical problem which i ubjec o numerical reamen. I imply ae ha he unique oluion of he analyical problem depend coninuouly on he problem daa. Thi, for inance, enable o eimae he effec of he modelling error and round-off error, in cae ha hey can be inerpreed a mall perurbaion in he inhomogeneiy. Moreover, he precie knowledge of iniial/erminal/boundary condiion, which are neceary and ufficien for he relaed IVP/TVP/BVP o be well-poed can alo be uilized in applicaion. I enable he uer o verify if he precribed boundary condiion are correcly aed and in cae ha hey are miing, he know how o correcly cloe he yem. The reuling IVP/TVP/BVP are hen uiable for he numerical reamen. Anoher rong moivaion for he preen analyi i he imporan cla of regular problem poed on he infinie inerval [, which can be relaed o he ingular problem of ype, 2 poed on, ]. Originally, uch problem ofen have he form, x = Nx + g, x, [,, bx, x =, 6 wih coninuou N, g and b. There are numerou applicaion of hi ype, where ake a role of ime, and here are everal poibiliie o olve problem 6 numerically. The olde mehod i o olve he problem on a finie inerval [, L], where L i appropriaely large and replace he boundary condiion bx, x = by bx, xl =, ee [8]. The diadvanage of hi approach are evere. I i ofen neceary o olve he problem for differen value of L in order o find ou he accuracy of he approximaion. 3

4 Moreover, i may be neceary o olve he problem on very large inerval o guaranee a reaonable qualiy of he numerical oluion, cf. [28], where L = O 3. To overcome hee difficulie, anoher developmen wa propoed, where he aympoically correc boundary condiion were derived and impoed a he righ boundary L [27, 29]. Thi mehod i by no mean raighforward and reul in highly nonlinear condiion. Anoher opion i o ue a coordinae ranformaion and olve he o-called free boundary formulaion, ee [6]. The main idea of hi mehod i a follow. Le u conider he BVP, Then, i free boundary formulaion read: x = g, x, x, [,, x = x, x = x. x ɛ = g, x ɛ, x ɛ, [, ɛ ], x ɛ = x, x ɛ ɛ = x, x ɛ ɛ = ɛ, where ɛ i an unknown parameer and ɛ. Noe ha hi mehod can be applied o ODE of a lea order wo provided ha he fir derivaive of x end monoonically o zero a infiniy, lim x =. Moreover, uually, we have o olve a erie of BVP for differen value of ɛ. In view of difficulie of approache baed on runcaion of he inerval of inegraion, anoher idea eem o be more promiing. I i baed on a ranformaion of he inerval [, o he finie domain, ]. The advanage of working on a finie mall inerval [, ] while dicreizing he analyical problem are eviden, however wih hi ranformaion, we uually inroduce a ingulariy a = and he problem daa become nonmooh. For he ranformaion = e, yem in 6 ake he form and hi mean ha he reuling ingulariy i of he fir kind, where he power of in he denominaor of he righ-hand ide i one. Such ingulariy can in general, be handled more efficienly, when compared o a ingulariy of he econd kind, where he power of in he denominaor of he righ-hand ide in i larger han one. A an example, we poin o a boundary value problem from a heory for he exploive cryallizaion of hin amorphou layer on a ubrae [6, 25, 26]. Here, he aim i o compue he cryallizaion rae and he emperaure diribuion of a cryallizaion fron propagaing hrough a hin layer of amorphou maerial on a ubrae. The original problem poed on a emi-infinie inerval [, when ranformed o [, ] uing = e, reul in a boundary value problem for a yem of wo nonlinear equaion. Uing he andard ranformaion [6, 2], = / +, inroduce a numerically le advanageou ingulariy of he econd kind. Moivaed by he above dicuion, we focu our aenion on he ingular nonlinear yem, x = M x + f, x,, ]. Our aim i o pecify he boundary condiion in 2 depending on he pecrum of M. We fir deal wih he cae of only negaive real par of eigenvalue of M and inveigae he aociaed IVP. Then, we udy he cae of only poiive real par of eigenvalue of M and analyze a erminal value problem TVP. Finally, a BVP i udied in he cae of boh poiive and negaive real par of eigenvalue of M. In all hree cae we inend o prove he exience of oluion which are a lea coninuou. The Banach Fixed Poin Theorem eem o be very helpful in dealing wih difficulie caued by he ingulariy a =. I provide boh, he exience and uniquene of a oluion of ubjec o uiable iniial, erminal or boundary condiion. However, cerain rericion have o be impoed on he lengh of he inerval where he unique olvabiliy i guaraneed. Thi mean ha we fir inveigae yem on a ufficienly mall inerval [, δ], < δ. The form of iniial, erminal or boundary condiion ha pecify he unique coninuou oluion on [, δ] depend on he pecral properie of he conan marix M. Then, having he exience and uniquene on [, δ], we inveigae a correponding IVP, TVP or BVP on [, ] by mean of he Leray-Schauder alernaive. The paper i organized a follow. Preliminarie are inroduced in Secion 3. Reul for he linear cae wih a conan coefficien marix are ummarized in Secion 4. The main Secion 5 i devoed o he analyi of he nonlinear problem. 4

5 We fir deal wih he cae of only negaive real par of eigenvalue of M. I urn ou ha he unique coninuou oluion on [, δ] i deermined by he following rucure of iniial condiion: Mx + f, x =. 7 Condiion ufficien for he olvabiliy of he IVP, 7 on [, ] are preened in Secion 5., ogeher wih he dicuion of he unique olvabiliy of he IVP, 7. Noe ha he form of he iniial condiion 7 follow from he requiremen ha he oluion x of i coninuou on he cloed inerval [, ] including he ingular poin =. Hence, we can inerpre 7 a he neceary condiion for o be he well-poed. The cae where all eigenvalue of he marix M have poiive real par i conidered Secion 5.2. In hi iuaion, each oluion of equaion on [, ] aifie condiion 7. Conequenly, here i no uniquely olvable IVP and herefore, we have o udy he yem ubjec o a erminal condiion which i choen for impliciy a x = c, c R n. 8 In Secion 5.2, condiion for he exience and uniquene of oluion o he TVP, 8 on [, ] are formulaed. Finally, in Secion 5.3, we admi he marix M o have a mixed pecrum wihou zero and purely imaginary eigenvalue. Baed on he reul of Secion 5. and Secion 5.2 for M wih only negaive and only poiive real par of eigenvalue, repecively, yem equipped wih he following boundary condiion i udied: NMx + N f, x =, Px = Pc, c R n, 9 where N and P are appropriaely defined projecion marice. We provide exience and uniquene reul for he BVP, 9. In Secion 6, we numerically imulae hree model problem covering he hree differen pecra of he marix M analyzed in he paper. Finally, concluion can be found in Secion Preliminarie Throughou he paper we ue R n and C n o denoe he n-dimenional vecor pace of real-valued and complex-valued vecor x, repecively, equipped wih he maximum norm, x = max{ x i ; i n}. Similarly, we denoe by R m n and C m n he pace of real-valued and complex-valued m n marice B, repecively, wih he norm defined a B = max{ m j= b i j ; i n}. Moreover, I R n n i he ideniy marix and Θ R n n i he zero marix. Le J [, ], hen CJ; R n i he pace of he real-valued funcion which are coninuou on J, while CJ;, and CJ; [, are he pace of coninuou poiive and nonnegaive funcion, repecively. We equip he pace C[, δ]; R n ], δ, ] wih he norm x δ := max{ x ; [, δ]} and he pace C[δ, ]; R n, δ, wih he norm x := max{ x ; [δ, ]}. Finally, C J; R n i he pace of real-valued vecor funcion which are coninuouly differeniable on J, CJ R n ; R n and CJ; R n m are he pace of real-valued vecor funcion and marix funcion, coninuou on he paricular e. The exience of oluion on he whole inerval [, ] of he IVP, 7, and he TVP, 8 i hown uing he Leray-Schauder alernaive [5] and by he pecial form of he Bihari inequaliy [5, 7], formulaed in he following lemma. Lemma 2 Bihari. Le δ, and u C[δ, ]; [,. Le w C[, ;, be nondecreaing, and If u aifie he inegral inequaliy where B, B 2 are poiive conan, hen w d =. u B + B 2 wu d, [δ, ], δ u G GB + B 2 δ, [δ, ], 5

6 where G i defined a and G i he invere of G. Gx = x w d, x [,, Noe, ha if we apply Lemma 2 wih u replaced by u + δ, we obain a ueful modificaion of he Bihari inequaliy. Lemma 3. Le δ, w and G be a in Lemma 2. If u aifie he inegral inequaliy where B, B 2 are poiive conan, hen u B + B 2 wu d, [δ, ], u G GB + B 2, [δ, ]. Anoher imporan ool ued in he proof i he Leray-Schauder alernaive, cf. [5, Cor. 8.]. Lemma 4 Leray-Schauder alernaive. Le Y be a Banach pace and K : Y Y a compleely coninuou operaor. Then he following alernaive hold: Eiher x = λk x ha a oluion for every λ [, ] or he e S = {x Y : x = λk x for ome λ, } i unbounded. 4. Linear yem wih conan coefficien marix A In hi ecion, we collec he mo imporan reul for he linear cae wih a conan coefficien marix A. We ue hee prerequiie for he inveigaion of he nonlinear yem in Secion 5. We fir conider he linear homogeneou yem x = A x,, ], wih a regular marix A R n n. If Φ i he fundamenal oluion marix of yem aifying Φ = I, hen Φ ha he form Φ = A A k ln k = expa ln =,, ]. k! Le A have he eigenvalue λ,..., λ m C, m n and le u denoe by J he Jordan canonical form of A. Moreover, le n,..., n m be he dimenion of he Jordan boxe J,..., J m correponding o he no necearily differen eigenvalue λ,..., λ m. Le E C n n be he aociaed marix of generalized eigenvecor of A, ha i, he marix ranforming A o i canonical form. Then, k= A = EJE, A = E J E, J = diagj,..., J m, J = diag J,..., J m. The baic properie of he fundamenal oluion marix are From, we ee ha he marice A and A aify A A = A A, A = I, A = A I,, ]. 2 A = A A, A = A A = A A I,, ]. 3 6

7 Chooe k {,..., m} and conider he eigenvalue λ k = σ k + iρ k. Le u denoe by Λ k he following n k n k marix: ln ln 2 ln 2!... n k n k! ln n k 2 n k 2! Λ k = ln...,, ] Then, we have J k = λ k Λ k, for, ], and hence Jk λk J k = = Λ k = σ k iρ k Λ k,, ]. Since ln / = ln for, ], we obain Λ k = n k Λ ln j k =,, ], j! j= J k = σ k Λ k, J k = σ k Λ k,, ]. 4 Clearly, from i follow lim + α ln j = for α >, j N, 5 σ k < lim + J k = and σk > lim + J k =. 6 Eigenvalue of A wih negaive real par. Aume ha all eigenvalue λ k of A have negaive real par Then, by and 6, we have Inegraion of 3 over [τ, ], ] give Leing τ + and uing 7, we obain and i follow by 2 λ k = σ k + iρ k, σ k <, k =,..., m. lim + A = Θ. 7 A τ A = A A I d. τ A I d = A A,, ], A I d = A. 8 The nex lemma i a pecial cae of Lemma 3 and Lemma 4 from [8]. Since i proof i hor in hi eing, we preen i here for a reader convenience. Lemma 5. Aume ha all eigenvalue of A have negaive real par. Then, lim + J k I k d =, J k I k d = n k j=, k =,..., m, 9 σ k j+ where I k R n k n k i he ideniy marix. 7

8 Proof. For k =,..., m, conider We fir how For =, i follow from 2 ha ψ k = n k j j= p= σ k ln p,, ]. 2 p! σ k j+ p J k I k d = ψk,, ]. 2 ψ k = n k j= σ k j+, and by 5 and 2, we obain lim ψ k =. + Uing 2, 4 and repeaed inegraion by par, we conclude for [τ, ], ] τ J k I k d= σ k iρ k Λ k d= τ τ n k σ k Finally, le τ end o zero. Then, 2 hold and hi compee he proof. Nex corollary i a direc conequence of Lemma 5 and. Corollary 6. Aume ha all eigenvalue of A have negaive real par. Then, J I d = max k m n k j= σ k j+, j= ln j d=ψ k ψ k τ. j! A I d E E max k m n k j=. 22 σ k j+ Lemma 7. Aume ha all eigenvalue of A have negaive real par and conider δ, ] and h C[, δ]; R n. Then, lim A I h d = A h Proof. Le l N and le u define u l = l A I h d, [, δ]. Clearly, u l C[, δ]; R n for l N. We now how ha lim u l A I h d = uniformly on [, δ]. 24 For [, δ], we have by 2 l l A I h d l A I h d = A I h δ d E E max k m l { ψ k l A I h d } h δ. Moreover, 9 implie ha lim l ψ k /l = for k =,..., m, and hence, 24 follow. Conequenly, he funcion i coninuou on [, δ] and herefore, due o 8, u = lim u = u = + A I h d A I d h = A h. 8

9 Eigenvalue of A wih poiive real par. Aume ha all eigenvalue λ k of A have poiive real par Then, by and 6, we have Inegraion of 3 over [, δ], ] give λ k = σ k + iρ k, σ k >, k =,..., m. lim + A = Θ. 25 δ A A = A A I d. We now muliply he la equaion by A and ue 2 o obain A A d = A I, δ, δ], 26 and, by 25, A lim + d = A. Nex lemma follow from [8, Lemma 7] and [9, Lemma 3.4]. Lemma 8. Aume ha all eigenvalue of A have poiive real par. Then, here exi a conan c > uch ha A d c,, ], 27 lim A + d = for each δ, ]. 28 δ Lemma 9. Aume ha all eigenvalue of A have poiive real par. Le δ, ]. Then, for, δ] Jk nk d j= σ j+ k, k =,..., m. 29 Proof. Chooe k {,..., m}. Then, by 2, 4, we can wrie for, δ] Jk δ d = σk n k ln j n k d = h j, 3 j! where Hence, h j = σ k σ k h = σ k Inegraion by par yield, for, δ] and j {,..., n k }, h j = σ k δ j δ σ k ln + j j! σ k and, due o 3 and he fac ha he fir erm i nonpoiive, we deduce j= j= ln j d, j =,..., n k. 3 j! σ k d σ k,, δ]. 32 σ k σ k j ln d, h j σ k h j,, δ]. 33 Conequenly, 29 follow from 3, 32, and 33. 9

10 A a direc conequence of Lemma 9 and, we have he following reul. Corollary. Aume ha all eigenvalue of A have poiive real par. Then, for δ, ] and, δ] J n k d max k m j= σ j+ k, A n k d E E max k m j= σ j+ k. 34 Lemma. Aume ha all eigenvalue of A have poiive real par and conider δ, ] and h C[, δ]; R n. Then, A lim + h d = A h. 35 Proof. Chooe ε >. Then here exi δ, δ] uch ha h h < ε 2c,, δ, 36 where he conan c i from 27. The coninuiy of h on [, δ] provide a conan c > wih According o 28 in Lemma 8, here exi δ 2 > uch ha For, δ], le u inroduce I := δ h h c, [, δ]. 37 A d < A h h d, I 2 := Le δ = min{δ, δ 2 }. By 27, 36-38, if, δ, hen I A h h d + δ ε 2c,, δ A h d A h. A h h d < ε 2c c + c ε = ε. 2c Conequenly, lim + I =. Moreover, by 26, I 2 A A I A δ h A A h. δ Finally, by 25 we have lim + I 2 = and he aemen 35 follow from he following inequaliy leing + A h d A h I + I 2,, δ]. 5. Exience and uniquene reul for he nonlinear problem In hi ecion we inveigae he original nonlinear yem, x = M x + From now on, we aume ha he eigenvalue λ k of M aify f, x,, ]. λ k = σ k + iρ k, σ k, k =,..., m.

11 According o, we have for A = M M = EJE, M = E J E, J = diagj,..., J m, J = diag J,..., J m, 39 where E C n n i he marix of generalized eigenvecor aociaed wih M. In wha follow, we denoe g, x := M Mx + f, x, [, ], x R n, 4 n k β := E E max, 4 k m σ k j+ where n k i he dimenion of he Jordan box J k, k =,..., m. Depending on he pecrum of M, we hall diinguih beween IVP, TVP, and wo-poin BVP for yem. 5.. Eigenvalue of M wih negaive real par IVP Aume ha he eigenvalue λ k of M aify j= λ k = σ k + iρ k, σ k <, k =,..., m, and conider yem ubjec o he nonlinear iniial condiion 7, Mx + f, x =. A a fir ep in he exience proof of a lea one oluion of he IVP, 7 on he inerval [, ], we how he exience of a unique oluion of he IVP, 7 on an inerval [, δ] [, ] for a ufficienly mall δ. On he bai of Lemma 7, where hτ = Mτ Mxτ + f τ, xτ and x C[, δ]; R n, we define an operaor H : C[, δ]; R n C[, δ]; R n by H x = and noe ha 8 and 23 imply M I M Mx + f, x d, [, δ], 42 H x = M f, x = lim + H x. 43 Lemma 2. Aume ha all eigenvalue λ k of M have negaive real par and conider δ, ]. A mapping x C[, δ]; R n i a fixed poin of he operaor H if and only if x i a oluion of he IVP, 7 on [, δ], or equivalenly, x C[, δ]; R n C, δ]; R n aifie equaion on, δ] and he iniial condiion 7. Proof. We ue noaion 4: g, x := M Mx + f, x, for [, ], x R n, and carry ou he ubiuion τ = in he righ-hand ide of 42. Then, for [, δ] M I g, x d = M τ M I gτ, xτ dτ. 44. Aume ha x i a fixed poin of H. Then due o he above ubiuion, x = M τ M I gτ, xτ dτ, and by he baic properie 2 and 3 of he fundamenal oluion marix M, x = M M I τ M I gτ, xτ dτ + M M I g, x = M x + f, x,, δ].

12 Clearly, by 43, x i coninuou a = and aifie condiion 7. So, x i a oluion of he IVP, 7 on [, δ]. 2. Le x be a oluion of yem on [, δ]. According o Definiion, x i coninuou on [, δ]. Muliplying by M, we obain M x = M x M M I x = M I g, x,, δ]. 45 Inegraing 45 over [τ, ], δ], we have M x τ M xτ = Leing τ + and uing 7, we ge Conequenly by 44, and by 8 x = τ M x d = M x M M I x d = M x = x = Therefore, x i a fixed poin of H, and he reul follow. τ M I g, x d, M I g, x d,, δ]., δ], M I g, x d = M g, x = M f, x. τ M I g, x d. Remark 3. In he cae ha all eigenvalue of M have negaive real par, any oluion x of yem on [, δ], δ, ], aifie he iniial condiion 7. Thi follow from he econd par of he proof in Lemma 2 and clarifie he form of a correcly poed iniial condiion 7. Lemma 4. Aume ha all eigenvalue λ k of M have negaive real par and le β in 4 be given. Aume ha a, and L, /β exi uch ha f, x f, y L x y, [, a], x, y R n. 46 Then, here exi δ, a] uch ha he IVP, 7 ha a unique oluion u on [, δ]. Proof. Since M i coninuou on [, ], here exi δ, a] uch ha M M 2 β L, [, δ]. Therefore, by 46, for [, δ] and x, y R n, M Mx y + f, x f, y L x y, L = 2 β + L. 47 Le C[, δ]; R n be a Banach pace equipped wih he norm x δ = max{ x : [, δ]} and le H be given by 42. By Lemma 2, a mapping u C[, δ]; R n i a oluion of he IVP, 7 on [, δ] if and only if u i a fixed poin of he operaor H. We how ha H i a conracion. Le x, y C[, δ]; R n and g be given by 4. Then, by 22 wih A = M, 4 and 47, we have for [, δ], H x Hy = M I g, x g, y d L M I x y d L β x y δ, and hence, H x Hy δ L β x y δ. Aumpion Lβ < implie L β < due o 47 and herefore, H i a conracion. Hence, he IVP, 7 ha a unique oluion u on [, δ]. 2

13 We are now ready o how he fir main reul. To hi end, applying Lemma 2 and Lemma 4, we conruc a coninuaion of he oluion u from Lemma 4 ono he inerval [δ, ]. Theorem 5. Aume ha all eigenvalue λ k of M have only negaive real par and le β be pecified by 4. Le a, and L, /β exi uch ha f aifie he Lipchiz condiion 46. Finally, aume ha here exi W > uch ha f, x W + ω x, [a, ], x R n, 48 where ω C[, ;, i nondecreaing, ω for [,, and Then, he IVP, 7 ha a lea one oluion on [, ]. Proof. By he Lipchiz condiion 46, we can ee ha d ω =. Denoe Q := W + max{ f, : [, a]}. Then 48 and 49 yield f, x f, + L x, [, a], x R n. 49 f, x Q + L x + ω x, [, ], x R n. 5 Denoe Q 2 := L + + max{ M M : [, ]}, and conider g pecified in 4. Then 5 yield g, x Q + Q 2 ω x, [, ], x R n. 5 By Lemma 4, here exi δ, a] uch ha he IVP, 7 ha a unique oluion u on [, δ]. We now prove ha here exi a lea one oluion v of yem which i a C -coninuaion of u ono [δ, ]. To hi end, for b R n, we dicu yem on [δ, ] ogeher wih he iniial condiion xδ = b. 52 Le C[δ, ]; R n be equipped wih he norm x = max{ x : [δ, ]} and le Φ be he fundamenal oluion marix of he yem x = M x, [δ, ], wih Φ = I. Then, and Φ, Φ C[δ, ]; R n n. So we can wrie Φ = M Φ, Φ = M, [δ, ], 53 Φ = max{ Φ : [δ, ]}, Φ = max{ Φ : [δ, ]}. 54 I i clear ha a mapping v i a oluion of he IVP, 52 on [δ, ] if and only if v i a oluion of he inegral equaion x = ΦΦ δb + Φ Φ g, x d in C[δ, ]; R n. Hence, we define an operaor K : C[δ, ]; R n C[δ, ]; R n a K x = ΦΦ δb + Φ δ δ Φ g, x d. In order o how ha K admi a fixed poin v, we apply Lemma 4. Due o he coninuiy of he vecor funcion g and he marix funcion Φ, Φ, i i no difficul o verify ha K i coninuou and K map bounded e ino bounded 3

14 e. Nex, le Ω C[δ, ]; R n be bounded and denoe µ := up{ x : x Ω}. Then, for δ < 2 and x Ω, we have K x 2 K x Φ 2 Φ Φ δ b + Φ 2 Φ Φ g, x d 2 + Φ 2 Φ g, x d Φ 2 Φ Φ δ b + r δ + Φ Φ r δ 2, where r = up{ g, x : [δ, ], x µ }. Hence he e {K x : x Ω} i equiconinuou on [δ, ]. Conequenly, K i compleely coninuou. Finally, we prove ha he e S = {x C[δ, ]; R n : x = λk x for ome λ, } i bounded in C[δ, ]; R n. Le x C[δ, ]; R n aify x = λk x for ome λ,. Then, by 5 and 54, x K x Φ Φ Q + Q 2 ω x b + d K + K 2 ω x d, [δ, ], δ δ where Le u define Combining he inequaliy wih Lemma 2, we obain K = Φ Φ b + Q δ δ, K 2 = Φ Φ Q 2 δ. Gv := v dξ ωξ, v. x K + K 2 ω x d, [δ, ], δ x G GK + K 2 δ, [δ, ], where G i he invere of he funcion G. Hence, he e S i bounded in C[δ, ]; R n. Conequenly, by Lemma 4, he IVP, 52 ha a oluion v on [δ, ]. Due o and 52, wih b = uδ, we have uδ = vδ and u δ = v δ which mean ha v i a C -coninuaion of u ono [δ, ]. A a reul, he mapping { u, [, δ], x = v, [δ, ], i a oluion of he IVP, 7. Corollary 6. Le Lipchiz condiion 46 hold on [, ], which mean ha a = in Theorem 5. Then, he IVP, 7 ha a unique oluion on [, ]. Proof. The exience of a unique oluion u of he IVP, 7 on ome inerval [, δ], δ, ], follow from Lemma 4. Since he Lipchiz condiion hold on he whole inerval [, ], he oluion u can be uniquely exended a a oluion of ono [, ] Eigenvalue of M wih poiive real par TVP Aume ha all eigenvalue λ k of M have poiive real par, λ k = σ k + iρ k, σ k >, k =,..., m, and le u inveigae yem wih he erminal condiion 8, x = c, c R n. 4

15 In order o prove he exience of a lea one oluion of he TVP, 8 on [, ], we fir how he exience of a unique oluion x of yem on [, δ], for a ufficienly mall δ, ], uch ha x aifie he erminal condiion xδ = b, b R n. 55 Conider g pecified in 4, ha i, g, x = M Mx + f, x, for [, ], x R n. Due o Lemma, where h = g, x and x C[, δ]; R n, we define an operaor H : C[, δ]; R n C[, δ]; R n by M δ b M g,x d,, δ], H x = 56 lim + H x = M f, x, =. Lemma 7. Aume ha he eigenvalue λ k of M have poiive real par and le δ, ]. A mapping x C[, δ]; R n i a fixed poin of he operaor H if and only if x i a oluion of he TVP, 55 on [, δ], ha i, x C[, δ]; R n C, δ]; R n aifie equaion on, δ] and he erminal condiion 55. Proof.. Le x be a fixed poin of H. Then xδ = b, and by baic properie 2 of he fundamenal marix M, x = M M I δ M b M M I M I g, x d + g, x = M x + f, x,, δ]. 2. Le x be a oluion of he TVP, 55. Then muliplying by M and ubiuing g by 4, we ee ha x aifie 45 which inegraed over [, δ], δ] yield M M x = b g, x d,, δ]. δ Since x i coninuou on [, δ], i follow from 25 and 35, According o 56, x i a fixed poin of H. x = lim + x = M g, x = M f, x. 57 Remark 8. In he cae of only poiive real par of he eigenvalue of M each oluion x of on [, δ], δ, ], aifie he iniial condiion 7. To ee hi, we muliply by M and inegrae over [, δ], δ]. Therefore, x aifie M x = xδ δ M g, x d,, δ]. Thu, x = M f, x, ee 57 in he la par of he proof of Lemma 7. Lemma 9. Aume ha he eigenvalue λ k of M have poiive real par and le β be given via 4. Moreover, le here exi a, and L, /β uch ha he Lipchiz condiion 46 hold. Then, here exi δ, a] uch ha for each b R n he TVP, 55 ha a unique oluion on [, δ]. In addiion, hi oluion aifie he iniial condiion 7. Proof. The argumen are imilar o hoe from he proof of Lemma 4. We ake he pace C[, δ]; R n, where δ, a] i uch ha 47 hold and define an operaor H by 56. In order o prove ha H i a conracion, we conider x, y C[, δ]; R n and g from 4. By 34 wih A = M and 47 we obain, H x Hy = for, δ], and M g, x g, y d L H x Hy = lim + H x Hy L β x y δ. M x y d L β x y δ, Hence, H x Hy δ L ρ x y δ. The aumpion Lβ < implie L β < due o 47. Therefore, H i conracive and conequenly, he TVP, 55 ha a unique oluion on [, δ]. By Remark 8, hi oluion aifie 7. 5

16 We can now how our econd main reul characerizing he exience of a oluion of he TVP, 8 on [, ]. Theorem 2. Aume ha he eigenvalue λ k of M have poiive real par and le β be pecified in 4. Le here exi a, and L, /β uch ha f aifie he Lipchiz condiion 46. Finally, aume ha here exi W > uch ha he funcion f aifie 48. Then, for each c R n, he TVP, 8 ha a lea one oluion on [, ]. Moreover, hi oluion aifie he iniial condiion 7. Proof. We argue imilar o he proof of Theorem 5 and find Q, Q 2, uch ha he funcion g pecified in 4 aifie 5. By Lemma 9 here exi δ, a] uch ha for each b R n he TVP, 55 ha a unique oluion on [, δ]. We fir prove ha for uch δ and each c R n, here exi a lea one oluion v of he TVP, 8 on he inerval [δ, ]. Le Φ be he fundamenal oluion marix given in 53. Then, Φ, Φ C[δ, ]; R n n and we can ue 54. Subiuing g ino, we ee ha a mapping v i a oluion of he TVP, 8 on [δ, ] if and only if v i a oluion of he inegral equaion x = Φc Φ Φ g, x d in C[δ, ]; R n. Hence, we define an operaor K : C[δ, ]; R n C[δ, ]; R n a K x = Φc Φ Φ g, x d. 58 Uing he ame argumen a in he proof of Theorem 5, we conclude ha K i compleely coninuou. Finally, aume ha x C[δ, ]; R n aifie x = λk x for ome λ,. Then, by 5 and 54, x K x Φ c + Φ Q + Q 2 ω x d K + K 2 ω x d, [δ, ], where Le u define Combining he inequaliy wih Lemma 3, we obain K = Φ c + Φ Q δ δ, K 2 = Φ Φ Q 2 δ. Gv := v dξ ωξ, v. x K + K 2 ω x d, [δ, ], x G GK + K 2 δ, [δ, ], where G i he invere of funcion G. Hence, he e S = {x C[δ, ]; R n : x = λk x for ome λ, } i bounded in C[δ, ]; R n and we can apply he Leray-Schauder alernaive from Lemma 4. Conequenly, he TVP, 8 ha a oluion v on [δ, ]. Le u chooe b = vδ. Then, i follow from Lemma 9 ha a unique oluion of problem, 55 on [, δ] exi uch ha uδ = vδ. Due o we have u δ = v δ which mean ha v i a C -coninuaion of u on [δ, ]. A a reul, he mapping { u, [, δ], x = v, [δ, ], i a oluion of he TVP, 8 on [, ]. In addiion, by Lemma 9, u aifie 7. Corollary 2. Le he Lipchiz condiion 46 hold on [, ] which mean ha a = in Theorem 2. Then, for each c R n he TVP, 8 ha a unique oluion on [, ]. Moreover, hi oluion aifie he iniial condiion 7. Proof. Chooe c R n and le δ be a in Lemma 9. Then, he TVP, 8 ha a unique oluion v on [δ, ] due o he Lipchiz condiion 46 which i now aified for [, ]. Alo, for b := vδ, Lemma 9 provide a unique oluion u of he TVP, 55 on [, δ] wih uδ = vδ. Due o, u δ = v δ, and hu, x i a oluion of he TVP, 8 and aifie 7, a in he final par of he proof of Theorem 2. The uniquene of he oluion x follow from he uniquene of u and v. 6

17 5.3. Mixed pecrum of M BVP Conider he marix funcion M from yem and aume ha he eigenvalue λ k of M have nonzero real par. Le marice N, P R n n repreen projecion ono he ubpace X, X + R n, where X i aociaed wih he eigenvalue of M having negaive real par, X + i aociaed wih he eigenvalue of M having poiive real par, and N : R n X, P : R n X +, X X + = R n, Nx + Px = x, x R n. 59 Due o 39, N = EÑE, P = E PE. 6 Here Ñ R n n i he diagonal marix wih one a he poiion where he marix J, ee 39, ha he eigenvalue wih negaive real par and zero enrie elewhere. Similarly, P R n n i he diagonal marix wih one a he poiion correponding he poiion of he eigenvalue wih poiive real par in he marix J and zero enrie elewhere. Then, Ñx + Py = max{ Ñx, Py }, x, y R n. 6 Remark 22. By [8, Lemma 8], we know ha he marice N and P commue wih he marice M and M. For he proof ee [32, Lemma 9]. According o he previou ecion, he rucure of he correcly poed boundary condiion depend on he pecrum of he marix M. In Secion 5. he cae of only negaive real par of eigenvalue of M wa dicued and he IVP, 7 wa udied. Secion 5.2 wa concerned wih he cae of only poiive real par of eigenvalue of M and he TVP, 8 wih arbirary c R n wa of inere. On he bai of hee reul we now inveigae yem wih a mixed pecrum of M ubjec o he boundary condiion 9 of he form NMx + N f, x =, Px = Pc, c R n. A in Secion 5. and 5.2, we fir dicu he unique olvabiliy of an auxiliary BVP on an inerval [, δ] for a ufficienly mall δ. Hence, for b R n and δ, ], we inroduce he boundary condiion NMx + N f, x =, Pxδ = Pb, 62 and define he operaor N δ : C[, δ]; R n C[, δ]; X, P δ : C[, δ]; R n C[, δ]; X + N δ x = Nx, P δ x = Px, [, δ]. 63 Again, conider g given via 4: g, x = M Mx + f, x, for [, ], x R n, and define an operaor H : C[, δ]; R n C[, δ]; R n by 42 and an operaor H 2 : C[, δ]; R n C[, δ]; R n by 56. Conequenly, H and H 2 read, noe ha g, x = f, x, H x = H 2 x = M I g, x d, [, δ], 64 δ M b M g,x d,, δ], M g, x, =, 65 repecively. Conequenly, N δ H + P δ H 2 : C[, δ]; R n C[, δ]; R n. Lemma 23. Aume ha he eigenvalue λ k of M have nonzero real par and le δ, ]. Then, a mapping x C[, δ]; R n i a fixed poin of he operaor N δ H + P δ H 2 if and only if x i a oluion of he BVP, 62 on [, δ], ha i, x C[, δ]; R n C, δ]; R n aifie equaion on, δ] and he boundary condiion 62. Proof.. Aume ha x i a fixed poin of N δ H + P δ H 2. Since x = N δ H + P δ H 2 x = N δ H x + P δ H 2 x, 7

18 and by 59, N δ x + P δ x = x, we can ee ha x aifie N δ x = N δ H x, P δ x = P δ H 2 x. 66 Due o 44, 63 and 64, for [, δ], N δ x = N M I g, x d = N M τ M I gτ, xτ dτ, and a in Par of he proof of Lemma 2 having in mind Remark 22, we conclude M N δ x f, x = N x +,, δ]. The condiion Nx = N M f, x follow from 43 and 64. A in Par of he proof of Lemma 7, we deduce from 65 and Remark 22 ha M P δ x f, x = P x +,, δ], and Pxδ = Pb. Conequenly, x i a oluion of he BVP, 62 on [, δ]. 2. Le x be a oluion of yem on, δ]. Now, we follow he argumen from he Par 2 of he proof of Lemma 2, inegrae 45 over [τ, ], δ] and obain N δ x = N M τ M xτ + M M I g, x d. Leing τ + and uing 7, 44, and 64, we derive he fir equaliy in 66. Now, aume in addiion ha x aifie he erminal condiion 55 and inegrae 45 over [, δ], δ] a in he Par 2 of he proof of Lemma 7. Then, we have P δ x = P Finally, i follow from 25 and 35, δ M b τ M g, x d,, δ]. P δ x = lim + P δ x = P M g, x, hi mean ha he econd equaliy in 66 hold. Hence, x i a fixed poin of N δ H + P δ H 2. Remark 24. Le he eigenvalue λ k of M have nonzero real par and le x be a oluion of yem on [, δ], δ, ]. Then x aifie 7. To how hi aemen, we ue he argumen from he econd par of he proof of Lemma 2 and obain Nx = N A in Remark 8, we have M I g, x d,, δ], Px = P M xδ Nx = N M f, x. M g, x d,, δ]. A in he la par of he proof of Lemma 7, we conclude Px= P M f, x and he condiion 7 follow. We now proceed wih he proof of he exience of a unique oluion of he BVP, 62 on [, δ] for a ufficienly mall δ >. 8

19 Theorem 25. Aume ha all eigenvalue λ k of M have nonzero real par and le β be pecified in 4. Le here exi a, and L, /β uch ha he Lipchiz condiion 46 hold. Then, here exi δ, a] uch ha for each b R n he BVP, 62 ha a unique oluion on [, δ]. In addiion, hi oluion aifie he iniial condiion 7. Proof. A in he proof of Lemma 4, we chooe he pace C[, δ]; R n, where δ, a] i uch ha 47 hold. Chooe b R n and conider he operaor N δ and P δ given in 63 and he operaor H and H 2 defined in 64 and 65. By Lemma 23, in order o prove ha a unique oluion of he BVP, 62 on [, δ] exi, i i ufficien o how ha he operaor N δ H + P δ H 2 i a conracion on C[, δ]; R n. Uing 39, 4, 47, 6, and 6, we can deduce for [, δ] and x, y C[, δ]; R n, N δ H + P δ H 2 x N δ H + P δ H 2 y = N M I g, x g, y M d + P g, x g, y d = EÑE E J I E g, x g, y J d + E PE E E g, x g, y d E Ñ J I E g, x g, y J d + P E g, x g, y d { = E max Ñ J I E g, x g, y d, P J E g, x g, y } d { max Ñ J I d, P J } d E E L x y δ. Conequenly, uing 22, 34, and 4, we obain n k N δ H + P δ H 2 x N δ H + P δ H 2 y δ max k m σ j= k j+ E E L x y δ = βl x y δ. By 47, he aumpion βl < implie βl < and hence, N δ H +P δ H 2 i a conracion on C[, δ]; R n. Therefore, he BVP, 55 ha a unique oluion on [, δ]. By Remark 24, hi oluion aifie 7. In order o move from he BVP, 62 on [, δ] o he BVP, 9 on [, ], we require he addiional aumpion, P f, x = P f, Px, [δ, ], x R n. 67 Theorem 26. Aume ha he eigenvalue λ k of M have nonzero real par and le β be given in 4. Le here exi a, and L, /β uch ha he Lipchiz condiion 46 hold. Finally aume ha f aifie 48 and condiion 67. Then, for each c R n he BVP, 9 ha a lea one oluion on [, ]. Thi oluion aifie he iniial condiion 7. Proof. By Theorem 25 here exi δ, a] uch ha for each b R n he BVP, 62 ha a unique oluion on [, δ]. Conider uch δ and Φ from 53. A in he beginning of Secion 5.3, ake he ubpace X and X + aociaed wih he eigenvalue of M and he projecion N : R n X and P : R n X +. Operaor N δ and P δ are given in 63. Similarly, inroduce he operaor N : C[δ, ]; R n C[δ, ]; X, P : C[δ, ]; R n C[δ, ]; X + N x = Nx, P x = Px, [δ, ]. Sep. Define an operaor K 2 : C[δ, ]; R n C[δ, ]; R n by 58, or equivalenly, K 2 x = Φc Φ 9 Φ g, x d.

20 Applying he argumen from he proof of Theorem 2 o he operaor P K 2, we ee ha here exi i fixed poin w C[δ, ]; X +. Tha i, P K 2 w = w and we have Differeniaion now yield w = P Φc Φ w = P Φ c Φ Φ g, w Φ g, w d, [δ, ]. d + Pg, w, [δ, ], and uing 4 and 53, we ee ha w C[δ, ]; X + i a oluion of he TVP on [δ, ], w = M w + P f, w, [δ, ], w = Pc. 68 Sep 2. Conider he BVP, 62 on [, δ] and chooe Pb = wδ. Then, by Theorem 25 here exi a unique oluion u of he BVP, 62 on [, δ] and Puδ = wδ. 69 In addiion, u aifie he iniial condiion 7. Sep 3. Define an operaor K : C[δ, ]; R n C[δ, ]; R n by K x = ΦΦ δuδ + Φ δ Φ g, w + x d, where w C[δ, ]; X + i from Sep and uδ R n i from Sep 2. Applying he argumen from he proof of Theorem 5 o he operaor N K, we ee ha here exi i fixed poin z C[δ, ]; X. Hence, N K z = z and we have z = N ΦΦ δuδ + Φ Φ g, w + z d, [δ, ]. 7 Differeniaing 7, we obain for [δ, ] z = N Φ Φ δuδ + Φ δ δ Φ g, w + z d + Ng, w + z. Uing 53, 4, and 7, we can ee ha z C[δ, ]; X i a oluion of he IVP z = M z + N f, w + z, [δ, ], zδ = Nuδ, 7 on [δ, ]. Le v := w + z. Then, by 59, 69, and 7 Moreover, v belong o C[δ, ]; R n, and by 67, Therefore he equaion in 7 can be wrien in he form uδ = Puδ + Nuδ = wδ + zδ = vδ. 72 Pv = w, Nv = z, [δ, ], 73 P f, v = P f, Pv = P f, w, [δ, ]. 74 z = M Nv + N f, v, [δ, ], 2

21 and, by 74, he equaion in 68 can be rewrien a w = M Adding he la wo equaion o each oher reul in v = M Pv + P f, v, [δ, ]. v + f, v, [δ, ]. 75 Therefore, v i a oluion of equaion on [δ, ]. Due o, 72, and 75, we have u δ = v δ and o, v i a C -coninuaion of u on [δ, ]. A a reul, he mapping { u, [, δ], x = v, [δ, ], i a oluion of equaion on [, ]. Since he condiion in 68 can be due o 73 expreed a Pv = Pc and u aifie 7 according o Sep 2, x aifie 9 a well a 7. Corollary 27. Le he Lipchiz condiion 46 hold on he whole inerval [, ]. Thi mean ha in Theorem 26, a =. Then, for each c R n he BVP, 9 ha a unique oluion on [, ]. Thi oluion aifie he iniial condiion 7. Proof. Le δ be from Theorem 25 and chooe c R n. Since he Lipchiz condiion 46 i aified for [δ, ], here exi a unique oluion w of he TVP 68 on [δ, ] and w C[δ, ]; X +. By Theorem 25, here exi a unique oluion u of he BVP, 62 on [, δ] and Puδ = wδ. Similarly, ince he Lipchiz condiion 46 i aified for [δ, ], here exi a unique oluion z of he IVP 7 on [δ, ] and z C[δ, ]; X. Finally, { u, [, δ], x = w + z, [δ, ], i a oluion x of he BVP, 9 on [, ]. The uniquene of he oluion x follow ince u, w, and z are unique. Remark 28. Theorem 26 and Corollary 27 hold alo when if we replace aumpion 67 by N f, x = N f, Nx, [, δ], x R n Numerical imulaion of model problem We ue he open domain MATLAB code bvpuie2. [2] o olve he model problem deigned in Secion 6, ee [2] for bvpuie., he original verion of he code. The baic olver of he code i baed on polynomial collocaion, a widely ued and well-udied andard oluion mehod for wo-poin BVP []. Moreover, for ingular problem, many popular dicreizaion mehod like finie difference, Runge Kua or muliep mehod how order reducion, hu making compuaion inefficien and prohibiing aympoically correc error eimaion and reliable meh adapaion. Therefore, in our code developmen, we have choen collocaion a a high-order, robu, generalpurpoe numerical mehod. The above code are new verion of he general purpoe MATLAB code bvp, cf. [3], which ha already been uccefully applied o a variey of problem, ee for example [4, 7, 9, 2, 2]. The code i deigned o olve implici yem of differenial equaion whoe order may vary. Here, a an example, we conider he problem F, y 4, y 3, y, y, y =, <, 77 by 3, y, y, y, y 3, y, y, y =. 78 The order can alo be zero, which mean ha algebraic conrain which do no involve derivaive are alo admied. 2

22 Problem 77, 78 may alo include unknown parameer o be calculaed along wih he oluion y. In hi cae, appropriaely many addiional boundary condiion in 78 need o be precribed. For hi reaon he problem a hand i in cope of bvpuie and we can run he code wih no furher pre-handling. The numerical approximaion defined by collocaion i compued a follow: On a grid := {τ i : i =,..., L}, = τ < τ < τ L =, we approximae each componen of he analyical oluion by a piecewie defined collocaing funcion p := p i, [τ i, τ i+ ], i =,..., L, where we require p C q [, ] if he order of he underlying differenial equaion or he highe derivaive of he oluion componen i q. Here, p i are polynomial of maximal degree k + q which aify he yem 77 in he collocaion poin, { i, j = τ i + ρ j τ i+ τ i, i =,..., L, j =,..., k}, < ρ < < ρ k <. The aociaed boundary condiion 78 are alo precribed for p. Claical heory [] predic ha he convergence order i a lea Oh k, where h i he maximal epize, h := max i τ i+ τ i. The ame could be hown for he fir order problem wih a ingulariy of he fir kind. Quie ofen, even he uperconvergence order, in cae of Gauian poin Oh 2k, can be oberved in pracice. To make he compuaion more efficien, an adapive meh elecion raegy baed on an a poeriori eimae for he global error of he collocaion oluion ha been implemened. The general form of he example ued a an illuraion for he heory i x = M x + µ where µ > i a given parameer. Moreover, λ + m M = m 2, M = m 2 λ 2 + m 22 f, x,, ], λ, m λ i j C[, ], i, j =, 2. 2 We aume ha λ, λ 2 R, λ, λ 2, and define β := max {/ λ, / λ 2 }. The funcion f i choen o have he following form: f, x = p + Hr, x, 79 where p C[, ]; R 2, H C[, ]; R 2 2, and r C[, ] R 2 ; R 2 are conruced in uch a way ha he funcion µ f aifie he Lipchiz condiion 46 and he growh condiion Example The fir example ha wo verion. The eigenvalue of M, λ =, λ 2 = 2, are negaive and herefore, we formulae hee problem a IVP and relae Example. and.2 o Corollary 6 and Theorem 5, repecively. Example.: Here, we conider where x = M x + 3 f, x,, ], Mx + f, x =, 8 3 { + exp exp M=, M=, β=max, } =. exp2 2 + exp The funcion f ha he form pecified in 79 wih p = 2 /2, r, x = inx + x 2, H = cox + x 2 22 in co.

23 Since max [,] H = 2 and r, x r, y 2 x y, for [, ], x, y R 2, we conclude 3 f, x f, y x y, [, ], x, y R2. Hence, /3 f aifie he Lipchiz condiion 46 on [, ] and by Corollary 6, he IVP 8 ha a unique oluion on [, ], ee Figure, lef x x 2 x x Figure : Example. lef: Numerical oluion obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion ha been ued wih olerance Tol a = Tol r = 2. The number of ubinerval in he iniial and final meh wa 2. Thi mean ha in he meh adapaion raegy only he relocaion of poin in he iniial meh wa neceary o aify he olerance requiremen. No addiional grid poin have been added. The final meh, where every hird poin i hown, i alo included. One can ee ha he meh become dene cloe o he ingular poin becaue in he preen cae he nonmooh direcion field eem o influence he grid deniy in he viciniy of he ingular poin. Example.2 righ: Numerical oluion obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion wa no working due o he nonmooh daa, cf. α 2, and he problem wa olved on an equidian meh. The error eimae indicae a maximal global relaive error of.5 9. The oluion how a eep boundary layer cloe o =. The number of equidian ubinerval in he iniial and final meh wa 4. Example.2: In he econd verion of Example, we change he value of µ o µ = 5 2 and obain x = M x f, x,, ], Mx + 5 f, x =. 8 2 The marice M and H a well a he vecor p remain unchanged and β =. In f we now have inx + x r, x = α 2 x log + x + α cox + x x 2, x 2 log + x + x 2 where α =.5.5 2, <.5,, α 2 =.5 2,.5. We chooe a =.5. Then max [,.5] H < 2 and max [,.5] α = /8. Thu, r, x r, y /4 x y, for [,.5], x, y R 2. Conequenly, 5/2 f aifie he Lipchiz condiion 46 on [,.5]. In addiion 5/2 f aifie he growh condiion 48 on [.5, ] wih w = + 2 ln + 2 +, [,. Now, from Theorem 5 i follow ha he IVP 8 ha a oluion on [, ], ee Figure, righ Example 2 The econd example i alo given in wo verion. Now, he eigenvalue of M, λ =, λ 2 = 2, are poiive and Example 2. and 2.2 ake he form of TVP. Again, we relae Example 2. and 2.2 o Corollary 2 and Theorem 2, repecively. 23

24 Example 2.: The rucure of he problem and he daa funcion are imilar o hoe in Example.. We udy he TVP, M = x = M x exp exp exp2 2 + exp 2 f, x,, ], x =, M = {, β = max, } =. 2 2 Here f i given by 79, where p, r, H are from Example.. The olvabiliy follow from Corollary 2, cf. Figure x x Figure 2: Example 2.: Numerical oluion obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion ha been ued wih olerance Tol a = Tol r = 2. The number of ubinerval in he iniial and final meh wa 2. Thi mean ha in he meh adapaion procedure he meh poin have been merely relocaed o beer reflec he oluion behavior. The final meh, where every hird poin i hown, i alo included. Example 2.2: In he econd verion of Example 2, we change he value of µ o µ = 5/2 and obain x = M x f, x,, ], x =. 82 The marix M remain unchanged and β =. Funcion f i here deerminaed by p, H and r, x from Example 2.2. By Theorem 2, a oluion of he TVP 82 exi, cf. Figure x x 2 4 err x 6 err x Figure 3: Example 2.2: Numerical oluion lef obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion wa no working properly, and he problem wa olved on an equidian meh. The error eimae indicae a maximal global relaive error of The problem i hard o handle ince he daa are nonmooh, cf. α 2. The number of equidian ubinerval in he final meh wa 4. For illuraion, we include he error eimae for he abolue global error of he numerical oluion righ. 24

25 6.3. Example 3 We finally addre he cae of general BVP. Thi mean ha he eigenvalue of M have differen ign, λ = i negaive and λ 2 = 2 i poiive. Conequenly, he boundary condiion are wo-poin condiion involving boh boundarie = and =. The correponding heoreical reul for Example 3. and 3.2 are formulaed in Corollary 27 and Theorem 26, repecively. Example 3.: The ODE yem and he daa funcion bu H, are idenical o hoe in Example.. However, we conider a wo-poin BVP of he form x = M x + f, x,, ], x f, x =, x 2 =, 83 { + exp exp M =, M =, β = max, } =. exp2 2 + exp The funcion f ha again he form 79 wih p = 2 /2, r, x = inx 2 cox + x 2, H = in co. Since M coincide wih i Jordan canonical form J, we have by 6, N = Ñ, P = P, and λ x ÑMx =, Ñ 3 f, x = 3 f, x, Px =. x 2 Alo noe ha funcion f aifie 67. Thu, he correcly poed boundary condiion read: Ñ Mx+Ñ/3 f, x =, Px = Pc, c R 2. For c =, T hi amoun o λ x + 3 f, x =, x 2 = x + 3 f, x =, x 2 =, cf. 83. From Corollary 27, we conclude ha he oluion o he BVP 83 exi and i unique. The oluion of he BVP 83 can be found in Figure 4, lef. 3 x x x x Figure 4: Example 3. lef: Numerical oluion obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion ha been ued wih olerance Tol a = Tol r = 2. The number of ubinerval in he iniial meh wa 2, in he final meh 643. The almo equidian final meh, where every hird poin i hown, i alo included. Example 3.2 righ: Numerical oluion obained from bvpuie2. wih four Gauian collocaion poin. Meh adapaion ha no been ued. The number of equidian ubinerval in he final meh wa 4. The error eimae indicae a maximal global relaive error of Again he daa funcion are nonmooh, cf. α 2. Example 3.2: In he final problem, we change he value of µ o µ = 5/2 and obain x = M x f, x,, ], x f, x =, x 2 =