Internatonal Journal of Dfference Equatons ISSN 0973-6069, Volume 12, Number 1, pp. 145 160 2017) ttp://campus.mst.edu/jde A Dscrete Approac to Contnuous Second-Order Boundary Value Problems va Monotone Iteratve Tecnques Crstoper C. Tsdell Scool of Matematcs and Statstcs Te Unversty of New Sout Wales Sydney NSW 2052, Australa cct@unsw.edu.au Abstract Ts artcle nvestgates nonlnear, second-order dfference equatons subject to rgt-focal two-pont boundary condtons. Te partcular nterest s n dentfyng suffcent condtons under wc solutons to ts problem exst. Furtermore, f solutons to ts dscrete problem do exst, ten wat, f any, s ter relatonsp to solutons to te contnuous, rgt-focal analogue nvolvng second-order ordnary dfferental equatons? We sow ow te solutons to te dscrete problem can be appled to sow tat te contnuous problem does ave a soluton. Te metods used eren nvolve te constructon of sequences of vectors wc converge or ave a subsequence tat converges) to a soluton of te dscrete boundary value problem. In some cases tese convergent sequences are monotone. Te deas eren do not rely on a knowledge of nonlnear analyss and tus suc an approac may be accessble to a wder audence as tere s no relance on: fxed-pont teorems; upper and lower solutons; or topologcal degree. AMS Subject Classfcatons: 34B15. Keywords: Exstence of solutons, boundary value problems, successve approxmaton, monotone teraton, dfference equaton, ordnary dfferental equaton. 1 Introducton Ts paper consders te nonlnear, second-order dfference equaton x = f t 2, x, x ), = 1,..., n 1; 1.1) Receved June 20, 2016; Accepted September 2, 2016 Communcated by Martn Boner
146 C. C. Tsdell coupled wt te rgt-focal boundary condtons: x 0 = 0, x n and ts connectons wt te boundary value problem: = 0; 1.2) x = ft, x, x ), t [0, N]; 1.3) x0) = 0, x N) = 0. 1.4) Above, f : [0, N] D [0, N] R 2 R s a contnuous, nonlnear functon; N > 0 s a constant; te step sze s = N/n wt N/2; and te grd ponts are t = for = 0,..., n. Te dfferences are gven by { x+1 x x :=, for = 0,..., n 1, 0, for = n; x := { x x 1, for = 1,..., n, 0, for = 0; x := { x+1 2x + x 1, for = 1,..., n 1, 0, for = 0 or = n. Equatons 1.1), 1.2) are collectvely known as a dscrete, two-pont boundary value problem BVP) wt rgt-focal boundary condtons, wle 1.3), 1.4) are collectvely known as a contnuous, two-pont boundary value problem BVP) wt rgt-focal boundary condtons. In ts work, te partcular nterest s on dentfyng suffcent condtons under wc te solutons to te BVP 1.1), 1.2) wll exst and ow to approxmate tem. Furtermore, f solutons to te dscrete problem do exst, ten wat, f any, s ter relatonsp to solutons to te contnuous problem 1.3), 1.4)? A number of nterestng papers ave provded mportant advances n te above drectons. For example, Ganes [3], Lasota [6] and Myjak [7] were early contrbutors by employng fxed-pont approaces ncludng: contractve maps; a pror bounds on solutons; and lower and upper solutons. More recently, autors suc: as Henderson and Tompson [4,5]; Tompson [11], Tompson and Tsdell [12 14]; and Racůnková and Tsdell [8, 9, 16] ave mproved and extended several of te earler results, mostly va an approac nvolvng topologcal degree and fxed-pont teory. In contrast to te above, te metods used eren nvolve te constructon of sequences of vectors tat converge or ave a subsequence tat converges) to a soluton of 1.1), 1.2). In some cases tese convergent sequences are monotone. Ts approac s known as te monotone teratve tecnque and based on te metod of successve approxmatons. Te deas eren do not rely on a knowledge of nonlnear analyss and
BVPs and Monotone Iteratve Tecnques 147 tus suc an approac may be accessble to a wder audence as tere s no relance on: fxed-pont teorems; upper and lower solutons; or topologcal degree. Partcular nterest n te rgt focal boundary condtons 1.2) and 1.4) s motvated by bot abstract and appled problems. For example, Myjak [7, p. 122] presented an example sowng tat te tecnques n [7] cannot be appled to a BVP wt rgtfocal boundary condtons. Furtermore, rgt-focal condtons naturally appear n te matematcal descrpton of a number of nterestng penomena, suc as n beam analyss [1, Example 4.2]. Te two felds of dfferental equatons and dfference equatons provde a rc and natural framework to matematcally descrbe dynamcal penomena n contnuous tme and n dscrete tme, respectvely. Dfference equatons also fnd mportant uses n te numercal approxmaton of solutons to dfferental equatons. Tese two mportant applcatons of dfference equatons to modellng and approxmaton naturally motvate a deeper teoretcal study of te subject. Wen consdered sde-by-sde and compared, te matematcal teory of te two areas of dfferental equatons and dfference equatons can exbt strange connectons and nterestng dstnctons, especally concernng qualtatve propertes of solutons, as te followng motvatonal examples llustrate. Consder te lnear ntal value problem IVP): and ts dscrete analogue x = 2tx, t 1; 1.5) x1) = 1; 1.6) x = 2t x, = 1, 2, 3,... ; 1.7) x 1 = 1. 1.8) It s easy to see tat te soluton xt) = e 1 t2 to te contnuous IVP 1.5), 1.6) does not oscllate for x 1. However f > 1/2, ten all solutons to te dscrete IVP 1.7), 1.8) do oscllate at every pont n te sense tat x x +1 < 0 for = 1, 2, 3,..., wc may be verfed by rewrtng 1.7) as x +1 = x [1 2t ]. Consder te lnear BVP and ts dscrete analogue x = 2x, t [0, 4]; 1.9) x0) = 0, x4) = 1; 1.10) x = 2x 2, = 1,..., n 1; 1.11) x 0 = 0, x n = 1. 1.12)
148 C. C. Tsdell We see tat te contnuous BVP 1.11), 1.12) as a soluton of te form xt) = [sec4 2)] sn 2t). However for = 1 te dscrete BVP 1.11), 1.12) as no soluton, for n ts case 1.11) becomes x +1 = x 1 for = 1,..., n 1 and so we obtan 0 = x 0 = x 2 = x 4 wc contradcts te boundary condtons. Ts paper s organsed n te followng manner. Secton 2 contans some basc results tat wll needed trougout te paper. In Secton 3 some results are formulated tat ensure te exstence of solutons to 1.1) subject to 1.2). Furtermore, te deas yeld a computatonal procedure for approxmatng tese solutons. Fnally, Secton 4 presents a connecton between solutons to te dscrete problem and solutons to te contnuous problem. 2 Prelmnares In ts secton some basc results are provded tat wll be used n te man secton, keepng te paper somewat selfcontaned. Let b and c be postve constants. Consder te set R b,c := {t, u, v) [0, N] R 2 : t [0, N], u b, v c}. 2.1) Snce we wll make te assumpton tat f s contnuous on R b,c, we can always coose a constant M > 0 suc tat M max t,u,v) R b,c ft, u, v). 2.2) A soluton to 1.1) s a vector x = {x } n =0 R n+1 tat satsfes 1.1) for eac = 1,..., n 1 and wose grap les n R b,c, tat s, t, x, x /) R b,c, for = 1,..., n 1. A soluton to 1.1) s a contnuously twce-dfferentable functon x : [0, N] R, tat s, x C 2 [0, T ]), tat satsfes 1.1) for eac t [0, N] Te followng wellknown result reduces te study of BVPs to te study of equvalent ntegral/summaton equatons. Lemma 2.1. Let f : R b,c R be contnuous. Te dscrete BVP 1.1), 1.2) as te equvalent summaton equaton representaton x = Gt, t j )f t j, x j, x ) j, = 0,..., n, 2.3) were j=1 Gt, t j ) := { tj, for 1 j 1 n 1; t, for 1 j n 1. 2.4)
BVPs and Monotone Iteratve Tecnques 149 Smlarly, te contnuous BVP 1.3), 1.4) as te equvalent ntegral equaton representaton xt) = N 0 Gt, s)fs, xs), x s)) ds, t [0, N]. 2.5) Proof. Bot 2.3) and 2.5) are well known, for example, see [1, pp. 23 24] and can be verfed drectly. Te followng result establses some mportant propertes of G n 2.4), known as Green s functon. Lemma 2.2. Te functon G n 2.4) satsfes G 0 2.6) Gt, t j ) := Gt, t j ) Gt 1, t j ) 0, = 1,..., n, j = 1,..., n 1,2.7) Gt, t j ) N 2, = 0,..., n, 2.8) 2 j=1 Gt, t j ) N, = 1,..., n. 2.9) j=1 Proof. Altoug te proof nvolves smple computaton, we provde some detals for te beneft of te reader. Inequalty 2.6) s mmedate from te defnton of G. For = 0,..., n, we ave 1 Gt, t j ) = t j + j=1 j=1 j= t 1 = 2 j + t n ) j=1 = 2 2 1) + t n ) )] + 1 = t [n 2 t [n /2]. = t [N t /2] N 2 /2. Smlarly, 2.7) and 2.9) follow from { 0, for 1 j 1 n 1; Gt, t j ) :=, for 1 j n 1.
150 C. C. Tsdell Tus, for = 1,..., n, we ave Gt, t j ) = n ) j=1 = N t N. 3 Man Results Ts secton contans te man results on exstence and approxmaton of solutons to 1.1), 1.2). Teorem 3.1. Let f : R b,c R be contnuous and consder 1.1), 1.2). If MN 2 2b, MN c 3.1) ft, 0, 0) 0, for all t [0, N] 3.2) ft, u, v) ft, y, z), for all t [0, N], u y, v z 3.3) ten te dscrete BVP 1.1), 1.2) as at least one soluton x R n+1 wose grap les R b,c. Proof. Te basc dea of te proof s to defne a sutable sequence of vectors tat wll converge to a vector, wt ts lmt vector beng a soluton of 1.1), 1.2). Consder te summaton equaton 2.3) tat, by Lemma 2.1, s equvalent to 1.1), 1.2) and defne te sequence of vectors φ k) := φ k) 0,..., φ k) n ) for k = 0, 1, 2,... n a recursve fason va φ 0) = 0, = 0,..., n; 3.4) ) = Gt, t j )f t j, φ k) j, φk) j, = 0,..., n. 3.5) φ k+1) Frstly we sow tat our sequence of vectors φ k) s well defned for k = 0, 1,... by sowng: eac φ k) b for = 0,..., n; and eac φ k) / c for = 1,..., n. Ts means tat eac t, φ k), φ k) /) s n te doman of f for eac = 1,..., n 1 and k = 0, 1,.... We use proof by nducton. From te defnton of φ 0) t s easy to see tat φ 0) b for = 0,..., n; and φ 0) / c for = 1,..., n. Now assume, for some k 1 0, we ave φ k 1) b for
BVPs and Monotone Iteratve Tecnques 151 = 0,..., n; and φ k 1) φ k 1+1) / c for = 1,..., n. From 3.5), we ave, for = 0,..., n ) n 1 Gt, t j ) f t j, φ k 1) j, φk) j M Gt, t j ) MN 2 /2 from Lemma 2.2 and 3.1). Smlarly, for = 1,..., n φ k 1+1) Gt, t j ) f t j, φ k 1) j b M Gt, t j ) MN c, φk) j from Lemma 2.2 and 3.1). Tus, by nducton, we ave t, φ k), φ k) /) R b,c for eac = 0, 1..., n and k = 0, 1,... and so our sequence of vectors φ k) s well defned n 3.4), 3.5) for eac k = 0, 1,.... Furtermore, te above sows tat for eac k, te sequences of vectors φ k) and φ k) / are unformly bounded for = 0, 1,..., n and = 1,..., n respectvely. k+1) We now sow φ φ k) for k = 0, 1,... and φ k+1) / φ k) / for k = 0, 1,..., were we nterpret te nequalty between two vectors meanng te same nequalty olds between ter correspondng components. Once agan, we use nducton. For = 0,..., n consder = Gt, t j )f t j, 0, 0) φ 1) 0 = φ 0), were we ave used 2.6) and 3.2). Tus, φ 1) φ 0). In a smlar fason, for = 1,..., n, we ave ) φ 1) = 0 Gt, t j )f t j, 0, 0) = φ0).
152 C. C. Tsdell Now assume tat φ k1) φ k1 1) for some k 1 1, tat s, assume φ k 1) φ k 1 1) for = 0,..., n. Furtermore, assume φ k1) / φ k1 1) / for some k 1 1, tat s, assume φ k 1) / φ k 1 1) / for = 1,..., n. For eac = 0,..., n, we ave ) = Gt, t j )f φ k 1+1) Gt, t j )f = φ k 1), t j, φ k 1) j t j, φ k 1 1) j, φk1) j, φk1 1) j were we ave used 2.6) and 3.3). Tus, φ k+1) φ k) for k = 0, 1,.... Smlarly, for eac = 1,..., n, we ave ) φ k 1+1) = Gt, t j )f t j, φ k 1) j, φk1) j ) Gt, t j )f t j, φ k 1 1) j, φk1 1) j = φk 1), were we ave used 2.7) and 3.3). From te above, we conclude tat φ k) s a unformly bounded and nondecreasng sequence of vectors and so must converge to a vector φ, tat s lm φ k) = φ k for some φ R n+1. We fnally sow tat te above φ = φ 0,..., φ n ) R n+1 s actually a soluton to 1.1), 1.2). Snce eac φ k) b and eac φ k) / c, we must ave eac φ b and φ / c. Tus t, φ, φ /) R b,c for = 0,..., n. Furtermore, te contnuty of f on R b,c ensures tat ) f t, φ k), φk) f t, φ. φ ), as k for eac = 1,..., n. If we now take lmts n 3.5) as k, ten we obtan φ = Gt, t j )f t j, φ j, φ ) j, = 0,..., n; )
BVPs and Monotone Iteratve Tecnques 153 so tat our lmt vector φ s ndeed a soluton to 1.1), 1.2) n lgt of Lemma 2.2. Remark 3.2. Te proof of Teorem 3.1 provdes a computatonal tool for approxmatng or obtanng) solutons to 1.1), 1.2). For dscrete BVPs were te rgt-and sde does not feature x /, tat s wt f : R b R and x = ft, x ), = 1,..., n 1 3.6) R b := {t, u) [0, N] R 2 : t [0, N], u b} we ave te followng corollary to Teorem 3.1. Corollary 3.3. Let f : R b R be contnuous and consder 3.6), 1.2). If MN 2 2b, ft, 0) 0, for all t [0, N] ft, u) ft, y), for all t [0, N], u y ten te dscrete BVP 3.6), 1.2) as at least one soluton x R n+1 wose grap les R b. Proof. As te proof s vrtually dentcal to tat of Teorem 3.1 t s omtted. Example 3.4. Consder te dfference equaton x = 1 ) 5 t 2 x 3 x 1), = 1,..., n 1 3.7) 10 subject to 1.2) wt N = 1. Coose b = 1 and c = 1 to form R b,c and ten see tat M = 2/5. It s easy to verfy tat 3.1) and 3.2) old. Furtermore, f ft, p, q) denotes te rgt-and sde of 3.7), ten ft, p, q) s nonncreasng n bot p and q, so tat 3.3) olds. Tus, all of te condtons of Teorem 3.1 old and te exstence of soluton follows. Furtermore, ts soluton may be constructed as te lmt of te sequence defned recursvely n 3.4), 3.5) for te above f. Remark 3.5. Te proof of Teorem 3.1 can be smplfed f only exstence of solutons s sougt and not approxmaton). Te bounds φ k) b, φ k) / c 3.8) n te proof of Teorem 3.1 guarantee tat te sequence of vectors φ as at least one convergent subsequence and t can be furter sown tat any suc subsequence does converge to a soluton of 1.1), 1.2).
154 C. C. Tsdell However, as can be seen from te proof of Teorem 3.1, tere s more gong on. Te sequence generated n 3.4), 3.5) s also monotone and so must converge to a soluton of 3.4), 3.5) and tere s no need to searc for a convergent subsequence. Ts s a dstnct computatonal advantage over movng to subsequences. A result s now presented were te φ k) are bounded, but not monotone. However, as we sall see, t s easy to coose a par of monotone subsequences tat converge to a soluton from above and from below. Teorem 3.6. Let f : R b,c R be contnuous and consder 1.1), 1.2). If MN 2 2b, MN c 3.9) ft, 0, 0) 0, for all t [0, N] 3.10) ft, u, v) ft, y, z), for all t [0, N], u y, v z 3.11) ten te dscrete BVP 1.1), 1.2) as at least one soluton x R n+1 wose grap les R b,c. Proof. Te exstence part of te proof follows tat of Teorem 3.1 and so s omtted, k) owever, we pont out a few nterestng dfferences. Defne te sequence of vectors φ as n 3.4), 3.5). It can be sown tat φ k) s unformly bounded as n 3.8) and so must ave a convergent subsequence and, n turn, tat te lmt of ts subsequence s a soluton to 1.1), 1.2). However, φ k) s not a monotone sequence. On te oter and, t can be sown tat φ k) does possess subsequences tat are monotone. For example, nducton can be used to sow φ 2k) φ 2k+2) 3.12) φ 2k 1) φ 2k+1) 3.13) φ 2k) / φ 2k+2) / 3.14) φ 2k 1) / φ 2k+1) / 3.15) for k = 0, 1,... by nvokng 3.10) and 3.11). Te detals are omtted for brevty. In addton, t can be sown by nducton tat 1) k [ φ k) φ k+1) ] 0 1) k [ φ k) / φ k+1) /] 0 for eac k and so φ 2k) converges monotoncally to a soluton from above and converges monotoncally to a soluton from below. Corollary 3.7. Let f : R b R be contnuous and consder 3.6), 1.2). If φ 2k 1) MN 2 2b, ft, 0) 0, for all t [0, N] ft, u) ft, y), for all t [0, N], u y
BVPs and Monotone Iteratve Tecnques 155 ten te dscrete BVP 3.6), 1.2) as at least one soluton x R n+1 wose grap les R b. Remark 3.8. If t s known tat te dscrete BVP 1.1), 1.2) as, at most, one soluton and te condtons of Teorem 3.6 old, ten te even and odd subsequences n te proof of Teorem 3.6 wll converge to te same soluton from above and below, respectvely. Wt unqueness of solutons n mnd, te followng result s presented. Teorem 3.9. Let f : R b,c R be contnuous and consder 1.1), 1.2). If ft, u, v) < ft, y, z), for all t [0, N], u < y, v z 3.16) ten te dscrete BVP 1.1), 1.2) as, at most, one soluton x R n+1 wose grap les R b,c. Proof. Let p and q be two solutons to 1.1), 1.2). We sow tat p q. Let r := p q and assume tere s a j {0,..., n} suc tat r j = max =0,...,n [p q ] > 0. 3.17) Te frst boundary condton n 1.2) ensures j 0, wle te second boundary condton n 1.2) ensures tat f j = n, ten te maxmum r j also occurs at j = n 1. If j {1,..., n 1} ten te dscrete maxmum prncple gves and 0 r j 2 r j > 0, r j / 0 = p j q j 2 2 = ft j, p j, p j /) ft j, q j, q j /) > 0, were we ave used 3.16), and we reac a contradcton. Tus j / {1,..., n 1}. Combnng te above cases, we see tat r 0 for all = 0,..., n. A smlar argument to te above for te case r j < 0 sows tat r 0 for all = 0,..., n and so r = 0 for all = 0,..., n. Tus, p q and tere s, at most, one soluton. Corollary 3.10. Let f : R b R be contnuous and consder 3.6), 1.2). If ft, u) < ft, y), for all t [0, N], u < y ten te dscrete BVP 3.6), 1.2) as, at most, one soluton x R n+1 wose grap les R b.
156 C. C. Tsdell 4 A Dscrete Approac to Dfferental Equatons In ts secton, we form a relatonsp between solutons to te dscrete BVP 1.1), 1.2) and solutons to te contnuous BVP 1.3), 1.4). wc s based on te deas of Ganes [3]. We formulate a sequence of contnuous functons tat are based on te solutons to 1.1), 1.2) and furns some condtons under wc tey wll converge to a functon as 0, wt te functon beng a soluton to 1.3), 1.4). Te our approac uses te dscrete problem to generate exstence results for te contnuous problem. Te followng result nvolves a bound on te solutons and ter backward dfferences to 1.1), 1.2), wt te bounds beng ndependent of. We requre te followng notaton. Denote te sequence n m as m ; let 0 < m = N/n m ; and let t m = m for = 0,..., n. If 1.1), 1.2) as a soluton for = m and m m 0 tat we denote by x m := x m 0,..., x m n ), 4.1) ten we construct te followng sequence of contnuous functons from 4.1) va lnear nterpolaton to form x m t) := x m + xm +1 x m ) m t t m ), t m t t m +1; 4.2) for m m 0 and t [0, N]. Note tat x m t m ) = x m for = 0,..., n. Furtermore, defne v m := x m x m 1)/ and smlarly construct te sequence of contnuous functons v m on [0, N] by v m v m + vm +1 v m t t m ), for t m t t m t) := +1; m 4.3) v1 m, for 0 t t m 1. Lemma 4.1. Let f : [0, N] D [0, N] R R be contnuous and let R 0 and T 0 be constants. If 1.1), 1.2) as a soluton for m and m m 0 tat we denote by x m wt max =0,...,n xm R, m m 0 ; 4.4) max x m T, m m 0; 4.5) =1,...,n ten 1.3), 1.4) as a soluton x = xt) tat s te lmt of a subsequence of 4.2). Proof. Te proof s smlar to tat of [3, Lemma 2.4] and so s only sketced. For m m 0 consder te sequence of functons x m t) for t [0, 1] n 4.2). We sow tat te sequence of functons x m s unformly bounded and equcontnuous on
BVPs and Monotone Iteratve Tecnques 157 [0, 1]. For t [t m, t m +1] and m m 0, we ave x m t) x m + x m +1 x m ) t tm R + T N. Smlar calculatons sow tat v m s unformly bounded on [0, N]. For β, γ [0, N] and gven ε > 0, consder x m β) x m γ) x m +1 x m ) β γ m m T β γ < ε wenever β γ < δε) := ε/t. Tus, x m s equcontnuous on [0, N]. A smlar argument sows v m s equcontnuous on [0, N]. Te convergence teorem of Arzelà Ascol [10, p. 527] guarantees te sequence of contnuous functons x m = x m t) as a subsequence x km) t) tat converges unformly to a contnuous functon x = xt) for t [0, N]. Tat s, max t [0,N] xkm) t) xt) 0, as m Smlarly, v m = v m t) as a subsequence v km) t) tat converges unformly to a contnuous functon y = yt) for t [0, N]. Tat s, max t [0,N] vkm) t) yt) 0, as m. Furtermore, t can be sown tat x = y on [0, N]. Te contnuty of f ensures tat te above lmt functon wll be a soluton to 1.3), 1.4). Te next teorem, wc s motvated by [3, Teorem 2.5], requres te followng notaton. If 1.1), 1.2) as a soluton x for 0 < 0, ten we defne te contnuous functon xt, x) by xt, x) := x + x +1 x ) t t ), t t t +1 and defne te contnuous functon vt, x) by x x 1 vt, x) := + x +1 2x + x 1 2 t t ), for t t t +1 ; x 1 x 0, for 0 t t 1. 4.6)
158 C. C. Tsdell Teorem 4.2. Let f : [0, N] D [0, N] R R be contnuous and let R 0 and T 0 be constants. Assume 1.1), 1.2) as a soluton for 0 tat we denote by x wt max x R, max x =0,...,n =0,...,n T. 4.7) Gven any ε > 0 tere exsts a δ = δε) suc tat f δ, ten 1.3), 1.4) as a soluton x = xt) wt max xt, x) xt) ε 4.8) t [0,N] max vt, x) t [0,N] x t) ε. 4.9) Proof. Suppose, for some ε > 0, tere s a sequence m suc tat m 0 as m and for = m = N/n m 1.1), 1.2) as a soluton x m wt every soluton x = xt) to 1.3), 1.4) satsfyng at least one of max xt, x) xt) > ε 4.10) t [0,N] max vt, x) t [0,N] x t) > ε. 4.11) By assumpton, for m suffcently large, tere s a R 0 and T 0 suc tat te soluton x m to 1.1), 1.2) satsfes max =0,...,n xm R, max =0,...,n vm T. Tus, te condtons of Lemma 4.1 are satsfed, and so we obtan a subsequence x km) t) of x m t) and a subsequence v km) t) of v m t) tat converge unformly on [0, N] to a soluton x of 1.3), 1.4). Tus, te nequaltes 4.10) or 4.11) cannot old. We now relate te above abstract results to te deas from earler sectons. Teorem 4.3. Let te condtons of Teorem 3.1 or Teorem 3.6 old. Gven any ε > 0 tere s a δ = δε) suc tat f δ, ten 1.3), 1.4) as a soluton x tat satsfes 4.8) and 4.9). Proof. We sow tat te condtons of Teorem 4.2 are satsfed for R b,c = [0, N] D. Te soluton x to 1.1), 1.2) guaranteed to exst by Teorem 3.1 satsfes x b for = 0,..., n and x / c for = 1,..., n and so 4.7) olds wt R = b and T = c. Tus, all of te condtons of Teorem 4.2 old and te result follows. Example 4.4. Te dfferental equaton x = e x 4.12)
BVPs and Monotone Iteratve Tecnques 159 arses n certan problems from radaton, electroydrodynamcs and a range of oter problems nvolvng dffuson [1, p. 113]. We clam tat 4.12) as a soluton x = xt) for x [0, 1/2] satsfyng 1.4). Let ft, p) := e p. Coose b = ln 2 to form R b wt N = 1/2) so tat M = 2. It s easy to see tat ft, p) s nondecreasng n p, n fact, f s strctly ncreasng n p. Te condtons of Corollary 3.7 are satsfed and so te dfference equaton x 2 = e x, = 1,..., n 1 4.13) subject to 1.2) as at least soluton. Furtermore, ts soluton may be constructed as te lmt of te sequence defned recursvely n 3.4), 3.5) and te even subsequence wll converge monotoncally to te soluton from above, wle te odd subsequence wll converge to te soluton from below n vew of Remark 3.8. In addton, te condtons of Lemma 4.1, Teorem 4.2 and Teorem 4.3 old and so te contnuous BVP 4.13), 1.4) does possess a soluton and, gven any ε > 0, tere s a δ = δε) suc tat f δ, ten x satsfes 4.8) and 4.9). References [1] Baley, Paul B.; Sampne, Lawrence F.; Waltman, Paul E. Nonlnear two pont boundary value problems. Matematcs n Scence and Engneerng, Vol. 44 Academc Press, New York, 1968. [2] Ernstroem, Mats; Tsdell, Crstoper C.; Walén, Erk Asymptotc ntegraton of second-order nonlnear dfference equatons. Glasg. Mat. J. 53 2011), no. 2, 223 243. [3] Ganes, Robert. Dfference equatons assocated wt boundary value problems for second order nonlnear ordnary dfferental equatons. SIAM J. Numer. Anal. 11 1974), 411 434. [4] Henderson, J.; Tompson, H. B. Exstence of multple solutons for second-order dscrete boundary value problems. Comput. Mat. Appl. 43 2002), no. 10-11, 1239 1248. [5] Henderson, Jonny; Tompson, H. B. Dfference equatons assocated wt fully nonlnear boundary value problems for second order ordnary dfferental equatons. J. Dffer. Equatons Appl. 7 2001), no. 2, 297 321. [6] Lasota, A. A dscrete boundary value problem. Ann. Polon. Mat. 20 1968), 183 190.
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