ON BERNOULLI BOUNDARY VALUE PROBLEM

LE MATEMATICHE Vol. LXII (2007) Fsc. II, pp. 163 173 ON BERNOULLI BOUNDARY VALUE PROBLEM FRANCESCO A. COSTABILE - ANNAROSA SERPE We consider the boundry vlue problem: x (m) (t) = f (t,x(t)), t b, m > 1 x() = β 0 x (k) x (k) (b) x (k) () = β k+1, k = 0,...,m 2 where x(t) = (x(t),x (t),...,x () (t)), β i R, i = 0,...,, nd f is continuous t lest in the interior of the domin of interest. We give constructive proof of the existence nd uniqueness of the solution, under certin conditions, by Picrd s itertion. Moreover Newton s itertion method is considered for the numericl computtion of the solution. 1. Introduction In this pper we consider the following boundry problem: (1) x (m) (t)= f (t,x(t)), t b, m >1 (1b) x()=β 0, x (k) x (k) (b) x (k) ()=β k+1 k = 0,...,m 2 (1) where x(t) = (x(t),x (t),...,x () (t)), f is defined nd continuous t lest in the domin of interest included in [,b] R m ; [,b] R, nd β i R, i = Entrto in redzione 1 gennio 2007 AMS 2000 Subject Clssifiction: C0D1C3, PR0V4 Keywords: Bernoulli, Green function, Picrd s itertion,newton itertion

164 FRANCESCO A. COSTABILE - ANNAROSA SERPE 0,...,m 1. This problem is clled the Bernoulli boundry vlue ([4],[6]) problem. The boundry conditions in (1) (1b) re new nd it is esy to give them physicl nd engineering interprettions [1]; this is the motivtion of our investigtion. In [6] the uthors give non constructive proof of the existence nd uniqueness of the solution of (1) (1b), while in this work they prove the convergence of Picrd s itertion under certin conditions nd, therefore, supply constructive proof. The outline of the pper is the following: in section 2 we give the preliminries, in section 3 we investigte the existence nd uniqueness of the solution by Picrd s itertion; finlly, in section 4 we consider the Newton s itertions method for the numericl clcultion of the solution. 2. Definitions nd preliminries If B n (x) is the Bernoulli polynomil of degree n defined by [7] B 0 (x) = 1 B n(x) = nb n 1 (x) n 1 1 0 B n(x)dx = 0 n 1 (2) in recent pper Costbile [5] proved the following theorems. Theorem 1. Let f C (ν) [,b] we hve f (x) = f () + ν k=1 ( ) x h (k 1) S k f (k 1) R ν [ f ](x) (3) h k! where h = b, S k (t)=1b k (t) B k (0), f = f (), f (k) = f (k) (b) f (k) () R ν [ f ](x) = h(ν 1) ν! ( ( f (ν) (t) B ν ( x t h ) ( ))) t +( 1) ν+1 B ν dt h nd B m(t) = B m (t) 0 t 1, B m(t + 1) = B m(t) Theorem 2. Putting P ν [ f ](x) = f + ν k=1 ( ) x h (k 1) S k f (k 1) (4) h k!

ON BERNOULLI BOUNDARY VALUE PROBLEM 165 the following equlities re true P ν [ f ]() = f () f P ν [ f ](b) = f (b) f b P ν (k) P ν (k) (b) P ν (k) ()= f (k) (b) f (k) () f (k), k = 1,...,ν 1 (5) The conditions (5) in the previous equlities re clled Bernoulli interpoltory conditions nlogously to Lidstone interpoltory conditions [3],[4]. Theorem 3. If f C (ν+1) [,b] we hve where G(x,t) = 1 ν! [ (x t) ν + R ν [ f ](x) = G(x,t) f (ν+1) (t)dt ν k=1 S k ( x h ) h (k 1) k! ( ] ν )(b t) ν k+1 k 1 (6) with (x) k x k i f x 0 + = 0 i f x < 0 Theorem 4. For f C (ν) [,b] we hve R ν [ f ](x) hν 1 6(2π) ν 2 f (ν) (t) dt (7) For the following, we need Lemm 1.[6] If f C (ν) [,b] nd stisfies the homogeneous Bernoulli interpoltory conditions i.e: f () = 0 f (k) (b) f (k) (8) () = 0 k = 0,...,ν 2 putting M ν = mx f (ν) (t) t b the following inequlities hold f (k) (t) C ν,k M ν (b ) ν k 0 k ν 1 (9) where 1 C ν,0 = 3(2π) ν 2 1 C ν,k = 6(2π) ν k 2 k = 1,2,..,ν 1

166 FRANCESCO A. COSTABILE - ANNAROSA SERPE Proof >From (8) the expnsion (3) becomes [ ( ) ] f (t) = hν 1 t B ν B ν f (ν 1) R ν [ f ](t) (10) ν! h We lso hve from which t f (ν 1) (t) = f (ν 1) () + f (ν) (s)ds f (ν 1) f (ν 1) (b) f (ν 1) () M ν (b ) (11) Using the known inequlities in [7] B l (x) nd (7),(11) we hve from (10) l! 12(2π) l 2 l N, l 0, 0 x 1 f (t) hν M ν 3(2π) ν 2 tht is (9) for k = 0. With successive derivtion of (10) nd by pplying (8) we hve ( ) f (k) (t) = hν (k+1) (ν k)! f (ν 1) t B ν k + h hν (k+1) b ( ) t s f (ν) (t)b ν k ds k = 1,2,...,ν 1 (ν k)! h (12) nd pplying the previous inequlities we get f (k) (t) tht is (9) for k = 1,2,...,ν 1. hν k M ν 6(2π) ν k 2 k = 1,2,...,ν 1 3. Existence nd uniqueness To the boundry vlue problem (1) (1b) we ssocite the homogeneous boundry vlue problem x (m) (t) = f (t,x(t)), t b, m > 1 x() = x(b) = 0 x (k) x (k) (b) x (k) () = 0 k = 1,...,m 2 (13)

ON BERNOULLI BOUNDARY VALUE PROBLEM 167 From Theorem 3, the solution of the boundry vlue problem (13) is x(t) = G(t,s) f (s,x(s))ds (14) where G(t,s) is the Green function [8] defined by (6), with ν = m 1. The polynomil P [x](t) defined by (4) with x()=β 0, x (k) (b) x (k) ()= β k+1, k = 0,...,m 2, stisfies the boundry vlue problem: P (m) [x](t) = 0 P [x]() = β 0 P (k) P(k) (b) P(k) () = β k+1, k = 0,...,m 2 Therefore, the boundry vlue problem (1) (1b) is equivlent to the following nonliner Fredholm integrl eqution: x(t) = P [x](t) + Now, we hve the following results: Theorem 5.[6] Let us suppose tht G(t,s) f (s,x(s))ds (15) (i) k i > 0, 0 i re given rel numbers nd let Q be the mximum of f (t,x 0,...,x ) on the compct set [,b] D 0, where D 0 = (x 0,...,x ) : x i 2k i, 0 i }; (ii) mx P (i) [x](t) ki 0 i, where P [x](t) is the polynomil reltive to x s in (4); ( ) 1 ki (m i) (iii) (b ) Q C m,i 0 i. Then, the Bernoulli boundry vlue problem hs solution in D 0. Proof. The set } B[,b] = x(t) C () [,b] : x (i) 2 k i, 0 i is closed convex subset of the Bnch spce C () [,b]. Now we define n opertor T : C () [,b] C (m) [,b] s follows: (T [x](t)) = P [x](t) + G(t,s) f (s,x(s))ds (16) It is cler, fter (15), tht ny fixed point of (16) is solution of the boundry vlue problem (1) nd (1b). Let x(t) B[, b], then from (16), lemm 1, hypotheses (i),(ii),(iii) we find:

168 FRANCESCO A. COSTABILE - ANNAROSA SERPE () T B[,b] B[,b]; (b) the sets T [x] (i) (t) : x(t) B[,b] }, 0 i re uniformly bounded nd equicontinuous in [, b];on Bernoulli boundry vlue problem (c) T B[,b] is compct from the Ascoli - Arzel theorem; (d) from the Schuder fixed point theorem fixed point of T exists in D 0. Corollry 1. Suppose tht the function f (t,x 0,x 1,...,x ) on [,b] R m stisfies the following condition f (t,x 0,x 1...,x ) L + L i x i α i where L,L i 0 i re non negtive constnts, nd 0 α i 1. Then the boundry vlue problem (1) (1b) hs solution. Lemm 2 [6] For the Green function defined by (6), for ν = m 1 the following inequlities hold: G(t,s) g withon Bernoulli boundry vlue problem g = 1 ( ) ν! (b )m 1 + 2π2 m!. 3(2π 1) Proof. The proof follows from the known inequlities of Bernoulli polynomils nd from simple clcultions. Theorem 6.[6] Suppose tht the function f (t,x 0,x 1...,x ) on [,b] D 1 stisfies the following condition f (t,x 0,x 1...,x ) L + L i x i (17) where D 1 = (x 0,x 1...,x ) : x i mx +C m,i (b ) m g h ( L +C 1 θ C = mx t b P (i) t b [x](t) + ), 0 i L i P (i) [x](t) }

ON BERNOULLI BOUNDARY VALUE PROBLEM 169 ( ) θ = h g C m,i L i (b ) m i < 1, h = b (18) Then, the boundry vlue problem (1) (1b) hs solution in D 1. Proof. Let y(t) = x(t) P [x](t), so tht (1) nd (1b) is the sme s y (m) (t) = f (t,y(t)) y() = y(b) = 0 y (k) = 0 1 k m 2 whereon Bernoulli boundry vlue problem y(t) = y(t) + P [x](t), (19) y (t) + P [x](t),...,y () (t) + P () [x](t). Define M[,b] s the spce of m times continuously differentible functions stisfying the boundry conditions of (19). If we introduce in M[, b] the norm: y(t) = mx y (m) (t) t b then it becomes Bnch spce. As in theorem 5, it suffices to show tht the opertor T : M[,b] M[,b] defined by T [y](t) = G(t,s) f (s,y(s))ds mps the set ( )} L +C S = y(t) M[,b] : y hg 1 θ into itself. In order to demonstrte this, it is sufficient to utilise the conditions (17), lemm 1 nd lemm 2. The thesis follows from the ppliction of the Schuder fixed point theorem to the opertor T. Definition 1. A function x(t) C (m) [,b] is clled n pproximte solution of (1) (1b) if there exist non-negtive constnts δ nd ε such tht: mx t b x (m) (t) f (t,x(t)) δ P (i) mx t b [x](t) P(i) [x](t) ε Cm,i (b ) m i, 0 i m 1 (20) where P (i) [x](t) nd P(i) [x](t) re the polynomils defined by (5). The inequlity (20) mens tht there exists continuous function η(t) such tht: x (m) (t) = f (t,x(t)) + η(t)

170 FRANCESCO A. COSTABILE - ANNAROSA SERPE nd mx η(t) δ t b Thus the pproximte solution x(t) cn be expressed s: x(t) = P [x](t) + G(t,s) [ f (s,x(s)) + η(s)]ds In the following we shll consider the Bnch spce C () [,b] nd for y(t) C () [,b] the norm y is defined by: Cm,0 (b ) j y = mx mx y j (t) } 0 j C m, j t b Now we hve: Theorem 7.(Picrd s itertion)[2] With respect to the boundry vlue problem (1) (1b) we ssume the existence of n pproximte solution x(t) nd: (i) the function f (t,x 0,...,x ) stisfies the Lipschitz condition: m i f (t,x 0,...,x ) f (t,x 0,...,x ) L i x i x i on [,b] D 2 where D 2 = (x 0,...,x i ) : x j x ( j) (t) C N m, j C m,0, 0 j (b ) j (ii) θ < 1 (iii) N 0 = (1 θ) 1 (ε + δ) C m,0 (b ) m N Then, the following results hold: (21 ) there exists solution x (t) of (1) nd (1b) in } S(x,N 0 )= x C () [,b] : x x N 0 (21 b ) x (t) is, the, unique solution of (1) nd (1b) in S(x,N) (21 c ) the Picrd itertive sequence x n (t) defined by: x0 (t) = x(t) x n+1 (t) = P (t) + G(t,s) f (s,x n(s))ds n = 0,1,... converges to x (t) with: x x 0 θ n N 0 nd x x n θ(1 θ) 1 x 0 x n 1. }

ON BERNOULLI BOUNDARY VALUE PROBLEM 171 Proof. It suffices to show tht the opertor T :S(x,N) C (m) [,b] defined by T [x](t) = P [x](t) + G(t,s) f (s,x(s))ds where X(s) = (x(s),x (s),...,x () (s)), stisfies the conditions of the contrction mpping theorem. 4. Newton s itertion For n efficient numericl clcultion of the solution of problem (1) (1b) we cn consider Newton s itertion method. For our problem (1) (1b) the qusiliner itertive scheme is defined s: (22 ) x (m) n+1 (t) = f (t,x n(t)) + ( x (i) n+1 (t) x(i) n (t) ) f (t,x n(t)) x (i) n (t) (22 b ) xn+1 () = β 0 x (h) n+1 (b) x(h) n+1 () = β h+1, h=0,...,m 2, n=0,1,... where x 0 (t) = x(t) is n pproximte solution of (1) (1b). Theorem 8.(Newton s itertion) With respect to the boundry vlue problem (1) (1b) we ssume tht there exists n pproximte solution x(t), nd: (i) the function f (t,x 0,x 1,...,x ) is continuously differentible with respect to ll x i 0 i on [,b] D 2 ; (ii) there exist non-negtive constnts L i, 0 i m 1 such tht for ll (t,x 0,...,x ) [,b] D 2 we hve: f (t,x 0,...,x ) x i L i (iii) 3θ < 1 (iv) N 3 = (1 3θ) 1 (ε + δ) C m,0 (b ) m N Then, the following results hold: (23 ) the sequence x n (t) generted by the itertive scheme (22 ) (22 b ) remins in S(x,N 3 ).

172 FRANCESCO A. COSTABILE - ANNAROSA SERPE (23 b ) the sequence x n (t) converges to the unique solution x (t) of the boundry vlue problem (1) (1b). Proof. The proof requires the equlities nd the inequlities tht we hve previously determined nd is bsed on inductive rguments. REFERENCES [1] R.P. Agrwl - G. Akrivis, Boundry vlue problem occuring in plte deflection theory, Computers Mth. Appl. 8 (1982), 145-154. [2] R.P. Agrwl, Boundry vlue Problems for Higher Order Differentil equtions, World Scientific Singpore, 1986. [3] R.P. Agrwl - P.J.Y. Wong, Lidstone polynomils nd boundry vlue problems, Computers Mth. Appl. 17 (1989), 1377-1421. [4] F.A. Costbile - F. Dell Accio, Polynomil pproximtion of C M functions by mens of boundry vlues nd pplictions: A survey, J.Comput.Appl.Mth. doi: 10.1016/j.cm.2006.10.059, 2006. [5] F.A. Costbile, Expnsions of rel functions in Bernoulli polynomil nd pplictions, Conferences Seminrs Mthemtics University of Bri 273 (1999), 1-13. [6] F.A. Costbile - A. Bruzio - A. Serpe, A new boundry vlue problem, Pubbliczione LAN, Deprtment of Mthemtics, University of Clbri 18, 2006. [7] C. Jordn, Clculus of Finite Differences, Chelse Pu.Co., New York, 1960. [8] I. Stkgold, Green s Functions nd boundry vlue problems, John Wiley Sons, 1979.

ON BERNOULLI BOUNDARY VALUE PROBLEM 173 FRANCESCO A. COSTABILE Deprtment of Mthemtics, University of Clbri vi P. Bucci - Cubo 30/A - 87036 Rende(CS) Itly e-mil: costbil@unicl.it ANNAROSA SERPE Deprtment of Mthemtics, University of Clbri vi P. Bucci - Cubo 30/A - 87036 Rende(CS) Itly e-mil: nnros.serpe@unicl.it