We give a constructive proof of the existence and uniqueness of the solution, under certain conditions, by Picard’s iteration. Moreover Newton’s iteration method is considered for the numerical computation of the solution.We give a constructive proof of the existence and uniqueness of thesolution, under certain conditions, by Picard’s iteration. MoreoverNewton’s iteration method is considered for the numerical computation of the solution

Costabile, Francesco A.

Serpe, Annarosa

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXII (2007) – Fasc. II, pp. 163–173ON BERNOULLI BOUNDARY VALUE PROBLEMFRANCESCO A. COSTABILE - ANNAROSA SERPEWe consider the boundary value problem: x(m)(t) = f (t,x(t)), a≤ t ≤ b, m> 1x(a) = β0∆x(k) ≡ x(k)(b)− x(k)(a) = βk+1, k = 0, ...,m−2where x(t) = (x(t),x′(t), ....,x(m−1)(t)), βi ∈R, i= 0, ...,m−1, and f iscontinuous at least in the interior of the domain of interest.We give a constructive proof of the existence and uniqueness of thesolution, under certain conditions, by Picard’s iteration. MoreoverNewton’s iteration method is considered for the numerical compu-tation of the solution.1. IntroductionIn this paper we consider the following boundary problem:{(1a) x(m)(t)= f (t,x(t)) , a≤ t≤ b, m>1(1b) x(a)=β0, ∆x(k)a ≡ x(k)(b)−x(k)(a)=βk+1 k = 0, ...,m−2(1)where x(t) = (x(t),x′(t), ....,x(m−1)(t)), f is defined and continuous at leastin the domain of interest included in [a,b]×Rm; [a,b]⊂R, and βi ∈R, i=Entrato in redazione 1 gennaio 2007AMS 2000 Subject Classification: C0D1C3, PR0V4Keywords: Bernoulli, Green function, Picard’s iteration,Newton iteration164 FRANCESCO A. COSTABILE - ANNAROSA SERPE0, ...,m−1.This problem is called the Bernoulli boundary value ([4],[6]) problem.The boundary conditions in (1a)− (1b) are new and it is easy to givethem physical and engineering interpretations [1]; this is the motivationof our investigation.In [6] the authors give a non constructive proof of the existence anduniqueness of the solution of (1a)− (1b), while in this work they provethe convergence of Picard’s iteration under certain conditions and, there-fore, supply a constructive proof.The outline of the paper is the following: in section 2 we give the pre-liminaries, in section 3 we investigate the existence and uniqueness ofthe solution by Picard’s iteration; finally, in section 4 we consider theNewton’s iterations method for the numerical calculation of the solution.2. Definitions and preliminariesIf Bn(x) is the Bernoulli polynomial of degree n defined by [7]B0(x) = 1B′n(x) = nBn−1(x) n≥ 1∫ 10 Bn(x)dx= 0 n≥ 1(2)in a recent paper Costabile [5] proved the following theorems.Theorem 1. Let f ∈ C(ν)[a,b] we havef (x) = f (a)+ν∑k=1Sk(x−ah)h(k−1)k!∆ f (k−1)a −Rν [ f ](x) (3)whereh= b−a, Sk(t)=1Bk(t)−Bk(0) , fa= f (a), ∆ f (k)a = f (k)(b)− f (k)(a)Rν [ f ](x) =h(ν−1)ν!·∫ ba(f (ν)(t)(B∗ν(x− th)+(−1)ν+1Bν(t−ah)))dtandB∗m(t) = Bm(t) 0≤ t ≤ 1, B∗m(t+1) = B∗m(t)Theorem 2. PuttingPν [ f ](x) = fa+ν∑k=1Sk(x−ah)h(k−1)k!∆ f (k−1)a (4)ON BERNOULLI BOUNDARY VALUE PROBLEM 165the following equalities are truePν [ f ](a) = f (a)≡ faPν [ f ](b) = f (b)≡ fb∆P(k)ν ≡ P(k)ν (b)−P(k)ν (a)= f (k)(b)− f (k)(a)≡∆ f (k)a , k = 1, ...,ν−1(5)The conditions (5) in the previous equalities are called Bernoulli interpo-latory conditions analogously to Lidstone interpolatory conditions [3],[4].Theorem 3. If f ∈C(ν+1)[a,b] we haveRν [ f ](x) =∫ baG(x, t) f (ν+1)(t)dtwhereG(x, t) =1ν![(x− t)ν+−ν∑k=1Sk(x−ah)h(k−1)k!(νk−1)(b− t)ν−k+1](6)with(x)k+ ={xk i f x≥ 00 i f x< 0Theorem 4. For f ∈C(ν)[a,b] we have|Rν [ f ](x)| ≤ hν−16(2pi)ν−2∫ ba∣∣∣ f (ν)(t)∣∣∣dt (7)For the following, we needLemma 1.[6] If f ∈C(ν)[a,b] and satisfies the homogeneous Bernoulli in-terpolatory conditions i.e:{f (a) = 0f (k)(b)− f (k)(a) = 0 k = 0, ...,ν−2 (8)puttingMν = maxa≤t≤b∣∣∣ f (ν)(t)∣∣∣the following inequalities hold∣∣∣ f (k)(t)∣∣∣≤Cν ,k ·Mν ·(b−a)ν−k 0≤ k ≤ ν−1 (9)where Cν ,0 =13(2pi)ν−2Cν ,k =16(2pi)ν−k−2k = 1,2, ..,ν−1166 FRANCESCO A. COSTABILE - ANNAROSA SERPEProof >From (8) the expansion (3) becomesf (t) =hν−1ν![Bν(t−ah)−Bν]∆ f (ν−1)a −Rν [ f ](t) (10)We also havef (ν−1)(t) = f (ν−1)(a)+∫ taf (ν)(s)dsfrom which ∣∣∣∆ f (ν−1)a ∣∣∣≡ ∣∣∣ f (ν−1)(b)− f (ν−1)(a)∣∣∣≤Mν(b−a) (11)Using the known inequalities in [7]|Bl(x)| ≤ l!12(2pi)l−2 l ∈ N, l ≥ 0, 0≤ x≤ 1and (7),(11) we have from (10)| f (t)| ≤ hν ·Mν3(2pi)ν−2that is (9) for k = 0.With a successive derivation of (10) and by applying (8) we havef (k)(t) =hν−(k+1)(ν− k)! ∆ f(ν−1)a Bν−k(t−ah)+− hν−(k+1)(ν− k)!∫ baf (ν)(t)B∗ν−k(t− sh)ds k = 1,2, ...,ν−1(12)and applying the previous inequalities we get∣∣∣ f (k)(t)∣∣∣≤ hν−k ·Mν6(2pi)ν−k−2k = 1,2, ...,ν−1that is (9) for k = 1,2, ...,ν−1.3. Existence and uniquenessTo the boundary value problem (1a)−(1b) we associate the homogeneousboundary value problemx(m)(t) = f (t,x(t)), a≤ t ≤ b, m> 1x(a) = x(b) = 0∆x(k) ≡ x(k)(b)− x(k)(a) = 0 k = 1, ...,m−2(13)ON BERNOULLI BOUNDARY VALUE PROBLEM 167From Theorem 3, the solution of the boundary value problem (13) isx(t) =∫ baG(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 polynomial Pm−1[x](t) defined by (4) with x(a)=β0, x(k)(b)−x(k)(a)=βk+1, k = 0, ...,m−2, satisfies the boundary value problem:P(m)m−1[x](t) = 0Pm−1[x](a) = β0∆P(k)m−1 ≡ P(k)m−1(b)−P(k)m−1(a) = βk+1, k = 0, ...,m−2Therefore, the boundary value problem (1a)− (1b) is equivalent to thefollowing nonlinear Fredholm integral equation:x(t) = Pm−1[x](t)+∫ baG(t,s) · f (s,x(s))ds (15)Now, we have the following results:Theorem 5.[6] Let us suppose that(i) ki> 0, 0≤ i≤m−1 are given real numbers and let Q be the maximumof | f (t,x0, ...,xm−1)| on the compact set [a,b]×D0, whereD0 = {(x0, ...,xm−1) : |xi| ≤ 2ki, 0≤ i≤m−1};(ii) max∣∣∣P(i)m−1[x](t)∣∣∣≤ ki 0≤i≤m−1, where Pm−1[x](t) is the polynomialrelative to x as in (4);(iii) (b−a)≤(kiQ ·Cm,i) 1(m−i)0≤ i≤m−1.Then, the Bernoulli boundary value problem has a solution in D0.Proof. The setB[a,b] ={x(t) ∈C(m−1)[a,b] :∥∥∥x(i)∥∥∥∞≤ 2 · ki, 0≤ i≤m−1}is a closed convex subset of the Banach space C(m−1)[a,b].Now we define an operator T :C(m−1)[a,b]→C(m)[a,b] as follows:(T [x](t)) = Pm−1[x](t)+∫ baG(t,s) · f (s,x(s))ds (16)It is clear, after (15), that any fixed point of (16) is a solution of theboundary value problem (1a) and (1b).Let x(t) ∈ B[a,b], then from (16), lemma 1, hypotheses (i),(ii),(iii) wefind:168 FRANCESCO A. COSTABILE - ANNAROSA SERPE(a) TB[a,b]⊆ B[a,b];(b) the sets{T [x](i)(t) : x(t) ∈ B[a,b]} , 0≤i≤m−1 are uniformly boundedand equicontinuous in [a,b];On Bernoulli boundary value problem(c) TB[a,b] is compact from the Ascoli - Arzela theorem;(d) from the Schauder fixed point theorem a fixed point of T exists inD0.Corollary 1. Suppose that the function f (t,x0,x1, ...,xm−1) on [a,b]×Rmsatisfies the following condition| f (t,x0,x1...,xm−1)| ≤ L+m−1∑i=0Li |xi|αiwhere L,Li 0≤ i≤m−1 are non negative constants, and 0≤ αi ≤ 1.Then the boundary value problem (1a)− (1b) has a solution.Lemma 2 [6] For the Green function defined by (6), for ν = m− 1 thefollowing inequalities hold:|G(t,s)| ≤ gwithOn Bernoulli boundary value problemg=1ν!(b−a)m(1+2pi2m!3(2pi−1)).Proof. The proof follows from the known inequalities of Bernoulli poly-nomials and from simple calculations.Theorem 6.[6] Suppose that the function f (t,x0,x1...,xm−1) on [a,b]×D1satisfies the following condition| f (t,x0,x1...,xm−1)| ≤ L+m−1∑i=0Li |xi| (17)whereD1 = {(x0,x1...,xm−1) : |xi| ≤ maxa≤t≤b∣∣∣P(i)m−1[x](t)∣∣∣++Cm,i(b−a)mg ·h ·(L+C1−θ), 0≤ i≤ m−1}C = maxa≤t≤bm−1∑i=0Li∣∣∣P(i)m−1[x](t)∣∣∣ON BERNOULLI BOUNDARY VALUE PROBLEM 169θ = h ·g ·(m−1∑i=0Cm,iLi(b−a)m−i)< 1, h= b−a (18)Then, the boundary value problem (1a)− (1b) has a solution in D1.Proof. Let y(t) = x(t)−Pm−1[x](t), so that (1a) and (1b) is the same asy(m)(t) = f (t,y(t))y(a) = y(b) = 0∆y(k)a = 0 1≤ k ≤ m−2(19)whereOn Bernoulli boundary value problemy(t) = y(t)+Pm−1[x](t), y′(t)+P′m−1[x](t), .....,y(m−1)(t)+P(m−1)m−1 [x](t).Define M[a,b] as the space of m times continuously differentiable functionssatisfying the boundary conditions of (19). If we introduce in M[a,b] thenorm:‖y(t)‖∞ = maxa≤t≤b∣∣∣y(m)(t)∣∣∣then it becomes a Banach space. As in theorem 5, it suffices to show thatthe operator T :M[a,b]→M[a,b] defined byT [y](t) =∫ baG(t,s) · f (s,y(s))dsmaps the setS={y(t) ∈M[a,b] : ‖y‖∞ ≤ hg(L+C1−θ)}into itself. In order to demonstrate this, it is sufficient to utilise the con-ditions (17), lemma 1 and lemma 2.The thesis follows from the application of the Schauder fixed point theo-rem to the operator T.Definition 1. A function x(t)∈C(m)[a,b] is called an approximate solutionof (1a)− (1b) if there exist non-negative constants δ and ε such that:maxa≤t≤b∣∣x(m)(t)− f (t,x(t))∣∣≤ δmaxa≤t≤b∣∣∣P(i)m−1[x](t)−P(i)m−1[x](t)∣∣∣≤ ε ·Cm,i · (b−a)m−i, 0≤ i≤ m−1(20)where P(i)m−1[x](t) and P(i)m−1[x](t) are the polynomials defined by (5).The inequality (20) means that there exists a continuous function η(t)such that:x(m)(t) = f (t,x(t))+η(t)170 FRANCESCO A. COSTABILE - ANNAROSA SERPEandmaxa≤t≤b|η(t)| ≤ δThus the approximate solution x(t) can be expressed as:x(t) = Pm−1[x](t)+∫ baG(t,s)·[ f (s,x(s))+η(s)]dsIn the following we shall consider the Banach space C(m−1)[a,b] and fory(t) ∈C(m−1)[a,b] the norm ‖y‖ is defined by:‖y‖= max0≤ j≤m−1{Cm,0(b−a) jCm, j· maxa≤t≤b∣∣y j(t)∣∣}Now we have:Theorem 7.(Picard’s iteration)[2]With respect to the boundary value problem (1a)− (1b) we assume theexistence of an approximate solution x(t) and:(i) the function f (t,x0, . . . ,xm−1) satisfies the Lipschitz condition:| f (t,x0, . . . ,xm−1)− f (t,x0, . . . ,xm−1)| ≤m−i∑i=0Li |xi−xi| on [a,b]×D2where D2 ={(x0, . . . ,xi) :∣∣x j− x( j)(t)∣∣≤ N · Cm, jCm,0(b−a) j , 0≤ j ≤ m−1}(ii) θ < 1(iii) N0 = (1−θ)−1·(ε+δ )·Cm,0(b−a)m ≤ NThen, the following results hold:(21a) there exists a solution x∗(t) of (1a) and (1b) inS(x,N0)={x ∈C(m−1)[a,b] : ‖x− x‖ ≤ N0}(21b) x∗(t) is, the, unique solution of (1a) and (1b) in S(x,N)(21c) the Picard iterative sequence xn(t) defined by:{x0(t) = x(t)xn+1(t) = Pm−1(t)+∫ ba G(t,s) · f (s,xn(s))ds n= 0,1, . . .converges to x∗(t) with: ‖x∗− x0‖ ≤ θ n ·N0and‖x∗− xn‖ ≤ θ(1−θ)−1 · ‖x0− xn−1‖ .ON BERNOULLI BOUNDARY VALUE PROBLEM 171Proof. It suffices to show that the operator T :S(x,N)→C(m)[a,b] definedbyT [x](t) = Pm−1[x](t)+∫ baG(t,s) · f (s,X(s))dswhere X(s)=(x(s),x′(s), ....,x(m−1)(s)), satisfies the conditions of the con-traction mapping theorem.4. Newton’s iterationFor an efficient numerical calculation of the solution of problem (1a)−(1b)we can consider Newton’s iteration method. For our problem (1a)− (1b)the quasilinear iterative scheme is defined as:(22a) x(m)n+1(t) = f (t,xn(t))+∑m−1i=0(x(i)n+1(t)− x(i)n (t))· ∂ f (t,xn(t))∂x(i)n (t)(22b){xn+1(a) = β0x(h)n+1(b)−x(h)n+1(a) = βh+1, h=0, . . . ,m−2, n=0,1, . . .where x0(t) = x(t) is an approximate solution of (1a)− (1b).Theorem 8.(Newton’s iteration)With respect to the boundary value problem (1a)− (1b) we assume thatthere exists an approximate solution x(t), and:(i) the function f (t,x0,x1, . . . ,xm−1) is continuously differentiable withrespect to all xi 0≤ i≤ m−1 on [a,b]×D2;(ii) there exist non-negative constants Li, 0≤ i≤ m−1 such that for all(t,x0, . . . ,xm−1)∈ [a,b]×D2 we have:∣∣∣∣∂ f (t,x0, . . . ,xm−1)∂xi∣∣∣∣≤ Li(iii) 3θ < 1(iv) N3 = (1−3θ)−1(ε+δ ) ·Cm,0(b−a)m ≤ NThen, the following results hold:(23a) the sequence xn(t) generated by the iterative scheme (22a)− (22b)remains in S(x,N3).172 FRANCESCO A. COSTABILE - ANNAROSA SERPE(23b) the sequence xn(t) converges to the unique solution x∗(t) of theboundary value problem (1a)− (1b).Proof. The proof requires the equalities and the inequalities that we havepreviously determined and is based on inductive arguments.REFERENCES[1] R.P. Agarwal - G. Akrivis, Boundary value problem occuring inplate deflection theory, Computers Math. Appl. 8 (1982), 145 -154.[2] R.P. Agarwal, Boundary value Problems for Higher Order Differ-ential equations, World Scientific Singapore, 1986.[3] R.P. Agarwal - P.J.Y. Wong, Lidstone polynomials and boundaryvalue problems, Computers Math. Appl. 17 (1989), 1377-1421.[4] F.A. Costabile - F. Dell’Accio, Polynomial approximation of CMfunctions by means of boundary values and applications: A survey,J.Comput.Appl.Math. doi: 10.1016/j.cam.2006.10.059, 2006.[5] F.A. Costabile, Expansions of real functions in Bernoulli polyno-mial and applications, Conferences Seminars Mathematics Uni-versity of Bari 273 (1999), 1-13.[6] F.A. Costabile - A. Bruzio - A. Serpe, A new boundary value prob-lem, Pubblicazione LAN, Department of Mathematics, Universityof Calabria 18, 2006.[7] C. Jordan, Calculus of Finite Differences, Chelsea Pu.Co., NewYork, 1960.[8] I. Stakgold, Green’s Functions and boundary value problems, JohnWiley Sons, 1979.ON BERNOULLI BOUNDARY VALUE PROBLEM 173FRANCESCO A. COSTABILEDepartment of Mathematics, University of Calabriavia P. Bucci - Cubo 30/A - 87036 Rende(CS) Italye-mail: costabil@unical.itANNAROSA SERPEDepartment of Mathematics, University of Calabriavia P. Bucci - Cubo 30/A - 87036 Rende(CS) Italye-mail: annarosa.serpe@unical.it

ON BERNOULLI BOUNDARY VALUE PROBLEM

We give a constructive proof of the existence and uniqueness of the solution, under certain conditions, by Picard’s iteration. Moreover Newton’s iteration method is considered for the numerical computation of the solution

Francesco A. Costabile

Annarosa Serpe

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On Bernoulli boundary value problem

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ON BERNOULLI BOUNDARY VALUE PROBLEM

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