Existence and Uniqueness of Solutions to First Order Ordinary Differential Equations

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1 Existence nd Uniqueness of Solutions to First Order Ordinry Differentil Equtions Jesse Rtzkin Februry 1, Introduction These notes will prove tht there is unique solution to the initil vlue problem for wide rnge of first order ordinry differentil equtions (ODEs). The initil vlue problem we consider is du = F (x, u(x)), u() = b, (1) dx where F is given function, nd nd b re given rel numbers. A solution to this problem is function u(x) stisfying the differentil eqution u = F (x, u) with the proper initil condition u() = b. The result we will prove is Theorem 1 If F nd F u re continuous t (, b) then there is n ɛ > 0 such tht there is unique solution to (1) on the intervl ɛ < x < + ɛ. The rest of these notes re structured s follows. Section 2 contins some preliminry mteril regrding sequences of functions. Section 3 trnsforms the initil vlue problem (1) into n integrl eqution nd introduces the sequence of functions which will converge to the unique solution. Section 4 proves teh contrction mpping principle, which is interesting in its own right. We prove existence nd uniqueness in Section 5 by pplying the contrction mpping principle to the sequence we defined in Section 3. Finlly, we discuss some exmples in Section 6. I hve bsed these notes mostly on course notes I took s student [P, B]. One cn find user friendly, if brief, sketch of Picrd itertion in section 2.11 of [BP], nd more complete proof in [CL] (which is the stndrd reference for the theory of ODEs). 2 Preliminries This section contins some bsic tools we will need: the notion of distnce between two functions, notions of convergence of sequence of functions (both pointwise nd uniform), nd some consequences of such convergence. First we discuss the distnce between two functions. Let u(x) nd v(x) be function on the intervl 0 x 1. Wht should it men tht u nd v re close? It should men tht u(x) is close to v(x) for ll x [0, 1]. This motivtes the following definition. Definition 1 Let u(x) nd v(x) be continuous functions on [0, 1]. Then the C 0 (or L ) distnce between u nd v is u v := mx ( u(x) v(x) ). 0x1 1

2 There re severl comments. First observe tht u v is continuous function, so it chieves its mximum on the closed intervl [0, 1]. Also, there is nothing specil bout [0, 1]; we cn mke similr definition for ny closed intervl [, b] so long s < b. However, it is importnt tht we consider closed intervl, s the exmple u(x) = 1/x, v(x) = 1 on the open intervl (0, 1) illustrtes. Third, this distnce function stisfies ll the usul properties of distnce functions you re used to. Lemm 2 For ny two continuous functions u(x) nd v(x), u v 0, with equlity if nd only if u v. Symmetry holds: u v = v u. For ny three continuous functions u, v, w, we hve the tringle inequlity: u v u w + w v. Proof: The first property holds becuse the bsolute vlue of nonzero number is positive, nd only the bsolute vlue of 0 is 0. The second property holds becuse for ny two numbers nd b, b = b. To see the tringle equlity, let u v be mximized t the point x 0 [0, 1]. Then u v = u(x 0 ) v(x 0 ) u(x 0 ) w(x 0 ) + w(x 0 ) v(x 0 ) mx ( u(x) w(x) ) + mx ( w(x) v(x) ) 0x1 0x1 = u w + w v From clculus, you re fmilir with the ide of convergent sequence. A sequence of numbers { n } converges to, written lim n n =, if for ny ɛ > 0 there exists N such tht n > N implies n < ɛ. We hve two notions of convergent sequence of functions. Definition 2 Let {u n (x)} be sequence of continuous functions on the intervl [0, 1]. We sy u n converges to u(x) pointwise if lim n u n (x) = u(x) for ll x [0, 1]. We sy u n converges to u uniformly if lim n u n u = 0. Observe tht the pointwise limit of sequence of differentible functions does not need to be even continuous. Indeed, let u n (x) = x n for 0 x 1. Then pointwise u n u where { 0 x < 1 u(x) = 1 x = 1 We summrized some importnt properties in the following lemm. Lemm 3 Let the sequence {u n } of continuous functions on [0, 1] converge uniformly to u. Then {u n } lso converges to u pointwise, nd u is lso continuous. Before proving this lemm we mke two remrks. First, observe tht pointwise convergence does not imply uniform convergence. (Consider the exmple listed before the lemm.) Second, one cn strengthen this lemm (using essentilly the sme proof) to show tht the uniform limit of differentible functions is differentible. Proof: We first show tht uniform convergence implies pointwise convergence. Let x 0 [0, 1]. Then u n (x 0 ) u(x 0 ) mx 0x1 u n(x) u(x) = u n u 0, which implies u n (x 0 ) u(x 0 ). Next we show u is continuous. Choose x 0 [0, 1] nd let ɛ > 0. Choose N so tht if n > N then u n u < ɛ/3. Choose n n > N, nd let δ > 0 be such tht 2

3 if x x 0 < δ then u n (x) u n (x 0 ) < ɛ/3. Now pply the tringle inequlity twice, using tht u n (x) u(x) < ɛ/3 for ll x: u(x) u(x 0 ) u(x) u n (x) + u n (x) u n (x 0 ) + u n (x 0 ) u(x 0 ) ɛ 3 + ɛ 3 + ɛ 3 = ɛ. 3 Trnsformtion to n integrl eqution nd Picrd itertion There re two bsic steps to the proof of Theorem 1. The first step is to trnsform the initil vlue problem (1) into n integrl eqution. The second step is to construct sequence of functions which converges uniformly to solution of this integrl eqution. We define this sequence itertively, using scheme clled Picrd itertion (nmed fter Emile Picrd, ). First we consider the integrl eqution u(x) = b + F (t, u(t))dt. (2) Lemm 4 The function u stisfies the initil vlue problem (1) if nd only if it stisfies the integrl eqution (2). Proof: First suppose u stisfies (2). Then u() = b + nd, by the fundmentl theorem of clculus, du dx = d [ ] b + F (t, u(t))dt dx F (t, u(t))dt = b = d dx F (t, u(t))dt = F (x, u(x)). Now let u stisfy (1). Then, gin by the fundmentl theorem of clculus, u(x) = c + F (t, u(t))dt for some constnt c. Evluting t x =, we see this constnt c must be b. Next we define the sequence of functions which will converge to our solution of (2), t lest for x sufficiently close to. We strt with u 0 b, nd define the itertion scheme u n+1 (x) := b + F (t, u n (t))dt. It is useful to think of this sequence s certin opertion pplied repetedly to the originl function u 0 b. This opertion is given by Φ(u) := b + F (t, u(t))dt. Thus we hve defined Φ, which is function on the spce of functions. Such n object, function on the spce of functions, is usully clled functionl. This functionl Φ hs some very nice properties, which we will explore in the next two sections. 3

4 4 The contrction mpping principle Let (X, d) be spce equipped with distnce function d. Tht is, the function d = d(x, y) on pirs of points x, y X stisfies the properties listed in Lemm 2: d(x, y) 0 for ll x, y, with equlity if nd only if x = y. d(x, y) = d(y, x) d(x, y) d(x, z) + d(z, y). Definition 3 A sequence of points {x n } in such spce is clled Cuchy sequence if, given ɛ > 0 there is number N such tht n, m > N d(x n, x m ) < ɛ. Also, the spce (X, d) with distnce function is clled complete if ll Cuchy sequences converge. The rel line is complete, with the bsolute vlue function s its distnce function, is complete. However, the open intervl (0, 1) is not complete, s the exmple {x n = 1/n} illustrtes. The spce of continuous functions on closed intervl [, b], equipped with the C 0 distnce function defined bove, is complete. The proof of completeness is rther technicl, see chpter 7 of [R]. Roughly speking contrction is mp which decreses distnces. Definition 4 Let (X, d) be spce equipped with distnnce function d. A function Φ : X X from X to itself is contrction if there is number k (0, 1) such tht for ny pir of points x, y X we hve d(φ(x), Φ(y)) kd(x, y). It is importnt tht the constnt k is strictly less thn one. The following proposition is known s the contrction mpping principle. It is widely used in the fields of nlysis, differentil equtions, nd geometry. Proposition 5 Let Φ be contrction on complete spce (X, d) equipped with distnce function. Then Φ hs unique fixed point, tht is unique solution to the eqution Φ(x) = x. Proof: Pick ny x 0 X nd consider the sequence of itertes defined by x n+1 = Φ(x n ). We will show tht {x n } is Cuchy sequence. Let s first see tht the proposition follows once we estblish tht {x n } is Cuchy. Indeed, by completeness this sequence must converge to something, which we cll x = lim x n = lim n n Φn (x 0 ). Here Φ n mens we hve pplied the function Φ n times. Now pply Φ to both sides: Φ( x) = lim n Φn+1 (x 0 ) = lim n Φn (x 0 ) = x. Here we hve used the fct tht if limits exist then they re unique. Moreover, suppose there is nother fixed point, tht is nother ˆx such tht Φ(ˆx) = ˆx. Then, using the fct tht Φ is contrction nd both x nd ˆx re fixed points, d(ˆx, x) = d(φ(ˆx), Φ( x)) kd(ˆx, x). Since k < 1, this is only possible if d(ˆx, x) = 0, i.e. ˆx = x. We complete the proof by showing {x n } is Cuchy sequence. First observe tht, for ny n, d(x n, x n+1 ) = d(φ(x n 1 ), Φ(x n )) = kd(x n 1, x n ). 4

5 Applying this inequlity n times, we see d(x n, x n+1 ) k n d(x 0, x 1 ). Now pick ɛ > 0 nd let M := d(x 0, x 1 ). For n > m we hve d(x n, x m ) d(x m, x m+1 ) + d(x m+1, x m+2 ) + + d(x n 1, x n ) k m d(x 0, x 1 ) + k m+1 d(x 0, x 1 ) + k n 1 d(x 0, x 1 ) = M n 1 j=m = Mkm 1 k. k j Mk m k j Becude M is fixed number nd k < 1, we cn pick n N such tht m > N forces j=0 Mk m 1 k < ɛ, which in turn implies d(x n, x m ) < ɛ, which is the criterion for the sequence {x n } to be Cuchy sequence. 5 Convergence of the Picrd sequence In this section we will show the Picrd sequence converges using the contrction mpping principle. Recll tht we defined our function {u n } by the itertion scheme u 0 b, u n+1 = Φ(u n ) := b + F (t, u n (t))dt. The hert of the proof is series of estimtes, showing tht Φ decreses distnces, when distnces re computed using the C 0 distnce function on closed intervl, provided the intervl is smll enough. Let u 1 (x) nd u 2 (x) be continuous functions, nd consider Φ(u 1 ) Φ(u 2 ), s compred to u 1 u 2. We begin with some pointwise estimtes: Φ(u 1 )(x) Φ(u 2 )(x) = = F (t, u 1 (t))dt F (t, u 2 (t))dt (F (t, u 1 (t)) F (t, u 2 (t)))dt F (t, u 1 (t)) F (t, u 2 (t)) dt mx F u u 1(t) u 2 (t) dt x mx F u u 1 u 2. The consequence of this inequlity is, if we restrict to x < ɛ nd let M be the mximum of F/ u, the estimte Φ(u 1 ) Φ(u 2 ) ɛm u 1 u 2. Becuse F/ u is continuous, we cn choose ɛ smll enough so tht ɛm 1/2, so tht we hve Φ(u 1 ) Φ(u 2 ) 1 2 u 1 u 2. (3) 5

6 We re finlly redy to put everything together for the proof of Theorem 1. First observe tht solution of (2) stisfies Φ(u) = u, tht is solution is fixed point of Φ. Next, provided we restrict to x ɛ where ɛ > 0 is sufficiently smll, the inequlity (3) shows tht Φ is contrction on the spce of continuous functions on [ ɛ, + ɛ]. Becuse the spce of continuous functions on [ ɛ, +ɛ] is complete, we cn pply the contrction mpping principle to conclude tht Φ hs unique fixed point. In other words, the integrl eqution (2) hs unique solution u(x) on the intevl [ ɛ, + ɛ]. We conclude with some remrks. First, we see tht it is necessry to restrict to smll intervl x < ɛ in order for the proof to work. This restriction is clled short time existence. In other words, we hve shown tht solutions to n initil vlue problem exist for short time before nd fter the time of the initil condition. Is this rel restriction, or is it n rtifct of the proof? We will ddress this question in the next section. Next we remrk tht this is the typicl behvior of solutions which do not exist for ll x. More precisely, suppose u(x) solves (1) for x < x 1, but not on ny lrger intervl. Then we must hve u(x) s x x 1. (This fct is sometime clled the escpe lemm, nd its proof is beyond the scope of these notes.) Finlly, we note tht essentilly the sme proof works for first order systems, s well s the sclr equtions described bove. 6 Exmples We end these notes with some exmples illustrting when n initil vlue problem hs solutions, unique solutions, or long time solutions. For our first exmple, consider u = 2 u, choosing the positve brnch of the squre root function. Notice tht the right hnd side F (x, u) = 2 u doesn t hve continuous derivtive t u = 0, nd is not even well-defined for u < 0. As conseqence, we must hve u 0, which (from the eqution) implies u 0 s well. This is seprble equtions, nd the generl solution hs the form u(x) = (x + c) 2. However, this solution is only vlid for x c, where u 0. There is lso n equilibrium solution, nmely u(x) 0. Putting everything together, we see tht the initil vlue problem hs unique solution if u() > 0, multiple solutions if u() = 0 (the constnt solution nd the prbol), nd no solutions if u() < 0. This exmple illustrtes we cn t relx the hypotheses of Theorem 1. The right hnd side is continuous t u = 0, but it doesn t hve continuous derivtive. At tht point, there re solutions to the initil vlue problem but they re not unique. For out next exmple, we consider the eqution u = u 2. This time the right hnd side F (x, u) = u 2 is polynomil, so the initil vlue problem lwys hs unique solution. The generl solution of this eqution hs the form u(x) = (c x) 1 for some constnt c, nd there is constnt solution u(x) 0. This time we see tht the solution does not exist for ll x. Indeed, s x c the solution becomes unbounded, tht is u(x). This is the typicl behvior of solution to n ODE which does not exist for ll vlues of x: the solution must blow up ner finite vlue of x if it s not solution for ll vlues of x. References [BP] W. Boyce nd R. DiPrim. Elementry Differentil Equtions nd Boundry Vlue Problems. John Wiley & Sons,

7 [B] K. Bube Lecture Notes in Liner Anlysis. unpublished, 1996/7. [CL] E. Coddington nd N. Levinson. Theory of Ordinry Differentil Equtions. McGrw Hill, [P] C. Pugh. Lecture Notes in Rel Anlysis. unpublished, spring [R] W. Rudin. Principles of Mthemticl Anlysis. McGrw Hill,