Solutions of Nonlinear Integral Equations and their Application to Singular Perturbation Problems Citation Willett, Douglas Warren (1963) Solutions of Nonlinear Integral Equations and their Application to Singular Perturbation Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RP5F-1D73. https://resolver.caltech.edu/CaltechTHESIS:10082012-162104141 Abstract [See thesis for full Abstract] Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Erdélyi, Arthur Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Record Number: CaltechTHESIS:10082012-162104141 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10082012-162104141 DOI: 10.7907/RP5F-1D73 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7228 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 17 Oct 2012 16:56 Last Modified: 10 Jan 2024 00:51 Thesis Files Preview PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

Solutions of Nonlinear Integral Equations and Their Application to Singular Perturbation Problems Thesis by Douglas Warren vlillett In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 Acknowledgements I wish to thank Professor A. Erdelyi for introducing me to the subject, offering numerous suggestions and encouragement along the way, and tireless reading of the manuscript . My thanks are extended to taxpayers, who assisted me financially through the National Defense Education Act . Lastly, I thank Marie for her encouragement and sacrifice, Steve for exploding bad conjectures of mine, and Carolyn for handling well a tedious and difficult typing job . ABSTRACT Sufficient conditions for the existence and uniqueness, and estimates, of a continuous vector solution y(t) = u(t) + EL(t) y = y(t) to the integral equation M(s) f(s,y(s)) ds + Ef t are derived. t K(t,s) f(s,y(s)) ds, 0 A successive approximation technique involving a double sequence is used in the proof . This integral equation result is applied to the second order singular perturbation problem with differential equation Ex"+ p(t,e) x ' + q(t,e) x + r(t,e) + Eh(t,x,x',E) = o, O<t<l, and boundary conditions b (e) x(o,e) + b (e) x'(o,e) = £ (e), 0 1 2 c 1(e) x(l,E) + c 2 (e) x'(l,e) Conditions are established under which a certain sequence generated from this system is the basis for asymptotic expansions of a solution . The singular perturbation problem with differential equation Ex" + F(t,x,x' ,e) = 0 is also studied. and w = w(t,e) Under the assumptions that there exist functions and positive constants 1ew" + F( t, w, w, , e) 1 ~ As ~ Be, and certain relationships including s = s(E) A and B such that o ~ t ~ ,, 0 < E < E w and evaluated at the boundaries w' 0 , hold as E -> 0+ , "'e obtain an asymptotic expansion with leading term w(t,e) for a solution to this problem . 1. We are concerned with proving the existence and uniqueness of solutions for certain classes of ordinary differential equations which depend in a singular manner on a small positive parameter €. We are further concerned with describing these solutions by obtaining for them asymptotic expansions uniformly valid in the whole interval as More precisely, let P = P(€) P(€) -> 0 property that as be a positive function of € -> O+, and let t € € -> 0+ with the belong to the interval 00 I. The formal sum ~ x. ( t, €) • 1 ~ is said to be a uniform asymptotic expansion ~= with scale P(€) such that for € for x(t,€) if there exists a function A= A(t,€) -> 0+ and m = 1,2, ••• , m x(t,€) - ~ uniformly for t in I. ( 1.1) i=1 It is the nature of most perturbation problems, where the small parameter multiplies the highest derivative in the differential equation, to exhibit non-uniform convergence, as some point or points in the interval. singular perturbation problems . € -> O+, in the neighborhood of Such problems are usually called Most of the literature on singular perturbation problems has been concerned with the case when the non-uniformity occurs at one of the end points of the interval. It is the custom in this case to call the region near this end point a boundary layer in analogy with certain hydrodynamic phenomena. Most singular perturbation problems with linear differential equations exhibit boundary layers, and in this case theoretical means exist for determining when a particular end point is part of a boundary layer (see Wasow [1]). Usually, different analytic expressions are developed for the boundary layers and the rest of the interval (see, e.g., Levin and Levinson [2]). 2 In section 2 some results for integral equations are proven . These are developed in greater generality than needed for the specific applications in the following sections . This is done in the expectation that these results will be found useful in connection with other perturbation problems . Of fundamental concern in developing the theory for integral equations, which have the same solution as a given boundary value problem involving an ordinary differential equation, is the solution of Volterra integral equations . For such integral equations the convergence technique bas been examined in considerable detail for the linear case by Erdelyi [3], [4], and [5], and in some detail for the nonlinear case by Erdelyi [6] . Theorem 2 . 1 in this thesis is a result for nonlinear Volterra integral equations similar to Erdelyi ' s result in [6] , but differing enough to warrant a separate proof . Theorem 2.1 is used to get our main integral equation result Theorem 2 .2, which states conditions under which a Fredholm equation of the form y(t) = u(t) + E f K(t,s) f(s,y(s))ds , t e I, ( 1 . 2) I (more precisely, of the form 2 . 3) has a unique vector solution K(t,s) is allowed a discontinuity along t = s, and to satisfy a lipschitz condition with respect to The nature of the domain ~ y(t) f(s,y(s)) on I. is assumed y(s) . in which the lipschitz condition on f holds is very important in singular perturbation problems, because it is necessary for fJ to include functions y( t) = y( t, E) At the same time one desires to keep tJ compact . with non-uniformities , In the statement of Theorem 2 .2 the form of /:) is not specified to any great degree . Later, 3 when Theorem 2 .2 is applied to singular perturbation problems expected to have a boundary layer at the left endpoint of the interval, iJ is specialized to take this into account . The possibility exists that Theorem 2 .2 might be applied to problems having solutions with non- uniformities in the interior of the interval, by choosing the domain ~ in an appropriate manner . to have been done on such problems . Little work seems Lagerstrom [7] and his associates are presently engaged in research to produce examples of this nature . The literature on singular perturbation problems which exhibit boundary layers is quite extensive . Wasow [8] proves the existence of a solution and develops a single uniform asymptotic expansion for that solution in the case of a second order differential equation of the form ( 1.3) with ( 1.4) and a set of boundary conditions of the form x(t ,E) = £ , 0 ( 1.5) 0 Wasow •s theory applies only to problems that have solutions with at worst non- uniformities on the boundary of the interval . Erdelyi [ 9] proves the existence of a unique solution for a more general problem than 1 . 3, 1. 5, and he shows to some extent the behavior of this solution, as € -> 0+. Using integral equation techniques, Erdelyi is able to replace condition 1. 4 by the assumption that € -1 -~F - dx ' 2 is bounded . The boundary conditions for 4 the problem considered by Erdelyi are of the form 1.5, but with £ allowed to depend on 1 .£ 0 and €. A result of Wasow [1] is that most regular second order linear singular perturbation problems with solutions have a single boundary layer. Hence, it is only natural to expect problems involving an equation of the form Ex"+ p(t, E)x • + q(t,€)x + r(t,€) + €h(t, x, x ' ,€) = 0, ( 1.6) which is only "weakly nonlinear" , to have under very general conditions a solution with a boundary layer at the same point as the solution of the linear equation €x" + p(t,€)x ' + q(t,€)x + r(t,€) = 0 . ( 1. 7) In section 5 we treat the problem composed of equation 1 . 6 and boundary conditions b 1 (€)x(O,€) + b (€)x •(O,€) = £ (€), c 1(€)x(l , €) + c (€)x •(l,€) = .£ (€) . 0 1 2 2 ( 1.8) Sufficient conditions to guarantee the existence of a unique solution x = x(t,€) are stated, and uniform asymptotic expansions exhibiting a boundary layer are given for x and x1 • It is characteristic of these asymptotic expansions to consist of parts that may be computed by using only ordinary perturbation methods on problems having non- homogeneous first order linear differential equations and one boundary condition . We consider in section 6 the problem of the "strictly nonlinear" equation 1 . 3 with boundary conditions 1 .8 . Instead of considering this 5 problem as merely a "boundary layer problem", as Wasow does in [8] and Erdelyi does in [9] , we find conditions sufficient to guarantee the existence of a solution of the form X where w = w(t,E) = W + Ez, ( 1.9) is given so that z can be determined as a solution of a weakly nonlinear problem by the theory developed for such problems in section 5. c1 of class For a very general class of functions in t and of class cl in x and F, e.g., F(t,x,x 1 ,E) x', and E- 1 (f.F dX 12 = 0( 1) uniformly (see assumption 6ii for a precise statement of the conditions on F), we find it sufficient that w(t,E) satisfy II Ew( t ,E) + F(t,w(t,E), w1 (t,E),E) = 0(€) uniformly (1.10) and a similar weakened version of the boundary conditions . Nothing is said about the non-uniformities of determined any further than 1.10 w(t,E), nor is and the other general conditions assumed for it . One can show that Wasow [ 8] and Erdelyi [ 9] where w 0 = w0 (t) w(t,E) have taken is assumed to satisfy F(t, wo' •-r'' 0) = 0 0 ( 1.12) and and where J..L >I.e0 - w0 (o)l. - ( 1. 13) 6 sufficiently small, they show the existence of a function so that 1. 11 determines w satisfactorily. w (t,E), so 1 determined, can have only a boundary layer type of non-uniformity . the magnitude of Also , is limited enough to greatly restrict the generality ~ of the final result for w(t , E) . It seems certain that the integral equation technique developed in this thesis could be extended to obtain w( t,E) of the same nature as Wasow' s and Erdelyi ' s result, only for our more general problem. For boundary conditions of the form 1 • 8, one would choose ~ > I£ - 0 - b 1 w0 ( 0) - b2 w'0 ( 0) I and then would consider the weakly nonlinear differential equation, where "weakly" now means with respect to the new parameter ~, obtained by substituting 1 . 11 into 1. 10. The uniform asymptotic expansion we obtain for the general problem defined by 1.6 and 1 . 8 gives, when h =0, a uniform asymptotic expansion for the general second order linear equation with boundary values of the form 1. 8 . that In this case the expansion is obtained under the assumption p(t,E) is from the class c1 and q(t,E) is from the class C. 7 2. In what follows, t and s denote real variables confined to an interval I, which has t0 as left endpoint and point . t 0 = -~, then t is not included in then If t 0 is not included in 1 belong to I. R. is the set I. t Otherwise, {(t,s) : t € as right end- I, and if and 0 I, t t1 t 1 may or may not < s < t) ; 0 is a snall € positive parameter always assumed to be in the interval An n- dimensional vector will be thought of as an by xt will be meant the transpose of the n X 1 n X 1 matrix x. matrix, and Whenever we speak of matrices in general, we will mean scalars, column and row vectors, and square matrices compatible with these vectors . real numerical functions for elements . A= [aij(t)] All matrices will have As the norm of the matrix we take .2:. la.;J.(t)l . l.,J If all the a .. (t) l.J meant the matrix like manner . ( ... are integrable functions of t, by I .. (t)dt]. Jf a l.J A' (t) is defined in a The derivative is A(t)dt A partial ordering between matrices is defined component- wise by A < B if a. . ( t) < b . . ( t) l.J - l.J for all 1, j, and t. For later applications we shall need that our results in this section be stated with more precision than the estimates afforded by the norm defined above. vee t or wh ose Hence, for any matrix A define .th component l.S . 1.-- 2: j IAI to be the column Ia l.J .. ( t) I . For scalars this is the usual absolute value, and for a column vector x = (xi], lxl vector whose ith component is I is the scalar lx.l l. and lx t is the column 2: i Ix.l. l. 8 For any matrices A and B, 8' be the set of all vectors Let and satisfy the vector inequality vector d( t) . We say that A(t) x( t) which are continuous on lx(t)l < d(t) for some fixed positive integrable~ .! if f is I t 1- A(t)dt exists (is t + 0 I, A(t) ~ .! is locally integrable~ finite); A(t) is integrable over if for each t (t ,s); I , A(t,s) s; and A(t,s) ( t 0 't) as a function of A(t,s) is locally integrable over ~ t A(t,s) 0 belonging to each s and A(t 1- ,s)ds = lim t->t 1- t0 belonging to is locally int egrable ~ is integrable over the interval is integrable ~ ~ t 10 i f for It R if (2 . 1) A(t,s)ds exists . A function for each x = x(t) = x(t,e) e, o < e < e , the first 0 exist on I k ck(I) k with respect der ivatives of t on I. This will be denoted in the usual way by x e ck(I), x(t) e ck(I), or x(t,e) e ck(I) . o < e < e*, and that and that are continuous e* IP(e)l ~a IT(e)l so that p( €) = o ( T( €)) i f for any positive number exists another positive number o < e < e*(a) . x(t,e) T(e) we say that p(e) = O(T(e)) if there p(e) and exist positive constants a derivatives and x(t,e) if to For functions and these is a member of the class For vectors x(t, e) = O(y(t, e)) e*(a) so that x(t,e) and y(t,e) uniformly over I whenever a there IP(e ) l ~a IT(e)l whenever defined over I, we say 9 (or just "uniformly") if there exists positive constants so that lx(t,E)l ~ a ly(t,E)l for all and that x( t, E ) = o ( y( t, €) ) uniformly over i f for any positive number a E*(a) so that t in and 0 < E < E*; I I a and E* (or just ''uniformly" ) there exists another positive number lx(t,E)I ~ aly(t,E)I for all t in I and 0 < E < e*( a) . That these last definitions make sense for matrices follows from our previous definition of the absolute value and ordering of matrices. We will use the f ollowing well known result (see [10], p . 37): Lemma 2 . 1 . defined over I Let f(t), g(t), h(t), and k(t) and such that locally integrable over I. k(t)f(t), k(t)g(t) , and J t in k(s)f(s)ds , 0 and k(t) > 0 I, then f(t) ~ g(t) + h(t) exp( Jt k(s)h(s)ds) l t k(s)g(s)ds t 0 for all t in are t t for all k(t)h(t) If f(t) ~ g(t) + h(t) h(t) ~ o, be numerical functions t 0 I. Our first considerations will be for the unknown vector function y(t) in the nonlinear Volterra integral equation y(t) = u(t) + E i t K(t,s)f(s,y(s))ds. to (2 .2) 10 Sufficient conditions for the existence and uniqueness of a solution y(t) will be stated, and a bound on y(t) - u(t) will be obtained. The result for equation 2 .2 will then be used to find sufficient conditions to guarantee the existence of a unique solution y(t) to the nonlinear integral equation ~ y(t) = u(t) + EL(t) t 1- t0 E f M(t 1-, s )f(s,y(s))ds + (2 . 3) t K( t, s) f( s, y( s) )ds . to K in equations 2 .2 and 2.3 and LM compatible with the vectors in equation 2.3 are matrices y, u, and f . The first integral in equation 2 . 3 is defined in the manner of 2 . 1. u(t) E C(I); K(t,s) E C( ~ ); Assumption 2i . on is defined There exist nonnegative vector functions j(t) and a positive scalar function that I lf(t,x) - f(t,y)l I~ mt (t) lx - Yl IK( t, s ) I ~ j ( t) K( s ) . K(t) The lipschitz condition satisfied by imply that f(t,y) as a function of Assumption 2iii. and f(t,y) I X iJ . Assumption 2ii . and and o;(t) over I y all continuous on whenever f o;(t) ~ EK(t) m~ (t) j(t) t in is continuous on I. such and the continuity of m(t) o~(t) ~ K(t) llf(t,u(t)) II for all I and (t,x) , (t,y) E I X ~ JJ There exist locally integrable functions such that m(t) and 0 1 ( t) 0 ,, With 0 ( t) o '(s)ds o1(t) and 0 0 let (2.4) over Assumption 2iv. 6 (t) < d(t), t e I. 0 - Suppose £} and IR, y(t) e ; let T K(t,s)f(s,y(s)) be the mapping defined by Ty(t) = u(t) + E 1 t t A fixed point for is locally integrable (2.5) K(t,s)f(s,y(s))ds 0 T belonging to J1 will be constructed by the classical method of successive approximations, setting (2. 6 ) Although Erdelyi [6 ) works out for considerable generality the technique of taking successive approximations with the operator by our assumptions is not included. converge to a function bound for y*(t) T, the case covered We will prove here that the yk(t) satisfying 2 .2, and we will determine a y*(t) - u(t). Theorem 2 .1. Under assumptions 2i to 2iv, exists for all values of fixed point y*(t) as defined by 2 .6 k, T as defined by 2.5 has one and only one for which is locally integrable over yk(t) y*(t) e I, and J} and K(t) mt (t) IY*(t) I 12 (2 .7) Proof . that for The main part of the proof will be to show by induction k = 1, 2, • .. , yk( t) S , and exists and belongs to (2 .8) u(t) E defined in B by assumptions 2i and 2iv; hence, R. K( t, s) f( s, u( s)) if IK(t,s)f(s,u(s))l K(t,s)f(s,u(s)) t* < s < t where t 0 ( t * , t) t E I < t* < t involved are continuous on the fonn is locally integrable over is locally integrable over for each I where both ~ R. , , and i f is integrable over all intervals (see [11], p . 437) . Since all functions by assumption and since any interval of t* and latter requirement is certainly met . integrable over K(t,s)f(s,u(s)) is t are in I is bounded, the IK(t, s)f(s,u(s))l is locally !R , because IK(t,s)f(s,u(s))l ~ j(t)K{s)l lf(s,u(s))l I~ j(t)o~(s) by assumptions 2ii and 2iii , and also by assumption . j(t) o'(s) vie conclude that 0 y ( t) 1 (2 . 9) is integrable over ~ exists, and that (2 . 10) by equation 2 . 6 , inequality 2.9, and the definition of the triangle inequality to inequality 2 . 10, we obtain +j(t)Eo(t) . 0 0 ( t) . 0 Applying 13 Hence, y (t) E 1 J1 by assumption 2iv and the continuity of The induction statement has been verified for k = 1. Next, suppose that the induction statement holds for n ~ 1. u(t). k < n where From this hypothesis, assumptions 2ii and 2iii, and the definitions of 50 and o , we obtain 1 IK(t,s)f(s,yn(s))l ~ IK(t,s)l llf(s,yn(s))ll < n j(t)K(s) [ llf(s,u(s)) II + j: 1 n j ( t) K( s ) [ II f ( s , u ( s ) ) II + j: 1 llf(s,y}s)) - f(s,yj _1(s)) II ] < m t ( s ) Iy j ( s ) - y _1( s ) U~ n j ( t) K( s) [ II f( s, u( s)) II + Em T ( s) j ( s) 5 ( s) 0 L: [o (s))j- 1 1 j=1 J (j-1)! (2. 11) Inequality 2 . 11 implies that over IK(t,s)f(s,yn (s))l ~ , and it follows as before that integrable over ~ . We conclude that is locally integrable K(t,s)f(s,y (s)) n is locally yn+l(t) Hence, by exists . equation 2 .6, assumptions 2ii and 2iii, and inequality 2 . 8 for ,S j(t) E [o1(s)]n- 1 K( s ) m t ( s ) j ( s ) 50 ( s ) ( n -1} ! ds ' k = n, 14 which is inequality (2.8) for k = n + 1. n+1 ~ jyk(t) - Yk_1 (t)j < k=1 jyn+ 1 (t)l ~ lu(t)l + lu(t)l + j(t) E Hence, 50 (t) exp( 5 1(t)) = 6 0 (t) • By assumption 2iv, 1Yn+ 1 (t)l ~ d(t), and so yn+ 1 (t) E completely proves the induction statement for validity for k = n. all values of k = 1, 2, .••. k =n + 1 J1 . This given its We conclude that the induction statement holds for Inequality 2 .8 implies that ~ k=1 and since j(t), 50 (t) , and 5 (t) are all continuous functions on I, +co u(t) + ~ [yk(t) - yk_ (t)] converges 1 k=l 1 the telescoping infinite series absolutely and uniformly on every compact subinterval of I to a continuous +co function y*( t). Since y*(t) - yk(t) = ~ i=k [y. 1+ 1 (t) - y.(t)], 2.8 1 implies that +co 'y*(t)- yk(t)l -< +co ~ j(t) € Putting k = 0 5 ( t) in 2 . 12 produces 0 ~ j=k ~ j=k IY j+l ( t) - y.(t)l J [5 (t)]j 1 j! (2 . 12) 15 upon an application of the triangle inequality . It follows by assumption 2iv that We will show next that f(t,y) y*(t) is a fixed point for is a continuous function of f(t,y*(t)) ; lim T. Since y, we have (2 . 13) f(t,yk(t)) . k->oo In a manner similar to the derivation of 2 . 11, we get (2 . 14) for all k. The prime in 2 . 14 denotes a derivative with respect to Letting k -> oo s. in 2.14, we get by 2 . 13 that (2 . 15) Inequality 2 . 15 implies that over is locally integrable R , and it follows as before that integrable over R, K(t,s)f(s,y*(s)) is locally Because of 2 .14, J t lim k->oo j t K(t,s)f(s,yk(s))ds; t 0 t x(t) be any fixed point of is locally integrable over I. T in (2. 16) 0 by the Lebesgue dominated convergence theorem . Let K(t,s)f(s,y*(s))ds We conclude that ~ for which K(t) mt (t)lx(t) We need this last condition in order to 16 conclude from the assumptions that (2 . 17) Lemma 2 . 1 may be applied to 2 . 17 for the special case g( t) :: 0 in order to conclude that (2 . 18) Since nothing has been assumed to prevent mt (t) from being zero, we need to make one more computation with the true norms, i.e . , IY*(t) - x(t)l < E < E ft t IK( t, s) I II f( s, y* ( s)) - f( s, x( s)) II ds 0 Jt t IK( t, s ) I mt ( s ) Iy* ( s ) - x( s ) Ids ~ 0, t E I, 0 by 2 . 18 . y*(t) = x(t) Thus, for all t in I. The proof of Theorem 2 . 1 is now complete . Equation 2 . 3 will be considered next . In order to understand the be the set of basis for the considerations that follow, let DtO fU vectors I, M(t,s)f(s,x(s)) x(t) integrable over such that x(t) is defined over is ~ , and u*(t) = u(t) + Ux = u(t) + EL(t) ~ t 1- t0 M(t - ,s)f(s,x(s))ds 1 (2 . 19) 17 satisfies the assumptions of Theorem 2.1 for subset of vectors integrable over y(t) I. u(t) . J1 for which of iJ 1 be the Let K(t)mt (t)ly(t) j is locally vie consider the mapping S of if 0 S1 into given by Sx = z, where z = Tz + Ux. maps all of cfJ 0 to imply that iJ 0 and S (2 .20) Theorem 2 . 1 implies that into /t1 . S is single- valued and We will determine sufficient conditions has precisely one fixed point in the intersection of fJ, . The proof will be again by successive approximations, setting y~(t) = u(t), y!(t) = u(t) + U y~- 1 (t), y~(t) = u y~- 1 (t) + Ty~(t), k = 1, 2, ... '. (2.21) k = o, 1, •• • (2.22) and putting (2 .23) K(s) Assumption 2v . M(t,s) € C( ~ ), and as converges uniformly for s in I t -> t 1- , M(t,s)/K(s) to a function of s which is bounded in some non- degenerate subinterval of I some non-degenerate subinterval of with right endpoint Lemma 2 .2 . such that over I with left endpoint If assumption 2v holds and llv( s) II is integrable over v(s) I, then t 0 and in t 1• is a vector function is integrable 18 Proof. M~C;)) v( s) In order to show that is integrable over at , we must show the existence of the double limit lim M( lim t->t 1 - 't->t0 + t() ) v (s )ds . K s (2 .24) This will follow from the special boundness condition found in assumption 2v and the uniform convergence of M(t,s)/K(s), also part of assumption 2v . For the sake of brevity let the capital letter I with a subscript zero and any superscript denote a non- degenerate subinterval of left endpoint t , and let I 0 I with with a subscript one and any superscript denote a non- degenerate subinterval of I with right endpoint t • 1 We will prove the existence of the limits in 2 .24 by showing that for any positive number a , there exist intervals II such that v( s )ds II ,S a, v(s)ds - when s that s 1 <s 1 2 <s 3 0 s s <s • 4 Then, the left member of inequality 2 .25 is 4 are in I Suppose I 3 and 0 are in 0 and (2.25) 1 • bounded by IIM(s4,s)ll K( s) llv(s)llds + liM(s 4 , s) II II v( s ) I Ids • K( s) II v( s) II ds + (2 .26) 19 Assumption 2v implies that there exists intervals a constant c I 1 and 0 I 1 1 and such that 0 IIM(t,s)ll < c K(s) 0 when (t,s) 1 I 1l X I 1 • over I, there exist intervals I~ and (2 .27) Since is integrable llv(s) II f, included in I 1 and I 11 0 respectively, such that II v( s) II ds -< a/ 3c 0 when s, t I~ or are both in (2.28) f,. Substitution of inequalities 2 .27 and 2.28 into expression 2 .26 gives that the sum of the first two integrals 32 a is bounded by in when I 0 0 and Suppose that I t Because M( ~ s) s) interval are in s1 I0 1 t, II v( s ) II ds ~ c, • 0 converges uniformly at included in I 2 t -> t K( S) (s 4 ,s), (s ,s) € I~ X I . 3 2 .26 is bounded by a/3 when that intervals t' s) that M(K(s) -, there exists an < a (2. 30) 3c 1 It follows that the third integral in are in exist so that v( s) 1 such that 1 IIM(s 4,s) - M(s ,s) II 3 when (2 .29) · 1n · t egrable over 1s tO ~ I, . We conclude 0 2.25 holds, and hence, so . 20 Assumption 2vi. o~(t) function in t L( t) E C( I) , and there exists a locally integrable over such that I o~(t) ~ K(t) mt (t) jL(t)l for all I. We define = Jt t o~ ( s )ds . (2 . 31) 0 Assumption (2vii) . t -> t 1 - The limits as of o0 (t), o (t) , and 1 o (t) all exist . 2 Assumption (2vii) implies directly that [ o (t)exp( o (t))] ' 1 0 are integrable over I. [o (t)exp(o (t))] ' 1 2 and Hence, we obtain from assumption 2v and Lemma 2 .2 the existence of two bounded, continuous 50 ( t) -> ! liM( t t t zs)! I K{ s [ 5 (s)exp( o1 (s))]' ds, 0 (2 . 32) 0 52( t) ~ ( IIM(;t:lll [ o2 ( s )exp( 5 1( s))] ' ds . (2 . 33) 0 Let (2 . 34) 61 (t) Assumption (2viii) . for all t in I . E 52 (t 1) < 1 and 6 0 (t) + E 6 1(t) ~ d(t) 21 Theorem 2.2. ~~_.;_- Under assumptions 2i to 2viii, yk(t) 0 as defined by equations 2.21 and 2.22 exist for all values of as defined by 2 .20 bas one and only one fixed point IJ y( t) € and K( t )m t ( t) y(t) Iy( t) I is integrable over y~(t) and k, S for which I, (2.35) and ly(t) - u(t)l < [~0 (t) - u(t)] + € ~1 (t). Proof . on Once again the main part of the proof will be by induction k = 1, 2, ·• • • The induction statement is : 0 is locally integrable over y~(t) and I, 1 1 lyk(t) - yk- (t)l < [€ 52(t1)]k5 (t1) 0 solutions y~(t) € £1 , K(t)m~(t)ly~(t)l yk(t) exist for equations 2.22 and 2.21, 0 (2.36) - 0 € IL(t)l, (2 . 37) (2.38) and there exists ok(t) 0 satisfying inequality 2.23 and (2.39) (2.38) and (2 . 39) imply that by assumption 2viii . This means that assumption 2iv of Theorem 2 .1 is satisfied by the integral equation 2 .22 . It follows from Theorem 2 . 1 22 y~( t) . Thus, if that equation 2 .22 will have a unique solution inequalities 2 . 38 and 2 . 39 hold for a particular a unique solution k, then there exists y~(t) to equation 2 .22, y~(t) E iJ , K(t) J (t)ly!(t)l is locally integrable over y~(t) satisfies all the I, and in general conclusions of Theorem 2 . 1 . Consider the induction statement for in S by Theorem 2 . 1 . y~(t) Hence , is integrable over ~ is integrable over I , because k y~(t) = 1. exists and is will exist if M(t,s)f(s,y~(s)) K(s)l lf(s,y~(s ) )l I Inequality 2.15 implies that [o (s) exp(o (s))] ' 1 0 is integrable over Lemma 2 .2 implies then that M(t,s)f(s,y~(s)) I by assumption 2vii . t , s) K( s)f ( s , y*o( s ) ) = M(K(s) is integrable over t0 vu We obtain from 2.15 . the inequality t 1- ! IIM(t 1 -,s)llllf(s,y~(s))llds~ ! t t 0 t 1- IIM(t 1- ,s)ll K(s) , [o0 (s)exp(o 1 (s))] ds 0 It follows from the definit ion of t 1- J t IIM(t 1 - ,s)ll 50 (t) given in (2 . 32) that jjf(s,y~(s))i!ds~ 5 (t 1 ) . (2 . 40) 0 0 Hence, 0 ly\t) - y0 (t)j -< E 0 which is 2 . 37 for k = 1. 50 (t 1 ) Inequality 2.38 for from 2.41 by the triangle inequality . that 2 .23 and 2 . 39 hold, because Finally, IL(t)l, k =1 (2 . 41) follows directly o~(t) may be chosen so 23 J K( s ) II f ( s, y \ s ) ) II ds < t 0 t K( s ) II f ( s, u ( s ) ) II ds + 0 t J t K(s)l lf(s,y~(s)) - f(s,y~(s))l Ids~ 0 "'- (t) + J t K{s)mt (s) 0 t u 1y 1( ) s 0 0 - y ( s )I ds ' 0 0 and a2 (t) . by 2 . 41 and the definition of now conclude that y~(t) exists, locally integrable over statement for I. By a previous discussion we can y~(t) E Ja' , and K(t)mt (t) IY~(t) I is This completes the proof of the induction k = 1. Suppose next that the induction statement holds for and n > 1. solution Then for k = 1,2, ··· ,n k = n, 2 . 38 and 2 . 39 imply the existence of a y~(t) for equation 2 .22 when k = n. At the same time, the induction hypothesis assumes the existence of a solution these solutions are members of Jj ~(t) . and satisfy the integrability condition of Theorem 2 . 1, which implies uniqueness of solution . they must be the same function Both Hence, y~(t) . Since the existence of . Y~(t) satisfying 2 . 22 can be considered a direct result of Theorem 2 . 1, the inequalities included in the proof of Theorem 2 . 1 may be used here for 2 . 15 becomes y~(t) . For example, inequality 24 where the prime denotes a derivative with respect to s. It follows from assumption 2vii and inequality 2 . 39, which holds for k hypothesis, that I. [o:(s)exp(o 1(s))]' is integrable over =n by Hence K(s)l lf(s,y~(s))l I is integrable over M(t,s)f(s,y~(s)) IIM(t,s)ll llf(s,y~(s))ll are integrable over yn+ 1( t) as defined by equation 2 .21 exists . Further- <R, , which proves I, and by Lemma 2 .2, and a 0 more, for a proper choice of o (s), inequalities 2 . 39 and 2 .42 imply that (2 . 43) We obtain now from equations 2.21 and 2 .22 t 1- J t0 IIM(t 1 -, s) llmt (s) I Y~(s) -y~- 1 (s) Ids (2 . 44) and (2 . 45) by assumption 2ii. The integrals in inequalities 2.44 and 2 . 45 exist because of the part of the induction hypothesis Which says is locally integrable over I for k < n. K(t)mt(t)ly~(s)l 25 Lemma 2. 1 may be a pplied t'J 2 45 b E mt (s)j(s) F<i ve exp(o (s)) 1 It follows from ineouality 2.37 for (exp( o ( s)) 1 k =n that ~s K(r)m.,. (r) !y~{r) - y~- 1 (r) ldr] 1 < :) (2.46) Hence, t 1- J IIM(t 1 -,s)llmt(s)ly~{s) -y~- (s)lds ~50 (t 1 )(E52 (t 1 )]n. 1 (2 47) t ') Ineouality 2 37 f'Jr 2.47. E52 (t 1 ) < 1, 'Jne can prove the validity of inequality Because 2.38 for k =n + 1 k = n + 1 now follows from inequalities 2.44 and from the identity = u( t) + n+l L: k= l and the validity of 2.37 for k k 1 0 0 ( y ( t) - y - ( t)] k < n + 1. Next , ~t(s) l!f(s,yn+,(s)) II < K(s ) llf( s,u(s ) ) + 0 K(s) n+l k k 1 I jf(s,y0 (s)) - f(s,y0 - (s))l I< c0' (s) + k=1 ~ EB (t 1 ) 0 (2 .48) l - €52(t,) Inequality 2 . 48 implies that there exists a choice for both inequalities 2 .23 and 2 . 39 are satisfied . cn+l(t) 0 so that As a matter of fact, in all our considerations we could just as well choose for all n. This implies, as we have just shown by induction, that inequality 2 .23 holds for all values of for such a choice of k. Inequalities 2 . 39 and 2 . 43 hold trivially cn(t) . 0 By an earlier discussion in this proof, we conclude that y~+ 1 (t) exists satisfying equation 2 .22 and the conclusions of Theorem 2 . 1 hold for y~+ 1 (t) . For This completes the proof of the induction statement . k = 1, 2 , ... ' (2 . 49) follows in exactly the same way 2 . 46 was established for the particular case k = n. Since a solution y!(t) to equation 2 .22 exists for 27 k = 0,1, ••• , we may use 2 .22 and assumption 2ii to obtain Hence, inequalities 2 . 49 and 2 . 37 imply that for Since k = 1,2, ••• (2 . 50) E5 (t 1 ) < 1 by assumption (2viii) and since in general 2 II xII = II ( IxI ) II , it follows that +oo I: IIY!(t) - y!- (t)ll ~ 1 E5 (t ) 1 ~ 1-€02(t,) k=l [jiL(t)ll + Eo2 (t)exp(o 1 (t))llj(t)IIJ (2 . 51) +oo The right side of 2 . 51 is continuous on I, so I: k=l 1 I IY!(t)-y~- (t)l I is bounded on each compact subinterval of I . Thus, the telescoping +oo 1 series y~(t) + I: [y~(t) - y~- (t)] converges absolutely on I to k=l a continuous function y(t) . Inequality 2 . 50 gives further that ly(t) -y~(t)l < +oo I: j=k I Y~+ 1 (t) - y~(t)l ~ which is precisely inequality 2.35 in the conclusions . k = 0 Putting in 2 . 35 and applying the triangle inequality, we get ly(t)l ~ IY~(t)l + ~1(t) , or by inequality 2 . '7 of Theorem 2 . 1 . Assmnption (viii) gives now that J1 . Since y(t) € jy(t) - u(t)j ~ jy(t) -y~(t)l+ly*(t) -u(t)j by the triangle inequalit y, inequality 2 . 36 of the conclusions follows from 2 . 7 and 2 . 35 . We will prove next that y(t) is a fixed point for the mapping S defined by equation 2 .20, or what is the same, that y(t) is a solution to integral equation 2 . 3 . 2 .22 . In order to do this, let - > +co in equation k We will show that lim k- >too I t t 1 k M(t 1 -, s)f(s, y*(s))ds = 0 lim lim tr->t 1- k->too l t t k M(t, s)f(s,y*(s))ds 0 t 1- = ~ to M(t 1 -, s)f(s,y(s))ds 1 (2 . 52) and that t lim k->too L Since y~(t) K(t,s)f(s,y~(s))ds = 0 f K(t,s)f(s,y(s))ds . (2 . 53) 0 for n = 0,1, ••• is a unique solution by Theorem 2 . 1, inequality 2 . 15 found in the proof of Theorem 2 . 1 will be given as 29 inequality 2 . 43 when n = O, 1, ••• Letting n -> ~ in 2 . 43, one obtains (2 . 54) because f is a continuous function of y and lim y~(s) = y(s) . n->c;o The right side of inequality 2 . 54 is integrable over 2vii . Hence , K(t,s)f(s , y(s) ) I by assumption ~ • Further- is locally integrable over more, the interchange of limit and integral in 2 . 53 is valid by the Lebesgue dominated convergence theorem. Writing I I M(t,s)f(s,y~(s)) II < IIM(t, s)ll K(s) K( s) II f( s' y~( s) ) II ' we see by inequality 2 . 43 and the defini tions 2 . 32 and 2 . 33 of and 5 2 that I IM(t,s)f(s,y~(s))l I is bounded for all n 50 by an integrable function over ~ • Hence, the interchange of limits necessary to get the second equality in 2 . 52 is valid by the Lebesgue dominated convergence theorem . The only dependence of the first integral in 2 . 52 on k is through f(s,y~(s)) . In the convergence of the integral as t -> t 1 -,K(s)llf(s,y~(s))l is uniformly bounded with respect to lim by a function i ntegrable over k. The proof would proceed in a manner similar to the proof of Lemma 2 .2 . It follows I. This implies that k is uniform with respect to t->t, - 30 that lim and lim t->t 1 - may be interchanged to produce the first k->+oo equality of 2.52. The validity of 2.52 and 2.53 implies that y(t) is a solution of equation 2.3. In the proof of Theorem 2.2 there remains only to prove the uniqueness part of the conclusions. Suppose that equation 2 . 3, or, in other words, that mapping S, such that K(s)mt (s) lz(s)l m-r(t)ly(t)-z(t)l < Em.,.(t)IL(t)l f t J t + E mt (t) t z(t) z(t) is any solution of is any fixed point of the is integrable over I. Then, t 1- IIM(t 1 - ,s)ll llr(s,y(s)) - f(s,z(s))llds 0 IK(t,s)l llf(s,y(s)) - f(s,z(s))llds ~ 0 f E mT(t)IL(t)l t t 1- IIM(t 1 -, s)llmt-(s)ly(s) - z(s)lds + 0 1 K(s)mT(s) ly(s) - z(s) Ids • t E m1"(t)j(t) t (2 . 55) 0 The integral in 2. 55 exist by Lemma 2.2 and the integrability of K(s)mt(s) ly(s)l and K(s)mt (s) lz(s)l • Applying Lemma 2 . 1 to inequality 2 . 55, we obtain J t 1- t IIM(t 1 -, s) llmt (s) ly(s) - z(s) Ids , 0 Which yields with another application of Lemma 2 .1 mt (t)ly(t)-z(t)l = 0 . (2 . 56) 31 Inequality 2.55 is valid as a vector inequaltiy without the factor mT(t). Hence, it follows from 2.56 that y(t) = z(t) for all t in I. For the applications of Theorem 2 .2 to boundary value problems with second order ordinary di~ferential equations, it will be sufficient to use a two-dimensional ~orm of equation 2.3. vle need to further specify the "lipschitz function" Assumption 2ix. m( t) and tre domain fJ by specifying d( t). uT(t) = ut (t,€) = (u 1 (t,€),u (t,€)) 2 = 0(1) uniformly. Assumption 2ix is essentially a restriction on the transformations between the boundary value problem and its equivalent integral equation formulations . Only transforma.tionsyielding uniformly bounded u(t,€) can be used . Assumption 2x. for t in I uniformly, and cr(t,€) is a nonnegative numerical function defined and o < € < € 0 cr(t,€) ~ such that cr(t,€) € C(I), cr(t,€) = 0(1) l~(t,€)1 . For two new positive parameters ~ and v independent of t, define d~ (t) = (~,v + ~ and v cr(t,€)) . (2 . 57) can be considered measures of the size of the domain J1 The presence of cr( t, €) permits non- uniformities to occur in the second components of vectors in ~ • We define next (2 . 58) 32 w = w(~) where ~ and 6 is a nondecreasing numerical function of ~. defined in 2 . 4 and 2.34 respectively, let 1 For m given by 2 . 58, we actually have that 6 = 6(t,€; ~,v) o2 = 52 (t,€; ~,v) are functions also of ~ Assumption 2xi . o (t 1 ,€;~*,v*) and I, 2 in and and 6(t,€;~*,v*) and v. There exist positive constants such that t With ~*, exist for v* ) and €* 0 o < € < €* - 0 (2.60) and 6(t,€;~*,v*) = 0(1) uniformly . (2 .61) Assumptions 2ix to 2xi are sufficient to imply that 6 +€61<d, 0 when ~ v = €~, is sufficiently large, and € - is sufficiently small . More specifically, let ~ > ~ = sup limsup max ( Iu 1( t, €) I ,a ( t, €)) . t€I €->0+ Then, for ~ so- chosen, let >sup t€I ~ With € 0 ~ and (2 .62) ~ limsup ll6(t,€;~,v*) II €->0+ · (2 .63) fixed and satisfying 2 .62 and 2 .63, we require that be sufficiently small so that O<€ <min ( v* 0 - . ~ ' -~~ ~ ' €*)' 0 (2.64) 33 (2.65) 116( t, € ; IJ. , v*) II From the nature of 6 v 0 :s o < E < E0 . for 1{ (2.66 ) and 6 , one can show that 6 1 all 1J. and values IJ.*, v*. to make E5 (t ,E; IJ.,v) < 1, which is the reason for 2 . 65 . 2 1 property of 6 v provided it is assumed that it exists for one set of This, of course, depends on being sufficiently small € important here is that 6(t,E;IJ.,v) decreases, given that have from 2.64 exists for that t, E, and IJ. are fixed . v = Elt < v*, and hence, It follows from requirement 2 . 66 that 1r The other is nonincreasing as For we 6(t,E;IJ.,v) :S 6(t,E;IJ.,v*). >I l6(t,E;IJ.,v)l I for o < E < E0 • Hence, 2.64 implies that lu 1 I + v :S lu 1 I + IJ. -IJ. , lu 1 1 + v :S IJ. . in 2 . 62, it follows that and from our choice of Since a> 1~1 IJ. , we have I~ I + v <a + v, and so (2 . 67) Corollary. 2ix to 2xi and for d If asstunptions 2iv and 2viii are replaced by assumptions IJ. satisfies inequality 2 .62, then Theorem 2.2 is true and m given by 2 . 57 and 2 . 58 respectively and € sufficiently small. The restriction on IJ. is due to the way the assumptions have been formulated and is not an important requirement. lipschitz condition on f The domain in which the holds has been constructed about the zero vector rather than the vector u. 34 In this section we will be concerned with relating the second 3. order boundary value problem given by 2 Ey' + A(t,E)y + Eg(t,E) + E f(t,y,E) = 0, ( 3. 1) ( 3 .2) to an integral equation problem included in the theory of the previous section. All vectors, such as ( A= The interval I Ea 1(t,E) a 2 (t,<) ) Ea (t,E) 3 a (t,€) a are two-dimensional, and ( 3 . 3) 4 1 a 2 , a 4 E c (I); a 1, a la (t,E) I, 2 uniformly; and away from zero for (If f is taken to be the closed interval O<t < 1. Assumption 3i . are 0(1) y, g, and t in I E C(I); a 1, a , a2, a , a , a4 2 3 4 3 a (t,E) are positive and bounded 4 and 0 < € < E • 0 4 is negative, the substitution t* = 1 - t will change the problem to one of the type that we are considering.) The form of equati. on 3. 1 is not unique, i.e., higher order terms in E that are linear in y may be considered part of Ay or part of Under assumption 3i and the assumption that f € 2 f. satisfies a lipschitz condition of the type described in assumption 2ii with m given by 2.58, one can always determine A*(t,E) and f*(t,y,€) so that 35 A* and satisfy conditions of the same form as f* respectively, solution. Ey' + A*y = 0 and A and f has a special kind of fundamental We will assume that equation 3.1 is already written in the desired form. Assumption 3ii . Let Ey' + A(t,E)y = 0 have the fun~ntal solution H(t,€) = a(t,E)exp s(t,€) ' where a =-€ _, By assumption, rt J_ a 1 (r,E)dr - € s a _, I t a (r,E)dr) ds , s is positive and bounded away from zero and a 1 4 is uniformly bounded, so, o/•(t,E) < o , if € 0 4 t € I, is chosen sufficiently small . = < € < €0 ' For brevity we write exp o/(t,E) - o/(s,E) € 0 Then, ( 3.4) Assumption 3iii . g(t,e) E C(I) and The problem defined by differential equation 3.1 and boundary conditions 3.2 is equivalent to the problem of solving an integral equation of the type 2 . 3, where L(t,e) = -H(t,e) ( 3. 5) (3.6) ( 3 . 7) t + J: K(t,s,e)g(s,e)ds. ( 3 .8) Suppose that T 0 (e) = jb 1 + b 2 t3(o,e) I - 1 ( 3. 9) and (3.10) If (3 . 11) then det c H(1,E) ) ( b H(o,e) = 5 exp ( - E- 1 (3.12) 37 where (3.13) 3 . 11 is hardly a restriction in the light of equation 3.4 . It follows from equations 3. 5, 3 . 12, and 3 . 13 and a(o,€) = 0 that (3 . 14) where ~( t, €) We obtain by using = ( 1 ) € + t(t,o,€) IK(t, s,€)1 = O(~(t,€)) uniformly and assumption 3iii that I t IK(t, s,€)g(s , €) Ids~ 0 ltIK(t,s , €) I llg(s,€) lids 0 = 0(~) uniformly ( 3. 16 ) and where In order to simplify the estimate for lu(t,€)1, we assume that (3 . 18 ) and for some n. ( 3.1 9 ) 38 Then, g u(t,e)l o(1) ~ + where T = T(e) TU(t,e) uniformly, (3 .20) is a constant multiple of (3.21) It becomes clear now how to choose 2.58 so that ~ ~ a(t,E) ~ in I ~ and ~~(t,e)l and ~ ~ and 0 < € < € < 1 • o- a(t,e) in 2 . 57 and lu1(t,e)l for all t Let a(t,E) = [E + W(t,o,e)] a ( 3 .22) > a 0 > limsup T( € ) . € -> o+ ( 3-23) 0 and ~ Assumption 3iv. such that for d y(t,E)€ when (t,y) ~,v, and a0 and m defined by equations 2 . 57 and 2 . 58 with a given by 3.22 and when There exists positive constants ~ and a 0 satisfying 3.23, f(t,y(t,e),E) € C(I) fJ, f(t,o,e) = o, and llf(t,y,E) -f(t,z,E)II~ mt (t)ly- zl and Theorem~ and 3.19 hold . (t,z) are in I X ~ and 0 < € < e0 Let assumptions 3i to 3iv and relations 3 . 11, 3 . 18, If (3.24) and (3 .25) 39 then for sufficiently small y = y(t,E) to 3 . 1, 3 . 2 E there exists a unique solution such that ly(t,E)I = 0(~ T) uniformly (3 .26) and (3 .27) Proof . In the notation of section 2 , we may choose to all be constants for the present problem. j the lipschitz condition satisfied by f Since K, o;, and f(t,o,E) = o, gives llf(t,u,E)II <m-r- (t,E) lui , and hence, 3.14 implies that we may choose E- 1 52(t,E) to be a constant multiple of T1 (1+T0 Ib 1 l) ~(t, €) . It follows that 50 (t, E) = O(T) uniformly, o2 (t,E) = 0(T1 (1+T0 Ib 1 I)] uniformly, 50(t 1 ,E) = 0(TT2 ) , 52 (t 1, €) = O(T1T2 (1+T0 Ib 1 l)]. Hence, I j(~(t , €) - lu(t,E)I)I I = O(ET) = 0(€) and uniformly 40 We get that u(t,€) ~ 0(1) uniformly from 3.20 and T ~ 0(1). gives that € 52 (t 1,E) ~ o(1) . We conclude that the corollary to Theorem 2 .2 applies in the present case . from 2 . 36. 3.24 Relations 3 .26 and 3. 27 follow 41 4. In this section we will consider the second order scalar differential equation given by o, Tx =ex"+ x' + q(t,E)x + r(t,E) + Eh(t,x, x' , E) (4 . 1 ) along with boundary conditions of the form B[x] = b 1(E)x(o,e)+Eb (e)x•(o, E) = .£ ' 0 2 ( 4 .2) C[x] = c 1(e)x(1,E)+Ec (E)x ' (1,E) = 2 ( 4.3) The notation in 4 .2 and 4 . 3 is not meant to imply that Ec (E) 2 will later be assumed to be to allow us to identify vectors b O(E) . Eb (E) and 2 It is only a convenience = (b 1,b2 ) and c = (c ,c ) , 1 2 which will be used in applying the results of section 3 . Equation 4 . 1 is not a specialization of equation 1.6 but is the result of an elementary change of independent variable . We will show how the existence of a solution to the complete system 4 . 1, 4 .2, 4 . 3 depends on the existence of certain "approximate solutions" to systems that have a differential equation similar to 4 . 1 and only one boundary condition . Associated with the equation Tx = o is another equation of a similar type which we propose to call an "adjoint equation" of 4 . 1 . In order to define this adjoint equation of 4 . 1, let Tx = Px + Gx where Px is linear in x and of x higber than the first order . Furthermore, suppose that for the (x, Ex ' / r(t,E) is the function occurring in equation 4 . 1. It follows by the P that !} , Qx = r( t, E) + 0( E+e- tj E) vector linearity of in its derivatives and Qx contains no derivatives uniformly, where 42 v T(x + w) = v (Px + Pw) + v Q(x + w) = v (Px + Qx) + v rw + v [Q(x + w) - Gx] = v (Px + Qx) + [B(v,w)] I + w P*v + v [Q(x + w) - Gx] , where B(v,w) is the bilinear concomitant of the formal adjoint to P. v and w and v. w (4.4) can be determined as a function Tx = 0 The differential equation adjoint to defined to be the equation in v w( t, v) of w = w(t , v) and t and with respect to P is given by T* v = w P*v + v [Q(x + w) - Qx] = o, X where is We set B(v,w) = 0 and assume that P* (4.5) Tx = Tx(t,E) = 0. For the considerations that follow, we choose Px = Ex" + x ' + q (t,E) x . It follows that up to an arbitrary constant factor, which we have taken equal to unity . The equation *X T v =0 is now determined uniquely by 4.5. the arbitrary constant factor in w, it can be shown that general depends only upon the coefficient of The significance of the operator equation: T* X x' in Except for * Tv= 0 X in Px = 0 . is indicated in the following 43 (4.6) Assumption 4i . q(t,E), r(t , E) € C(I) and q(t , E) , r(t,E) = 0(1) uniformly . Define T 2 = T2 (E) = lc 1 I + E lc2 l , T = T (€) = 1 + T1T (1 + T 2 0 3 3 lb1 I), and "" t U (t,E) = (,.,.u 1 ( t , E), ""u2 ( t , E)) = ( l,E+e - t/E ) . ( 4.8) Assumption 4ii . There exist functions ~ = ~(t,E) and ~ = ~(t,E) 2 both in c (I) such that for a given positive bounded function y(E), which satisfies we have T~ = 0( r) uniformly, ~ -v = O[E - 1r "" ~ ( t,E ) ] uniformly, (4 . 10) (4 . 11) X (4.12) 44 and ( 4 . 13) Suppose that ~Z = -( Z t,E) = -( X t,E ) + e -t / E -V ( t , E) and the vector ~ the domain which satisfy is defined as in 2 • 57 . d ( t , E) ( 4. 14 ) However , ass1..1Ille that is now defined to be all the vectors x(t,E) in C(I) lx1(t,E) -;(t, E)I ~ d (t , E) and l~(t,E) -E; ' (t, E)I < 1 for all t in and o < E < E0 I Choose (4 . 15) where o 0 is a positive constant, and suppose that \.I. > 00 (4. 16 ) • For the vector y t = (y ,y ), define a scalar function 1 2 1 f(t,y,E) = h(t,y ,E- y , E) . 1 2 Assumption 4iii . rfJ , o , and 1..1. f(t,y,E) f by satisfies Assumption 3iv only with as just def ined . It is not necessary to assume is part of assumption 3iv . f(t , o,E) = h(t, o,o , E) = o, which However, this is not an important point. If we now substitute X= W into + Z Tx = o, we get the new equation ( 4.17) 45 Tx = Pw + T~ + Q( w+~) in the unknown w. for "' = 0 Qz ( 4 . 18) Similarly, we may obtain two boundary conditions w from the boundary conditions 4.2 and 4 . 3 . Theorem 3.1 will be used to prove that this new problem in the unknown w has a solution, which has a bound proJ.0rtional to Theorem 4 . 1. Suppose that assumptions 4i T T e- 1/€ = o(1) 0 y ( €) • 1 to 4iii hold . If (4 . 19) ' ( 4 .20) (4 .21) ( 4 .22) ~ y = 0(1) , ( 4 .23) r T = o( 1) , 3 (4 .24) and then for € sufficiently small the problem 4 . 1, 4 .2 , 4 . 3 has a unique solution x and ) ( x- ,.,.z =0 ( 4 .25) where ~' T , and ~ 3 are defined in 4 . 13, 4 . 7, and 4 .8 respectively . € X ' -€~ I Proof . In the application of Theorem 3. 1, let y " ( .:. ) ' ( 4.26) 46 ) -1 A(t,e) = g(t,e) = ( :q(t,<) ( f(t,y,e) = ( :~(t,<) ( 4 .27) ' 1 ( 4.28) ) ' 0 - -1 y +z',e) - -z',e) h(t,y 1+z,e - h(t,z, 2 ) ' ( 4 .29) -B[~]' (4 . 30) .ere e) = £1 - C[~] • (4 . 31) £*(e) = .e 0 0 In the notation of section 3, we get here that and ft = -€ - 1 J_ q(s,e)exp(s;t)ds • 0 Thus, T.l. (i = 0,1,2) and u(t,e) with Ti(i = 0, 1, 2) and ~(t,e) of section 3. defined in 4 .7 and 4 . 8 are identified It follows that relations 4 . 19 through 4 .22 correspond to relations 3. 11, 3 .25, 3. 19, and 3. 18 respectively. We will show next that 4 .23 corresponds to 3. 24 by proving T = 0(T y) . 3 From 4 . 31, 4 . 12, and 4 . 9, we obtain 47 T (1+T0 Ib 1 l)£t ~ 0(YT ) + l(c 1 +c2 )~(1,E)­ 1 3 Ec2 ~'(1,E) e- 1/E T (1+T Ib I)= 0(rT ) + 1 1 0 3 0[E- 1e- 1/E T ~ (1+T Ib l )] = O(yT ) • 0 1 3 1 4 . 30 and 4 • 13 imply For g(t,E) as defined in 4 .28, we get llg(t,E) II < IT~ I < IT~I + e-t/E - - I~'; I , X and hence, from 4 . 10 and 4 . 11 II g( t' €) II = <1+€ _, e -2t/ €) o< r) . It follows that a constant multiple of in assumption 3iii. that 4iii From the definition of T = O(T y) . 3 There remains to show that assumption 3iv. y(E) One obtains f may be taken for g(E) T given by 3 .21, we see as defined by 4 .29 satisfies f(t , o, E) = 0 immediately. Assumption implies that when y and z are in [} as defined in this section. T(E) = O(yT ) and yT = o(l) by 4 .24, we have 3 3 hence, for sufficiently small E, ~ ~ o(t,E) ~ T(E) Since T(E) = o(l), and 48 The assumptions of Theorem 3. 1 hold for the present problem, and we conclude that there exists a unique vector y where x = ( €:. ) = ' is a solution to the problem 4 . 1, 4 .2, 4 . 3. follows from 3 .26 of Theorem 3 . 1 and T = O(yT ) . 3 Relation 4 .25 49 5. Theorem 4 .1 reduces the problem of existence and asymptotic expansion of a solution of the second order equation Tx = 0 with two boundary conditions to a problem of finding asymptotic expansions and v and ~ Tx = 0 of certain solutions of the second order equations v =0 each with one boundary condition . x We will determine X analytic expressions that have the properties of X and v by a method involving only regular perturbation procedures for weakly nonlinear first order equations with one boundary condition . Because there exists only one boundary condition to satisfy, one is at first tempted to treat the higher order terms Ex" and Ev" as part of the perturbations in the successive approximation schemes . This will require that the weakly nonlinear term, which will also occur in the perturbation term, be analytic in all its variables . all the derivatives need to be bounded uniformly for Furthermore, o<E<E . 0 Such cannot be the case for very general circumstances for the weakly nonlinear term in the equation factor T*v = 0, because of the presence of the X e - t/E . Hence, we shall proceed in a different manner . by an integrating factor for and t, setting Ee E t/E x '(o,E) = 0. Ex" + x ' and then integrate between zero This gives x' + J: t es/E h(s,x(s,E) , x'(s,E),E)ds = 0. 0 Multiply Tx = 0 ( 5. 1) 50 Integration by parts of the second term in equation 5. 1 and subsequent multiplication of the equation by E- 1 e - t/E produces the integro- differential equation x'+p(t,E)x+E- 1 J r(s,E)ex:p(s~t) t ds + 0 f t[h(s,x(s,E),x ' (s , E),E) - p(s,E)x •(s,E)]ex:p(s;t)ds = 0, ( 5 .2) 0 where (5 . 3) Equation 5.2 will be called the perturbation equation for Tx = 0 . We formally substitute CX) X= ~(t,E) = ~ Ek~(t,E) k=o into the perturbation equation 5 .2, and set successively (5 .4) k = o, 1, .•• ' where Q_ (s,E) = r(s,E), k = o, 1' ... ' 1 (5 . 5) and k ~= ~ i=o Ei X. ~ • ( 5.6) 51 The constants of integration appearing in the general solutions of the equations comprising 5.4 represent exactly one degree of freedom for x. vle will use this freedom to determine ;( t, €) at t =1 so that The solutions,if they exist, to the equations in 5.4 subject to the boundary conditions 5.7 are given by , ~(t,E) = ~ exp( !t p(s,E)ds) + k = o, 1, ••• ' ( 5 .8) where ( 5 . 9) k = 1, 2, •.• (5 . 10) T;n (t,E) = n+l ~(t ,E), n=o, 1, •••• (5 .1 1) Also, € Suppose that n is specified and that ; (t,E) a formal expansion Vn (t,E) = n exists . Then, 52 for v in equation 'l\'v v = 0 X n and boundary condition 4 .2 x n constructed in the same way as was devised. may be For brevity, we will write If X oo = lim rr->oo X n exists, tren will represent an asymptotic expansion. We will state conditions under and "' v exist for all n and assume the role of "' X and n n+l y = € in section 4 . These same conditions will imply which "' X n "' v for that and "' v X(X) exist . (X) A' Suppose that q..J on I is the set of continuous vectors such that jy,(t,€) - x0 ct, €) I < ~ ' _, IY2(t,€) - x0' (t,E)j <n+ € n: + 0' € 0 _, for given constants ~ > 0' Assumption 5i . in x and x' y(t,€) = (5 . 12) -; (t,€) and cr 0 , which satisfy >0 and tt>O . For 0 < € < €0 , h(t,x,x 1 ,€) ~, 0 cr(t,€) = ~, and of class C in t, when is of class c2 (x,x')T € J)' and t € I; 53 b(t,x (t,E),x •(t,E),E) = 0(1) 0 uniformly; and 0 db d2 b OX' OXOX' ~ = 0( 1 ) , """"""! = 0 ( 1 ) , and d2 b - - = 0(€) 2 1 $! = 0( 1 )+0( x - x ) , ox 0 1 ( x,x 1 ) f uniformly when € Q. I IV dX and t € I . Let ~ 1 min(~,n+a be a fixed positive constant not larger than 0 ). Two applications of the mean value theorem and use of assumption 5i give lb(t,x,x', E) - b(t,z,z ' ,€)1 ~ m1 (1x- zl + lx '-z ' 1), when lx- x0 I+ lx' - x0• l and lz-z0 I+ lz' - z0' l ~ m 1 1• Let assumptions 4i and 5i bold . then for sufficiently small on - I, n xn(t, E) = Z k 5 .8, a function ~(t,€) in lim~~ (t,E) = ~·( t,€) n->oo n ~ n ~(t,E) where 1• If € and n = o, 1, .•. , ~ (t,E) € ~(t,E) k=o and are not greater than is a constant whose value depends in general on Lermna 2 ( 5. 14) exists is defined by equation c2 (I) exists so that lim~ (t,E )= ~(t,€) n->oo n uniformly for all t in I, I~n (t,€) - xo (t,E)I+I~n• (t,€) -x•(t,€)1 o -< ~ 1 , (5 . 16) l~(t,€) -x0 (t,E)I+I~• (t,€) -x•(t,€)1 ~ ~1' 0 #V Tx (t,E) = 0(€ n n+1 ) uniformly for t in I, (5.18) and T~(t,€) = o. Proof . - For any function (5 . 19) Q(t,€) define Q by 54 Q =sup I jQ(t,E)I 1. ( 5 .20) tEI From equation 5 .8 and 5. 9 we obtain ( 5 .21 ) for some constant R. Hence, follows from equation 5. 4, and so Q0 -< p x'0 + h(t,x0 ,x'0 ,E) = 0(1) by equation 5 . 15 and h(t,x ,x' ,E) = 0(1) 0 0 (5 . 22) uniformly . Define Q = Q( E) = 1 + (l+p)(l+T 1Ec 1) sup exp( 1 2 o<t<s<1 We will show by induction that for Jt s p(r,E)dr) . E sufficiently small ( 5 .23) and n = 0,1, . • • ( 5.24) and that equation 5 . 16 holds . Suppose that equations 5 . 16 and 5 .24 hold for n < k Then, and I~K+1 - x0 l + l~'K+1 - x0' I -< k+1 E i=l Ei(lx.l +lx!~ l) -< ~ and k > 0. 55 if € is fixed small enough to make 0 2EQ max ( ro1+p, l-! - 1Q ) ~ 1, 1 € can be so chosen because and h(t,x ,x~,E) 0 0 = 0(1) 0 EQ and uniformly . o < E< E 0 EQQ 0(€) by equation 5.15 are 0 ~+ (t,E) 1 We conclude that is defined, and n = k + 1 follows now from equation 5. 14 . Equation 5.24 for This completes the induction proof . T -x n -- O(En+ 1 ) un1·fo rmly f or Q = 0( 1) uniformly . n equation 5 . 11 and t for 1 +co I f o llows f rom We obtain Because 2 € Q(ro +p) ~ 1 from equation 5.25. · 1n o < € < € 0 , the infinite series t in I and o < € < € 0 there exists a function to a sum ~(t,E) lim ~n (t,E) = ~(t,E) n->co .... xn' , x, n and € n+1 we conclude that to a c 1 (I) in and From equation 5 . 11, we see that l ess than such that lim ~' ( t, € ) n- >co n €X n II lim En+ 1 ~ = 0 n n->co Because Ex" converges uniformly for function which must be = ~ ' ( t, € ) ~~~ ( t, €) • uniformly . is a linear combination of Qn . n It follows that l-! • 1 Hence , uniformly for t in I and o < € < € o lim T~ ( t, e ) = T lim~ (t,e) n--> oo n n->oo n The proof of the lemma is now complete . For n fixed, n = o, 1 , ... , oo, we will consider next ~n (t, e) . The convention ~oo (t,e) = ~(t,e) vle vlri te T* X ~s used i n what follows . in the form n ev" - v ' + q*(t,e)v + eh*(t,v,v ' ,e) , where q*(t,e) = q(t,e) - ad~ (t,x0 (t,e), x01 (t,e),e), X h*(t, v, v•,e) = e t/e ~ [h(t,x+ve n - t/e ~ ,x'n + ~ ,e )] + v ' e - t/e - € - 1ve - t/e ,e ) - h ( t , ~xn ,x' n Set k = 0, 1, • . • , n, and k = 1, • . • , n, where ( 5 .28) 57 1 J q*(s,€)exp(t- s) ds, t € ~1(t,€) = o, ~(t,€) = p*(t, €) vk(t,€)+€ -k[h*(t,~nk'~rik,€) - h*( t, ~ k 1 , ~ k 1 , €) ], n, n, - k = 0, 1, . • • , n . Here k v k i ~ € v . = ~ nk = i=o n1. i=o € i v., k = 0, . • . , n; and l. "' v = o. n, - 1 This will produce ~ vn = € n+1 * Q"'n X n The solution of the system of differentia l equations 5 .27 subject to the boundary conditions € _ 1d_rt [exp( k J t p*(r,€)dr)] ! 1~- (r , €)exp(s;r)dr ds , s 0 = 0, 1, 5 .28 is given by s 1 ( 5 . 30) . • . , n, where k = 1, ( 5. 31) ••• , n . Because of assumption 5i we obtain upon two applications of the mean value theorem the existence of a constant so that lh*(t,v, v ' , E) - h*(t,w,w ' , E) I < m2 [ lv '-w' I (5 . 32) when v, we c1 (I) v, w satisfy and I"'xn - xo +xe - t/e I 1;'-x n o' + ( Ex '-x ) e - 1e - t/el ~ n + € - 1 o "' ( 0 ~ t,€ ) (5 . 33) rxn - xo 1 and 1;'-x n o' 1 may be lJBde arbitrarily small uniformly for x. in n by choosing ~, 5 . 33 hold for all e of Lemma 5. 1 small . and Hence , the inequalities and w are _, - t/€ ~ n -~ + e - 1o "'~Iex'-x I € e 1 0 ( 5. 34) ~, sufficiently small, when v c 1(I) and satisfy for x, Asswnption 5ii . any positive number Let v (t,e) 0 be defined by equation 5 . 30 . a, there exists a number € 0 (a) 0 when t e I and o < e < e 0 0 For such that ], (a) . Define ( 5. 35) Lemma 5 .2 . 5 . 15 hold . Suppose t hat assumptions 4i, 5i, and 5ii and equation If (5 . 36) 59 then for sufficiently small E, ~ (t,E) exists on I, ~ (t,E) n n n k = ~ E vk(t,E) where vk(t,E) depends upon n and is given by 5.30, k=o and T* ~ (t,E) = 0[E~ (t,E)] 2 "' n X uniformly . (5 . 37) n Proof . One can show from the definition of h* 5. 15 and 5 . 36, which imply that v0 (t,E) and and from equations v0' (t,E) are 0(1) uniformly, that by using the mean value theorem twice and assunption 51 . the definition of Q* 0 that constant independent of Q* < Q E0- E and 1 It follows from uniformly, where "' Q is a u_ c n. Define ~ = sup o<s<t<1 exp J t p*(r,E )dr s and The bar in p* and in what follows has the meaning attached to it in equation 5.20 . Q* = 0(1), because T = 0(1) by assumption. 4 We will sketch the induction proof that for and k = o, E sufficiently small .. . , n l~nk(t,E)e-t/EI ~ 1! - 1-1 1 , IE~:UC(t,E) - "'vnk(t,E) Ie - t/E ~ (n~ )E + o "'~(t,E), 1 0 ~~(t,E)j ~ (ru2+p*)k Q*k Q €- l ~(t,E) . and (5 . 38) 60 The existence of ~ ( t,€ ) - =v v n nn follows from equation 5. 38 . = 0 ( Q*n-Q € -1 ~~ ) = 0 ( € - 1~~ ) uniformly for t Furthermore, in I and equation 5.29 imply equation 5.37 of the lemma . The first step in getting from i ~ k to i ~ k + 1 (k ~ 0) in the induction consists of using equation 5.30 and the induction hypothesis to obtain l ~(u,€ )1 ds ~ i = o, ... ' k . ( 5.39) Thus, by assumption 5ii - ' k lvn,k+ll ~ lv0 l + .z: ~=o € i+l lvi+ 1 1 < lv0 l + (5 . 40) for sufficiently small in equation 5. 39 for satisfied by vi+ 1 € v. ~+ 1 independent of k and n. The bound obtained and the first order differential equation give lv~+ 1 I < P*lv. 1 I + sup IQt(s,€)1 ~ ... ~+ t<s<1 ... i = o, 1, ... ' k . (5.41) 61 It follows by assumption 5ii that I EV,.,n,k+1 1 lv 0 - ....V n,k+1 I < < + I + O(E) < o0 , for sufficiently small E independent of n and k. ~+ (t,E) 1 Hence, exists, and + E- J t o f 1 ~.)t,E)(T4 f l ~(s,E) 1 I e - s/E ds + 0 sup IG{(r , E) jds)] ~ (m2 +P*)( 1+ s<r<1 t sup IG{(s,E) je- s/E ds)] • s<r<l o The complete induction statement for k + 1 The lemma is essentially proven if n follows . is finite . For n = oo the additional problem of considering the convergence of ; 00 (t,E) gives no difficulty, because equations 5 . 40 and 5 . 41 are then valid for +oo Ei v ! (t,E) ~ i=o k -> oo. = ~00 (t,E) are uniformly convergent in ~ i=o Hence, equation 5 .29 gives that as = 0,1, •. • +oo Ei v.(t,E) and Both i E ';" (t €) ook ' converges uniformly in The end result is that there exists a function ~(t,E) in c2 (I) such that *"" =O . Tvv X 00 I. I 62 Theorem~ · Suppose that assumptions 4i, 51, 5ii 4 . 19 to 4 .22, 5.15, 5.36 and i n (t , E) Choose and~ (t,E) 1 if Let n n = 0, 1, ..• ,+oo, be specified, be given by Lemmas 5 . 1 and 5 .2 respectively . n y(E) = En+ infinite . hold . and equations n 1 is finite and y(E) = e - /E if n is If (5 . 42) t hen for x = x(t, E) E sufficiently small a unique solution exists for equation 4 . 1 and boundary conditions 4 .2 and 4 . 3, and x(t,E) = ~ (t,E)+ e - t/E ~ (t,E) + O(rT ) n n 3 uniformly for X 1 t in I, ( t,E ) = -X 1 ( t , E) + e - t/E -V 1 ( t,E ) - E- 1e - t/E -V ( t,E ) n n n + O[E - 1 rT ~(t,E)] 3 uniformly for t in I. (5 . 44) Proof . This theorem is an immediate consequence of Lemmas 5 . 1 and 5. 2 and Theorem 4 . 1. The choice for makes equation 4 .24 a consequence and 5 . 15 and the definition of y (E) satisfies equation 4 . 9 and of equation 4 . 20 . Using equations 4 .20 T , we get that equation 5 . 42 implies 2 the validity of equation 4 .23 in the present case. Assumption 4ii holds here because of the conclusions of Lemmas 5 . 1 and 5 .2 and the choice for y. Assumption 4iii holds because of assumption 5i . by equations 5. 32 and 5 . 34 . This is shown 6. The results of the previous section lvill be reformulated in terms of the singular perturbation problem with differential equation ey" + F(t, y,y•,e) = 0 (6 . 1 ) and boundary conditions C[y] B[y] = 1 ' 0 (6 .2) vle will state sufficient conditions to guarantee the existence of a solution y of the form y = W+ X s(E) where the existence of ~ = ~(e) x is given by the theory of section 5, and is a positive function of ~ e such that = 0( €) . (6 . 3) ( t , y( t, E) ,y ' ( t,e) ,e) In what follows let the full argument and its derivatives be denoted by [y] = [y(t,e)]. Assumption 6i . There exists a function such that w(t, e) e c2 (I) Assumption 6ii . IY' - w{t,e)l~ "'1-l = O(e ~- 1 ) oF and of F w = w(t,e) of ew" + F[w] = 0(~) t and e uniformly . and y ' when ly-w(t,e)l + 2 2 o F o F -1 o F for some constant 1-l · E and -2 ' ayay J ' ' -ay •2 oy F uniformly for is of class c2 ,.., in y 2 1-l and t ly-w( t , E) I+ IY' - •1{ t , e) I < "' I·J in [ oF [w] E C1 (I) ; cyr oF (w), Oy oF ( w), and ( J;~ w] ) ' 7fY' oF = 0( 1) uniformly; and cyr [ w] ~ 1l > 0 when t E I and o < E < e , Oy hr] E C(I) and 0 for some constant 1'1 · 64 The functions in equations 4 . 1, 4 .2, and 4.3 become in the notation of the present problem the following: q <t, E ) = <a;~ [ n- ~ "'1, 2 w r(t,e) =<a;~ [w]) - [ 2 (ew" + F[w]) E- 1, [ <a;~ [w])' E sX' + 1 2 1 2 2 h(t,x,x',E) = (e / E / a;~ [w]) - oF oF F[w+Ex] - F[w] - dYf [w] ;x• - dy [w] Ex } , .£0 = 0 0 .£1 = 01 -1 - B[ w]) E ' C[w]) E- 1 . ( 6 . 4) In the context of the present problem, functions T.~ (i = o, q*, p*, x0 , v0 , and ... , 4 ) can be defined from the equationsof 6 . 4 in the same way as found in sections 4 and 5 . Assumption 6iii . For £0 and w, equations 5 . 15 and 5 . 36 hold. Assumption 6iv. a0 < -~ E s- 1 when .£ defined in 6 .4 and depending on 1 Also, equations 4. 19 to 4 .22 hold . There exists a constant t EI and 0 a 0 such that !v(t,e)l < 0 < E < E0 . Assumption 6i v is a requirement that ll is not too small . From this a another number e0 (a) condition we get for any positive number that lv0 (t,e)l + ae < a 0 ( 6 . 5) and t/E e ' (6 .6 ) such when t E I o < E < E (a) . and 0 Equation 6 . 5 is part of assumption 5ii, which must be verified to hold for the present problem . Assumption 5i holds for h defined in 6 . 4, if ~ ~- lxl + lx 1 I < 1 · Thus, assumption 5i holds for lx- x0 I < ~ if n , and ~, a are determined so that 0 ~ when t E I and ~ and then fix + lx0 I + n + E - 1a o < E < E > a0 and 0 Let • n > 0. 6 . 6 when a > ~ + Ix0 I + n + for t in I and a o < E < E • 0 0 a 0 be determined by assumption 6iv, Inequality 6 . 7 will result f r om inequality + Ix 1 I, 0 becattse x 0 and x1 0 are bounded This follows from equation 5 . 15, which is assumed to hold for the present problem . Assumptions 5i and 5ii have been shown to be valid . i-le conclude that all the assumptions of Theorem 5 . 1 are true here . Hence, t he following theorem has been proven . Theorem 6 . 1. Suppose that assumptions 6i to 6iv hold, and let be given (n = o, 1, .•. , oo) . If y =e - 1/E . n is infinite, define n · ~s Let f ~n~ · · t e, d e f ~ne · n 1 y = En+ , and i f "'z n (t,e )= --n x (t,E )+e -t/ € "' vn ( t,e ) be given by Theorem 5. 1 wher e the fUnctions of the equations 4 . 1, 4 .2, 4.3 66 are given in 6 .4 . Then, if (1 + T 0 lb 1 I)( lc 1 I + T T Ib 1 j) ~ T Y = 0( 1) , 2 0 2 the problem consisting of e~uation 6 . 1 and boundary conditions 6 .2 has a solution y(t,E) for t in I and € sufficiently small, and y(t,E) = w(t,E) + ~(€) ~n(t , E) + 0(~ y T ) 3 uniformly, 1 y•(t,E) = w•(t,E) + ~(€) ~~(t,E) + 0(E- srT ~(t,E) ] 3 uniformly . REFERENCES [ 1] W. Wasow, On the asymptotic solution of boundary value problems containing a parameter. J . Math. Physics 23 (1944), 175- 183. [2] J. J. Levin and N. Levinson, Singular perturbations of nonlinear systems of differential equations and an associated boundary layer equation. J. Rtnl. Mech . and Anlys. 3 (1954), 247-270 . [ 3] A. Erdelyi, Singular Volterra integral equations and their use in asymptotic expansions . MRC Technical S'LITllllB.ry Report No. 194 . Mathematics Research Center, U.S . Army, U. of Wisconsin, Madison, 1-lis . ( 1960) . 87 pp . (4] A. Erdelyi, An expansion procedure for singular perturbations. Accademia delle Scienze de Torino [ 5] 95 (1960-61), 1- 22 . A. Erdelyi, An example in singular perturbations . Monatshefte f. Math. 66 ( 1962), 123-128. [6] A. Erdelyi{ A result on non- linear Volterra integral equations . (To appear J P . Lagerstrom et al . , An example of singular perturbations. (Unpublished notes-from the seminar in applied mathematics Spring 1962, California Institute of Technology, Pasadena, Calif . ~ [8] Wasow, Singular perturbations of boundary value problems for nonlinear differential equations of the second order, Comm. Pure Appl. Math. 9 (1956), 93- 113 . \of . A. Erdelyi, On a nonlinear boundary value problem involving a small parameter . (To appear) . [ 10] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1953 . [ 11 ] T . M. Apostol, Mathematical Analysis . Reading, Mass .: Addison- Wesley, 1957 .