RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS FOR FIRST ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper we study the convergence of the approximate solutions for the following first order problem u’(t) f(t,u(t));t [O,T],au(O)-bu(to) =c,a,b > 0, a +b > 0,t0 (0,T]. Here fI R tt is such that exists and is a continuous function for some k > 1. Under some additional conditions on -, we prove that it is possible to construct two sequences of approximate solutions converging to a solution with rate ofconvergence of order k.


INTRODUCTION
The method of upper and lower solutions is a well-known theoretical procedure to prove the existence of a solution for a given nonlinear problem.Under additional conditions, it is possible to apply the monotone iterative technique that provides a constructive scheme for the solutions.Moreover one can use the monotone iterates to give error bounds.For practical purpose it would be interesting to know the order of convergence ofthose monotone sequences of approximate solutions.
We recall that for a given Barmch space (E, II), and a convergent sequence {x,} z in E, it is said that the order of convergence is k 1, 2, if there exist g > 0 and no N such that IIx./a roll -< Xllz xll vn _> no, When k 1 (k 2) we say that the convergence is linear (quadratic).
It is not difficult to see [1], [2] that the convergence fthe sequence of the approximate solutions given by the monotone iterative technique is very slow.Indeed, that convergence is linear but in general, not quadratic.Under some convexity conditions, the method of quasilinearization [3], [4], [5] provides a monotone increasing sequence converging uniformly and quadratically to the solution.
It would be important to have some general methods leading to monotone sequences converging to a solution with order of convergence k > 2.
To be specific, let us consider the following boundary value problem u'(t)=f(t,u(t)), au(O)-bu(to)=c, tI=[0,T], T>0, (1.1) where f I x R ]R is a continuous function, to E (0, T], and a, b >_ 0, with a + b > 0 As usual, we say that a E C (I) is a lower solution for the problem (1.1) if Similarly,/ C (1) is an upper solution for the problem (I.I) if If t0 T, it is proved in [6] that a _</ on I implies that there exists at least one solution u of (I I), with u e [c,/] (v E C(1); a(t) _< v() _</(t), t 6 I) Moreover, if there exists MI > 0 such that a be -M1t > 0 and f(t, u) 4-M1u is nondecreasing in u [a(t),/(t)], I, ( then it is possible to construct two monotone sequences {a,) and {/), which start in a and / respectively, and converge uniformly to the extreme solutions and of (I.I) on [a,/].We insist that, in general, the order of convergence of the monotone sequences is at most k I in the space E C (1) with the usual uniform norm.
Now we obtain an extension of this result to the problem (I.I) as follows.First, we prove that for some m > 0(a, b _> 0, a + b > 0, a be -' > 0) if u' + mu >_ 0 in I and au(O) bu(to) >_ 0 then u >__ 0 in I. To prove this, we use Lemma 2.3 in [6] and we obtain that u _> 0 in [0, t0].Thus, u(to) _> 0 which implies, using again Lemma 2.3 in [6], that u _> 0 in It0, T] and, in consequence, u _> 0 in I. Now, we define a0 a, =/, and for n >_ 1, a, and/, are given as the unique solution of the following linear problem: u' + MlU f(t, rl) + Mrl, au(O) bu(to) c (1.3) with r] a,_ and r] =/5,,_ respectively.Condition (1.2) implies that a <_ cq s c,, _< _< / s/5 and the two sequences converge uniformly to the extremal solutions of (1.1) On the other hand, note that if exists and it is continuous in n {(t, u) e x .; ,(t) < u </(t)}, then condition on f in (1.2) is equivalent to the following requirement: O..f (t, u) > M1 (t, u) 6 f.
(1.4)Recently [7] the method of quasilinearization was generalized for the initial :alue problem (b 0) by not demanding f(t, u) to be convex in u for t E I but imposing the following less restrictive condition there exists M2 > 0 such that f(t, u) + M2u 2 is convex in u for any t 6 I.
(1.5)If exists and it is continuous for every (t, u) 6 f, then (1.5) holds and it is equivalent to O2f(t,u)> -2M2, (t,u) 6f2 (16) Ou2 If this last condition is satisfied, then there exists a nondecreasing sequence starting at the lower solution and converging uniformly and quadratically to the unique solution of the initial value problem [7].
These results are extended in Ill, where two monotone sequences of approximate solutions are constructed, one nondecreasing starting at the lower solution and the other one nonincreasing starting at the upper solution, that converge uniformly to the unique solution of the initial value problem and the order of convergence is k provided that there exists ---, and it is continuous in f Note that in this case there exists Mk > 0 such that i)k f (t, u) > (k!)Mk (t, u) f. ( t:3u k The periodic boundary value problem (u(0)= u(T)) is considered in [2].There the authors construct a sequence {c,} which converges quadratically to a solution u [a,/3] of the periodic problem.In this case, they suppose that f satisfies (1.6) and a(T) < 6 < 0, where (,())d, In this paper we study problem (1.1) and we construct two monotone sequences which converge to the extremal solutions in [a,/] of (1.1) provided that there exists and it is a continuous function in and that for each [a, ] a-be a(t) > 6 > 0. (1.9) The following result from [8] is the basic tool to prove our main result.THEOREM 1.1.If there exist a _</3 lower and upper solutions respectively for the problem (1.1), then there exists a solution u [a,/] of (1.1).
We finally note that we generalize previous known results. 2. MAIN RESULT Now, we obtain, in the following result, that if there exists a continuous function in f and condition (1.9) is verified, then it is possible to construct two sequences which converge to the extremal solutions and of (1.1) rapidly, that is, the order of convergence is k.THEOREM 2.1.Suppose that there exist a _</3 lower and upper solutions respectively for the problem (1.1).
If there exists k _> 1 such that is continuous in f2, and if condition (1.9) is verified, then there exist two monotone sequences {a.} and {/,,} with a0 a and & =/3, which converge uniformly to the extremal solutions b and of (1.1) in [a,/3].This convergence is of order k.
PROOF.We first note that problem (1.1) has, by Theorem I. 1, a solution in [a, ].Let us denote 7 such a solution.
To construct the sequence {a,}, let t I and a(t) <_ v <_ u <_/(t).We first note that for a given tEI: where X(t) E Iv, u].Now, since exists and is continuous in f, (1.7) is verified.
We conclude, using again Theorem 1.1, that problem u'(t) g(t,u(t),an(t)), I, au(O) bu(to) c (2.5) has a solution an+x Jan, 7].The so obtained sequence {an} is nondeereasing and bounded in C '1 (I), whence it converges in C(I) to some continuous function [a, 7].
That is, p [a, 7] is a solution of (1.1).Since 7 is an arbitrary solution of (1.1) it is clear that is the minimal solution of (1.1) in [a,/], which exists by (1.9).Now, we prove that the convergence is order k.For it, using (2.1), we have that '(t) f(t, ( Furthermore.a(t) < wn(t), for all n N and e I.
B' (A B) '-A'-1-B we can write that Finally, using that for all A, B 6 R, 2=0 w'+(t) p(t)wn+,(t) <_ where Ck N + M > 0 In consequence, it is verified tt where a.(t) f p.(a)da.Thus, w.+,(t) e a"(> wn+(O)+Ok e-"('>w(a)da ( Now, using expression (2.7) for to d the ui aw.+(O) bw+(to), since a,b 0, we conclude that Now, due to the Nct that , w 0 n , d using te eresin f , endition (1 9) implies that there ests N ch that e(tl > /2 > 0 for 1 n no.Thus, o where A is a positive conat.Note that the prous inequities hold since there ests in fl d they e eominuous nions for 1, k.Thus, since a [a,], we have m ere ests a constt D > 0 such tt I(t)l 5 D for 1 n .
To coronet the sequence {} we define the follog nion: A. CABADA, J. J. NIETO AND S. I-IEIKKILA Here Mk and N are nonnegative constants given by (1.7) and (2.6) respectively Thus, it is easy to see that h(t, u, v) > f(t, u), for all E I and a(t) < u < v < (t).
In consequence, 7 < ,, < _1 < =/ for n > 1, and {/} converges uniformly to , where is the maximal solution in [a, ] of (1.1).Now, the definition of h, expression (2.1) and inequalities (1 7) and (2.6) imply that the convergence of { , } to is of order k.F'I REMARK 2.1.Condition (1.9) may seem very restrictive but, as we will see in the following exhrnple, in some cases it is a fundamental condition.Let us consider the problem u'(t) f(u(t)), E I, u(0) u(to), with f defined by f(u) u if u < 0 and f(u) 0 otherwise.Note that a 1/2 and/ 0 are lower and upper solutions respectively for this problem.
Analogously to the example given in [2] we show that the sequences obtained via the monotone method converge linearly but not quadratically to the unique solution u 0. If we use the function g (for k 2) as in Theorem 2.1 (see formula (2.2)) we obtain that ctn+l (2 V/')an.Clearly, there exists a constant ) > 0 such that II+all <_ ,1111 if and only if c,,+ _< (x/ 2)/).This last inequality does not hold.
For it we say that cr is a lower solution for (3.1) if d(t) < f(t,a(t)), t I, aa(t )-ba(T) < c.
Analogously, we define an upper solution by reversing the previous inequalities.This case can be reduced, by a simple change of variable, to that considered in preceding sections, as we will see in the following result.

THEOREM 3 . 1 .
If there exist tr and/ lower and upper solutions respectively of (3.1) on I, with < a, there exists a continuous function in { (t, x); E I, (t) < x < or(t) } and f satisfies b-ae (tt) > 6 > 0, (3.2) with 0 defined as