A GENERALIZED UPPER . AND LOWER SOLUTIONS METHOD FOR NONLINEAR SECOND ORDER . ORDINARY DIFFEINTIAL EQUATIONSx

The purpose of ths paper s to study a nonlinear boundary value problem of second order when the nonlinearity s a Carathodory function. It s shown that a generalzed upper and lower solutions method is vald, and the monotone teratve technique for finding the mnmal and maximal solutions s developed.


I. INTRODUCTION
We shall, in this paper, develop the method of upper and lower solutions  We first note that the classical arguments of [2] for f continuous are no longer valid since if u is a solution of (P), then u" needs not to be continuous but only u" (5 Ll(0, Tr). Here we extend classical and well-known results when f is continuous (see [2]) to the case when f is a Carathodory function.
If we choose a o=a1=c 0=cl=0, then the boundary u'(0) = u'(Tr) = 0. Thus, we have the Neumann boundary value problem conditions read ," = f(t, ,), ,'(o) = ,,'(-) = o. (N) We shall consider in Sections 2 and 3 this simpler boundary value problem so as to clearly bring out the ideas involved. On the other hand, there is no additional complication in studying (P) instead of (N). We list the corresponding results for (P) in Section 4.
Finally, in Section 5 and following the ideas developed in previous sections, we present the method of upper and lower solutions for the boundary value problem (P) when a0, a 1 > 0 and b o, b >_ 0. In particular, we do so for the Dirichlet problem Let us assume that f:IxRR is a Carathodory function, that is, f(.,u) is measurable for every u (5 R and f(t, is continuous for a.e. t (5 I. Moreover, we suppose that for every/ > 0 there exists a function hh R (5 L(I) with If(t,u) <_ h(t) for a.e. (5 I and every u --< R. Let E = {u (5 w2'l(I) u'(0)= u'(r)= 0} with the norm of w2'l(/)and F = LI(I) with the usual one. We shall denote by I[" I! E and I1" [I the norms in E and F, respectively. By a solution of (N) we mean a function u (5 E satisfying the equation for a.e. t (5 I. Now, suppose that a, fl (5 W2'1(I) are such that a(t) < fl(t), t (5 I. Then, relative to (N) we shall consider the following modified problem ,,"(t) = a(t,,,(t)) u(t) + p(t, (t)),,,'(o) = ,,'() = o, where (t) fo< g(t, u) = f(t, p(t, u)) and p(t, u) = u for a(t) _< u < (t) (t) for u > (t). We note that g is a Carathodory function and that the Neumann problem (N) is
(2.6) and Similarly, E W2'l(I) is an upper solution for (N)if '(t) > f(t, (t)) for a.e. t I '(0) _< 0 _</'(Tr). (2.8) We are now in a position to prove the following result which shows that the method of upper and lower solutions is still valid when f is a Carathodory function.
Theorem 2.1: Suppose that a, 1 e W2'1(I) are lower and upper solutions for (N), respectively, such that a(t) < fl(t) for every I. Then there exists at least one solution u of (N) such that a(t) < u(t) < (t) for every I.

Proof:
We first note that any solution u of (N) such that a _< u _</ is also a solution of (2.2). On the other hand, any solution u of (2.2) with c < u _</ is a solution of (N). We shall show that any solution u of (2.2) is such that a < u _< fl on I and that (2.2) has at least one solution. Now, let u be a solution of (2.2). We first show that c(t) _< u(t), for every t E I. If a(t) > u(t) for every t I, then -u"(t)=f(t,(t))-u(t)+(t) for a.e. t I. Thus we obtain the following contradiction 0 0 0 Thus, there exists t 1 E I with a(tt)_< u(tx). Now, suppose that there exists t'E I such that a(t') > u(t'). Set to = a-u and let t o E I, tO(to) = maz{to(t): I). We first suppose that; t o (0, r) and t o < t (the case t o > t t is similar Now, if t o = 0, then '(0)_< 0 and we get that 9'(0)= cd(0)>_ 0 and '(0)= 0. As before, there exists t 2 > 0 such that (t2)= 0 and (t)> 0 for every t [0,t2) and ' is increasing on [0, t2) which contradicts that (t2) = 0. The case t o = 7r is analogous.
This shows that c(t) < u(t) for every t q I and by the same reasoning we obtain that u(t) < (t) for every t I. We next prove that (2.2) has at least one solution.
In order to apply the well-known theorem of Leray-Schauder, define the operators L: E--.F and N: F---.r by Lu = u" + u and Nu = g(., u(. )) + p(-, u(. )) respectively. Note that L is continuous, one-to-one, and onto. where H = i-L-: F--F and i: EF is the canonical injection. H is continuous and compact since w2'l(I) is compactly imbedded into LI(I). Let 7 = min{cr(t):t I} and di = maz{(t):t e I}. if u is a solution or (2.2), then lu(t) < R = maz{7,5} for every t e I. Taking into account this, condition (2.1), and that a(t) < p(t, u(t)) < 3(t) for every I, we have I I ANu I I II h I! + 2 = C.
In consequence, if u is a solution of (2.9) we have that !! u I I E C. I I H II, where C is a constant independent of X E [0, 1] and u E F. Thus, we have proved that all the solutions of (2.11) are bounded independent of A [0, 1] and we can conclude that (2.11) with A = 1, that is (2.2), is solvable. This concludes the proof of the theorem. For M E F with M(t)> 0 for a.e. I and r/6 F we shall consider the following Neumann boundary value problem The operator L (defined in the proof of Theorem 2.1) is continuous, one-to-one and onto. Thus, by the open mapping theorem, its inverse L-1 is continuous. For cr (5 F, let L-lr = u be the unique solution of the linear problem u" + Mu = or, u'(O) = 0 = u'(Tr).

Tile MONOTONE
If a, are lower and upper solutions for (N) respectively, let us introduce the following condition in order to develop the monotone method: There exists M ( F with M(t) >_ 0 for a.e. t ( I and we have that for a.e. t e I and for every u, v e R such that a(t) < u < v <_ o"(t) = u"(t) >_ f(t,(t))-M(t)(t) + M(t)u(t) + f(t, fl(t)) >

M(t)[13(t)r/(t)]-M(t)o(t) + M(t)u(t) = M(t)o(t).
By Lemma 3.1 we can conclude that a(t) _< 0 for every ( I, that is, u < fl on I. The proof that c < u is similar. To show that validity of (3.5), let = Kr h -Kr/2. ( I and, in consequence, we obtain that Kr h _< Kr/2 on I. Suppose that and fl are lower and upper solutions, respectively, of (N) such that < 13 on I and (3.3) holds. Then, there exists monotone sequences {an} and {/3n} with o = c, o =t3, a n < flm for every n, m N and tim a_ = r, ldrnt3 n = p uniformly on I. Here, r and p are respectively the minimal and maximal solutions of (N) between and t3 in the sense that if u is a solution with a <_ u < fl on I, then r < u <_ p on I.
Taking into account property (3.5) we see that a 1 = Ka 0 < Ka x = a 2 and, by induction, that a n<a n+ for every nEN. Similarly, defining o = and Dn =KDn-x we have that /n + 1 < /n, n N. Combining properties (3.4) and (3.5) we see that a < a n _< Dm -< fl for every n, m N. Therefore, the sequence {an} is uniformly bounded and increasing and it has a pointwise limit, say r(t), t I. We now prove that r is a solution of (N). Choose R > 0 such that lan(t) <_ R for every n NI, t E I. The sequence {a} is bounded in r since -a(t) = M(t)an(t + f(t, an_ l(t))+ M(t)a n l(t) 3) and (2.4), that is, r is a solution of (N).
Using the same integral representation for the solutions of (N) we get that {fin} converges uniformly to a solution p of (N) and it is obvious that a < r _< p < .
Finally, if u is a solution of (N) with a_< u <_ / on I, then a _< Ku = u <_ fl. By induction we get that a n <_ u <_ fin for every n Ni which implies that r < u _< p and concludes the proof of the theorem. and an upper solution of (P) if the reversed inequalities hold in (4.1) and (4.2).
If we know the existence of upper and lower solutions for (P), then we can guarantee the existence of a solution for (P). Theorem 4.1: Assume that a and fl respectively, such that a(t) < 3(t) for every I. the problem (P) such that a(t) <_ u(t) <_ 13(t), t I. are lower and upper solutions of (P) Then there exists at least one solution u for In order to develop the monotone method, we need the following result which is analogous to Lemma 3.2. Proof: If (t) > 0 for every E I, then "(t) > 0 for a.e. t I and 9' is strictly increasing on I and '(0) < '(r). However, B(0) < 0 and B(a') < 0 implies that '(0) > 0 and '(r)<_ 0 which is a contradiction. Now, reasoning as in the proof of Lemma 3.2 we see that there is no t q I with (t) > 0. This allows us to show the validity of the monotone iterative technique for the boundary value problem (P). the minimal and maximal solutions respectively, of (P) between cr and 13.
Here, r and p are Proof: For a _< q _< fl, we solve the boundary value problem