NONRESONANCE CONDITIONS FOR FOURTH ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

This paper is devoted to the study of the problem U(4) f(t, u, u’, u", u’"), ,(0) ,,(2), ,,’(0) ,’(2), u"(o) u"(), u"’(o) ,’"(2). We assume that f can be written under the form /(t, u, u’, u", u’") f2(t, u, u’, u", u’")u" + fl (t, U, U’, U", U"t)U’ +fo(t, U, U’, U", U"’)U + r(t, u, u’, u", u’") where r is a bounded fltnction. We obtain existence conditions related to uniqueness conditions for the solution of the linear problem U(4) au + b/zH, u(O) ,(), ,’(0) u’(9.), u"(O) ,"(2), u"’(O) ,,’"(2).

(1.4) Y. Yang proves the existence of a solution under the assumption that If(t, u, v)l < alul + blvl + c, with a+b<l.\Vit l lc' st.' lndary colliis, Y. Ymg 23 l alo estallild tlm cxisWwe ff a solution fear tle u (') g(l, u, u', u", u"')uh (l, u, ', u",   msmnig that, flr U R4, h(t, l]) is bounded and therc exists k E N sucl that k <infg(l U) < supg(,U) < (k+ 1) a.e.on[0,], wlmre the notation means that tlm inequality is strict on a subset of positive measure.This lmt condition can be clearly read a non-interference condition of tlm nonlinearity with respect to the spectrum of the operator u u(), subject to the boundary conditions (1.4).Recently, M.   Del Pino and R. Manevich [5] have extcnded Y. Yang's [22] result.They prow, the existen of a solution for (1.3), (1.4), suming that If(, , ,) ( + Z)l alul + blvl + c, the numbers a, fle R and a, b, c R+ bein such that k4 k l for allkeN*, The aim of this paper is to provide analogous results for the periodic boundary vMue problem (1.1), (1.2).Those results will however be obtained by a method of prf different from that of the above-mentioned papers.
LEMMA 1.The problem (1.6), (1.7) has a nontrivial solution if d only if there exists k e N such that k 4 a-bk.
For k e N, we will call the set L {(a, b) e R 2 k a-bk2} an eigenline of (1.6), (1.7).
The condition (1.5) then means that the point (a, 3) does not lie on y eigenline.
We will treat problem (1.1), (1.2) by separating two ces, depending on the form of f.In the first ce, discussed in section 3, we will sume that f c be written under the form f(, , ', ", '") ((, , ')u')' + h(, , ', ", '") + r(, , ', ", '").(.8) where r(t, u, u , u', um) is a bounded function.We will be able to prove the existence of a solu- tion under the assumption that the point (g(t, x, y), h(t,x, y, z, w)) always lies in a rectangle with sides parallels to the axes and which does not intersect y eigenline.This kind of result c be considered as an extension of the results of Y. Yang [2] (with obvious modifications, sin the boundary conditions e different).It is worthwhile to adapt that result in the ce where the point (g(t,x,y),h(t,x,y,z,w)) is located at the left of M1 eigenlines.That situation is Mso deMt with i scctiCm 3; tlw rcsflts l;laicd tlcrc generMizc results of C. Gupta aM .1.Mawlfin [13].
Tim above results Iav, ,t sil)lc g'.clric lescripli in t,rms of cigcnlincs; however, they have the drawback to rely ( 11c dcc()l)siti() (1.8) f f, for wlficl wc are mable to ve practical hypot]w.sesensuring ils cxistecc.T]crcforc, in section 4, we wrk with a different decomposition of f, namely where r(t, u, u , u", um) is a boundcd fimction.Such a decomposition can be obtained, for istam, on tle basis f an hypothesis of the type That kind of condition hs been used in M. Del Pino and R. Manevich [5], C. Fabry and F. Munyamarere [6], Y. Yang [22].Unfortunately, the existence conditions obtned with the decom- position (1.9) e not nice those obtnl with (1.8).In particular, we e unable to answer a question raised by M. Del Pino and R. Man,erich can the existence of a solution be proven under the hypothesis that the point (f0, f) lies in a rectangle which does not intersect the eigenlines ?
In the sequel, we will refer to the case where f admits the decomposition (1.8) the "symmetric ce", whcre the decomposition (1.9) will be referred to s the "nonsymmetric ce".
They prove the existence of a solution, assuming that, for some n N, n a(t) liminf f(t, u) limsup f(t, u) b(t) (n + 1) .(1.10) Their proof relies on the use of a coercive quadratic fo.More precisely, they use the ft that the Sobolev space H admits a decomposition H with dim < such that, for all , fi , we have =[' b(t)] =[' a(t)U] 6]] + for rome 6 > 0.
In section 2, we introduce abstract version of such type of hothesis to obtain existence result for the general nonlinear equation Lu-Nu; that result will then be applied to the periodic fourth order boundary value problem.
At the end of this introduction let us fix some notations.We will use the following spies: Vc will often ()lly writ.t;Ck, L , W :'v, II '.Tim nornm on llose spaces are defined s usual; Fiscally we will say timt a fitwtitm f [0, 2] x R R satisfies LP-Carahodory conditions if 1) for each u R , f is emsurable i t; 2) ft,r almost every [0, 2], f is continu(,us in u; 3) for every R > 0, there cxists a flmcti()n h.n LP(0,2) such that, ]f(t, u)] h,n(t) a.c.

ABSTRACT RESULTS
I this section, we first.I)resct an existccc result for tim abstract nonlinear equation Lu-Nu;   that result is a generalization of Theorem 3 in C. De Coster, C. Fabry and P. Habets [4].
Let H be a Hilbcrt space, X and V be normed spaces such that H is continuously embedded in W, the dual of V. We denote by (., .) the pairing in V. Let A, B X V be linear, symmetric opcrators such that for all u X V (Au, u) <_ (Bu, u).
Denote by .(X,V) the set of linear, symmetric operators S X V such that, for all u X V <Au, u> <_ <Su, u> <_ <Bu, u>.
The theorem below uses coincidence degree arguments; for a presentation of that theory and the definition of L-compacity and L-complete continuity, the reader is referred to J. Mawhin [14].THEOREM 2. Let L dora L C H 71X V be a linear, symmetric, Fredholm operator of index zero and N" X V be a L-completely continuous operator.Assume that (a) the operators A, B are L-compact; (b) the bilinear forms .,/3,:" dom L x dom L R respectively defined by .4(u,v) (Au, v), I(u, v) (Bu, v>, ft.(u, v)  (Lu, v) admit continuous extensions to H x H denoted by .4,/3,; (c) for every K > 0, there exists M > 0 such that if u is a solution of A+B If, moreover, (i) there exist D(., .) a positive definite bilinear form and a decomposition H -/ @/ with dim / < oc such that, for any fie/, fi e/, [(, ) (, )]-[(, )-A(, )1 _> v( + , + ); (2.1) (ii) the operator N admits the decomposition Nu-G(u)u + Q(u)   where for all u X, G(u) '(X, V), the bilinear form (x) defined on dom L x dom L by j(x)(u, v) (G(x)u, v) admits a continuous extension to H x H denoted by G(x) and there exists R > 0 such that, for all u dom L, u + 5 with fi /, / and ]lulln _> R, (Q( + ), -> < z)( + e, + e). (2.2)   Then there exists at least one solution u of Lu-Nu.l)llOOF.Iiy tile c()il('il('('t: degree tleory, we oly lave to find an a priori bound in X for the solutions of ., N,,, ( ) u, I0, I.

(.3)
By lypothesis (c), it, is ,.,g1 t,o 1)rove Assume by cont,radictim that t,lere is a solution (A, u) f (2.3) with Ilull R. Let us write u t+ witl [I and fi [t, let N AN+(I-A)(A+B)/2, Multiplyig (2.3) by Ft-, we obtain using (2.1) which gives a contradiction and proves that The existence of a decomposition H @ [I such that (2.1) holds can be obtained by means of the following proposition which is proved in C. De Coster, C. Fabry md P. Habets [4].PROPOSITION 3. t, H be a vector space and ,, B be real, bilinear, symmetric forms on H. Assume that (i) there is some m R such that, C is a scal product which makes H a Hilbert space; (ii) B-A is positive definite, i.e.
Vits of Theorem 2 d Proposition 3 can be written, in whi& the operators belonng (X, V) satisfy a one-sided condition only.THEOREM 4. Let L dom L C H X V be a linear edholm operator of index zero d B X V a line L-compact operator.Consider the set (X, V) of operators S" X V such that for all X a V <Su, u> <Bu, u>.
The prf is similar to that of Theorem 2.
REMARK.Notice that the linearity and the symmetry of the operators of (X, V) e not required.
The following proposition is proved in C. De Coster, C. Fabry d P. Habets [4].PROPOSITION 5. t H be a vector spa and , B, be real, biline, syetric for on H. Assume that (i) there is some m R such that is a sc product which makes H a Hilbert spe; (ii) is positive definite; (iii) for y sequence (u)} such that u} u, one h (u, u) V(u, u).
Then the following equivalences hold" (a) the eigenvalues A of the problem Vv e H, (u, v) B(u, v) A(u, v)   e all strictly positive; (b) there exists 5 > 0 such that for any u H, one h ( )(u, u) c(u, (c) for y rl biline syetfic form 8 on H such that 8 B, u 0 is the only solution of Vv e H, (u, v) 8(u, v).
It is ey to see that L is a Fredholm operator of index zero d h a compact generized inverse from L into H.. It is not difficult to prove that A, B are L-compact and N H. L will be L-completely continuous if we prove that it is continuous d maps bounded sets into bounded sets.For that purpose, we will use the following result Let X be a metre space, (f) a suence in X and f X be such that for any subsuence of (f.) the ists a sub-subsuence which converges to f Then, the initial suence convenes to f.So, let (u.) C H. be such that u, converges to some u in the H.-no, d let (u.) be a subsequence of (u.).Consider the sequence .(t)f(t, u.(t), u(t), u(t),-'" u. (t)) and the subs quence (=) which corresponds to (u.).As (u) converges to u in the H.-norm, there exists a sul)-sfl)s('.(luc('(.'(.%) of (u,,) mwl tlmt fot' 0, 1,2,3, (ii) timre exists h L(0,2r) such that lu(')(t)l < h(t) on [0 2r] (see, for example, Brezis [3,   p. 58l).By l,ypotl,esis (II1), wc l,avc ,% (t) (t) a.c. on [0, 21, where (t) f(t, t(t), u'(t), u"(t), '"(t)).
Moreover, from 0c l/3-bCmd on tl, sub-subsequewe (u) wc deduce a C2-bomd on i, and, by (I1), (I12) and (II3), we can find a function 9 L such that I,% (t)l 9(t).The continuity of N follows from the besgue dominaUxl convergence theorem.Moreover, we eily deduce from the structure of N that it maps bounded ses into bounded ses.Now we will prove that hypothesis (c) of Thcrem 2 is satisfied.By the continuous injection of H 2 into C and tle hypothesis on f, we have that, if there exists K > 0 such ha for all solution of (2.3) we have I111 < K, then there exists M M(K) such that Ilull, M. It is ey o conclude tha there exists M2 such flint Ilull M=.

AT THE LEFT OF ALL EIGENLINES
In this subsection, we will consider problem (3.1)-(3.2) with f as in (3.3), but we will assume only one-sided conditions on h and g.Unfortunately, in that case we cannot have a u'-dependence in f.This is due to the fact that we cannot find m H bound on the solutions of (2.3) from an H2-bound, as in Theorem 6.
As in the previous case we will give some simple hypotheses which ensure that condition (3.9) is satisfied.
REMARK.We can observe that condition (3.13) is equivalent to for all k E ll 0 < k 4 + k .
(3.13) 4. NON-SYMMETRIC CASE In the previous section, we have assumed that the function f admits a symmetric decomposi- tion of the form (3.3).It does not seem easy to give practical hypothesis which ensure that such a decomposition exists.On the opposite, it is easy to find practical conditions under which f has a decomposition of the form f(t,u,u' u")= A(t,u,u', + f(t,u, + fo(t,u,u', u")u" u' u")u' u")u + r(t, u, u' u") with r bounded.For a given function u, the operator v f(., u(.), u'(.), u"(.))v" is not symmet- ric; that difficulty is dealt with below by treating that operator as a "perturbation" of a linear symmetric operator.For the sake of simplicity, we will assume that f 0, since the presence of the corresponding term introduce only technical difficulties.As in section 3, we will consider separately the case where the function f stays asymptotically between two eigenlines and the case where it stays asymptotically at the left of all eigenlines.
+ Z'[(l + ).'a + q' (p-a(t) ).l (4.6)By (4.5), (4.6) and the we lower semi-continuiW of the norm we have u 0. Then we know that u converges wetly in H and in the C-norm to 0 d that converges to 0 in the H-no.By (4.6), we s that u-" nverges to 0 in the L%norm d then we have that u converg to 0 in the H%norm, which contricts the fact that [u]]a 1.Consequently, hypothesis (i) of [(h(t, u, u', q)u" + r(t, u, u")](a e) < ( + ) u ''2 + + ) u.
REMARK. 1) In the case where a ---0, the result is still true provided that we replace (4.2)   by a strict inequality.
2) In the case where b--0, we obtain the same conditions as in the symmetric case.
3) As the following example shows, the above result is not contained in the results of Del Pino- Manasevich [5].Consider the case where p -2, q -4,a 13/15 and b E ]1/15,2/15].The existence of a solution can be proven by our theorem with 1, whereas the results of Del Pino-Manasevich do not apply in that case.

AT THE LEFT OF ALL EIGENLINES
In this subsection, we come back to the situation where the nonlinearity can be considered to be asymptotically "at the left of all eigenlines", leading to one-sided existence conditions.We consider two different situations, depending on the regularity of the limiting functions.THEOREM 12. Let f [0, 2n] x ]Ra ]R be a function which satisfies hypothesis (H4) of Theorem 9. Assume that there exists a E R, b > 0, q R, a,7 Ll((0,27r),]R+) such that for all (u, v, w) iR a and a.e.
The proof follows by arguments similar to those of Theorem 11.
A slightly different result can be obtained, assuming more regularity on .