SOME RESULTS ON BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

Existence r,sdts fir a second order bo,mdary vahw p,l)l,nz fir fiuwtiCmal differential ('qati()IlS, are giwn. The rcsflts ar( })asc(t tmth(' nmlin('ar Alterna.tiv, ()f Leray-S('haaMcr and rely .m a lmni t>mls on slut,ims. Th's' rcstlts arc geu'alizations f receipt rr,sdts flom wdinary differential equations aIll ctmil)lcte Cmr ,ar)ier rcstlts Cm the same

The above inequalities follows by essentially the saxne reasoning as in lexnmas 2.3 and 2.4 d [].Obviously in the case T 2 we cm me the inequMit,y () instead of (b).blAIN

RESULTS
Now we present our tnain remtlt on the existence of solutions of the BVP (E)-(BC).
Then the BVP (E)-(BC) has at least one solution.

CONCLUDING REMARKS
In [4] the BVP (E)-(BC) has been studied under the following conditions, which are briefly reproduced here.
We also remind tha.t: The BVP (E)-(BC) has at least one solution if (.4)) and (A) h,)d, [4.The following questions aze immediately isen.
Has the BVP (E)-(BC) a solution if I (A,) and (Ha) (or (H'a) hold ?
The answer in all of the above questions is positive.Indeed the cases 2) and 3) are obvious, since every one of conditions (H), (A2) and .43gives independently a priori bound on x or x.Some comments are needed for the case 1 ).By taking the inner product of integrating by parts over [0, T] and using (At) we get < It o)I I,1: -k This implies the existence of he Imd M nd the rest ff the lr[ is esseially he ane a We sunmarize the abwe discussion in the follwing THEOREM 3.1.The BVP (E)-(BC) has at least ne slutirn if ne of the filhwing pairs cnditions hlds: 1 (A) and (A) 2) (A,) and (A.I 3) (A) and (H)(or (H'e) 4) (H)and (Ae} 5) (H) and (A.) 6) (H) and (He) (or (Hz)