TWO-POINT BOUNDARY VALUE PROBLEMS INVOLVING REFLECTION OF THE ARGUMENT

Two point boundary value problems involving reflection of the argument are studied. The nonlinearity involved is allowed to cross asymptotically any number of eigenvalues of the associated linear eigenvalue problem as long as those crossings take place in subsets of sufficiently small measure.

Differential equations with reflection of the argument represent a particular case of functional differential equations whose arguments are involutions.Important in their own right, they have applications in the investigation of stability of differential difference equations.Initial value problems for equations with involutions have been considered in numerous papers.A survey of results in this direction is given in [I].However, research on boundary value problems for such equations is developed yet insufficiently.Wiener and Aftabizadeh initiated the study of problem (I.i) in the case that f 0 and g(t,x,y) is bounded on [-I,I] l x l in [2].The methods used in this paper are similar to the ones used by Gupta-Mawhin [3] for periodic solutions of Lienard's differential equations.We mention that in addition to using the classical spaces C([-I,I]), ck([-l,l]) and Lk(-l,l) of continuous, k-times continuously differentiable or measurable real functions the k-th power of whose absolute value is Lebesgue integrable we use the space HI(-l,l) defined by HI(-I,I) {x" [-I,I] I x is abs.cont.on [-I,i] and L 2 x'
Assume, now, that the conclusion of the lemma is not true.Then we can find a sequence {Xn} in HI(-1, I) and x in HI(-1, i) such that IXnl i, for all n, Xn x in C([-l,l]), x x H n in HI(-1,1) with x (-I) x (I) 0 for all n and n n 0 <.BF(Xn <--nl for all n 1,2 for a.e.t in [-I,I] with strict inequality holding on a subset of [-I,I] of positive measure.Let 6(F 0) be given by lemma i.Then for all x HI(-1,1) with x(-1) x(1) Using the fact that HI(-1,1) C([-l,l]) and the Wirtinger inequalities, [4], as well as lema I, we get 2 2 -4'too '-'" REMARK I. Clearly the best value for 6(0) is , so that when r 0 0, F 0, x(1) O.
LEMMA 3. Let F F 0 F + F=o be as in Lemma 2 and 6(F 0) be given by Lemma I.
Then for all measurable real functions p on [-I,I] such that p(t) F(t) a.e.
on I-l,1], all continuous f: R I and all x Hl(-l,1) with x' absolutely continuous on [-I,I] and x(-l) x(1) (-i,i) with x' absolutely-continuous and x (-l) x(1) O, we get on integrating by parts that REMARK 2. We observe that the main ingredient in Lemmas 1,2, and 3 is that the x HI(-I,I) satisfy Wirtinger inequalities.Now, since it is easy to see that for O, where k >. 0 is given, (or x'(-l)-hx(-l) =0, where h >. 0 is given, x(1) 0) Wirtinger type inequalities hold, analogues of Lemma I, 2, 3, with f E 0, can be obtained.

EXISTENCE THEOREMS
Let f: be a continuous function and let g-[-i,I] x x be such that g(-,x,y) is measurable for each x, y e and g(t,-,') is continuous on I x l for a.e.t [-I,I].Assume, moreover, that for each r > 0 there exists er LI(-I'I) such that Ig(t,x,y)l <" er(t) for a.e.t in [-I,I], x in [-r,r] and all y e .We say that such a g satisfies Caratheodory's conditions.
THEOREM I. Assume that there exists F LI(-I,I) such that (3.1) where uniformly a.e. in t [-I I] and y .Suppose that F F 0 + F + F 2 L 1 -.
L L==(-I I) F LI(-I,I) and F 0 (-I,I) are such that Fo(t) < for a.e.t in [-I,I] with strict inequality on a subset of [-I,I] of positive measure and 4 Irl + -]rlJ < 6(r0)' where 6 (F O) is as determined in Lemma I.
Let, now, x(t) be a possible solution of (3.6) for some e [0,I].Since, now, (1-x)(r(t)+n) + %-fl(t,x(t),x(-t)) r(t) + n for a.e.t in [-I,I] we have on integrating by parts the equation obtained by multiplying the equation in (3.6) by -x(t) and applying

L
and hence Ixl C2' where C 2 is some constant independent of X [0,11.Thus equation (3.I) has at least one solution for each e e LI(-1,1).
x Let N [2-] so that r + < 2 and let r > 0 be such that gl(t,x,y (-1,1) such that lim sup g(t,x,y) r(t) uniformly a.e. in t [-I,I] and all y in .Suppose that r < 2 and that there exist real numbers a, A, r and R with a R, r < 0 < R such that for a.e.
t in [-I,i] and y in , g(t,x,y) A when x R and g(t,x,y) a when x r.
Then the boundary value problem (3.9) has at least one solution for each e : LI(-I,I) so that for a.e.t in [-I,I] I gl(t,x,y) : (A a) 0 for all x R and y in (3.13) gl(t,x,y) (a A) 0 for all x r and y in and there is an 0 <_-71(t,x,y) < r(t) + n a LI(-I,I) such that lh(t,x,y) < (t) for a.e.t in [I,i] and all x, y in .
If, now, x(t) R for all t in [-I,I] we have using (3.13), (3.14) which contradicts the assumption that R > 0. Similarly, x(t) S r for all t in [-I,I] leads to a contradiction.Hence, there exists a T in [-I,I] such that r < x(T) < R.
Next, it is easy to write explicitly the solution x(t) with jl x(t)dt 0 of the boundary value problem x"(t) y(t), x'(-l) x'(1) 0 for y e LI(-I,I) Il y(t)dt O. From this it is easy to deduce the existence of a 6 > 0 such with Ill x(t)dt and x'(-l) x'(1) 0 that for every x e clt-l,l, with (t) x(t) This completes the proof of the theorem.
COROLLARY 3. Let g: [-I,I] x ]R I satisfy Caratheodory's conditions and assume that there exists F(t) LI(-I,I) such that lim sup g(t,x) < F(t) uniformly a.e.ixl x r(t)dt < 2 and that there exist real numbers in t [-I,I].Suppose that F -I a, A, r, R with a < A, r < 0 < R such that for a.e.t in [-I,I], g(t,x) A when x R and g(t,x) a when x r.
Then the boundary value problem x"(t) + g(t,x(t)) REMARK 4. In case g(t,-) is monotonically-increasing for a.e.t in [-I,I] see Mawhin [7] for the boundary value problem (3.24) for a result similar to Corollary 3, above.
The existence of a solution for the boundary value problem "(t) + g(t,x(t),x(-t)) e(t) (3.25) x'(-l) O, x'(1) + hx(1) O, (h > 0 given) can be obtained in a similar manner as for (3.9).In fact the following theorem is true, whose proof we omit as it is very similar to the proof of Theorem 2.
THEOREM 3. Assume that there exists F(t) LI(-I,I) such that lim sup g(t,x,y) <. r(t) uniformly a.e. in t [-I,I] and all y in I. Suppose ixlo x that F < and that there exist real numbers a, A, r, R with a < A, r < o < R such that for a.e.t in [-I,I] and y in , g(t,x,y) A when x R and g(t,x,y) <. a when x < r.
Then the boundary value problem (3.25) has at least one solution for each L e (-I,I).
,... for eigen-values and the linear eigen-value problem C I,I) with x(-l) L L L Then (3 I) has at least one solution for each e (-I,I).PROOF Let n =7 [(ro) -7 Irl Irl ]-Then, there exists r > 0 such L w L that for a.e.t in [-I,I] all x with Jx Z r and y .lim sup g(t,x,y) < r(t) (3.2) ixl x TWO-POINT BOUNDARY VALUE PROBLEMS g(t,x,y) < F(t) B Define YI" [-l,1] x R x I l by

3 2 .
with  Ixl rl, all y in and a.e.t in [-I,I].Proceeding as in the proof of Theorem I we can write the equation (3.15) in (a-A) .<e I < (A-a).Clearly the equation in (3.9) is equivalent to the equation (Assume that there exists r(t) ,x,y) g(t,x,y) : (a + A) and el: [-I,I] by el(t) e(t) : (a + A) inserting (3.20) in (3.18) we get (3.22)C.P. GUPTA Now, by the fact that there exists a T [-I,I] with r < x(T) < R we have that t x'(s)ds[ < max (-r R) + 2[x'[ L <= max (-r,R) / 2lx"l L using (3.21).Hence x(t)dt[ < [(x(t)ldt < 2 max(-r R) 4[x"[ L Finally inserting (3.23) in(3.22)we get that there exists a constant 01 > 0 independent of % [0,1] such that Ix"l z o L Using this in(3.20)we then deduce the existence of a constant 0 one solution for each e L2(-I,I, with 2a e(t)dt < 2A.