BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH REFLECTION OF THE ARGUMENT

Linear and nonlinear boundary value problems for differential equations with reflection of the argument are considered.


I. INTRODUCTION.
In [1][2][3][4] a method has been discovered for the study of functional differential equations whose argument deviations are involutions.Important in their own right, they have applications in the investigation of stability of differential-difference equations.Differential equations with involutions can be transformed by differen- tiation to higher order ordinary differential equations and, hence, admit of point data initial or boundary conditions.Initial value problems for such equations have been studied in numerous papers.However, boundary value problems even for differ- ential equations with reflection of the argument have not been considered yet.
The purpose of this paper is to discuss existence and uniqueness of solutions of y" f(x, y(x), y(-x)), (I.I) where f e C[ [-a, a] 2. PRELIMINARY RESULTS.
First, we prove a sequence of lemmas for the linear case, y"(x) a(x)y(x) + b(x)y(-x) + c(x), which are needed in order to prove our results for the general equations of the form (I.).
Before we proceed further, we present some results without proof, which help to simplify the proofs of our results.
LEMMA 2.1.[5, pp.182]We are now in a position to state our results.fb a Since y(a) y(b)  PROOF.On multiplying (2.1) by y(x), and integrating the result from a to b, we find, because of (3.1), Applying Lemma 3.1 and the Cauchy-Schwartz inequality, we get a 0 Lemma 3.2 and the above inequality then imply inequality (3.2), LEMMA 3.4.Suppose a(x) and b(x) satisfy all conditions of Lemma 3. (3.2) PROOF.First, we show the uniqueness.Suppose u(x) and v(x) are solutions of (2.1), (3.2).Let R(x) By Lemma 3.3, R(x) O, which implies u(x) v(x).So problem (2.1), (3.2) has a unique solution.To prove the existence, let u(x) and v(x) be solutions of the following initial value problems We notice that u(x) and v(x) exist and are unique.Moreover, v(b)   O, because if v(b) O, then from v(a) O, (ii) and Lemma 3.3 we have v(x) 0 which contradicts v'(a) i.Therefore, by linearity, Y2-u(b) defines the solution of the problem (2.1), (3.2).Proof is complete.
Let us now, consider the second order linear functional differential equation y"(x) C[-a, a], a > O.We shall show that, under certain condi- tions on a(x) and b(x), equation (3.3) with a boundary condition has a unique solu- tion on [-a, a] and obtain an estimate for such solution.
The function z(x) will be a solution of (3.15), (3.16) provided s, t satisfy sY2(a) + tY4(a) A 2 Yl(a) Y3(a), If A Y2(a)Y"4(a) Y2(a)Y4(a) # O, a unique solution of the preceding linear system can be found, and the corresponding function z(x) then is the unique solution of (3.15), (3.16).However, if A O, then Y2(a) Y4(a)   y(a) y(a) p (constant).
We can assume that p # O, because if p O, then Y2(a) 0 and by means of Taylor's formula it can be shown that the solution of (4) A(x)-"'+ B(x) has the property Y2'(-a) 0, contradicting the original assumption y(-a) i.
Now, we consider the general equation (i.I) and prove the following theorems.
Then from (3.29) and the estimates on G(x,t) and G (x,t), it follows that x 1 1 1 lTy(x)[ i lyl+Y01 +a IYl-Y01 +7 azM' and IT'y(x) I-< a IYl-Y01 + aM.C[[-a, a], R] by Ascoli's theorem.The Schauder's fixed point theorem then yields a fixed point of T, which is a solution of (I.i), (1.2), thus completing the proof of the theorem.
All of these considerations show that T is completely continuous by Ascoli's theorem.
Schauder's fixed point theorem then yields a fixed point of T, which is a solution of (I.i), (3.34).
Thus, T is completely continuous by Ascoli's theorem.Schauder's fixed point theorem then yields a fixed point of T, which is a solution of (1.1), (1.3).