FUNCTIONAL DIFFERENTIAL EQUATIONS A RECIPROCITY PRINCIPLE

The functional differential equations proposed for solution here are mainly ordinary differential equations with fairly general argument deviations. Included among them are equations with involutions and some with reflections of the argument. Solutions will be obtained by quadratures in terms of implicitly defined functions. They have a wide range of applicability from the stability theory of differential-difference equations to electrodynamics and biological models.

are involutions as in [1-5] and effective procedures are now known for dealing with them.In [6], Wiener and Aftabizadeh gave existence and uniqueness theorems for boundary-value problems problems for reflection of the argument in some cases of linear and nonlinear equations.In some instances, however, the solutions are given in abstract terms or are made to depend on the solutions of equations of high-order.We seek more concrete results in some instances and more general in others.

MAIN EQUATION AND RESULTS.
The central theme of this paper and its major motivation was provided by the equation y' y(g(x)), for given g(x) (2.1) This equation was suggested for solution by W. R. Utz in [7].
We now state our main THEOREM (2.1).Let g(x) be continuous, I-i and onto, then the solution of (2.1) can be found by quadratures.We write (2.1) in parametric form as -Xand u,V have replaced y' and y respectively in a new variable, t.The symbol, is used for composition of functions.
Let V Al(t ), X Bl(t).Then from the above system, U, we have u AlogoB and also The differential equation is solvable by separation or variables in a familiar way where h is an arbitrary (2.6) (2.7) function.We write it as a system @ u K u hoVoX In system K substitute as before u A(t), V B(t).(2.8) equate corresponding parts of (2.4), (2.8) so that when (2.8) holds, (2.4) must also hold.In addition, we require that the solutions of the differential equations (2.1) and (2.5) coincide.This gives us the set of equations A B A. ogoB A B B oh-oA, AzQB BoA-ohoB (2.9) The first three equations refer to the insides of corresponding pairs of parentheses in (2.4) and (2.8) and the last one ensures that the solutions are identical for (2.1) and (2.5).
Eliminating the functions A and Bi from (2.9) gives -i BogoB h (2.10) A reciprocity relation exists between g,h which is the reason for the title of this paper.The solution process by steps is as follows: (1) Choose B; (2) find h from (2.10); (3) find A by (2.6).With the values of h and B, (4) find B from the equation, B B-oh-oA and find the solution of (2.1) from its value, Q AloB

BoB
This completes the proof. 3. THE NESTING EQUATION.
We turn now to the equation y' k(y(g(x))) (3.1)where k,g are both functions with inverses.This last equation has been called a nesting equation.In the same way as before we have the THEOREM 3. Now k,g are given and so (3.5) tells us how to choose, h, once B is chosen arbitrarily.
Consider the equation y'(x) A(x)y + B(x)y(g(x)) (4.1) for which we have the following THEOREM 4.1.Let A(x), B(x) be integrable in finite terms and let g(x) be one- to-one and onto.Then (4.1) is solvable by quadratures.for some Q.Proceed as before.
In spite of the attractive result in section 2 was unable to go past the scalar case.

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Sketch of) Proof: Multiply both sides of (4.1) by exp(-f A(x)) as usual to bring it