From: "Arthur L. Rubin" <216-5888@mcimail.com>
Subject: Re: Q: help with a functional equation:
Date: Thu, 25 Mar 1999 07:23:36 +0700
Newsgroups: sci.math
Keywords: differential-difference equations

Nils Chr. Framstad wrote:
> 
> I would like to find the solutions to the following equation in the
> unknown function f:
> 
> f(x + q/f'(x)) - f(x) = K   (q > 0 and K are arbitrary (real) constants)
> 
> The obvious candidates of the form Ax + B and B + C log (x+D), and also
> functions made from pasting such, but are there more? Preferably I'd
> want every piecewise differentiable solution, but I'd be more than happy
> with every strictly decreasing C1 function defined on the positive
> half-line...

OK, I'll try.  Let g = f^-1 (the inverse function of f).  Then the
equation transforms to g(y+K) = g(y) + q g'(y).  (If K = 0 , 
then g is constant, which doesn't correspond to a possible solution 
of the original problem.)

If g is defined on a half-line (meaning, in addition to f being 
strictly monotone on the positive half-line, that f->+/-infinity as 
x goes to infinity), then we can attempt to solve this 
differential/difference equation.  One attempt at a solution 
involves the possible representation of g(x) as 
Sum_k P_k(y) exp(lambda_k y), where the P_k are polynomials, (or, 
when necessary, polynomials in y and log(y), etc.), and the lambda_k 
are a decreasing (increasing if f was decreasing) (in real part) 
sequence of complex numbers.  Solving for possible values of lambda, 
we find that the "inditial" equation is exp(K lambda) = 1 + q lambda. 
The real solutions of this are 0 and a unique lambda_0 (which is 
non-zero unless K=q).   Hence the only solutions with real lambda are 
of the form g(y) = A + B exp(lambda_0 y), or A + B y (if q = K), which 
correspond to the solutions you gave.  However, there are other complex 
solutions of the inditial equation for lambda; if 
Re(lambda') < lambda_0, you can construct monotone solutions as 
y -> +infinity of the form 
g(y) = A + B exp(lambda_0 y) + Re(C exp(lambda' y)); while if 
Re(lambda') > lambda_0, you can construct monotone solutions of the 
same form as y->-infinity.  I don't remember where the solutions
of the inditial equations actually reside in the complex plane, so 
I can't tell you what form additional solutions would have, but
they do exist if K>0.  It appears that there are no solutions for
K<=0, as the inditial equation has only 0 for a real root, 
corresponding to all solutions defined on an infinite interval
being non-monotone somewhere on that interval.

If g is real-analytic with radius of convergence >|K| somewhere, we 
get the same result by equating coefficients of the power series.

We do need to show that f(x) cannot go to a finite limit as
x -> +infinity, but I'll leave that as an exercise to the reader.

> 
> I will probably face similar equations later, so a textbook reference
> would be appreciated as well.

I have a text on differential/difference equations somewhere...but I
can't find it.  I find the series and asymtotic methods described
in _Differential Equations_, Harry Hockstadt (New York: Holt Rinehart 
Winston, 1963-1964) helpfull.

--  
Arthur L. Rubin   216-5888@mcimail.com