I'm not a professional mathematician, but I've struggled to find people who can address this question. I have a semi-colloquial question, and I'm not an expert at it, I don't know how to properly phrase the question, but I'm searching for an understanding.
For a simple delay differential equation $x'(t) = x(t-a),$ one possible solution $c_0\exp(t*W(a)/a)$ where $W$ is the Lambert-W function.
However, when I read about delay equations, and if I understand correctly, it turns out the solution can potentially include any generally continuous history function, not just exponentials. Unlike typical ordinary differential equations, the ordinary Banach contraction theorem doesn't quite guarantee uniqueness the same way, if at all.
But, this exponential function is a nice result, and sometimes other people make a point that it satisfies the delay equation.
So, with respect to practical applications in science, how reliable is the exponential solution? When does it make sense to pose as a solution to the delay equation? When is it unique?
Is it perhaps the case that if you assume a smooth or analytic solution, then the exponential function is a unique smooth solution? I'm just confused and unsure because no professor I've talked to understands these kinds of equations.
