ON THE SOLUTION TO THE SECOND PART OF HILBERT’S FIFTH PROBLEM: A VIEW FROM CAUCHY PROBLEMS, GIVING TWO-UNKNOWN PARTIAL DIFFERENTIAL, DIFFERENCE, AND FUNCTIONAL EQUATIONS IN BIJECTIVE CORRESPONDENCE(°)
As has been often the case with Hilbert’s problems, so too the second part of Hilbert’s fifth problem (HP#5.2 for short) gives rise to unexpected implications and by-products as shown by this paper. The direct (vs. converse, see below) HP#5.2 is the following question about the functional equation (FE for short) unknown functions and solutions:
How far, the FE solutions obtained with the differentiability of the unknown functions (hence, via the PDE/ODE resulting from the differentiation of the FE) are similar to, or different from, those obtained without differentiability?
The solution to the direct and converse HP#5.2, see “®” and “¬”, respectively, in (*) and (**) below, is that
For the class of PDE Cauchy problems (IVPs for short) and the bijectively corresponding, actually equivalent, FE IVPs we consider, both in two-unknown functions given by Riemann-integral operator solutions, such solutions remain the “same” - without differentiability - only in so far, or in the sense, that the Riemann-integral operator and the continuous arbitrary function occurring the said solutions are replaced with the summation operator and the arbitrary sequence occurring in the solutions to the corresponding non-differentiable FE IVP, i.e., the following natural correspondences hold
(*)
(**)
between the Riemann-integral operator solutions to the pairs of equivalent PDE and differentiable FE IVPs considered, and the summation operator solutions to the bijectively corresponding non-differentiable FE IVP, which turns out to be a difference equation IVP in two-unknown functions.
Cauchy problems or IVPs for FEs are not dealt with in the literature, and the converse HP#5.2 either, whereas, as the literature shows, the attempts at solving the direct HP#5.2 (e.g., for the Cauchy and other FEs ) paradoxically imply the differentiability, instead of eliminating it.
partial differential equations, difference equations, functional equations, mean-value.
