On second look, it seems that I was using the Holder semi-norm and not the Holder norm. The argument I employed was not quite correct.

perhaps one should take $||f_{n_k}- f||_\alpha$ with the Holder $\alpha$ norm on an arbitrary compact subset of the original set.

I should also add that in the solution setting I am interested in (see the word bounded), locally Lipschitz nonlinearities are sufficient for uniqueness. Of course as you have pointed out in the references above, in more general solution settings this is not true.

I should also say thankyou. I was aware of the Haraux and Weissler paper (probably the closest thing I have seen to the questions I have considered), albeit as you mention it does have a slightly different flavour ($L_p$ setting). Nonetheless, I can continue my reference search now.

Good suggestions.

However, I would still appreciate any kind of reference to a similar piece of work, as it would make the introduction of the paper a tad nicer to read. At the moment, I still have little idea as to whether or not this has even been done before.

I could but it took me (for the first type of result) about 10 pages to construct (the second took a further 10), so perhaps not here. I will happily send you a copy when it is all written up (in a slightly more presentable form) though.

I ask again though, can anybody send me a reference to a similar piece of work! I already know how it is done, I just want a piece of work to reference in the introduction of the paper.

At Willie, not that low. It is continuous, but obviously not Lipschitz as then a uniqueness result would hold.

Sorry Michael Renardy, quite simply, consider a problem which has non-unique solutions to begin with (which I hope is relatively trivial for you). You are correct in that is the starting point though and non-uniqueness must occur to get spatial inhomogeneity. When it is all written up nicely I'll send you a copy.

I should also comment that I have already constructed spatially inhomogeneous solutions to these type of problems. I am merely looking for references for other works which have (if they exist) also bothered to do so ... and maybe find out why they bothered to construct them too.

This is merely an existence result, no guarantee is given with regard to spatial inhomogeneity. For all we know, the contraction mapping based existence result will just give solutions to the first order ode $u_t=f(u)$ with $u(0)=0$.