Application of biopesticides is a globally rising phenomena on yearly
basis, and the use of traditional insecticides is on the decline. North
America uses the largest percentage of the biopesticide market share at
44 %, followed by the Europe with 20 %, each South and Latin American
countries with 10 %, and about 6 % in Asia and India. However biopesticide
growth is projected at 10 % annually; it is highly variable among the
regions constrained by factors such as regulatory hurdles, public and

We review the solutions of some functional equations for which the differentiability was the aim of study in the beginning of the last century. By decomposing these functions on the Schauder basis (which are only Lipschitz of order 1), we determine the exact Hölder regularity (even when it exceeds 1) and thus prove that this regularity changes widely from point to point. We also determine the Sobolev (or Besov) spaces to which these functions belong.

The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the multi-fractal formalism has been proved to be valid for self-similar functions. But, multi-fractals encountered in physics or image processing are not exactly self-similar. For this reason, we extend the validity of the multi-fractal formalism for a class of some non-self-similar functions.

Selfsimilar functions can be written as the superposition of similar structures, at different scales, generated by a function g. Their expressions look like wavelet decompositions. In the case where g is regular, the multifractal formalism has been proved for the corresponding selfsimilar function, for Hölder exponents smaller than the regularity of g.

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{ö}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness'' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis.