Abstract
In this work, we present some properties of the pseudo differential operator .and the hypotheses proposed on the class of symbols are stated to obtain the regularity needed .Hence the minimal decreasing rays are studied to prove the existence of a total system of eigenvectors .Also these results Are applied for a cubic family of non-linear eigenvalues problem. Moreover, a non-linear eigenvalue problem is studied with a family of quasihomogenous
operators, for which we benefit from the facilities obtained by the symbols of pseudo differential operators on the calculation of the parametrix to obtain a results that implies to demonstrate the existence of the non-trivial eigenvalues for a triple type of quasi-homogenousity for a family of non-linear eigenvalue problems.
