In this study, we analyze convection-pressure split Euler flux functions which contain weakly hyperbolic convection subsystems. A system is said to be a weakly hyperbolic if all eigenvalues are real with no complete set of linearly independent (LI) eigenvectors. To construct an upwind solver based on the flux difference splitting (FDS) framework, we require to generate a full set of LI eigenvectors. This can be done through the addition of generalized eigenvectors which can be computed from the theory of Jordan canonical forms. Once we have a complete set of LI generalized eigenvectors, we construct upwind solvers in the convection-pressure splitting framework. Since generalized eigenvectors are not unique, we take extra care to ensure no direct contribution of generalized eigenvectors in the final formulation of both the newly developed numerical schemes. The first scheme is based on Zha & Bilgen type splitting approach, while the second is based on Toro & Vázquez splitting. Both schemes are tested on several benchmark test problems on 1-D and on some typical 2-D test problems which involve shock instabilities. The concept of generalized eigenvector based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.