The main purpose of this work is divided into five categories: First, is to devote the study of the generalized linear eigenvalue problems with some new extended theorems. Moreover some real life problems are presented in which its mathematical formulation lead to the generalized linear eigenvalue problems. Second, some approximate methods are utilized for solving the generalized linear eigenvalue problems associated with the integral and integro-differential equations (with or without delay). Third^ a modified method is purposed for solving the integral and integro-differential equations of the first and second kinds based on the existence of the eigenvalues for the corresponding integral operator. Fourth, the nonlinear eigenvalue problems with the applications on the integral and integro-differential equations are studied Finally a new approach for solving the linear system of n-th order ordinary differential equations with constant matrix coefficients is introduced.