The main theme of this work can be divided into three categories, which summarized as follows: First, give some theorems which ensure the existence and uniqueness for the solution of the nonlinear Sturm-Liouville boundary value problem (without parameters) and apply these theorems on some real life applications. Second, study the necessary conditions for the existence of the solutions for the nonlinear Sturm-Liouville eigenvalue problem and devote this study to some real life problems. Third, some approximation methods are presented to solve the nonlinear eigenvalue problems related to the differential and the integro-differential equations, namely Tonti's approach, the collocation method and Galerkin's method with some illustrative examples. For each discussed example, we write a program in MathCAD software package and the results were shown in tables.