Investigation and Solution of Eigenvalue Problems Involving Nonlinear Differential Equations Using Continuous Multiplicative Differential Equations and Method of Green Function

Investigating and solving eigenvalue problems, which is the main goal of Sturm-Liouville theory, generally emphasizes linear differential equations. This theory presents the solutions of eigenvalue problems in the form of infinite series of eigenfunctions of the spectral problem. Investigating and solving eigenvalue problems includes nonlinear differential equations from the unsolved problems of this theory. In this article, at first we try to generalize the basic concepts and basic definitions of continuous multiplicative calculus, including the definitions of continuous multiplicative derivative and continuous multiplicative integral. Then we present the important properties of these two operations along with their basic formulas. After that, recalling the eigenvalue problems, especially the Sturm-Liouville theory, generalize it to analysis and non-Newtonian calculus including continuous multiplicative differential equations. How to calculate eigenvalues and eigenfunctions is presented in general for eigenvalue problems including multiplicative differential equations.. Finally, in a few examples, we present the solutions of eigenvalue problems involving nonlinear differential equations by finding their eigenvalues and eigenfunctions. It should be noted that the above method for solving boundary value problems involving nonlinear differential equations including parameters is devoted only for authors.