Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 191, pp. 1-12.

Title: Basic results on nonlinear eigenvalue problems of fractional order
 
Author: Mohammed Al-Refai (United Arab Emirates Univ., Al Ain, UAE)

Abstract:
 In this article, we discuss the basic theory of boundary-value problems
 of fractional order $1 < \delta < 2$ involving the Caputo derivative.
 By applying the maximum  principle, we obtain necessary conditions for
 the existence of eigenfunctions, and show analytical lower and upper bounds
 estimates of the eigenvalues. Also we obtain a sufficient condition
 for the non existence of ordered solutions,  by transforming
 the problem into equivalent integro-differential equation.
 By the method of lower and upper solution, we obtain a general
 existence and uniqueness result: We generate two well defined monotone
 sequences of lower and upper solutions which converge uniformly to the
 actual solution of the problem.
 While some fundamental results are obtained, we leave others as 
 open problems  stated in a conjecture.

Submitted November 20, 2011. Published October 31, 2012.
Math Subject Classifications: 34A08, 34B09, 35J40.
Key Words: Fractional differential equations; boundary-value problems;
           maximum principle; lower and upper solutions; 
	   Caputo fractional derivative