Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 152, pp. 1-19.

Title: Non-existence of solutions for two-point fractional and third-order
       boundary-value problems
        
Author: George L. Karakostas (Univ. of Ioannina, Greece)

Abstract:
 In this article, we provide sufficient conditions for the
 non-existence of solutions of the boundary-value problems with fractional
 derivative of order $\alpha\in(2,3)$  in the  Riemann-Liouville sense
 $$\displaylines{
 D_{0+}^{\alpha}x(t)+\lambda a(t)f(x(t))=0,\quad t\in(0,1),\cr
 x(0)=x'(0)=x'(1)=0,
 }$$
 and in the Caputo sense
 $$\displaylines{
  ^CD^{\alpha}x(t)+f(t,x(t))=0,\quad t\in(0,1),\cr
 x(0)=x'(0)=0, \quad x(1)=\lambda\int_0^1x(s)ds;
 }$$
 and for the third-order differential equation
 $$
 x'''(t)+(Fx)(t)=0, \quad \hbox{a.e. }t\in [0,1],
 $$
 associated with three among the following six conditions
 $$
 x(0)=0,\quad x(1)=0,\quad x'(0)=0, \quad x'(1)=0,
 \quad x''(0)=0,  \quad x''(1)=0.
 $$
 Thus, fourteen boundary-value problems  at resonance and six
 boundary-value problems at non-resonanse are studied.
 Applications of the results are, also, given.

Submitted October 3, 2012. Published June 28, 2013.
Math Subject Classifications: 34B15, 34A10, 34B27, 34B99.
Key Words: Third order differential equation; two-point boundary-value problem;
           fractional boundary condition; nonexistence of solutions.