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.