Electron. J. Diff. Equ., Vol. 2013 (2013), No. 152, pp. 1-19.

Non-existence of solutions for two-point fractional and third-order boundary-value problems

George L. Karakostas

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.

Show me the PDF file (334 KB), TEX file, and other files for this article.

George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr

Return to the EJDE web page