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.

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

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