Existence Results for a Fractional Boundary Value Problem via Critical Point Theory
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 47-64
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In this paper, we consider the following boundary value problem \[ \left\lbrace \begin{array}{lll} D_{T^{-}}^{\alpha } (D_{0^{+}}^{\alpha } (D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha } u(t))) ) = f(t, u(t)), \quad t \in [0, T], \\ u(0)= u(T)= 0\\ D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(0))= D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(T))= 0, \end{array} \right. \] where $0 \alpha \le 1$ and $f\colon [0, T]\times \mathbb {R} \rightarrow \mathbb {R} $ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.
In this paper, we consider the following boundary value problem \[ \left\lbrace \begin{array}{lll} D_{T^{-}}^{\alpha } (D_{0^{+}}^{\alpha } (D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha } u(t))) ) = f(t, u(t)), \quad t \in [0, T], \\ u(0)= u(T)= 0\\ D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(0))= D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(T))= 0, \end{array} \right. \] where $0 \alpha \le 1$ and $f\colon [0, T]\times \mathbb {R} \rightarrow \mathbb {R} $ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.
Classification :
26A33, 34B15, 58E05
Keywords: Existence results; fractional differential equation; boundary value problem; critical point theory; minimization principle; Mountain pass theorem; Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions
Keywords: Existence results; fractional differential equation; boundary value problem; critical point theory; minimization principle; Mountain pass theorem; Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions
@article{AUPO_2015_54_1_a3,
author = {Boucenna, A. and Moussaoui, T.},
title = {Existence {Results} for a {Fractional} {Boundary} {Value} {Problem} via {Critical} {Point} {Theory}},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
pages = {47--64},
year = {2015},
volume = {54},
number = {1},
mrnumber = {3468600},
zbl = {1354.34013},
language = {en},
url = {http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a3/}
}
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%0 Journal Article %A Boucenna, A. %A Moussaoui, T. %T Existence Results for a Fractional Boundary Value Problem via Critical Point Theory %J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica %D 2015 %P 47-64 %V 54 %N 1 %U http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a3/ %G en %F AUPO_2015_54_1_a3
Boucenna, A.; Moussaoui, T. Existence Results for a Fractional Boundary Value Problem via Critical Point Theory. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 47-64. http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a3/