Geodesic
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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 
@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/}
}
TY  - JOUR
AU  - Boucenna, A.
AU  - Moussaoui, T.
TI  - Existence Results for a Fractional Boundary Value Problem via Critical Point Theory
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2015
SP  - 47
EP  - 64
VL  - 54
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a3/
LA  - en
ID  - AUPO_2015_54_1_a3
ER  - 
%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/

[1] Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. International Journal of Bifurcation and Chaos 22, 1250086 (2012), 1–17. | DOI | MR | Zbl

[2] Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Computers and Mathematics with Applications 62 (2011), 1181–1199. | DOI | MR | Zbl

[3] Bai, C.: Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem. Electonic Journal of Differential Equations 136 (2013), 1–12. | MR | Zbl

[4] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam, 2006. | MR | Zbl

[5] Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners. Springer-Verlag, New York, 2011. | MR | Zbl

[6] Friedman, A.: Foundations of Modern Analysis. Dover Publications, New York, 1982. | MR | Zbl

[7] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Newark, NJ, 1993. | MR

[8] Diethlem, K.: The Analysis of Fractional Differential Equations. Springer, New York, 2010. | MR

[9] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986. | MR | Zbl

[10] Mawhin, J., Willem, M.: Critical point theory and hamiltonian systems. Springer, New York, 1989. | MR | Zbl

[11] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, 1999. | MR | Zbl

[12] Brezis, H.: Analyse fonctionnelle, théorie et applications. Massons, Paris, 1983. | MR | Zbl