Article
Full entry |
PDF
(0.3 MB)
Feedback
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
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
Summary:
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.
References:
[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 10.1142/S0218127412500861 | MR 2926062 | Zbl 1258.34015
[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 10.1016/j.camwa.2011.03.086 | MR 2824707 | Zbl 1235.34017
[3] Bai, C.: Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem. Electonic Journal of Differential Equations 136 (2013), 1–12. MR 3084616 | Zbl 1295.34007
[4] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam, 2006. MR 2218073 | Zbl 1092.45003
[5] Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners. Springer-Verlag, New York, 2011. MR 2722059 | Zbl 1214.35025
[6] Friedman, A.: Foundations of Modern Analysis. Dover Publications, New York, 1982. MR 0663003 | Zbl 0557.46001
[7] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Newark, NJ, 1993. MR 1347689
[8] Diethlem, K.: The Analysis of Fractional Differential Equations. Springer, New York, 2010. MR 2680847
[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 0845785 | Zbl 0609.58002
[10] Mawhin, J., Willem, M.: Critical point theory and hamiltonian systems. Springer, New York, 1989. MR 0982267 | Zbl 0676.58017
[11] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, 1999. MR 1658022 | Zbl 0924.34008
[12] Brezis, H.: Analyse fonctionnelle, théorie et applications. Massons, Paris, 1983. MR 0697382 | Zbl 0511.46001
Search
Partner of