Existence of nontrivial solutions for a system of fractional advection-dispersion equations. (English) Zbl 1427.34011
The authors consider a system of symmetric fractional equations of the form:
\[\begin{gathered}
\frac{d}{dt}\left(\frac{1}{2}(_0\mathcal D_t^{-\beta_i}u_i')(t)+\frac{1}{2} (_t\mathcal D_T^{-\beta_i}u_i')(t)\right)+F'_{u_i}(t,u_1(t),\dots,u_n(t))=0 \quad a.e.\quad t\in [0,T], \\ u_i(0)=u_i(T)=0
\end{gathered}\]
for \(1\leq i \leq n\), \(0\leq \beta _i <1\), and \(_0\mathcal{D}_t^{-\beta_i}\), \(_t\mathcal{D}_T^{-\beta_i}\) are the left and right Riemann-Liouville fractional integrals, respectively. \(F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n \) is a Caratheodory-type function.
Some conditions on \(F\) are given which guarantee the existence of a non-trivial solution. The main tool is a critical point theorem by G. Bonanno and G. D’Aguí [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3–4, 1977–1982 (2010; Zbl 1200.34020)].
Some conditions on \(F\) are given which guarantee the existence of a non-trivial solution. The main tool is a critical point theorem by G. Bonanno and G. D’Aguí [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3–4, 1977–1982 (2010; Zbl 1200.34020)].
Reviewer: Zdzisław Dzedzej (Gdansk)
MSC:
| 34A08 | Fractional ordinary differential equations |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional advection-dispersion equation; weak solution; critical point theory; anomalous diffusion; variational methodCitations:
Zbl 1200.34020References:
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