The Applications of Critical-Point Theory to Discontinuous Fractional-Order Differential Equations

Abstract

We consider a fractional equation involving the left and right Riemann–Liouville fractional integrals and with Sturm–Liouville boundary-value conditions. We establish the variational structure of the problem and, by using critical-point theory, the existence of an unbounded sequence of solutions is obtained.

Three solutions for a class of fractional impulsive advection‐dispersion equations with Sturm‐Liouville boundary conditions via variational approach

“…Remark 2.1. From Tian and Nieto, 10 we know that the fractional derivative space Ẽis the space of functionsũ ∈…”

“…Proof. (8) and s|ũ| 2 ≤ (S(t)ũ,ũ) ≤s|ũ| 2 , are used to prove this Lemma, the proof is similar to that for Lemma 5.4 in Tian and Nieto, 10 and we omit it here.…”

Ground state solutions of a fractional advection–dispersion equation

“…However, the traditional second‐order advection–diffusion equation cannot accurately simulate these physical phenomena of anomalous diffusion. Therefore, some scholars have extended the classical advection–diffusion equation, that is, put forward the fractional advection–dispersion equation 6–10 . The fractional advection–dispersion equation can not only simulate anomalous diffusion but also simulate turbulence, chaotic dynamics of classical conservative systems and other physical phenomena.…”

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

“…The extensive application of fractional calculus has made people pay increasing attention to solving fractional differential equations with boundary or initial conditions, and the main methods used are fixed point theory, coincidence degree theory, the variational method, the monotone iterative method, and the upper and lower solution method (see other studies [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and the references therein).…”