Three Solutions for Impulsive Fractional Boundary Value Problems with \({\mathbf {p}}\)-Laplacian


The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a p-Laplacian and Dirichlet conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u(t))+|u(t)|^{p-2}u(t)= \lambda f(t,u(t)), &{}t\ne t_j,\ \ t\in (0,T),\\ \varDelta (D^{\alpha -1}_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u))(t_j)=I_j(u(t_j)),\\ u(0)=u(T)=0, \end{array}\right. } \end{aligned}$$

where \(\alpha \in (\frac{1}{p}, 1]\) and \(p > 1\). Their approach is based on variational methods. The main result is illustrated with an example.

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Graef, J.R., Heidarkhani, S., Kong, L. et al. Three Solutions for Impulsive Fractional Boundary Value Problems with \({\mathbf {p}}\)-Laplacian. Bull. Iran. Math. Soc. (2021).

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  • Fractional p-Laplacian
  • Impulsive effects
  • Three solutions
  • Variational methods

Mathematics Subject Classification

  • 26A33
  • 34B15
  • 34K45