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

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    John R. Graef & Lingju Kong

  2. Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, 67149, Iran

    Shapour Heidarkhani & Shahin Moradi

Authors
  1. John R. Graef
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  2. Shapour Heidarkhani
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  3. Lingju Kong
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  4. Shahin Moradi
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Correspondence to John R. Graef.

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Communicated by Majid Gazor.

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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. 48, 1413–1433 (2022). https://doi.org/10.1007/s41980-021-00589-5

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