Electron. J. Differential Equations, Vol. 2018 (2018), No. 34, pp. 1-13.

Variational methods for Kirchhoff type problems with tempered fractional derivative

Nemat Nyamoradi, Yong Zhou, Bashir Ahmad, Ahmed Alsaedi

Abstract:
In this article, using variational methods, we study the existence of solutions for the Kirchhoff-type problem involving tempered fractional derivatives
$$\displaylines{ M \Big(\int_{\mathbb{R}} |\mathbb{D}_+^{\alpha, \lambda} u (t)|^2 dt\Big) \mathbb{D}_-^{\alpha, \lambda} (\mathbb{D}_+^{\alpha, \lambda} u (t)) = f (t, u (t)), \quad t \in \mathbb{R},\cr u \in W_\lambda^{\alpha, 2} (\mathbb{R}), }$$
where $\mathbb{D}_{\pm}^{\alpha, \lambda} u (t)$ are the left and right tempered fractional derivatives of order $\alpha \in (1/2,1]$, $\lambda> 0$, $W_\lambda^{\alpha, 2} (\mathbb{R})$ represent the fractional Sobolev space, $f \in C (\mathbb{R} \times \mathbb{R}, \mathbb{R})$ and $M \in C (\mathbb{R}^+, \mathbb{R}^+)$.

Submitted September 6, 2017. Published January 24, 2018.
Math Subject Classifications: 26A63, 34A38, 35A45.
Key Words: Tempered fractional calculus; Kirchhoff type problems; Variational methods.

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Nemat Nyamoradi
Department of Mathematics
Faculty of Sciences
Razi University
Kermanshah 67149, Iran
email: neamat80@yahoo.com
Yong Zhou
Faculty of Mathematics and Computational Science
Xiangtan University, Hunan 411105, China
email: yzhou@xtu.edu.cn
Bashir Ahmad
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group
Faculty of Science, King Abdulaziz University
Jeddah 21589, Saudi Arabia
email: bashirahmad_qau@yahoo.com
Ahmed Alsaedi
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group
Faculty of Science, King Abdulaziz University
Jeddah 21589, Saudi Arabia
email: aalsaedi@hotmail.com

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