Topological Methods in Nonlinear Analysis

Existence of solutions for a Kirchhoff type fractional differential equations via minimal principle and Morse theory

Nemat Nyamoradi and Yong Zhou

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In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff type fractional differential equation: \begin{equation*} \begin{cases} \displaystyle M \bigg (\int_{\mathbb{R}} (|{}_{- \infty} D_t^\alpha u (t)|^2 + b (t) |u(t)|^2 )\, d t \bigg) \\ \qquad \cdot ({}_tD_\infty^{\alpha} ({}_{- \infty} D_t^\alpha u (t) ) + b(t) u (t)) = f (t, u (t)), & t \in \mathbb{R}, \\ u \in H^\alpha (\mathbb{R}), \end{cases} \end{equation*} where $\alpha \in ({1}/{2},1)$, ${}_tD_\infty^{\alpha}$ and ${}_{- \infty} D_t^\alpha$ are the right and left inverse operators of the corresponding Liouville-Weyl fractional integrals of order $\alpha$ respectively, $H^\alpha$ is the classical fractional Sobolev Space, $u \in \mathbb{R}$, $b \colon \mathbb{R} \to \mathbb{R}$, $\inf\limits_{t \in \mathbb{R}} b (t) \gt 0$, $f \colon \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M\colon \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.

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Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 617-630.

First available in Project Euclid: 21 March 2016

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Nyamoradi, Nemat; Zhou, Yong. Existence of solutions for a Kirchhoff type fractional differential equations via minimal principle and Morse theory. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 617--630. doi:10.12775/TMNA.2015.061.

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