Topological Methods in Nonlinear Analysis

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

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 21 March 2016

https://projecteuclid.org/euclid.tmna/1458588654

Digital Object Identifier
doi:10.12775/TMNA.2015.061

Mathematical Reviews number (MathSciNet)
MR3494961

Zentralblatt MATH identifier
1360.34017

Citation

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. https://projecteuclid.org/euclid.tmna/1458588654

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