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.