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Multiplicity of Solutions for Kirchhoff Fractional Differential Equations Involving the Liouville-Weyl Fractional Derivatives
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) ( IF 0.3 ) Pub Date : 2020-06-01 , DOI: 10.3103/s1068362320010069
Ali Ashraf Nori , Nemat Nyamoradi , Nasrin Eghbali

Abstract

In this paper, by using variational methods and critical point theory we investigate the existence of solutions for the following fractional Kirchhoff-type equation:

$$\begin{cases}S\left(\int\limits_{\mathbb{R}}{|{}_{-\infty}D_{t}^{\alpha}u(t)|}^{2}dt\right){}_{t}D_{\infty}^{\alpha}({}_{-\infty}D_{t}^{\alpha}u(t))+l(t)u(t)=f(t,u(t)),\quad t\in\mathbb{R},\\ u\in H^{\alpha}(\mathbb{R}),\end{cases}$$

where \(\alpha\in(\frac{1}{2},1]\), \({}_{-\infty}D_{t}^{\alpha}\) and \({}_{t}D_{\infty}^{\alpha}\) are the left and right Liouville-Weyl fractional derivatives of order \(\alpha\) on the whole axis \(\mathbb{R}\), respectively, \(u\in\mathbb{R}\), \(l:\mathbb{R}\to\mathbb{R}\) is continuous and has a positive minimum, \(f\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\), and \(S:\mathbb{R}^{+}\to\mathbb{R}^{+}\) is a continuous function.



中文翻译:

涉及Liouville-Weyl分数阶导数的Kirchhoff分数阶微分方程的解的多重性

摘要

在本文中,通过使用变分方法和临界点理论,我们研究了以下分数基希霍夫型方程的解的存在性:

$$ \ begin {cases} S \ left(\ int \ limits _ {\ mathbb {R}} {| {} _ {-\ infty} D_ {t} ^ {\ alpha} u(t)|} ^ {2 } dt \ right){} _ {t} D _ {\ infty} ^ {\ alpha}({} _ {-\ infty} D_ {t} ^ {\ alpha} u(t))+ l(t)u (t)= f(t,u(t)),\ quad t \ in \ mathbb {R},\\ u \ in H ^ {\ alpha}(\ mathbb {R}),\ end {cases} $ $

其中\(\ alpha \ in(\ frac {1} {2},1] \)\({} _ {-\ infty} D_ {t} ^ {\ alpha} \)\({} _ {吨} d _ {\ infty} ^ {\阿尔法} \)的左和右刘维-外尔的顺序分数衍生物\(\阿尔法\)对整个轴\(\ mathbb {R} \),分别,\( u \ in \ mathbb {R} \)\(l:\ mathbb {R} \ to \ mathbb {R} \)是连续的,并且具有正的最小值\(f \ in C(\ mathbb {R} \ times \ mathbb {R},\ mathbb {R})\)\(S:\ mathbb {R} ^ {+} \ to \ mathbb {R} ^ {+} \)是一个连续函数。

更新日期:2020-06-01
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