Positive solutions to boundary value problems of p-Laplacian with fractional derivative

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• 作者：Xiaoyu Dong ; Zhanbing Bai ; Shuqin Zhang
• 关键词：conformable fractional derivative ; singular Green’s function ; fixed point theorems on cone ; p ; Laplacian operator
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1566KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

In this article, we consider the following boundary value problem of nonlinear fractional differential equation with p-Laplacian operator: $$\begin{aligned}& D^{\alpha}\bigl(\phi_{p}\bigl(D^{\alpha}u(t) \bigr)\bigr)= f\bigl(t, u(t)\bigr), \quad0< t< 1, \\& u(0)= u(1)= D^{\alpha}u(0)= D^{\alpha}u(1)=0, \end{aligned}$$ where \(1<\alpha\leq2\) is a real number, \(D^{\alpha}\) is the conformable fractional derivative, \(\phi_{p}(s)=\vert s\vert ^{p-2}s\), \(p>1\), \(\phi_{p}^{-1}=\phi_{q}\), \(1/p+1/q=1\), and \(f:[0, 1]\times[0,+\infty)\to[0,+\infty)\) is continuous. One of the difficulties here is that the corresponding Green’s function \(G(t, s)\) is singular at \(s= 0\). By the use of an approximation method and fixed point theorems on cone, some existence and multiplicity results of positive solutions are acquired. Some examples are presented to illustrate the main results.Keywordsconformable fractional derivativesingular Green’s functionfixed point theorems on conep-Laplacian operatorMSC34B1835J0534A081 IntroductionRecently, differential equations have been of great interest. Integer order differential equations with p-Laplacian have been subject to a lot of research [1, 2]. Now, many people pay attention to the existence and multiplicity of solutions for boundary value problems of fractional differential equations with p-Laplacian by the use of techniques of nonlinear analysis [3–6], upper and lower solutions method [7, 8], coincidence degree [9], Banach contraction mapping principle [10], etc.