Uniqueness and properties of positive solutions for infinite-point fractional differential equation with p-Laplacian and a parameter

Volume 10, Issue 10, pp 5156--5164

Publication Date: 2017-10-12

http://dx.doi.org/10.22436/jnsa.010.10.03

Authors

Li Wang - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P. R. China
Chengbo Zhai - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P. R. China

Abstract

Using new methods for dealing with an infinite-point fractional differential equation with p-Laplacian and a parameter, we study the existence of unique positive solution for any given positive parameter \(\lambda\), and then give some clear properties of positive solutions which depend on the parameter \(\lambda>0\), that is, the positive solution \(u_\lambda^{*}\) is continuous, strictly increasing in \(\lambda\) and \(\lim_{\lambda\rightarrow +\infty}\|u_\lambda^*\|=+\infty,\lim_{\lambda\rightarrow 0^+}\|u_\lambda^*\|=0.\) Our analysis relies on some new theorems for operator equations \(A(x,x)=x\) and \(A(x,x)=\lambda x\), where \(A\) is a mixed monotone operator.

Keywords

Uniqueness, positive solution, \(p\)-Laplacian, infinite-point fractional differential equation, mixed monotone operator

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