# 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