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

Wang, Li ¹ ; Zhai, Chengbo ¹

¹ School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P. R. China

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 10, p. 5156-5164.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.10.03

Classification : 26A33, 34B18, 34B27
Keywords: Uniqueness, positive solution, \(p\)-Laplacian, infinite-point fractional differential equation, mixed monotone operator

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