Abstract and Applied Analysis

Positive Solutions of Fractional Differential Equation with $p$-Laplacian Operator

Abstract

The basic assumption of ecological economics is that resource allocation exists social optimal solution, and the social optimal solution and the optimal solution of enterprises can be complementary. The mathematical methods and the ecological model are one of the important means in the study of ecological economics. In this paper, we study an ecological model arising from ecological economics by mathematical method, that is, study the existence of positive solutions for the fractional differential equation with $p$-Laplacian operator ${{\mathcalbf{D}}_{\mathbf{t}}}^{\beta }\left({\phi }_{p}\left({{\mathcalbf{D}}_{\mathbf{t}}}^{\alpha }x\right)\right)\left(t\right)=f\left(t,x\left(t\right)\right)$, $t\in \left(\mathrm{0,1}\right)$, $x\left(\mathrm{0}\right)=\mathrm{0}$, $x\left(\mathrm{1}\right)=ax\left(\xi \right)$, ${{\mathcalbf{D}}_{\mathbf{t}}}^{\alpha }x\left(\mathrm{0}\right)=\mathrm{0}$, and ${{\mathcalbf{D}}_{\mathbf{t}}}^{\alpha }x\left(1\right)=b{{\mathcalbf{D}}_{\mathbf{t}}}^{\alpha }x\left(\eta \right)$, where ${{\mathcalbf{D}}_{\mathbf{t}}}^{\alpha },{{\mathcalbf{D}}_{\mathbf{t}}}^{\beta }$ are the standard Riemann-Liouville derivatives, $p$-Laplacian operator is defined as ${\phi }_{p}\left(s\right)={\left|s\right|}^{p-2}s,p>1$, and the nonlinearity $f$ may be singular at both $t=0,1$ and $x=0.$ By finding more suitable upper and lower solutions, we omit some key conditions of some existing works, and the existence of positive solution is established.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 789836, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511817

Digital Object Identifier
doi:10.1155/2013/789836

Mathematical Reviews number (MathSciNet)
MR3039143

Zentralblatt MATH identifier
1273.91351

Citation

Ren, Teng; Chen, Xiaochun. Positive Solutions of Fractional Differential Equation with $p$ -Laplacian Operator. Abstr. Appl. Anal. 2013 (2013), Article ID 789836, 7 pages. doi:10.1155/2013/789836. https://projecteuclid.org/euclid.aaa/1393511817