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 ${{\mathcal{D}}_{\mathit{t}}}^{\beta }({\phi }_p({{\mathcal{D}}_{\mathit{t}}}^{\alpha }x))(t)=f(t,x(t))$, $t\in (\mathrm{0,1})$, $x(\mathrm{0})=\mathrm{0}$, $x(\mathrm{1})=ax(\xi )$, ${{\mathcal{D}}_{\mathit{t}}}^{\alpha }x(\mathrm{0})=\mathrm{0}$, and ${{\mathcal{D}}_{\mathit{t}}}^{\alpha }x\left(1\right)=b{{\mathcal{D}}_{\mathit{t}}}^{\alpha }x\left(\eta \right)$, where ${{\mathcal{D}}_{\mathit{t}}}^{\alpha },{{\mathcal{D}}_{\mathit{t}}}^{\beta }$ 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 $t=\mathrm{0,1}$ 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.