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Positive Solutions of Fractional Differential Equation with p‐Laplacian Operator

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  • Teng Ren
  • Xiaochun Chen

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 Dtβ(φp(Dtαx))(t)=f(t,x(t)), t ∈ (0,1), x(0) = 0, x(1) = ax(ξ), Dtαx(0)=0, and Dtαx1=bDtαxη, where Dtα,Dtβ are the standard Riemann‐Liouville derivatives, p‐Laplacian operator is defined as φp(s) = |s|p−2s, 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.

Suggested Citation

  • Teng Ren & Xiaochun Chen, 2013. "Positive Solutions of Fractional Differential Equation with p‐Laplacian Operator," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:789836
    DOI: 10.1155/2013/789836
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    References listed on IDEAS

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    1. Min Jia & Xin Liu & Xuemai Gu, 2012. "Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional‐Order Integral Boundary Value Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Min Jia & Xin Liu & Xuemai Gu, 2012. "Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional-Order Integral Boundary Value Problem," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-21, October.
    3. Almeida, Caio & Vicente, José, 2009. "Are interest rate options important for the assessment of interest rate risk?," Journal of Banking & Finance, Elsevier, vol. 33(8), pages 1376-1387, August.
    4. Jinhua Wang & Hongjun Xiang, 2010. "Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p‐Laplacian Operator," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    5. Xinguang Zhang & Lishan Liu & Benchawan Wiwatanapataphee & Yonghong Wu, 2012. "Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-16, May.
    6. Xinguang Zhang & Lishan Liu & Benchawan Wiwatanapataphee & Yonghong Wu, 2012. "Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    7. Jinhua Wang & Hongjun Xiang, 2010. "Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p -Laplacian Operator," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-12, September.
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