Abstract and Applied Analysis

Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

Dongyuan Liu, Zigen Ouyang, and Huilan Wang

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Abstract

We consider the following state dependent boundary-value problem D0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0, 0<t<1; y(0)=0, ηy(σ(1))=y(1), where Dα is the standard Riemann-Liouville fractional derivative of order 1<α<2, 0<η<1, p0, 0<β<1, β+1-α0 the function g is defined as g(t,u):[0,1]×[0,)[0,), and g(0,0)=0 the function f is defined as f(t,u):[0,1]×[0,)[0,)σ(t), τ(t) are continuous on t and 0σ(t), τ(t)t. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.

Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 263748, 11 pages.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1429105004

Digital Object Identifier
doi:10.1155/2015/263748

Mathematical Reviews number (MathSciNet)
MR3303263

Citation

Liu, Dongyuan; Ouyang, Zigen; Wang, Huilan. Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 263748, 11 pages. doi:10.1155/2015/263748. https://projecteuclid.org/euclid.aaa/1429105004


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References

  • V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 10, pp. 3337–3343, 2008.
  • V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods \textquotesingle Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
  • Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 7-8, pp. 3249–3256, 2009.
  • V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of fractional Dynamic Systems, Cambridge Scientiffic, Cambridge, UK, 2009.
  • N. Kosmatov, “Integral equations and initial value problems for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 7, pp. 2521–2529, 2009.
  • D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis, vol. 69, no. 11, pp. 3692–3705, 2008.
  • R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1095–1100, 2010.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  • Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 2, pp. 916–924, 2010.
  • Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.
  • S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 85–95, 2008.
  • M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006.
  • S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters A, vol. 370, no. 5-6, pp. 379–387, 2007.
  • M. M. El-Borai, “Exact solutions for some nonlinear fractional parabolic partial differential equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 150–153, 2008.
  • J. L. Wu, “A wavelet operational method for solving fractional partial differential equations numerically,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 31–40, 2009.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1993.
  • M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007.
  • E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlocal fractional differential equations,” The Electronic Journal of Qualitative Theory of Differential Equations, vol. 3, pp. 1–11, 2008.
  • S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 11, pp. 5545–5550, 2009.
  • Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
  • A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003.
  • C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 611–621, 2004.
  • M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 7-8, pp. 2391–2396, 2009.
  • S. Q. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136–148, 2003.
  • C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 4, pp. 677–685, 2006.
  • W. H. Deng, “Smoothness and stability of the solutions for nonlinear fractional differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1768–1777, 2010.
  • D. Delbosco, “Fractional calculus and function spaces,” Journal of Fractional Calculus, vol. 6, pp. 45–53, 1994.
  • R. P. Agarmal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, UK, 2001.
  • A. Granas, R. B. Guenther, and J. W. Lee, “Some general existence principles in the Caratheodory theory of nonlinear differential systems,” Journal de Mathématiques Pures et Appliquées. Neuvième Serie, vol. 70, no. 2, pp. 153–196, 1991.
  • C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010. \endinput