Abstract and Applied Analysis

Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

Qiuping Li, Shurong Sun, Ping Zhao, and Zhenlai Han

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Abstract

We discuss the initial value problem for the nonlinear fractional differential equation L ( D ) u = f ( t , u ) ,    t ( 0,1 ] ,    u ( 0 ) = 0 , where L ( D ) = D s n - a n - 1 D s n - 1 - - a 1 D s 1 , 0 < s 1 < s 2 < < s n < 1 , and a j < 0 , j = 1,2 , , n - 1 , D s j is the standard Riemann-Liouville fractional derivative and f : [ 0,1 ] × is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 615230, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168370

Digital Object Identifier
doi:10.1155/2012/615230

Mathematical Reviews number (MathSciNet)
MR2947733

Zentralblatt MATH identifier
1247.35189

Citation

Li, Qiuping; Sun, Shurong; Zhao, Ping; Han, Zhenlai. Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 615230, 14 pages. doi:10.1155/2012/615230. https://projecteuclid.org/euclid.aaa/1365168370


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References

  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Ca, USA, 1999.
  • D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012.
  • A. E. M. El-Mesiry, A. M. A. El-Sayed, and H. A. A. El-Saka, “Numerical methods for multi-term fractional (arbitrary) orders differential equations,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 683–699, 2005.
  • I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674–684, 2009.
  • Y.-K. Chang, V. Kavitha, and M. Mallika Arjunan, “Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order,” Nonlinear Analysis A, vol. 71, no. 11, pp. 5551–5559, 2009.
  • A. Chen, F. Chen, and S. Deng, “On almost automorphic mild solutions for fractional semilinear initial value problems,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1318–1325, 2010.
  • D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
  • V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis A, vol. 69, no. 8, pp. 2677–2682, 2008.
  • V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007.
  • V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008.
  • J. D. Ramírez and A. S. Vatsala, “Monotone iterative technique for fractional differential equations with periodic boundary conditions,” Opuscula Mathematica, vol. 29, no. 3, pp. 289–304, 2009.
  • F. A. McRae, “Monotone iterative technique and existence results for fractional differential equations,” Nonlinear Analysis A, vol. 71, no. 12, pp. 6093–6096, 2009.
  • S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis A, vol. 71, no. 5-6, pp. 2087–2093, 2009.
  • 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.
  • S. Q. Zhang, “Existence of positive solution for a singular initial value problem for a fractional differential equation,” Acta Mathematica Sinica, vol. 50, no. 4, pp. 813–822, 2007.
  • V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008.
  • A. Arikoglu and I. Ozkol, “Comments on “Application of generalized differential transform method to multi-order fractional differential equations”, Vedat Suat Erturk, Shaher Momani, Zaid Odibat [Commun Nonlinear Sci Numer Simul 2008;13:1642–54],” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1737–1740, 2008.
  • D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,” Journal of Physics A, vol. 43, no. 38, article 385209, p. 7, 2010.
  • S. Sun, Y. Zhao, Z. Han, and M. Xu, “Uniqueness of positive solutions for boundary value problems of singular fractional differential equations,” Inverse Problems in Science and Engineering, vol. 20, no. 3, pp. 299–309, 2012.
  • Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011.
  • Y. Zhao, S. Sun, Z. Han, and Q. Li, “Positive solutions to boundary value problems of nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 390543, 16 pages, 2011.
  • W. Feng, S. Sun, Z. Han, and Y. Zhao, “Existence of solutions for a singular system of nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1370–1378, 2011.
  • S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.
  • 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.
  • Y. Zhao, S. Sun, Z. Han, and Q. Li, “The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2086–2097, 2011.
  • S. Sun, Q. Li, and Y. Li, “Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations,” Computers and Mathematics with Applications. In press.
  • G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 6, pp. 1019–1024, 2012.
  • G. Wang, “Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2425–2430, 2012.