Skip to main content
SpringerLink
Log in

Existence and uniqueness for a class of multi-term fractional differential equations

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we consider the initial value problem for the nonlinear fractional differential equations

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} D^\alpha u(t)=f(t,u(t),D^{\beta _1}u(t),\ldots ,D^{\beta _N}u(t)), \quad &{}t\in (0,1],\\ D^{\alpha -k}u(0)=0, \quad &{}k=1,2,\ldots ,n, \end{array} \right. \end{aligned}$$

where \(\alpha >\beta _1>\beta _2>\cdots \beta _N>0\), \(n=[\alpha ]+1\) for \(\alpha \notin \mathbb {N}\) and \(\alpha =n\) for \(\alpha \in \mathbb {N}\), \(\beta _j<1\) for any \(j\in \{1,2,\ldots ,N\}\), D is the standard Riemann–Liouville derivative and \(f:[0,1]\times \mathbb {R}^{N+1}\rightarrow \mathbb {R}\) is a given function. By means of Schauder fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution are obtained,respectively. As an application, some examples are presented to illustrate the main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B. V, Amsterdam (2006)

    MATH  Google Scholar 

  2. Zhou, Yong: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  4. Chang, Y., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. 71, 5551–5559 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Anping, Chen, Fulai, Deng, Siqing: On almost automorphic mild solutions for fractional semilinear initial value problems. Comput. Math. Appl. 59, 1318–1325 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delboso, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)

    Article  MathSciNet  Google Scholar 

  7. Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lakshmikantham, V., Vatsala, A.S.: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11, 395–402 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Han, Z., Pan, Y., Yang, D.: The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter. Appl. Math. Lett. 36, 1–6 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, Q., Sun, S., Zhao, P., Han, Z.: Existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations. Abstract Appl. Anal. 2012 Article ID 615230, 1–14 (2012)

  11. McRae, F.A.: Monotone iterative technique and existence results for fractional differential equations. Nonlinear Anal. 71, 6093–6096 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Y Zhou: Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, 2015

  13. Lu, H., Han, Z., Sun, S.: Multiplicity of positive solutions for Sturm–Liouville boundary value problems of fractional differential equations with p-Laplacian. Bound. Value Probl. 2014(26), 1–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Erturk, V.S., Momani, S.: Application of generalized differential transform method to multi-order fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 13, 1642–1654 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Arikoglu, A., Ozkol, I.: 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;13:1642C54]. Commun. Nonlinear Sci. Numer. Simul. 13, 1737–1740 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 70, 2521–2529 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deng, J., Ma, L.: Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23, 676–680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deng, J., Deng, Z.: Existence of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 32, 6–12 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Su, X.: Boundary value problem for a coupled systerm of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)

    Article  MathSciNet  Google Scholar 

  20. El-Mesiry, A.E.M., El-Sayed, A.M.A., El-Saka, H.A.A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations. Appl. Math. Comput. 160, 683–699 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Momani, S.: A numerical scheme for the solution of multi-order fractional differential equations. Appl. Math. Comput. 182, 761–770 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 674–684 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80(1), 101–116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bhrawy, A.H., Abdelkawy, M.A.: A fully spectral collocation approximation for multi-dimensional fractional Schrodinger equations. J. Comput. Phys. 294, 462–483 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35, 5662–5672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, S.: Existence of positive solution for a singular initial value problem for fractional differential equation. Acta Math. Sin. Chin. Ser. 50, 813–822 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Sun, S., Zhao, Y., Han, Z., Xu, M.: Uniqueness of positive solutions for boundary value problems of singular fractional differential equations. Inverse Prob. Sci. Eng. 20(3), 299–309 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, Y., Sun, S., Lei, Z., Li, Q.: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(4), 2086–2097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Feng, W., Sun, S., Han, Z., Zhao, Y.: Existence of solutions for a singular system of nonlinear fractional differential equations. Comput. Math. Appl. 62(3), 1370–1378 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhai, C.B., Xu, L.: Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simulat. 19, 2820–2827 (2014)

    Article  MathSciNet  Google Scholar 

  32. Zhao, Y., Sun, S., Han, Z., Zhang, M.: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 217(16), 6950–6958 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Zhang, S.: Monotone iterative method for initial value problem involving Remann–Liouville fractiaonal derivatives. Nonlinear Anal. 71, 2087–2093 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bai, Z.B.: Eigenvalue intervals for a class of fractional boundary value problem. Comput. Math. Appl. 62, 3253–3257 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Arara, A., Benchohra, M., Hamidi, N., Nieto, J.J.: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 72, 580–586 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, S., Li, Q., Li, Y.: Existence and uniqueness of solutions for a coupled system of multi term nonlinear fractional differential equations. Comput. Math. Appl. 64(10), 3310–3320 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, X., Han, Z., Li, X.: Boundary value problems of fractional \(q\)-difference Schroinger equations. Appl. Math. Lett. 46, 100–105 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Han, Z., Hongling, L., Zhang, C.: Positive solutions for eigenvalue problems of fractional differential equation with generalized \(p\)-Laplacian. Appl. Math. Comput. 257, 526–536 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research is supported by the Natural Science Foundation of China (61374074, 61374002), and supported by Shandong Provincial Natural Science Foundation (ZR2013AL003).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, Jinan, 250022, Shandong, People’s Republic of China

    Qiuping Li, Chuanxia Hou, Liying Sun & Zhenlai Han

  2. Quancheng College, University of Jinan, Penglai, 265600, Shandong, People’s Republic of China

    Qiuping Li

Authors
  1. Qiuping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanxia Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liying Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhenlai Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenlai Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Q., Hou, C., Sun, L. et al. Existence and uniqueness for a class of multi-term fractional differential equations. J. Appl. Math. Comput. 53, 383–395 (2017). https://doi.org/10.1007/s12190-015-0973-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-015-0973-8

Keywords

Mathematics Subject Classification

Search

Navigation