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Existence results for multi-point boundary value problem to singular fractional differential equations with a positive parameter

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Abstract

This paper is concerned with a multi-point boundary value problem for a generalized Laplacian fractional differential equation with both singular weight and positive parameter. By means of Guo-Krasnoselskii and Schauder’s fixed point theorems, we discuss the number of positive solutions in terms of different ranges of the positive parameter. Moreover, two examples are given to illustrate the main results.

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References

  1. Agarwal, R.P., O’Regan, D., Stan\({\check{e}}\)k, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)

  2. Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916–924 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, T., Liu, W., Hu, Z.: A boundary value problem for fractional differential equation with \(p\)-Laplacian operator at resonance. Nonlinear Anal. 75, 3210–3217 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, T., Xu, X.: On the number of positive solutions for a four-point boundary value problem with generalized Laplacian. J. Fixed Point Theory Appl. 23, 1–22 (2021)

    Article  MathSciNet  Google Scholar 

  6. Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)

    Book  MATH  Google Scholar 

  7. Devi, A., Kumar, A., Baleanu, D., Khan, A.: On stability analysis and existence of positive solutions for a general non-linear fractional differential equations. Adv. Differ. Equ. 2020, 1–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. El-Shahed, M.: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007, 1–8 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular \(p\)-Laplacian fractional differential equation system with several parameters. Nonlinear Anal. Model. Control. 23, 182–203 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, L., Liu, L., Feng, Y.: Uniqueness of iterative positive solutions for the singular infinite-point \(p\)-Laplacian fractional differential system via sequential technique. Nonlinear Anal. Model. Control. 25, 786–805 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Han, Z., Lu, H., 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 

  12. Jiang, W.: Solvability of fractional differential equations with \(p\)-Laplacian at resonance. Appl. Math. Comput. 260, 48–56 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Jong, K.S.: Existence and uniqueness of positive solutions of a kind of multi-point boundary value problems for nonlinear fractional differential equations with \(p\)-Laplacian operator. Mediterr. J. Math. 15, 1–17 (2018)

    Article  MathSciNet  Google Scholar 

  14. Jong, K.S., Choi, H.C., Ri, Y.H.: Existence of positive solutions of a class of multi-point boundary value problems for \(p\)-Laplacian fractional differential equations with singular source terms. Commun. Nonlinear Sci. Numer. Simulat. 72, 272–281 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jong, K.S., Choi, H.C., Jang, K.J., Pak, S.A.: A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method. Appl. Numer. Math. 160, 313–330 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jong, K.S., Choi, H.C., Kim, M.C., Kim, K.H., Jo, S.H., R.O.: On the solvability and approximate solution of a one-dimensional singular problem for a p-Laplacian fractional differential equation Chaos, Soliton. Fract. 147, 110948 (2021)

  17. Krasnoselskii, M.A.: Positive Solutions of Operator Equation, MR 31: 6107. Noordhoff, Gronignen (1964)

    Google Scholar 

  18. Kartsatos, A.G.: Advanced Ordinary Differential Equations. Mancorp Publishing, Florida (1993)

    MATH  Google Scholar 

  19. Lee, Y.H., Xu, X.: Existence and multiplicity results for generalized Laplacian problems with a parameter. Bull. Malays. Math. Sci. Soc. 43, 403–424 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leibenson, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg SSSR. 9, 7–10 (1983)

    MathSciNet  Google Scholar 

  21. Li, D., Chen, F., Wu, Y., An, Y.: Multiple solutions for a class of \(p\)-Laplacian type fractional boundary value problems with instantaneous and non-instantaneous impulses. Appl. Math. Lett. 106, 106352 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Y., Qi, A.: Positive solutions for multi-point boundary value problems of fractional differential equations with \(p\)-Laplacian. Math. Meth. Appl. Sci. 39, 1425–1434 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, Y., Jiang, W.: Existence and nonexistence of positive solutions for fractional three-point boundary value problems with a parameter. J. Funct. Space. 2019 (2019)

  24. Liu, X., Jia, M., Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem with \(p\)-Laplcaian operator. Appl. Math. Lett. 65, 56–62 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luca, R.: Positive solutions for a system of fractional differential equations with \(p\)-Laplacian operator and multi-point boundary conditions. Nonlinear Anal. Model. Control. 23, 771–801 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lv, Z.W.: Existence results for \(m\)-point boundary value problems of nonlinear fractional differential equations with \(p\)-Laplacian operator. Adv. Differ. Equ. 2014, 1–16 (2014)

    Article  MathSciNet  Google Scholar 

  27. Ma, D.X.: Positive solutions of multi-point boundary value problem of fractional differential equation. Arab J. Math. Sci. 21, 225–236 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Nyamoradi, N., Tersian, S.: Existence of solutions for nonlinear fractional order \(p\)-Laplacian differential equations via critical point theory. Fract. Calc. Appl. Anal. 22, 945–967 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shen, C., Zhou, H., Yang, L.: Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter. Math. Meth. Appl. Sci. 39, 4504–4511 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, H.: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 281, 287–306 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, F., Liu, L., Wu, Y.: A numerical algorithm for a class of fractional BVPs with \(p\)-Laplacian operator and singularity-the convergence and dependence analysis. Appl. Math. Comput. 382, 125339 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Xu, X., Lee, Y.H.: Some existence results of positive solutions for \(\varphi \)-Laplacian systems. Abstr. Appl. Anal. 2014, 1–11 (2014)

    MathSciNet  Google Scholar 

  33. Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou, B., Zhang, L., Addai, E., Zhang, N.: Multiple positive solutions for nonlinear high-order Riemann-Liouville fractional differential equations boundary value problems with p-Laplacian operator. Bound. Value Probl. 2020, 1–17 (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Xianghui Xu is supported by the Natural Science Foundation of Shandong Province of China (ZR2019BA032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

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  1. School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, Shandong, People’s Republic of China

    Tingzhi Cheng & Xianghui Xu

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  1. Tingzhi Cheng
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  2. Xianghui Xu
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Correspondence to Xianghui Xu.

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Cheng, T., Xu, X. Existence results for multi-point boundary value problem to singular fractional differential equations with a positive parameter. J. Appl. Math. Comput. 68, 3721–3746 (2022). https://doi.org/10.1007/s12190-021-01690-y

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  • DOI: https://doi.org/10.1007/s12190-021-01690-y

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