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Topological Degree Theory and Ulam’s Stability Analysis of a Boundary Value Problem of Fractional Differential Equations

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Frontiers in Functional Equations and Analytic Inequalities

Abstract

In this article, we study the existence and uniqueness of positive solution to a class of nonlinear fractional order differential equations with boundary conditions. By using fixed point theorems on contraction mapping together with topological degree theory, we investigate some sufficient conditions in order to obtain the existence and uniqueness of positive solution for the considered problem. Further we also investigate different kinds of Ulam stability for the considered problem. Moreover, we also provide an example to justify the whole results.

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References

  1. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)

    Book  MATH  Google Scholar 

  2. A.A. Kilbas, O.I. Marichev, S.G. Samko, Fractional Integrals and Derivatives (Theory and Applications) (Gordon and Breach, Switzerland, 1993)

    MATH  Google Scholar 

  3. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (Elsevier, Amsterdam, 2006)

    Google Scholar 

  4. V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems (Cambridge Academic, Cambridge, 2009)

    MATH  Google Scholar 

  5. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

    MATH  Google Scholar 

  6. I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering (Academic, New York, 1999)

    Google Scholar 

  7. V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, HEP, Heidelberg, Germany, 2010)

    Book  MATH  Google Scholar 

  8. I. Podlubny, Fractional differential equations. An introduction to fractional derivative, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, vol. 198 (Academic, San Diego, 1999)

    Google Scholar 

  9. M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391–2396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Ahmad, S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. M. El-Shahed, J.J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 59, 3438–3443 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Belarbi, M. Benchohra, A. Ouahab, Existence results for functional differential equations of fractional order. Appl. Anal. 85, 1459–1470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for differential inclusions with fractional order. Adv. Stud. Contemp. Math. 12(2), 181–196 (2008)

    MathSciNet  MATH  Google Scholar 

  15. M. Benchohra, J.R. Graef, S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87, 851–863 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)

    MathSciNet  MATH  Google Scholar 

  17. K. Shah, R.A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions. Differ. Equ. Appl. 7(2), 245–262 (2015)

    MathSciNet  MATH  Google Scholar 

  18. K. Shah, H. Khalil, R.A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. J. Chaos Solitons Fractals 77, 240–246 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. C.F. Li, X.N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Wang , Y. Zhou, W. Wei, Study in fractional differential equations by means of topological degree methods. Numer. Funct. Anal. Optim. 33(2), 216–238 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. R.A. Khan, K. Shah, Existence and uniqueness of positive solutions to fractional order multi-point boundary value problems. Commun. Appl. Anal. 19, 515–526 (2015)

    Google Scholar 

  22. K. Shah, A. Ali, R.A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems. Bound. Value Probl. 2016, 43 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26, 103–107 (2010)

    MathSciNet  MATH  Google Scholar 

  24. M. Obloza, Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 13, 259–270 (1993)

    MathSciNet  MATH  Google Scholar 

  25. J. Wang, X. Li, Ulam-Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015)

    MathSciNet  MATH  Google Scholar 

  26. S. Tang, A. Zada, S. Faisal, M.M.A. El-Sheikh, T. Li, Stability of higher order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 9, 4713–4721 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Lijun, D. Wang, G. Wang, Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects. Appl. Math. Comput. 268, 186–200 (2015)

    MathSciNet  MATH  Google Scholar 

  28. I. Stamova, Mittag-Leffler stability of impulsive differential equations of fractional order. Q. Appl. Math. 73(3), 525–535 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.C. Trigeassou, et al., A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3), 437–445 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Wang, L. Lv, W. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)

    MathSciNet  MATH  Google Scholar 

  31. K. Shah, D. Vivek, K. Kanagarajan, Dynamics and stability of Ψ-fractional pantograph equations with boundary conditions. Bol. Soc. Paran. Mat. 22(2), 1–13 (2018)

    Google Scholar 

  32. A. Zada, S. Faisal, Y. Li, On the Hyers–Ulam stability of first order impulsive delay differential equations. J. Funct. Spaces 2016, 6 pages (2016)

    Google Scholar 

  33. A. Zada, O. Shah, R. Shah, Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)

    MathSciNet  MATH  Google Scholar 

  34. T.M. Rassias, On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. S.M. Ulam, Problems in Modern Mathematics (Wiley, New York, 1940)

    MATH  Google Scholar 

  36. S.M. Ulam, A Collection of Mathematical Problems (Interscience, New York, 1960)

    MATH  Google Scholar 

  37. P. Kumama, A. Ali, K. Shah, R.A. Khan, Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations. J. Nonlinear Sci. Appl. 10, 2986–2997 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependece for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)

    Google Scholar 

  39. F. Haq, K. Shah, G. Rahman, M. Shahzad, Hyers-Ulam stability to a class of fractional differential equations with boundary conditions. Int. J. Appl. Comput. Math. 2017, 1–13 (2017)

    MathSciNet  Google Scholar 

  40. J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81, 50–56 (2018)

    Google Scholar 

  41. A. Zada, S. Ali, Y. Li, Ulam’s type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Difference Equ. 2017, 317 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis (Springer, New York, 2011)

    Book  MATH  Google Scholar 

  44. J. Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. D.H. Hyers, G. Isac, T.M. Rassias, Stability of Functional Equations in Several Variables (Birkhäuser, Basel, 1998)

    Google Scholar 

  46. S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis (Hadronic Press, Palm Harbor, 2001)

    MATH  Google Scholar 

  47. S.M. Jung, Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. S.E. Takahasi, T. Miura, S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation y′ = λy. Bull. Korean Math. Soc. 39, 309–315 (2002)

    Google Scholar 

  49. D.S. Cimpean, D. Popa, Hyers-Ulam stability of Euler’s equation. Appl. Math. Lett. 24, 1539–1543 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Wang, Y. Zhou, Mittag-Leffler-Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25, 723–728 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Wang, L. Lv, Y. Zhou, Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces. J. Appl. Math. Comput. 38(1–2), 209–224 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  53. K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics (Springer, New York, 2010)

    Google Scholar 

  54. F. Isaia, On a nonlinear integral equation without compactness. Acta. Math. Univ. Comenianae. 75, 233–240 (2006)

    MathSciNet  MATH  Google Scholar 

  55. S.M. Ulam, A Collection of the Mathematical Problems (Interscience, New York, 1960)

    MATH  Google Scholar 

  56. T.M. Rassias, On the stability of functional equations and a problem of Ulam. Acta. Appl. Math. 62, 23–130 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors declare that there does not exist any conflict of interest. Further, this work has been supported by the National Natural Science Foundation of China (11571378).

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Authors and Affiliations

  1. Department of Mathematics, Government Postgraduate Jehanzeb College Saidu Sharif Mingora Swat, Saidu Sharif, Khyber Pakhtunkhwa, Pakistan

    Amjad Ali

  2. Department of Mathematics, University of Malakand, Chakadara Dir (L), Khyber Pakhtunkhwa, Pakistan

    Kamal Shah

  3. Department of Mathematics, Sun Yat-sen University, Guangzhou, China

    Yongjin Li

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  1. Amjad Ali
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  2. Kamal Shah
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Correspondence to Yongjin Li .

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Editors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

  2. Pedagogical Department E. E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, Athens, Greece

    John Michael Rassias

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Ali, A., Shah, K., Li, Y. (2019). Topological Degree Theory and Ulam’s Stability Analysis of a Boundary Value Problem of Fractional Differential Equations. In: Anastassiou, G., Rassias, J. (eds) Frontiers in Functional Equations and Analytic Inequalities. Springer, Cham. https://doi.org/10.1007/978-3-030-28950-8_4

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