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Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives

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Abstract

As a follow-up to the inherent nature of Caputo-Hadamard fractional derivative (CHFD) and the Hadamard fractional derivative ( HFD), little is known about some asymptotic behaviors of solutions. In this paper, a system of fractional differential equations including two types of fractional derivatives the CHFD and the HFD is investigated. The leading derivative is of an order between zero and two whereas the nonlinearities may contain fractional derivatives of an order between zero and two. Under some reasonable conditions, we prove that solutions for the system with nonlinear right hand sides approach a logarithmic function, logarithmic decay and boundedness as time goes to infinity. Our approach is based on a generalized version of Gronwall-Bellman inequality and appropriate desingularization techniques, which we prove. In addition, several manipulations and lemmas such as a fractional version of L’Hopital’s rule are used. Our results are illustrated through examples.

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References

  1. Aljoudi, S., Ahmad, B., Alsaedi, A.: Existence and uniqueness results for a coupled system of Caputo-Hadamard fractional differential equations with nonlocal Hadamard type integral boundary conditions. Fractal Fract. 4, 1–15 (2020)

    Google Scholar 

  2. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    MathSciNet  Google Scholar 

  3. Anastassiou, G.A.: Opial type Inequalities involving Riemann-Liouville fractional derivatives of two functions with applications. Math. Comput Modelling. 48, 344–374 (2008)

    MathSciNet  Google Scholar 

  4. Atangana, A., Băleanu, D.: New fractional derivative with non-local and non-singular kernel. Therm. Sci. 20, 757–763 (2016)

    Google Scholar 

  5. Babusci, D., Dattoli, G., Sacchetti, D.: The Lamb-Bateman integral equation and the fractional derivatives. Fract. Calc. Appl. Anal. 14, 317–320 (2011). https://doi.org/10.2478/s13540-011-0019-3

    Article  MathSciNet  Google Scholar 

  6. Babusci, D., Dattoli, G., Sacchetti, D.: Integral equations, fractional calculus and shift operators. arXiv preprint arXiv:1007.5211 (2010)

  7. Băleanu, D., Mustafa, O.G., Agarwal, R.P.: On the solution set for a class of sequential fractional differential equations. J. Phys. A: Math. Theor. 43, 385209 (2010)

    MathSciNet  Google Scholar 

  8. Băleanu, D., Mustafa, O.G., Agarwal, R.P.: Asymptotically linear solutions for some linear fractional differential equations. Abstr. Appl. Anal. 2010, 1–8 (2010)

    MathSciNet  Google Scholar 

  9. Băleanu, D., Mustafa, O.G., Agarwal, R.P.: Asymptotic integration of (\(1+\alpha \))-order fractional differential equations. Computers Math. Appl. 62, 1492–1500 (2011)

    MathSciNet  Google Scholar 

  10. Băleanu, D., Agarwal, R.P., Mustafa, O.G., Coşulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A: Math. Theor. 44, 055203 (2011)

    MathSciNet  Google Scholar 

  11. De Barra, G.: Measure Theory and Integration. Woodhead Publishing Limited, Cambridge (2003)

    Google Scholar 

  12. Cohen, D.S.: The asymptotic behavior of a class of nonlinear differential equations. Proc. Amer. Math. Soc. 18, 607–609 (1967)

    MathSciNet  Google Scholar 

  13. Constantin, A.: On the asymptotic behavior of second order nonlinear differential equations. Rend. Math. Appl. 13, 627–634 (1993)

    MathSciNet  Google Scholar 

  14. Constantin, A.: On the existence of positive solutions of second order differential equations. Ann. di Mat. Pura ed Appl. 184, 131–138 (2005)

    MathSciNet  Google Scholar 

  15. Dannan, F.M.: Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations. J. Math. Anal. Appl. 108, 151–164 (1985)

    MathSciNet  Google Scholar 

  16. Garra, R., Polito, F.: On some operators involving Hadamard derivatives. Integr. Transf. Spec. Funct. 24, 773–782 (2013)

    MathSciNet  Google Scholar 

  17. Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to selfsimilar protein dynamics. Biophys. J. 68, 46–53 (1995)

    Google Scholar 

  18. Graef, J.R., Grace, S.R., Tunç, E.: Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives. Fract. Calc. Appl. Anal. 20, 71–87 (2017). https://doi.org/10.1515/fca-2017-0004

    Article  MathSciNet  Google Scholar 

  19. Jarad, F., Abdeljawad, T., Băleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 1–8 (2012)

    MathSciNet  Google Scholar 

  20. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Asymptotic behavior of solutions to nonlinear fractional differential equations. Math. Model Anal. 21, 610–629 (2016)

    MathSciNet  Google Scholar 

  21. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems. Electron. J. Differ. Equ. 2016, 1–14 (2016)

    MathSciNet  Google Scholar 

  22. Kassim, M.D., Tatar, N.-E.: Stability of logarithmic type for a Hadamard fractional differential problem. J. Pseudodiffer. Oper. Appl. 11, 447–466 (2020)

    MathSciNet  Google Scholar 

  23. Kassim, M.D., Tatar, N.-E.: Convergence of solutions of fractional differential equations to power-type functions. Electron. J. Differ. Equ. 2020, 1–14 (2020)

    MathSciNet  Google Scholar 

  24. Kassim, M.D., Tatar, N.-E.: Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives. Fract. Calc. Appl. Anal. 24, 483–508 (2021). https://doi.org/10.1515/fca-2021-0021

    Article  MathSciNet  Google Scholar 

  25. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  26. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problem-I. Appl. Anal. 78, 153–192 (2001)

    MathSciNet  Google Scholar 

  27. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Birkhauser Basel, Berlin (2009)

    Google Scholar 

  28. Kusano, T., Trench, W.F.: Global existence of second order differential equations with integrable coefficients. J. London Math. Soc. 31, 478–486 (1985)

    MathSciNet  Google Scholar 

  29. Kusano, T., Trench, W.F.: Existence of global solutions with prescribed asymptotic behavior for nonlnear ordinary differential equations. Ann. di Mat. Pura ed Appl. 142, 381–392 (1985)

    Google Scholar 

  30. Li, C., Li, Z.: Stability and logarithmic decay of the solution to Hadamard-Type fractional differential equation. J. Nonlinear Sci. 31, 1–60 (2021)

    MathSciNet  Google Scholar 

  31. Li, M., Wang, J.: Analysis of nonlinear Hadamard fractional differential equations via properties of Mittag-Leffler functions. J. Appl. Math. Comput. 51, 487–508 (2016)

    MathSciNet  Google Scholar 

  32. Liang, Y., Wang, S., Chen, W., Zhou, Z., Magin, R.L.: A survey of models of ultraslow diffusion in heterogeneous materials. Appl. Mech. Rev. 71, 1–16 (2019)

    Google Scholar 

  33. Lipovan, O.: On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations. Glasg. Math. J. 45, 179–187 (2003)

    MathSciNet  Google Scholar 

  34. Iomin, A., Mendez, V., Horsthemke, W.: Fractional Dynamics in Comb-Like Structures. World Scientific Publishing Co. Pte. Ltd, Singapore (2018)

  35. Magin, R.L., Abdullah, O., Băleanu, D., Xiaohong, J.Z.: Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. Magn. Reson. 190, 255–270 (2008)

    Google Scholar 

  36. Medved, M.: On the asymptotic behavior of solutions of nonlinear differential equations of integer and also of non-integer order. Electron. J. Qual. Theory Differ. Equ. 2012, 1–9 (2012)

    Google Scholar 

  37. Medved, M.: Asymptotic integration of some classes of fractional differential equations. Tatra Mt. Math. Publ. 54, 119–132 (2013)

    MathSciNet  Google Scholar 

  38. Medved, M., Pospišsil, M.: Asymptotic integration of fractional differential equations with integrodifferential right-hand side. Math. Model. Anal. 20, 471–489 (2015)

    MathSciNet  Google Scholar 

  39. Michalski, M.W.: Derivatives of Non Integer Order and Their Applications. Institut Matematyczny, Poland (1993)

    Google Scholar 

  40. Metzler, R., Schick, W., Kilian, H.G., Nonennmacher, T.F.: Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)

    Google Scholar 

  41. Mustafa, O.G., Rogovchenko, Y.V.: Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. Nonlinear Anal. Theory Methods Appl. 51, 339–368 (2002)

    MathSciNet  Google Scholar 

  42. Pachpatte, B.G.: Inequalities for Differential and Integral Equations. Academic Press, San-Diego (1998)

    Google Scholar 

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

    Google Scholar 

  44. Rao, S.N., Msmali, A.H., Singh, M., Ahmadini, A.A.H.: Existence and uniqueness for a system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions. J. Funct. Spaces. 2020, 1–10 (2020)

    MathSciNet  Google Scholar 

  45. Rogovchenko, Y.V.: On asymptotics behavior of solutions for a class of second order nonlinear differential equations. Collect. Math. 49, 113–120 (1998)

    MathSciNet  Google Scholar 

  46. Rogovchenko, S.P., Rogovchenko, Y.V.: Asymptotics of solutions for a class of second order nonlinear differential equations. Univ. Iagellonicae Acta Math. 38, 157–164 (1998)

    MathSciNet  Google Scholar 

  47. Sabatier, J., Agrawal, O.P., Machado, J.T.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    Google Scholar 

  48. Saengthong, W., Thailert, E., Ntouyas, S.K.: Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional differential equations with two point boundary conditions. Adv. Differ. Equ. 2019, 1–16 (2019)

    MathSciNet  Google Scholar 

  49. Sousa, J.V.C., Oliveira, E.C.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    MathSciNet  Google Scholar 

  50. Tong, J.: The asymptotic behavior of a class of nonlinear differential equations of second order. Proc. Amer. Math. Soc. 84, 235–236 (1982)

    MathSciNet  Google Scholar 

  51. Trench, W.F.: On the asymptotic behavior of solutions of second order linear differential equations. Proc. Amer. Math. Soc. 14, 12–14 (1963)

    MathSciNet  Google Scholar 

  52. Waltman, P.: On the asymptotic behavior of solutions of a nonlinear equation. Proc. Amer. Math. Soc. 15, 918–923 (1964)

    MathSciNet  Google Scholar 

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Acknowledgements

The author is thankful to the reviewers for their careful reading and suggestions.

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  1. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34151, Saudi Arabia

    Mohammed D. Kassim

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Kassim, M.D. Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives. Fract Calc Appl Anal 27, 281–318 (2024). https://doi.org/10.1007/s13540-023-00235-3

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  • DOI: https://doi.org/10.1007/s13540-023-00235-3

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