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Impulsive integro-differential systems involving conformable fractional derivative in Banach space

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Abstract

The existence of the mild solution to the impulsive integro-differential equation with a Conformable Fractional differential operator of order \(0<\gamma \le 1\) has been established in this article by considering both classical and non-local conditions. The idea of fixed point theorems, operator semigroups generated by the linear part of the equation, and nonlinear functional analysis are used to produce existence findings. We incorporated illustrations to validate our findings.

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All data generated and analyzed during this study are included in this manuscript.

References

  1. Podlubny I (1999) Fractional differential equations. Academic Press, London

    Google Scholar 

  2. Miller KS, Ross B (1993) An introduction to fractional calculus and fractional differential equations. John Wiley and Sons, Newyork

    Google Scholar 

  3. He JH (1998) Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 167:57–68

    Article  ADS  MathSciNet  Google Scholar 

  4. El-Sayed AMA (1996) Fractional order wave equation. Int J Theor Phys 35:311–322

    Article  MathSciNet  Google Scholar 

  5. He JH (1999) Some applications of nonlinear fractional differential equations and their approximations. Bullet Sci Technol 15:86–90

    Google Scholar 

  6. Khan M, Ullah S, Kumar S (2021) A robust study on 2019-n COV outbreaks through non-singular derivative. Eur Phys J Plus 136:1–20

    Article  Google Scholar 

  7. Delbosco D, Rodino L (1996) Existence and uniqueness for nonlinear fractional differential equations. J Math Anal Appl 204:609–625

    Article  MathSciNet  Google Scholar 

  8. El-Borai MM (2002) Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solit Fract 14:433–440

    Article  ADS  MathSciNet  Google Scholar 

  9. Cheng Y, Guozhu G (2005) On the solution of nonlinear fractional order differential equations. Nonlinear Anal Theory Methods Appl 310:26–29

    Google Scholar 

  10. El-Borai MM (2004) Semigroups and some nonlinear fractional differential equations. Appl Math Comput 149:823–831

    MathSciNet  Google Scholar 

  11. Kexue L, Jigen P, Junxiong J (2012) Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J Funct Anal 263:476–510

    Article  MathSciNet  Google Scholar 

  12. Furati KM, Kassim MD, Tatar N (2012) Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput Math Appl 64:1616–1626

    Article  MathSciNet  Google Scholar 

  13. Bonilla B, Rivero M, Rodriguez-Germa L, Trujillo JJ (2007) Fractional differential equations as alternative models to nonlinear differential equations. Appl Math Comput 187:79–88

  14. Khalil R, Al Horani M, Yousef A, Sababheh M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70

    Article  MathSciNet  Google Scholar 

  15. Gokdogan A, Emrahual EC (2016) Existence and uniqueness theorem for sequential linear CF differential equations. Miskolc Math Notes 17:267–279

    Article  MathSciNet  Google Scholar 

  16. Atraoui M, Bouaouid M (2021) On the existence of mild solutions for nonlocal differential equations of the second order with CF derivative. Adv Diff Equ 2021:1–12

    Article  Google Scholar 

  17. Laxmikantham V, Bainov D, Simeonov PS (1989) The theory of impulsive differential equations. World Scientific, Singapore

    Book  Google Scholar 

  18. Dishlieva K (2016) Impulsive differential equations and applications. J Appl Computat Math 1–6

  19. Shah V, George RK, Sharma J, Muthukumar P (2018) Existence and uniqueness of classical and mild solutions of generalized impulsive evolution equations. Int J Nonlinear Sci Num Solut 19:775–780

    Article  MathSciNet  Google Scholar 

  20. Kataria HR, Patel PH (2020) Congruency between classical and mild solutions of Caputo fractional impulsive evolution equation on Banach Space. Int J Adv Sci Technol 29,3s:1923–1936

    Google Scholar 

  21. Kataria HR, Patel PH (2018) Existence and uniqueness of nonlocal Cauchy problem for fractional differential equations on Banach space. Int J Pure Appl Math 120:10237–10252

    Google Scholar 

  22. Shah V, Sharma J, Georege RK (2023) Existence and uniqueness of classical and mild solutions of fractional Cauchy problem with impulses. Malaya J Mat 11(01):66–79

    Article  MathSciNet  Google Scholar 

  23. Ur Rehman MA, Ahmad J, Hassan A, Awrejcewicz J, Pawlowski W, Karamti H, Alharbi FM (2022) The dynamics of a fractional-Order mathematical model of cancer tumor diseases. Symmetry 14(2022):1–28

    Google Scholar 

  24. Kumar S, Chauhan RP, Momani S, Hadid S (2020) Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Num Methods Part Diff Equ 1–27

  25. Bayrak MA, Demir A, Ozbilge E (2022) On the numerical solution of conformable fractional diffusion problem with small delay. Num Methods Part Diff Equ 38:177–189

    Article  MathSciNet  Google Scholar 

  26. Kataria HR, Patel PH, Shah V (2020) Existence results of noninstantaneous impulsive fractional integro-differential equation. Demonstr Math 53:373–384

    Article  MathSciNet  Google Scholar 

  27. Deng K (1993) Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J Math Anal Appl 179:630–637

    Article  MathSciNet  Google Scholar 

  28. Zhou Y, Jiao F (2010) Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl 11:4465–4475

    Article  MathSciNet  Google Scholar 

  29. Byszewski L (1991) Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem. J Math Anal Appl 162:494–505

    Article  MathSciNet  Google Scholar 

  30. Balachandran K, Kiruthika S, Trujillo JJ (2011) On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput Math Appl 62:1157–1165

    Article  MathSciNet  Google Scholar 

  31. Gao Z, Yang L, Liu G (2013) Existence and uniqueness of solutions to impulsive fractional integro-differential equations with nonlocal conditions. Appl Math 4:859–863

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Acknowledgements

Authors wish to acknowledge Department of Mathematics, National Institute of Technology Puducherry for organizing the International Conference on Fractional Calculus: Theory, Applications and Numerics and giving us opportunity to present research paper at the conference.

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Authors did not receive any funding from any sources for this research work.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, The M. S. University of Baroda, Vadodara, 390 002, India

    Haribhai R. Kataria

  2. Department of Mathematics, Faculty of Science, Krishna School of Science, Drs. Kiran & Pallavi Patel Global University (KPGU), Vadodara, 391 243, India

    Prakashkumar H. Patel

  3. Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara, 390 001, India

    Vishant Shah

Authors
  1. Haribhai R. Kataria
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  2. Prakashkumar H. Patel
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  3. Vishant Shah
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Contributions

HK contributed in critical revision of the manuscript and intellectual inputs and supervision of the research work; PP was responsible for conceptualization, analysis, interpretation of data, and original drafting of manuscript; VS for analysis as well as simultaneous drafting of manuscript; all authors read and approved final manuscript.

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Correspondence to Prakashkumar H. Patel.

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Kataria, H.R., Patel, P.H. & Shah, V. Impulsive integro-differential systems involving conformable fractional derivative in Banach space. Int. J. Dynam. Control 12, 56–64 (2024). https://doi.org/10.1007/s40435-023-01224-3

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  • DOI: https://doi.org/10.1007/s40435-023-01224-3

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