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Existence Results for Nonlinear Hilfer Pantograph Fractional Integrodifferential Equations

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Abstract

The main aim of this paper is to study the existence and uniqueness solutions for the nonlinear Hilfer pantograph fractional differential equations. This paper initiates with the persistence of the nonlinear Hilfer pantograph fractional differential equation. Also, it extended to the fractional integrodifferential equation. The premises are attained by using the fixed-point theorem. Ultimately, numerical examples are furnished to demonstrate our outcomes.

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Data Availability

No data were used to support this study.

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Acknowledgements

The authors are thankful to the referees for the improvement of the paper. The authors W.Shatanawi, and T. Abdeljawad would like to thank Prince Sultan University for the support through TAS research lab.

Author information

Authors and Affiliations

  1. Department of Mathematics, PSG College of Technology, Coimbatore, India

    B. Radhakrishnan & T. Sathya

  2. Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh, 11671, Saudi Arabia

    M. A. Alqudah

  3. Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586, Riyadh, Saudi Arabia

    W. Shatanawi & T. Abdeljawad

  4. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    T. Abdeljawad

  5. Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Korea

    T. Abdeljawad

  6. Department of Mathematics and Applied Mathematics School of Science and Technology, Safako Makgatho Health Science University, Ga-Rankuwa, South Africa

    T. Abdeljawad

Authors
  1. B. Radhakrishnan
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  2. T. Sathya
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  3. M. A. Alqudah
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Correspondence to B. Radhakrishnan or T. Abdeljawad.

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Appendix A

Appendix A

1.1 Solution Representation for Problem (3.1)

Consider the nonlinear Hilfer pantograph fractional differential equation (3.1).

$$\begin{aligned} D_{0^+}^{\alpha , \beta } w(t)= & {} {\mathscr {F}}(t,w(t), w(\lambda t)),\\ \text{ taking } {\mathbb {I}}_{0^+}^{\alpha } \text {on both sides}\\ {\mathbb {I}}_{0^+}^{\alpha }D_{0^+}^{\alpha , \beta } w(t)= & {} {\mathbb {I}}_{0^+}^{\alpha } {\mathscr {F}}(t,w(t), w(\lambda t)\\ {\mathbb {I}}_{0^+}^{\gamma }D^{\gamma }w(t)= & {} {\mathbb {I}}_{0^+}^{\alpha }{\mathscr {F}}(t,w(t), w(\lambda t)) \\ w(t)-\frac{{\mathbb {I}}_{0^+}^{1-\gamma }w(0) }{\Gamma (\gamma )}t^{\gamma -1}= & {} {\mathbb {I}}_{0^+}^{\alpha }{\mathscr {F}}(t,w(t), w(\lambda t))\\ w(t)= & {} \frac{w_{0}t^{\gamma -1}}{\Gamma (\gamma )}+\frac{1}{\Gamma (\alpha )}\int _{0}^{t}(t-s)^{\alpha -1}{\mathscr {F}}(s,w(s), w(\lambda s)) ds. \end{aligned}$$

Hence w(t) is a solution of problem (3.1).

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Radhakrishnan, B., Sathya, T., Alqudah, M.A. et al. Existence Results for Nonlinear Hilfer Pantograph Fractional Integrodifferential Equations. Qual. Theory Dyn. Syst. 23, 237 (2024). https://doi.org/10.1007/s12346-024-01069-x

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