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Integrable Solutions for Gripenberg-Type Equations with m -Product of Fractional Operators and Applications to Initial Value Problems

Author

Listed:
  • Ateq Alsaadi

    (Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi Arabia
    These authors contributed equally to this work.)

  • Mieczysław Cichoń

    (Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
    These authors contributed equally to this work.)

  • Mohamed M. A. Metwali

    (Department of Mathematics, Faculty of Sciences, Damanhour University, Damanhour 22514, Egypt
    These authors contributed equally to this work.)

Abstract

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m -product of fractional operators on a half-line R + = [ 0 , ∞ ) . We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L 1 N -solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.

Suggested Citation

  • Ateq Alsaadi & Mieczysław Cichoń & Mohamed M. A. Metwali, 2022. "Integrable Solutions for Gripenberg-Type Equations with m -Product of Fractional Operators and Applications to Initial Value Problems," Mathematics, MDPI, vol. 10(7), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:7:p:1172-:d:786689
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    References listed on IDEAS

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    1. Sen, Mausumi & Saha, Dipankar & Agarwal, R.P., 2019. "A Darbo fixed point theory approach towards the existence of a functional integral equation in a Banach algebra," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 111-118.
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    Cited by:

    1. Mohamed M. A. Metwali & Shami A. M. Alsallami, 2023. "On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces," Mathematics, MDPI, vol. 11(18), pages 1-13, September.

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