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On Erdélyi–Kober fractional Urysohn–Volterra quadratic integral equations

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  • Darwish, Mohamed Abdalla

Abstract

A very general nonlinear singular integral equation is introduced, namely u(τ)=f1(τ,u(τ))+βf2(τ,u(τ))Γ(γ)∫0τsβ−1k(τ,s,(Au)(s))(τβ−sβ)1−γds,0≤τ≤1,β > 0 and 0 < γ < 1. The above equation is called Erdélyi–Kober fractional Urysohn–Volterra quadratic integral equation. The main goal is to show that the above equation has solutions in C[0, 1] and these solutions are nonnegative and nondecreasing on [0, 1]. By means of a measure of noncompactness and Darbo fixed point theorem we prove our main results. In the end of the paper, we give an example to show that our assumptions of our abstract results are rather easy to verify.

Suggested Citation

  • Darwish, Mohamed Abdalla, 2016. "On Erdélyi–Kober fractional Urysohn–Volterra quadratic integral equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 562-569.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:562-569
    DOI: 10.1016/j.amc.2015.10.040
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    References listed on IDEAS

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    1. Mohamed Abdalla Darwish & Kishin Sadarangani, 2015. "On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order," Mathematische Nachrichten, Wiley Blackwell, vol. 288(5-6), pages 566-576, April.
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    Cited by:

    1. Alyami, Maryam Ahmed & Darwish, Mohamed Abdalla, 2020. "On asymptotic stable solutions of a quadratic Erdélyi-Kober fractional functional integral equation with linear modification of the arguments," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).

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    1. Alyami, Maryam Ahmed & Darwish, Mohamed Abdalla, 2020. "On asymptotic stable solutions of a quadratic Erdélyi-Kober fractional functional integral equation with linear modification of the arguments," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).

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