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Asymptotic Behavior of Solutions of Nonlinear Fractional Differential Equations with Caputo-Type Hadamard Derivatives

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Abstract

In this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form

$$^{C,H}D_a^rx\left( t \right) = e\left( t \right) + f\left( {t,x\left( t \right)} \right),\,\,\,\,a >1,$$

where r = n+α–1, α ∊ (0,1), and n ∊ℤ+. They also apply their technique to investigate the oscillatory and asymptotic behavior of solutions of the related integral equation

$$x\left( t \right) = e\left( t \right) + \int_a^t {{{\left( {\ln \frac{t}{s}} \right)}^{r - 1}}} \,\,k\left( {t,s} \right)f\left( {s,x\left( s \right)} \right)\frac{}{s},\,\,\,a > 1,\,\,\,r\,\,is\,as\,above.$$

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    John R. Graef

  2. Department of Engineering Mathematics Faculty of Engineering, Cairo University, Orman, Giza, 12613, Egypt

    Said R. Grace

  3. Department of Mathematics, Faculty of Arts and Sciences Gaziosmanpasa University, Tokat, 60240, Turkey

    Ercan Tunç

Authors
  1. John R. Graef
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  2. Said R. Grace
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  3. Ercan Tunç
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Correspondence to John R. Graef.

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Graef, J.R., Grace, S.R. & Tunç, E. Asymptotic Behavior of Solutions of Nonlinear Fractional Differential Equations with Caputo-Type Hadamard Derivatives. FCAA 20, 71–87 (2017). https://doi.org/10.1515/fca-2017-0004

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