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Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line

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Abstract

Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem:

$$\begin{array}{l} {D^\alpha }u\left( t \right) = u\left( t \right)\varphi \left( {t,\,u\left( t \right)} \right),\,t \in \left( {0,\infty } \right),\,1 \alpha \le 2, \\ \begin{array}{*{20}{c}} {{{\lim }^{2 - \alpha }}} \\ {t \to 0} \\ \end{array}u\left( t \right) = a,\,\begin{array}{*{20}{c}} {{{\lim }^{t - \alpha }}} \\ {t \to \infty } \\ \end{array}u\left( t \right) = b,\, \\ \end{array}$$

where Dα is the standard Riemann-Liouville fractional derivative, a, b are nonnegative constants such that a + b > 0 and φ(t, s) is a nonnegative continuous function that is required to satisfy an appropriate condition related to a class 𝜅α satisfying suitable integrability condition. We also give a global behavior of such solution.

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Authors and Affiliations

  1. Mathematics Department, King Saud University College of Science, P.O.Box 2455, Riyadh, 11451, Saudi Arabia

    Imed Bachar

  2. College of Sciences and Arts Rabigh Campus, King Abdulaziz University, P.O.Box 344, Rabigh, 21911, Saudi Arabia

    Habib Mâagli

  3. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092, Tunis, Tunisia

    Imed Bachar

Authors
  1. Imed Bachar
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  2. Habib Mâagli
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Correspondence to Imed Bachar.

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Bachar, I., Mâagli, H. Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line. FCAA 19, 1031–1049 (2016). https://doi.org/10.1515/fca-2016-0056

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  • DOI: https://doi.org/10.1515/fca-2016-0056

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