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Solvability of some fractional differential equations in the Hölder space \({\mathcal {H}}_{\gamma }(\mathbb {R_+})\) and their numerical treatment via measures of noncompactness

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Abstract

We study the following fractional boundary value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }\upsilon (t)+f(t,\upsilon (t))=0,\quad \alpha \in (1,2],\quad 0<t<+\infty , \\ \upsilon (0)=0,\quad D^{\alpha -1}\upsilon (\infty )=\lambda \int _{0}^{\tau } \upsilon (t)\hbox {d}t. \end{array}\right. } \end{aligned}$$

The goal of this paper is to bring forward a new family of measures of noncompactness and prove a fixed point theorem of Darbo type in the Hölder space \({\mathcal {H}}_{\gamma }(\mathbb {R_+})\). Moreover, we provide an example which supports our main result and in carrying out an proximate solution for the mentioned example with high precision we apply several numerical methods.

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Authors and Affiliations

  1. Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran

    Hojjatollah Amiri Kayvanloo, Mohammad Mehrabinezhad & Farzaneh Pouladi Najafabadi

  2. Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

    Mohammad Mursaleen

  3. Aligarh Muslim University, Aligarh, 202002, India

    Mohammad Mursaleen

Authors
  1. Hojjatollah Amiri Kayvanloo
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  2. Mohammad Mursaleen
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  3. Mohammad Mehrabinezhad
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  4. Farzaneh Pouladi Najafabadi
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Correspondence to Mohammad Mursaleen.

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Communicated by Hamid Pezeshk.

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Amiri Kayvanloo, H., Mursaleen, M., Mehrabinezhad, M. et al. Solvability of some fractional differential equations in the Hölder space \({\mathcal {H}}_{\gamma }(\mathbb {R_+})\) and their numerical treatment via measures of noncompactness. Math Sci 17, 387–397 (2023). https://doi.org/10.1007/s40096-022-00458-0

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