Solvability of some fractional differential equations in the Hölder space Hγ(R+) and their numerical treatment via measures of noncompactness

Abstract

We study the following fractional boundary value problem: $$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\alpha }\upsilon (t)+f(t,\upsilon (t))=0,\alpha \in (1,2],0<t<+\infty ,\hfill \\ \upsilon (0)=0,D^{\alpha -1}\upsilon (\infty )=\lambda {\int }_0^{\tau }\upsilon (t)\text{d}t.\hfill \end{array}\right)\end{array}$$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.