New Caputo fractional differential equation of higher order on unbounded interval and solvability in the new Banach space

The concept of Darbo's fixed point theorem together with the notion of measures of noncompactness is a useful tool for considering the existence of solution for analyzing differential equations. The focal point of this investigation is the existence of solutions for Caputo fractional differential equations with boundary conditions in a Banach space. In this study, first we define a new Caputo fractional differential equation of higher-order on unbounded interval and then define a new Banach space (C(R+,C[a,b]),& Vert;.& Vert;C phi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C(\mathbb {R_+}, C[a,b]), \ \Vert .\Vert _{C_{\phi }})$$\end{document} and determine the compact subsets and a new regular measures of noncompactness on this space. Further, we investigate the solvability of new Caputo fractional differential equation by applying the technique of measures of noncompactness in conjunction with the Darbo's fixed point theorem. Finally, the applicability of our main result is shown by constructing two examples.