Measures of noncompactness to solve a new fractional differential equation of higher order on unbounded domain in the new banach space

Abstract

In this paper, we introduce a new Banach space equipped with a weighted norm depending on a function, and we characterize the compact subsets of this space. Based on this characterization, we define a new measure of noncompactness adapted to this functional framework. Using this measure of noncompactness together with Darbo’s fixed point theorem, we study the solvability of a higher-order Caputo fractional differential equation on an unbounded domain with nonlocal boundary conditions. The main novelties of this work are the introduction of the weighted Banach space, which is particularly suitable for problems on unbounded domains; the construction of a new measure of noncompactness that captures both the lack of equicontinuity and the lack of uniform decay at infinity; and the analysis of a new class of higher-order fractional boundary value problems. Finally, we provide a concrete example to illustrate the applicability of our main result.