Measures of Noncompactness to Solve a New Fractional Differential Equation of Higher Order on Unbounded Domain in the New Banach Space

In this paper, we introduce a new Banach space (C(ℝ_+, L^p(ℝ_+)), ‖·‖ _C_ϕ) equipped with a weighted norm depending on a function ϕ , 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 C(ℝ_+, L^p(ℝ_+)) , 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.