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

The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations

“…Based on the MNC, Darbo's fixed point theorem demonstrates the existence of fixed points for a condensing map and become a significant extension of the Schauder fixed point theorem. Recently, some of generalized Darbo's fixed point theorems have been established based on the concept of generalized MNC (see [6]) and applied to the existence results for solving various types of nonlinear differential equations, nonlinear functional integral equations as well as their infinite systems (see, e.g., [2,9,12,13]). This paper focuses on fixed point theorems for condensing operators on Fréchet spaces with respect to vector-valued MNC.…”