Existence of a Unique Monotone Solutions for a Fractional Differential Equation in L^1 Using a Measure of Noncompactness

Abstract

In this paper, we investigate the existence and uniqueness of monotone solutions for a class of fractional differential equations in the Banach space . The considered problem involves fractional-order terms with multiple non-independent components, which allows us to formulate the problem in an appropriate integral form. By constructing a suitable operator associated with the corresponding integral equation, we establish the existence and uniqueness of solutions under appropriate assumptions on the nonlinear terms and the fractional operators. The proof is mainly based on the theory of measures of noncompactness and an appropriate fixed-point theorem.

Consequently, the existence of a unique monotone and non-decreasing solution in  is established.