Studying variable-order caputo fractional differential equations through non-compactness and fixed point theorem

Intoduction:Fractional calculus is a branch of mathematical analysis that studies the possibility of extending the order of the differentiation and integration operators to a noninteger order.In this study we establish new results of existence, using advanced mathematical tools such as the Kuratowski measure of noncompactness and fixed point theorems.

Main Results: In this research, we studied the boundary value problem.

(1)

where is a continuous function, and is the caputo fractional derivative of variable order.

Let be a partition of the interval and let be a PWCF with respect to , i.e,

Using (H1) the BVP(1) becomes:
where are constants with and . Using (H1), the BVP(1) becomes
(2)
with boundary conditions

Lemma: Assuming that is a continuous function, there exists a number , such that

The solution of the integral equation is given by

(3)

where

solves (2).

Theorem: Under Lipschitz conditions: , where and with and provided inequality condition :
(4)
holds. Then, ( 2) possesses at least one solution in $\Pi_{\vartheta}$. The proof is established using the measure of noncompactness and Darbo's fixed point theorem, which leads to the existence results for BVP(1), and the uniqueness is proved using the Banach fixed point theorem.

Conclusion: This Study is a valuable contribution to the expanding field of fractional calculus, in which we skillfully employed the Darbo’s Fixed Point Theorem in conjunction with the Kuratowski Measure of Non Compactness, which is a valuable method.