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.
