SOLVABILITY OF NONLOCAL HILFER FRACTIONAL MATRIX BOUNDARY VALUE PROBLEMS WITH <inline-formula><tex-math id="M1">$P$</tex-math></inline-formula>-LAPLACIAN AT RESONANCE IN <inline-formula><tex-math id="M2">$\mathbb{R

In this paper, the solvability of boundary value problems for a class of nonlinear Hilfer fractional differential equations at resonance in $ \mathbb{R}^n $ is studied. In the past, research on matrix boundary value problems has consistently been conducted within the context of linear differential equations. The main contribution of this paper is the extension of linear problems to nonlinear ones. We begin by defining two Banach spaces endowed with appropriate norms and constructing suitable operators in these Banach spaces. Subsequently, by using the extension for the continuous theorem, certain sufficient conditions for the solvability of the problem are obtained. Finally, an example is provided to verify the effectiveness of our main results.