New class of Caputo fractional differential equations and solvability in double sequence space <inline-formula><tex-math id="M1">$ m^2(\phi) $</tex-math></inline-formula>

First, we define a new class of Caputo fractional differential equations of order $ 3<\jmath\leq4 $. Then, we show that $ m^2(\phi) $ is a Banach space, and we define a new Hausdorff measure of noncompactness (MNC) in this space. Then, by the new MNC, we discuss the existence of solutions of infinite systems of a new class of fractional differential equations in double sequence space $ m^2(\phi) $. Finally, we present an example to show the efficiency of our results.