solvability of the infinite system of hadamard-type boundaryvalue problem in the double sequence space

In this work, we are interested in representing the solution of hadamard type fractional differential equation by introducing the concept of double sequence space 2c (∆). after that, we construct the hausdorff measure of non-compactness on the space 2c (∆). furthermore, we see the existence of a solution of hadamard-type fractional differential equation on the space 2c (∆). after that, we demonstrate an example to see the applicability of our results.