Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.