Our research work is mainly based on applicability of rich theory of functional analysis to analyze the existence, uniqueness and continuous dependence on initial data of the solutions of the evolution equations of integral and fractional orders with non-local conditions. Our work can be divided into four major parts: In the first part, we consider the non-local evolution equations of integral order and having operators with dense domain. Because of operators having dense domain, we use the theory of semi- group for our analysis. In the second part, we focus our attention on evolution...