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Perforce Acknowledgements 1 Introduction and Preliminaries 1.1 Introduction 1.2 Literature Survey 1.3 Preliminaries 1.3.1 Semigrouups 1.3.2 Fractional Powers of Operators 1.3.3 Functional Differential Equations 1.3.4 Neutral Differential Equations 1.3.5 Fractional Calculus 1.4 An Outline of Dissertation 2 Semilinear integro-differential equations with nonlocal conditions-¢ñ 2.1 Introduction 2.2 Preliminaries and Assumptiors 2.3 Main Results 2.4 Modified Results 2.5 Application 3 Semilinear integro-differential equations with nonloeal conditions-¢ò 3.1 Introduction 3.2 Preliminaries and Assumptions 3.3 Main Results 4 Non-autonomous nonlinear integro-differential equations with infinite delay 4.1 Introduction 4.2 Preliminaries and Assumptions 4.3 Existence of Mild Solution 4.4 Existence of Maximal Solution 4.5 Theoretical Application 4.6 Example 5 Second-order integro-differential equation with non-autonomous op-erator 5.1 Introduction 5.2 Preliminaries 5.3 Main Results 5.4 Applications 6 On a nonlinear abstract neutral differential equations with a deviated argument 6.1 Introduction 6.2 Preliminaries and Assumptions 6.3 Existence of Local Solutions 6.4 Existence of Global Solutions 6.5 Example 7 Fractional differential equations-¢ñ 7.1 Introduction 7.2 Preliminaries 7.3 Local Existence of Mild Solutions 7.4 Global Existence 7.5 Application 8 Fractional differential equations-¢ò 8.1 Introduction 8.2 Preliminaries and Assumptions 8.3 Approximate Equations and Solutions