ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1 Introducing the Even-More General Semi-Linear Cauchy Problem (Semi-CP) 165 7.2 New Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.3 Behind the Scenes: Issues and Resolutions Arising in the Study of (Semi-CP) 168 7.4 Lipschitz to the Rescue! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.5 Gronwall’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.6 The Existence and Uniqueness of a Mild Solution for (Semi-CP) . . . . . . . . . . . . . 176 7.7 Dealing with a Perturbed (Semi-CP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
The models introduced in Chapter 6 are all examples of what happens when the external forces appearing in a system of differential equations or a higher-order differential equation involve nonlinear interactions among the components of the solution vector. Our present goal is to develop an extension of the theory formulated for (Non-CP) in Chapter 5 that enables us to handle large classes of such nonlinear forcing terms. The theory developed will subsume the theory of Chapters 4 and 5 as special cases.
