The method of lines applied to nonlinear nonlocal functional differential equations (Q624582)

Using the method of lines, the author proves the existence and uniqueness of a strong solution for the following nonlocal functional differential equation in a real reflexive Banach space \(X\): \[ u'(t)+Au(t)=f(t,u(t),u_{t}), \quad t\in (0,T];\qquad h(u_{0})=\phi \quad\text{on }[-\tau,0] \] where \(0<T<\infty \), \(\phi \in \mathcal{C}_{0}:=\mathcal{C}([-\tau ,0],X)\), \( \tau >0,A:D(A)\subset X\rightarrow X\) is a single-valued and \(m\)-accretive nonlinear operator, \(f:[0,T]\times X\times \mathcal{C}_{0}\rightarrow X\) satisfies a local Lipschitz-like condition, \(h:\mathcal{C}_{0}\rightarrow \mathcal{C}_{0}\) and there exists a Lipschitz continuous \(\chi \in \mathcal{C }_{0}\) such that \(h(\chi )=\phi \) and \(\chi (0)\in D(A)\).