A method of lines for a nonlinear abstract functional evolution equation

Abstract

Let X X be a real Banach space with X ∗ {X^\ast } uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E) \[ u ′ ( t ) + A ( t ) u ( t ) = G ( t , u t ) , u 0 = ϕ , t ∈ [ 0 , T ] . u\prime (t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T]. \] The operators A ( t ) : D ⊂ X → X A(t):D \subset X \to X are m m -accretive and G ( t , ϕ ) G(t,\phi ) is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators A ^ ( t ) \hat A(t) and the corresponding use of nonlinear evolution operator theory.