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1 Journal of Applied Analysis

1 2000

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On the convergence of the method of lines for quasi-nonlinear functional evolutions in Banach spaces

Jin B. J., Kim J. K.

Journal of Applied Analysis

2000

Vol. 6, nr 1

105-127

This paper is concerned with the existence of global limit solutions for the quasi-nonlinear functional evolution problem x`∈ A(t, xt) x+ G(t, xt, Ltx),t ∈ [0,T], (FDE, φ) x0=φ, where A(t, ψ 1) and G(t , ψ 1,Lt ψ 2) are defined, with respect to ψ 1, on a subspace of the space PC([-r, 0],X) of all piecewise continuous functions f : [--r, 0] → X. An appropriate subspace of PC([--r, t],X) is the domain of definition of the nonlinear operators Lt, t ∈ [0,T]. The operators A(t, ψ)x are w-dissipative and Lipschitz - like in (t, ψ ) which are more general conditions than those of Karsatos-Liu. The operators G and Lt are Lipschitzian mappings on their respective domains. Moreover, we investigate the uniqueness and strong solution for such problem.