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We establish a sufficient condition for the existence, uniqueness and global uniform asymptotic stability of a C 0 -solution for the nonlinear delay differential evolution equationis the infinitesimal generator of a nonlinear semigroup of contractions, f : R + × C([ −τ, 0 ]; D(A)) → X is continuous and g : C b ([ −τ, +∞); D(A)) → C([ −τ, 0 ]; D(A)) is nonexpansive.

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We prove the continuity of the C 0-solution with respect to the right-hand side and the initial nonlocal condition to the nonlinear delay differential evolution equation u (t) ∈ Au(t) + f (t, u t), t ∈ R + , u(t) = g(u)(t), t ∈ [ −τ, 0 ], where τ > 0, X is a real Banach space, A is an m-dissipative operator, f : R + × C([ −τ, 0 ]; D(A)) → X is Lipschitz continuous with respect to its second argument and g : C b ([ −τ, +∞); D(A)) → C([ −τ, 0 ]; D(A)), is nonexpansive.

Abstract. We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C 0 -solution.