Existence Results for Differential Inclusions with Nonlinear Growth Conditions in Banach Spaces

In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is, x(·)(t) ∈ a.e. on I, x(t) ∈ S, ∀t ∈ I, x(0) = x₀ ∈ S, (∗), where S is a closed subset in a Banach space X, I = [0, T], (T > 0), F : I × S → X, is an upper semicontinuous set-valued mapping with convex values satisfying F(t, x) ⊂ c(t)(||x|| + ||x|| (p)K, ∀(t, x) ∈ I × S, where p ∈ ℝ, with p ≠ 1, and c ∈ C([0, T], ℝ(+)). The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.