We study Brezis-Nirenberg type theorems for the equationwhere Ω is a bounded domain in R N , g(x, ·) is increasing and f is a dissipative nonlinearity. We apply such theorems for studying existence and multiplicity of positive solutions for the equationwhere q > 0, p > 1 and λ > 0.

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Abstract. In this paper, we study the convergence of the sequence defined bywhere 0 ≤ αn ≤ 1 and T is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

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In this paper, we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a Banach space. We prove that if S is an amenable semigroup and S=[T t : t # S] is a nonexpansive semigroup on a closed, convex subset C in a uniformly convex Banach space E such that the set F(S) of common fixed points of S is nonempty, then there exists a nonexpansive retraction P from C onto F(S) such that PT t =T t P=P for each t # S and Px # co[T t x : t # S] for each x # C. In this case, there exists a net [A : ] of finite averages of S such that for each t # S and for each bounded subset B of C, lim : &A : T t x&A : x&=0 and lim : &T t A : x&A : x&=0 uniformly for x # B. Also, if the norm of E is Fre chet differentiable, then for each x # C, Px is the unique common fixed point in s # S co[T ts x : t # S]. Furthermore, if [+ : ] is an asymptotically invariant net of means, then for each x # C, [T + : x] converges weakly to Px.