Let, and be three sequences of nonnegative numbers such that (2.10).
Let,, and be three sequences of nonnegative numbers satisfying the recursive inequality (1.14).
Let { a n }, { b n } and { c n } be three sequences of nonnegative real numbers with ∑ n = 1 ∞ b n < ∞ and ∑ n = 1 ∞ c n < ∞.
Let { δ n }, { β n }, and { γ n } be three sequences of nonnegative numbers such that δ n + 1 ≤ β n δ n + γ n for all n ∈ N. If β n ≥ 1 for all n ∈ N, ∑ n = 1 ∞ ( β n − 1 ) < ∞ and γ n < ∞, then lim n → ∞ δ n exists.
Let { a n }, { b n }, and { c n } be three sequences of nonnegative real numbers satisfying a n + 1 ≤ ( 1 − c n ) a n + b n c n, ∀ n ≥ 1, where { c n } ⊂ ( 0, 1 ) such that (i) c n → 0 and ∑ n = 1 ∞ c n = + ∞, (ii) either lim sup n → ∞ b n ≤ 0 or ∑ n = 1 ∞ | b n c n | < + ∞.
To prove the nonemptiness, let (uin W_{p}(Omega)), and let ({s_{n}}_{ngeq1}), ({r_{n}}_{ngeq1}), ({t_{n}}_{ngeq1}) be three sequences of step functions such that s_{n}rightarrow u, qquad r_{n}rightarrownabla u, qquad t_{n}rightarrow Delta u and |s_{n}|leq|u|, qquad |r_{n}|leq|nabla u|, qquad |t_{n}|leq|Delta u| quad mbox{a.e. on } Omega.
