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and assume there exist constants R > 0, α ≥ 0 such that.

For a given, assume there exist a nonnegative vector and a vector such that (4.6) and (4.7) hold with.

To present our main results, we assume there exist constants 0 < a, b, c, d with a < b < d such that.

For a given assume there exist positive vectors and such that (4.6) and (4.7) hold and the set point is asymptotically stable with constant control for all.

By using Corollary 3, inequality (19) and noting ηk+1- η k ≤ α, we can get the following corollary: Corollary 6. Assume there exist constants λ > 0, integer α > 1 and q ≥ e2λα, such that the following inequalities hold.

We assume there exist positive constants (mathbf{K}_{0}), (mathbf{T}_{0}) such that biglVert mathbf{F}_{n}(m,epsilon bigrVert _{(beta,mu)} lemathbf {K}_{0}frac{1}{mathbf{T}_{0}^{n}} for every (0le nle N_{0}) and (epsilonin D 0,epsilon_{0})).

where and we assume there exists such that and.

Indeed, assume there exists k ∈ [ 0, 1 ) such that (1.1) holds.

Assume there exists another vertex (uin p_{xy}) that dominates (vin R).

Without loss of generality, we assume there exists such that (4.3).

In this case, assume there exists (jin{1,2,3}) such that (Delta _{j2}neq0).